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Independent University of Moscow French-Russian Laboratory "J.-V. Poncelet"

International Workshop

IDEMPOTENT AND TROPICAL MATHEMATICS AND PROBLEMS OF MATHEMATICAL PHYSICS

G.L. Litvinov, V.P. Maslov, S.N. Sergeev (Eds.)

Organizing committee: G.L. Litvinov, V.P. Maslov, S.N. Sergeev, A.N. Sobolevski i Web-site: http://www.mccme.ru/tropical07 E-mail: tropical07@gmail.com

Moscow, August 25-30, 2007

Volume I I

Moscow, 2007


Litvinov G.L., Maslov V.P., Sergeev S.N. (Eds.) Idempotent and tropical mathematics and problems of mathematical physics (Vol. II) - M.: 2007 - 116 pages This volume contains the proceedings of an International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007. 2000 Mathematics Subject Classification: 00B10, 81Q20, 06F07, 35Q99, 49L90, 46S99, 81S99, 52B20, 52A41, 14P99

c 2007 by the Independent University of Moscow. All rights reserved.


CONTENTS

Ultrasecond quantization superfluidity in nanotubes

of

a

classical

version

of

Victor P. Maslov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Policy iteration and max-plus finite element method

David McCaffrey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Using max-plus convolution to obtain fundamental solutions for differential equations with quadratic nonlinearities

Wil liam M. McEneaney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Polynomial quantization on para-hermitian symmetric spaces from the viewpoint of overgroups: an example

Vladimir F. Molchanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
The structure of max-plus hyperplanes

V. Nitica and I. Singer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Image processing based on a partial differential equation satisfying the pseudo-linear superposition principle

E. Pap and M. Strboja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Tropical analysis on plurisubharmonic singularities

Alexander Rashkovskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Minimal elements and cellular closures over the max-plus semiring

Serge Sergeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 i
Semiclassical quantization of field theories

Oleg Yu. Shvedov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Convex analysis, transportation and reconstruction of peculiar velocities of galaxies

Andre Sobolevski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 i i
The Weyl algebra and quantization of fields

Alexander V. Stoyanovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


Polynomial quantization on para-hermitian pseudo-orthogonal group of translations

spaces

with

Svetlana V. Tsykina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
The horofunction boundary

Cormac Walsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Quantization as approximate description of a diffusion process

(in Russian) Evgeny M. Beniaminov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Idempotent systems of nonlinear equations and computational problems arising in electroenergetic networks (in Russian)

A.M. Gel'fand and B.Kh. Kirshteyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Classical and nonarchimedean amoebas in extensions of fields

(in Russian) Oksana V. Znamenskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Generalization of ultra second quantization for fermions at non-zero temperature (in Russian)

G.V. Koval and V.P. Maslov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
` Contact classification of Monge-Ampere equations

(in Russian) Alexey G. Kushner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
On amoeba of discriminant of algebraic equation (in Russian)

Evgeny N. Mikhalkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Universal algorithms solving discrete Bellman systems of equations over semirings (a computer demonstration)

(in Russian) A.V. Chourkin and S.N. Sergeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
The curves in C2 whose amoebas determine the fundamental group of complement (in Russian)

Roman Ulvert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 List of participants and authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112


Ultrasecond quantization

5

Ultrasecond quantization of a classical version of sup erfluidity in nanotub es1 Victor P. Maslov
1. In order to distinguish the classical theory in its modern understanding from the quantum theory, it is necessary to modify (somewhat) the ideology habitual to physicists, for whom the classical theory is simply the whole body of physics as it existed in the 19th century before the appearance of quantum theory. Actually, the correct meaning is that the classical theory is the limit of the quantum one as h 0. Thus, Feynman correctly understood that spin is a notion of classical mechanics. Indeed, it is obtained via a rigorous passage from quantum mechanics to classical mechanics [1]. In a similar same way, the polarization of light does not disappear when the frequency is increased, and is therefore a property of geometric rather than wave optics, contrary to the generally accepted belief, which arose because the polarization of light was discovered as the result of the appearance of wave optics. Consider a "Lifshits hole", i.e. a one-dimensional SchrЕ odinger equation with potential symmetric with respect to the origin of coordinates with two troughs. Its eigenfunctions are symmetric or antisymmetric with respect to the origin. As h 0 this symmetry remains, and since the square of the modulus of the eigenfunction corresponds to the probability of the particle to remain in the troughs, it follows that in the limit as h 0, i.e., in the "classical theory", for energies less than those required to pass over the barrier, the particle is simultaneously located in two troughs, although a classical particle cannot pass through the barrier. Nevertheless, this simple example shows how the ideology of the "classical theory" must be modified. To understand this paradox, one must take into consideration the fact that the symmetry must be very precise, up to "atomic precision", and that stationary state means a state that arises in the limit for "infinitely long" time. When we deal with nanotubes whose width is characterized by "atomic" or "quantum" dimensions, then new unexpected effects occur in the "classical" theory. Thus, already in 1958 [2], I discovered a strange effect of the standing longitudinal wave type in a slightly bent infinite narrow tube, for the case in which its radius is the same everywhere with atomic precision.
1The work has b een supp orted by the joint RFBR/CNRS grant 05-01-02807 and

by the RFBR grant 05-01-00824.


6

Victor P. Maslov

It was was impossible at the time to implement this effect in practice, which would have allowed to obtain a unimode laser, despite A.M.Prokhorov's great interest in the effect. 2. Now let us discuss the notion known as "collective oscillations" in classical physics and as "quasiparticles" in quantum physics. In classical physics, it is described by the Vlasov equation for selfcompatible (or mean) fields, in quantum physics, by the Hartrey (or the Hartrey-Fock) equation. (1) Variational equations depend on where (i.e., near what solutions of the original equation) we consider the variations. For example, in [3, 4, 5] we considered variations near a microcanonical distribution in an ergodic construction, while in [7, 8, 9, 10] this was done near a nanocanonical distribution concentrated on an invariant manifold of lesser dimension, i.e., not on a manifold of constant energy but, for example, on a Lagrangian manifold of dimension coinciding with that of the configuration space. (2) Let us note the following crucial circumstance. The solution of the variational equation for the Vlasov equation does not coincide with the classical limit for variational equations for the mean field equations in quantum theory. Consider the mean field equation in the form t (x) = t h2 + Wt (x) t (x), 2m V (x, y )|t (y )|2 dy ,

ih (1)

-

Wt (x) = U (x) +

with the initial condition |t=0 = 0 , where 0 belongs to W2 (R ) and 2 satisfies dx|0 (x)| = 1. In order to obtain asymptotics of the complex germ type [11] one must write out the system consisting of the Hartrey equation and its dual, then consider the corresponding variational equation, and, finally, replace the variations and by the independent functions F and G. For the functions F and G, we obtain the following system of equations:

i (2)

2 H F t (x) 2 H = dy F t (y ) + Gt (y ) ; (x) (y ) t (x) (y ) Gt (x) 2 H 2 H -i = dy F t (y ) + Gt ( y ) . t (x) (y ) (x) (y )

The classical equations are obtained from the quantum ones, roughly i speaking, by means of a substitution of the form = e h S (the VKB


Ultrasecond quantization
i

7

method), = e h S , where S = S , = (x, t) C , S = S (x, t) C . To obtain the variational equations, it is natural to take the variation not only of the limit equation for and , but also for the functions S and S . This yields a new important term of the equation for collective oscillations. Let us describe this fact for the simplest example, which was studied in N.N.Bogolyubov's famous paper concerning "weakly ideal Bose gas" [12]. Suppose U = 0 in equation (1) in a three-dimensional cubical box of edge L, the wave functions satisfying the periodicity condition (i.e., the problem being defined on the 3-torus with generators of lengths L, L, L). Then the function (3) (x) = L
-3/2 i/h(px-t)

e

,

where p = 2 n/L, n is an integer, satisfies the equation (1) for (4) = p2 + L-3 2m dxV (x).
()

For = 2 n/L, n a nonzero integer, consider the functions F and G() (x) given by F
()t ()t

( x)

(5) here

(x) = L- (x) = L

3/2

e e

i h i h

|(p+)x+( -)t|

, ;

G

-3/2

|(-p+)x+( +)t|

- = (6) =

(p + )2 p2 - + V + V , 2m 2m p2 (p - )2 - + V + V , 2m 2m V = L-
3

| |2 - | |2 = 1, From the system (6), we find (7) = -p +
i

dxV (x)e

i h

x

.

2 + V 2m

2 2 - V .

In this example u = e h s(x,t) , u = e- h , where s(x, t) = px + t, while the variation of the action for the vector u, u equals x + t.

s(x,t)


8

Victor P. Maslov Under a more accurate passage to the limit, we obtain V V0 = L-3 dxV (x)

. Thus, in the classical limit, we have obtained the famous Bogolyubov relation (7). In the case under consideration u(x) = 0 and, as in the linear SchrЕ odinger equation, the exact solution coincides with the quasiclassical one. In the paper [10], the case u(x) = 0 is investigated, and it turns out that the relation similar to (7) is the classical limit as h 0 of the variational equation in this general case. The curve showing the dependence of on is known as the Landau curve and determines the superfluid state. The value cr for which superfluidity disappears is called the Landau critical level. Bogolyubov explains the superfluidity phenomenon in the following terms: "the `degenerate condensate' can move without friction relatively to elementary perturbations with any sufficiently small velocity" [10, p. 210]. However, there is no Bose-Einstein condensate whatever in these mathematical considerations, it is just that the spectrum defined for < cr is a positive spectrum of quasiparticles. This means it is metastable (see [13]). The Bose-Einstein condensate is not involved here, it is only needed only to show that it would be wrong to believe that this argument works for a classical liquid, as one might think from the considerations above. Indeed, for example, the molecules of a classical nondischarged liquid are, as a rule, Bose particles. For such a liquid, one can write out the N particle equation, having in mind that each particle (molecule) is neutral and consists of an even number l of neutrons. Thus each ith particle is a point in 3(2k + l)-dimensional space, where k is the number of electrons, xi R6k+3l , depends on the potential u(xi ), xi R6k+3l and we can consider the N -particle equation for xi , i = 1, . . . , N , with pairwise interaction V (xi - xj ). 3. However, there is a purely mathematical explanation of this paradox. The thing is that Bogolyubov found only one series of points in the spectrum of the many particle problem. Landau wrote "N.N.Bogolyubov recently succeeded, by means of a clever application of second quantization, in finding the general form of the energy spectrum of a Bose-Einstein gas with weak interaction between the particles" ([14, p. 43]). But this series is not unique, i.e., the entire energy spectrum was not obtained. In 2001, the author proposed the method of ultra second quantization [15]; see also [16], [17], [18], [19], [20], The ultra second quantization of the SchrЕ odinger equation, as well as its ordinary second quantization, is


Ultrasecond quantization

9

a representation of the N -particle SchrЕ odinger equation, and this means that basically the ultra second quantization of the equation is the same as the original N -particle equation: they coincide in 3N -dimensional space. However, the replacement of the creation and annihilation operators by cnumbers, in contrast with the case of second quantization, does not yield the correct asymptotics, but it turns out that it coincides with the result of applying the Schroeder variational principle or the Bogolyubov variational method. For the exotic Bardin potential, the correct asymptotic solution coincides with the one obtained by applying the ultra second quantization method described above. In the case of general potentials, in particular for pairwise interaction potentials, the answer is not the same. Specifically, the ultra second quantization method gives other asymptotic series of eigenvalues corresponding to the N -particle SchrЕ odinger equation, and these eigenvalues, unlike the Bogolyubov ones (7), are not metastable. It turns out that the main point is not related to the Bose-Einstein condensate, but has to do with the width of the capillary (the nanotube) through which the liquid flows. If we consider a liquid in a capillary or a nanotube of sufficiently small radius the velocity corresponding to metastable states is not small. Hence at smaller velocities the flow will be without friction. The condition that the liquid does not flow through the boundary of the nanotube is a Dirichlet condition. It yields a standing wave, which can be regarded as a pair particle-antiparticle: a particle with momentum p orthogonal to the boundary of the tube, and an antiparticle with momentum -p. We consider a short action pairwise potential V (xi - xj ). This means that as the number of particles tends to infinity, N , interaction is possible for only a finite number of particles. Therefore, the potential depends on N in the following way: V
N

= V ((xi - xj )N

1/3

).

If V (y ) is finite with support V , then as N the support engulfs a finite number of particles, and this number does not depend on N . h As the result, it turns out that for velocities less than min(cr , 2mR ), where cr is the critical Landau velocity and R is the radius of the nanotube, superfluidity occurs. Now let me present my own considerations, which are not related to the mathematical exposition. Viscosity is due to the collision of particles: the higher the temperature, the greater the number of collisions. In a


10

Victor P. Maslov

nanotube, there are few collisions, and only with the walls, and those are taken into account by the author's series. It is precisely this circumstance, and not the Bose-Einstein condensate, which leads to the weakening of viscosity and so to superfluidity. What I am saying is that the main factor in the superfluidity phenomenon, even for liquid helium 4, is not the condensate, but the presence of an extremely thin capillary [21], [22]. It seems to me that a neutral gas like argon could be used for a crucial experiment.

Bibliography
[1] V. P. Maslov, Perturbation Theory and Asymptotic Methods. Nauka, Moscow, 1978 (in Russian). [2] V. P. Maslov, Doklady AN SSSR ,1958, 123(4), 631-633 (in Russian). [3] V. P. Maslov, Quasi-Particles Associated with Lagrangian Manifolds and (in the Ergodic Case) with Constant Energy Manifolds Corresponding to Semiclassical Self- Consistent Fields. V. Russian J. Math. Phys., 1996, 3 (4), 529-534. [4] V. P. Maslov. Russian J. Math. Phys., 1996, 4(1), 117-122 [5] V. P. Maslov. Russian J. Math. Phys., 1996, 4(2), 266-270 [6] V. P. Maslov, A. S. Mishchenko, Quasi-classical asymptotics of quasi-particles. Sbornik: Mathematics, 1998, 189(6), 901-930. [7] V. P. Maslov. Russian J. Math. Phys., 1995, 2(4), 527-534. [8] V. P. Maslov. Russian J. Math. Phys., 1995, 3(1), 123-132. [9] V. P. Maslov. Russian J. Math. Phys., 1995, 3(2), 271-276. [10] V. P. Maslov. Russian J. Math. Phys., 1995, 3(3), 401-406. [11] V. P. Maslov, The Complex-VKB Method for Nonlinear Equations. Moscow, Nauka, 1977 (in Russian). [12] N. N. Bogolyub ov, On the theory of superfluidity. - In: Selected Works of N.N. Bogoliubov, Kiev, Naukova dumka, 1970, vol. 2, pages 210-224 (in Russian). [13] V. P. Maslov, O. Yu. Shvedov, The Complex Germ Method for Multiparticle and Quantum Field Theory Problems, Moscow, Editorial URSS, 2000 (in Russian). [14] L. D. Landau, On the theory of superfluidity.- In: Collected Papers of L.D. Landau, Moscow, Nauka, 1969, vol.2, pages 42-46 (in Russian). See also: Phys Rev., 75, 884, 1949. [15] V. P. Maslov, Quantization of Thermodynamics and Ultrasecond Quantization. Moscow, Institute of Computer Sciences, 2001 (in Russian). [16] V. P. Maslov, Generalization of the Second Quantization Method to the Case of Special Tensor Products of Fock Spaces and Quantization of Free Energy. Functional Analysis and Its Applications, Oct. 2000, 34(4), 265-275. [17] V. P. Maslov, Super-second quantisation and entropy quantisation with charge conservation. Russian Math. Surveys, 2000, 55 (6), 1157-1158. [18] V. P. Maslov, Some Identities for Ultrasecond-Quantized Operators. Russian J. Math. Phys., 2001, 8(3), 309-321. [19] V. P. Maslov, Quantization of Thermodynamics, Ultrasecondary Quantization and a New Variational Principle. Russian J. Math. Phys., 2001, 8(1), 55-82


Policy Iteration & Max-Plus FEM

11

[20] V. P. Maslov, Ultratertiary quantization of thermodynamics. Theor. and Math. Phys., 2002, 132(3), 1222-1232. [21] V. P. Maslov, Dependence of the Superfluidity Criterion on the Capil lary Radus. Theor. and Math. Phys., 2005, 143(3), 741-759. [22] V. P. Maslov, Resonance between One-Particle (Bogoliubov) and Two-Particle Series in a Superfluid Liquid in a Capil lary. Russian J. Math. Phys., 2005, 12(3), 369-379

Policy iteration and max-plus finite element metho d David McCaffrey
1. Intro duction We consider the finite horizon differential game
T

(1.1) v (x, T ) = inf sup
a(.) b(.) 0

1 1 2 x(s)2 + a(s)2 - b(s)2 ds + (x(T )) 2 2 2

over tra jectories (x(.), a(.), b(.)) satisfying x(s) = f (x(s)) + g (x(s))a(s) + h(x(s))b(s), x(0) = x, where x(s) X Rn , a(s) U Rm , b(s) W Rr .This problem arises, for example, as the differential game formulation of a well-known class of non-linear affine H control problems - see [5, 6, 7, 4] for details. In particular it is known that the value function v (x, t) for the finite horizon problem is a (possibly non-smooth) solution to the Hamilton-Jacobi-Isaacs equation (1.2) H (x, v / x) = v / t

with initial condition v (x, 0) = (x) for (x, t) X Ч (0, T ], where the Hamiltonian is defined as 1 1 2 H (x, p) = min max p (f (x) + g (x)a + h(x)b) + x2 + a2 - b2 . a b 2 2 2 Note this Hamiltonian is non-convex in p. Suppose we choose some feedback function a(x) and, on any solution ^ tra jectory (x(.), a(.), b(.)), define the control input a(s) = a(x(s)) for all ^ s. We can then define fa (x, b) = f (x) + g (x)a(x) + h(x)b and la (x, b) = ^ ^ ^ 2 12 1 x + 2 a(x)2 - 2 b2 , and consider the finite horizon optimal control problem ^ 2
T

(1.3)

va (x, T ) = sup ^
b(.) 0

la (x(s), b(s))ds + (x(T )) ^


12

David McCaffrey

over tra jectories (x(.), b(.)) satisfying x(s) = fa (x(s), b(s)), x(0) = x. In ^ this case, the value function va (x, t) satisfies the Hamilton-Jacobi equation ^ (1.4) Ha (x, va / x) = va / t ^ ^ ^ with initial condition va (x, 0) = (x) for (x, t) X Ч (0, T ], where the ^ Hamiltonian is defined as Ha (x, p) = maxb {pfa (x, b) + la (x, b)} . Note that ^ ^ ^ this Hamiltonian is convex in p for all x. A max-plus analogue of the finite element method (FEM) is set out in [1] for the numerical computation of the value function va solving this ^ convex optimal control problem (1.3). In this note, we set out a policy iteration algorithm for the solution of the non-convex differential game (1.1). This involves the use of the max-plus FEM to solve (1.4) for a given fixed control feedback a(x) in the value determination step of the algorithm, and then a QP to improve the control feedback in the policy improvement step.We show here that the algorithm converges. It can also be shown that the approximation error on the converged solution is of order t + x(t)-1 , the same order as that obtained in [1] for the errors associated with the max-plus FEM. We do not give details of this result here, due to limited space. 2. The Max-Plus Finite Element Metho d In the following, let S t denote the evolution semi-group of the PDE (1.2). This associates to any function , the function v t = v (., t) where v is the t value function of the differential game (1.1). Similarly, let Sa denote the ^ evolution semi-group of the PDE (1.4) for some fixed feedback function t a(.). This associates to any function , the function va = va (., t) where ^ ^ ^ va is the value function of the otimal control problem (1.3). Maslov [3] ^ t observed that the semi-group Sa is max-plus linear. We now briefly review ^ the max-plus finite element method (FEM) set out in [1] for the numerical computation of va ^ Let Rmax denote the idempotent semi-ring obtained from R, with its usual order , by defining idempotent addition as a b := max(a, b) and ? multiplication as ab := a + b. Then let Rmax := Rmax {+}, with the convention that - is absorbing for the mutiplication. ? ? For X a set, we consider the set RX of Rmax valued functions on X. max ? max with respect to componentwise addition This is a semimodule over R (u, v ) - u v , defined by (u v )(x) = u(x) v (x), and componentwise scalar multiplication (, u) - u, defined by (u)(x) = u(x), where ? ? ? u, v RX , Rmax and x X. Note that the natural order on RX max max arising from the idempotent addition, i.e. the order defined by u v


Policy Iteration & Max-Plus FEM

13

u v = v , corresponds to the componentwise partial order u v u(x) v (x) for all x X. ? ? Now let X and Y be sets and consider an operator A : RY RX max max ? max valued functions on Y to Rmax valued functions on X. Such ? from R ? ? an operator is called linear if , for all u1 , u2 RY and 1 , 2 Rmax , max ? A(u1 1 u2 2 ) = A(u1 )1 A(u2 )2 . Given some Rmax valued function ? a RX ЧY on X Ч Y , we are then interested in the linear operator A : max ? ? ? RY RX with kernel a which maps any function u RY to the max max max ? function Au RX defined, in terms of the normal arithmetic operations max on R, by (2.1) Au(x) = sup {a(x, y ) + u(y )}
y Y

Then, as shown in the references cited in [1], this kernel operator A is ? ? residuated, i.e. for any v RX , the set {u RY : Au v } has a max max ? ? maximal element. The residual map A# : RX RY then takes any max max ? ? v RX to this maximal element in RY defined, again in terms of the max max normal arithmetic operations on R, as the function (2.2) (A# v )(y ) = inf {-a(x, y ) + v (x)}
xX

? The next notion to be introduced, for a kernel operator B : RY max ? X , is that of pro jection on the image imB of B . The pro jector is denoted Rmax ? ? ? PimB and is a map RX RX defined for all v RX by PimB (v ) = max max max max{w imB : w v }. Again as shown in the references cited in [1], this pro jector on the subsemimodule imB can be expressed as a composition PimB = B B # of B and its residual B # . If b(x, y ) denotes the kernel of B , then this formula can be expressed in the normal arithmetic of R, as (2.3) B B # (v )(x) = sup b(x, y ) + inf (-b( , y ) + v ( ))
y Y X max X

? ? Given a kernel operator C : R RZ with kernel c(z , x), we can max ? ? consider the transposed operator C : RZ RX with kernel c (x, z ) = max max ? c(z , x). We can then define a dual pro jector on the Rmin -subsemimodule -imC in terms of P -imC (v ) = min{w -imC : w v } for all v ? RX . Then, as above, this pro jector can be expressed as a composition max P -imC = C # C which, in the normal arithmetic of R, has the form (2.4) C
#

C (v )(x) = inf

z Z

-c(z , x) + sup (c(z , ) + v ( ))
X

Now we can define the max-plus FEM for approximating the value t function va = va (., t) for the optimal control problem (1.3). Let Y = ^ ^


14

David McCaffrey

{1, . . . , I }, X = Rn and Z = {1, . . . , J }. Consider a family {w1 , . . . , wI } ? of finite element functions wi : X Rmax , and a family {z1 , . . . , zJ } of ? max . The vectors = (i )i=1,...,I RI ? test functions zj : X R max and ? ? J can be considered as Rmax valued functions on Y ч = (чj )j =1,...,J Rmax and Z respectively. So, as above in equation (2.1), we can define max-plus ? ? ? ? kernel operators W : RY RX and Z : RZ RX with kernels max max max max W =col(wi )1iI and Z =col(zj )1j J . The action of W, which plays the role of operator B above, is as follows W (x) = sup {wi (x) + i }
iY

? ? while Z gives rise to the transposed operator Z : RX R max plays the role of operator C above, and acts as follows


Z max

which

(Z v )j = sup {zj (x) + v (x)} = zj |v
x X

where .|. denotes the max-plus scalar product. Then from equations (2.3) and (2.4), we can give the specific form of the corresponding two pro jectors (2.5) (2.6) P P
imW

(v )(x) (v )(x)

= =

sup wi (x) + inf (-wi ( ) + v ( ))
i Y X

-imZ



j Z

inf

-zj (x) + sup (zj ( ) + v ( ))
X

0 To start the algorithm off, we approximate the initial data va = ^ 0 with the maximal element va in the space imW spanned by the finite ^ 0 element functions. The approximation of va is denoted with a subscript h ^ and takes the form 0 vah (x) = (W 0 )(x) = sup wi (x) + ^ i Y 0 i

where the coefficients 0 are determined from the residuation of W given i in formula (2.2) as (2.7) 0 = inf (-wi (x) + (x)) . i
x X

As an induction assumption, suppose that at time step q t we have a vector of coefficients qt giving an approximation i v
q t a ^ q t ah ^

(x) = sup wi (x) +
iY

q t i

q of v by the maximal element vat in the space imW . Then the ^ (q +1)t (q +1)t approximation vah of va at the next time step can be calculated ^ ^


Policy Iteration & Max-Plus FEM as
(q +1)t

15

q vah (.) = PimW P -imZ Sa t vah t (.) ^ ^ ^ The coefficients of this approximation are given, from equations (2.5) and (2.6), by



i

(q +1)t

= inf

X

-wi ( ) + inf

j Z

-zj ( ) + sup zj ( ) + Sa t v ^ X

q t ah ^

( )

It is shown in [1] that vah is the maximal element in the space imW ^ spanned by the finite element functions which satisfies zj |v
(q +1)t ah ^ (q +1)t zj |Sa t v ^ q t ah ^

(q +1)t

for each test function zj . So vah is the maximal solution to a max-plus ^ variational formulation of the semi-group equation. If (see Section 3.3 of [1]) we further approximate the semi-group action t q t Sa vah by ^ ^
~^ Sa t v q t ah ^

(x) = sup wi (x) +
iY

q t i

+ tHa (x, wi / x) ^

then i (2.8) i

(q +1)t

can be written explicitly as =
X

(q +1)t

inf

-wi ( ) + inf

j Z

-zj ( ) + sup (zj ( )
X

+ sup wk ( ) +
kY

q t k

+ tHa ( , wk / x| ) ^

Finally, choose two sets (xi )iY and (xj )j Z of discretisation points, ^ ^ 2 c ^ and take the finite element functions to be wi (x) = - 2 x - xi 2 , for some fixed Hessian c, and test functions to be zj (x) = -a x - xj 1 , for some ^ fixed constant a. Then it is shown in Theorem 22 of [1] that the error T T vah - va = O(t + x(t)-1 ), where x is the maximal radius of ^ ^ the cells of the two Voronoi tessellations centred on the points (xi )iY and ^ (xj )j Z respectively. ^ 3. Policy Iteration with Max-Plus FEM in the Value Determination Step Now let p denote the cycle index within the policy iteration algorithm, and let q {0, . . . , N - 1} denote the time step index, so that the full time horizon T is divided into N equal steps of length t, i.e. T = N t, with the q th step running from q t to (q + 1)t. We restrict consideration of time-dependent feedback control policies a(x, t) to those in the form of


16

David McCaffrey

sequences of N constant-in-time policy components a0 (x), . . . , aN -1 (x) , and we then further restrict our choice of the individual policy components to functions aq (.) chosen from the set A = {a(.) : X U } of functions which are locally constant with respect to x on cells of the Voronoi tessellation VY centred on the origins (xi )iY of the finite element functions ^ wi . So suppose, as an induction hypothesis, that on iteration p, we have a set of constants {aqi } for q {0, . . . , N - 1} and i Y . These give rise to a p fixed policy ap which, for a given q , takes the form aq (x) = ap p ч(x) Y is the index of the cell of the Voronoi tessellation VY x. Note, the process can be initiated, for p = 0, by choosing value ai (say zero) such that aq (x) = ai for all q {0, . . . , N - 0 all x cell i of VY , where cell i is the one centred on the origin element wi .
q ч(x)

, where containing some fixed 1} and for xi of finite ^

3.1. Value Determination Step. The max-plus FEM outlined above t can be applied to approximate the value function vap solving the optimal control problem (1.3) with fixed strategy ap . The coefficients of the expansion of this approximation, with respect to the finite elements wi , are obtained as follows. For q = 0 and i Y , the coefficients 0 = 0 defined pi i in (2.7) above. Then, using (2.8), for q {0, . . . , N - 1} and i Y we get pi
(q +1)t

=

X

inf

-wi ( ) + inf

j Z

-zj ( ) + sup (zj ( )
X

+ sup wk ( ) +
k Y

q t pk

+ tHaq ( , wk / x| ) p
q ч( )

Note that in the Hamiltonian Hap we apply the policy aq ( ) = ap where p q ч( ) Y is the index of the cell of the Voronoi tessellation VY containing . The above can be re-arranged to give pi
(q +1)t

=

j Z

inf

- wi |zj + sup
k Y

q t pk

+ sup (zj ( )
X

+wk ( ) + tHaq ( , wk / x| ) p For a given policy a, let Tj
ka

= sup (zj ( ) + wk ( ) + tHa ( , wk / x| ))
X


Policy Iteration & Max-Plus FEM

17

In the normal max-plus FEM, the Tj ka terms can be calculated offline. This would be difficult in the application of max-plus FEM to policy iteration, since we don't know the policies a in advance. The relevant a for each p iteration is known at the start of that iteration and so, in principle, the next set of Tj ka terms for a given a could be calculated at the start of that iteration. However, this would be slow. An alternative is to approximate the Tj ka online by ~ Tj
ka

= zj |wk + tHa

opt jk

, wk / x|

opt jk

opt where j k = arg sup zj |wk = arg sup (zj ( ) + wk ( )) . Note, this approx~ imation T is presented in [1], where it is shown in Theorem 22 that the reT T sulting error estimate on the max-plus FEM deteriorates to vah - va = O( t + x(t)-1 ). So, finally, the coefficients of the expansion of the t approximation to the value function vap for fixed strategy ap are given by

(3.1)

pi

(q +1)t

= inf

j Z

~ - wi |zj + sup qt + Tj pk
kY

kaq p

3.2. Policy Improvement Step. For each i and q , there exists ?(iq ) j Z which achieves the inf in (3.1), so that (3.2) pi
(q +1)t

= - wi |z? + sup j
k Y ap q

q t pk

~j + T?k

aq p

~j For each k , the Hamiltonian within T?k

opt is evaluated at ?k , and so the j q ч(?k) j

strategy aq applied in the Hamiltonian term takes the value ap , where p ?k ) Y is the index of the cell of the Voronoi tessellation VY containing ч(j opt ?k . j The policy improvement can be formulated for test functions given by zj (x) = -a x - xj 1 for some constant a. Here, due to lack of space, we ^ consider only the special case where the constant term a in the test functions zj (x), so that they are therefore defined as (3.3) zj = 0 at x = xj ^ - otherwise

opt Then we have ?k = x? for all k Y and ч(?k ) = ч(?) Y is the ^j j j j index of the cell of the Voronoi tessellation VY containing x? . It follows ^j

that aq (x? ) = ap is the policy value applied in the Hamiltonian term p ^j j ~?kaq for all k . So every term T?kaq uses the same policy value aqч(?) ~j in Tj p p p for all k Y within the supkY operation in (3.2).

q ч(?) j


18 ? Now let k (iq ) = arg sup
(q +1)t pi k Y

David McCaffrey in (3.2), so that
a
q p

q ~j ? = - wi |z? + pt + T?k ? j k Y

Then we can improve the policy aq in cell ч(?) of V j p (3.4) sub ject to (3.5)
q ~j ? pt + T?ka ? k q t pk

by taking

~j ? min T?k
aU

a

~j + T?k

a

for all k Y . This optimisation is feasible since the current policy value q ч(?) j satisfies ap
q t ? pk

~? + T?k j

a

q ч(?) j p

q ~ pk t + T?k j

ap

q ч(?) j

for all k Y . ~j ? Let a = arg minaU T?ka sub ject to the constraints (3.5). In cell with ? ?) of the Voronoi tessellation VY , take new policy index ч(j aq p
+1

(x) = ap

q ч(?) j +1

:= a ?

for all x cell with index ч(?). Note that for each q , there may be some j remaining cells of VY whose indices = ч(?(iq )) for any i Y . In these cells j we leave the policy at time step q unchanged, i.e. if ч is the index of such a cell, then for all x cell with index ч aq p
+1

( x) = a

q ч p

Then the resulting new policy ap+1 = {aqi } is an improvement on the old p+1 q qt one ap = {aqi } in the sense that the corresponding vat and va+1) h , i.e. p ph (p the approximations to the value functions which solve the optimal control problem (1.3) with fixed policies ap+1 and ap respectively, satisfy (3.6) v
q t a(p+1) h

v

q t ap h

for all q {0, . . . , N } . To see this, note first that the policy improvement is unique. If, for a given q , there are two i giving rise to the same ?(iq ), then these both j result in the same policy improvement aq +1 (x) = a in cell ч(?) since the ? j p ~?kaq in (3.2) does not depend on i. term Tj p


Policy Iteration & Max-Plus FEM

19

t Next, suppose with a view to induction on q , that qp+1)i qt for pi ( all i. Then



(q +1)t pi

q ~j ? = - wi |z? + pt + T?k ? j k

a a a ?

q p q ч(?) j p

= - wi |z? + j

q pt ? k

~? + T?k j

q ~j ? - wi |z? + pt + T?k ? j k q t pk

= - wi |z? + sup j
k Y

~j + T?k

a ?



- wi |z? + sup j
k Y

q t (p+1)k q t (p+1)k

~j + T?k ~j + T?k

a ? aq +1 p

= - wi |z? + sup j
k Y

~j since every term T?k So pi
(q +1)t

aq +1 p

uses the same policy value in the same cell ч(?). j

inf - wi |z
j

j

+ sup
k Y

q t (p+1)k

~ + Tj

kaq p

+1

=

(q +1)t (p+1)i

Since this holds for all i, then it follows that for any x X sup (p
i (q +1)t (q +1)t +1)i

+ wi (x) sup pi
i

(q +1)t

+ wi (x)

i.e. va(p+1) h (x) vap h (x). So by induction, after noting that from (2.7), the coefficients at the initial time step q = 0 satisfy 0p+1)i = 0 = pi (
t q qt q 0 , it follows that qp+1)i pi t and va+1) h (x) vat (x) for all q . i ( ph (p Finally by taking b = 0 in (1.3), and by restricting our choice of t initial data to functions 0, we can see that vap 0 for all p. Since t t vap h - vap = O( t + x(t)-1 ), it follows that t ap h

(q +1)t

(3.7)

v

-K ( t + x(t)-1 )

for some K > 0. Hence, the above policy iteration algorithm converges to a time-dist N cretised finite element approximation vh (.) = vh t (.), . . . , vh t (.) to the t value function v of the differential game (1.1).


20 3.3. QP Optimisation. form Ha (x, p) = =

David McCaffrey The Hamiltonian appearing in (1.4) has the

max {pfa (x, b) + la (x, b)}
b

1 1 1 p(f + g a) + x2 + a2 - 2 phhT p 2 2 2 ~?ka has the specific form and for zj given by (3.3), Tj ? ~j ? ^j T?ka = wk (x? ) + tHa x? , wk / x|x? ? ? ^j ^j The policy improvement optimisation set out in (3.4) and (3.5) can then be formulated as min tHa x? , wk / x|x? ^j ? ^j
aU

sub ject to tHa x? , wk / x|x? ^j ? ^j
q t pk q - pt + wk (x? ) - ^j ? k

^j -wk (x? ) + tHa x? , wk / x|x? ? ^j ^j for all k Y . This can be simplified down to the following QP min
a U

wk 1 ? g a + a2 x 2

sub ject to wk wk ? - x x ga 1 1 (wk - wk ) + qt - ? t t pk 1 wk wk ? - 2 2 x x wk wk ? - f + x x + for all k Y , and evaluated at x? . ^j Bibliography
[1] M. Akian, S. Gaubert and A. Lakhoua, "The Max-Plus Finite Element Metho d for Solving Deterministic Optimal Control Problems: Basic Prop erties and Convergence Analysis", to appear in SIAM J. Contr. & Opt., preprint at arXiv:math.OC/0603619, March 2006. [2] A.J. Hoffman and R.M. Karp, "On Nonterminating Stochastic Games", Management Sci., 12, 359-370,1966. [3] V.P. Maslov, "On a new principle of sup erposition for optimisation problems", Russ. Math. Surveys , 42(3), 43-54, 1987.
T q t ? pk

hh

T

wk wk ? - x x


Policy Iteration & Max-Plus FEM

21

[4] D. McCaffrey, "Geometric existence theory for the control-affine H problem", J. Math. Anal. & Applic., 324, 682-695, 2006. [5] P. Soravia, "H control of nonlinear systems: differential games and viscosity solutions", SIAM J. Contr. & Opt., 34(3), 1071-1097, 1996. [6] A.J. van der Schaft, "On a state space approach to nonlinear H control", Syst. & Contr. Lett., 16, 1-8, 1991. [7] A.J. van der Schaft, "L2 gain analysis of nonlinear systems and nonlinear state feedback H control", IEEE Trans. Autom. Control , 37(6), 770-784, 1992.

Using max-plus convolution to obtain fundamental solutions for differential equations with quadratic nonlinearities1 Wil liam M. McEneaney
1. Intro duction We first consider time-invariant differential Riccati equations (DREs) of the form . (1.1) Pt = F ( Pt ) = A Pt + Pt A + C + P t Pt where C is symmetric and = is symmetric, nonnegative definite with at least one positive eigenvalue. Throughout, we assume that all of the matrices are n Ч n. We suppose one has initial condition, P0 = p0 where p0 is also symmetric. The Daivson-Maki approach uses the Bernoulli substitution to create a linear system of two matrices, each of the same size as Pt , thus leading to a fundamental solution. We obtain a completely different form of fundamental solution, with a particularly clear controltheoretic motivation. The new approach will be constructed through a finite-dimensional semigroup defined by this fundamental solution. The forward propagation of the fundamental solution is naturally defined by this operation through the semigroup property. We will consider linear/quadratic control problems parameterized by z I n , and the value functions associated with these control problems are R propagated forward by a max-plus linear semigroup, which we denote as S . The space of semiconvex functions is a max-plus vector space (moduloid) [10], [4], [2], [3], [7]. Working in the semiconvex-dual space, S has a
Research partially supported by NSF grant DMS-0307229 and AFOSR grant FA9550-06-1-0238.
1


22

William M. McEneaney

semiconvex-dual operator, B which takes the form of a max-plus integral operator with kernel, B (x, y ), taking the form of a quadratic function. The matrix, , defining this quadratic kernel function will be the fundamental solution of the DRE. We will define a multiplication operation ( -multiplication) with the semigroup property, specifically t+ = t , where the operation involves inverse, multiplication and addition n Ч nmatrix operations (in the standard algebra). We will also define an exponentiation operation ( -exponentiation) such that t = 1 t . The solution - - of (1.1) will be obtained by Pt = D 1 t D p0 where the D and D 1 operators are descended from the semiconvex dual and its inverse. It is important to note that the fundamental solution approach has the benefit that one only solves once for t , even if one wishes to solve the DRE for a variety of initial conditions. This approach may be extended to a class of quasilinear, first-order PDEs, yielding a fundamental solution for a class of such PDEs. More specifically, we consider PDEs 0 = - Pt + A P - B P + 1 P P 2 . on the domain [0, T ] Ч L where L = [0, L]. For simplicity, we consider the scalar case, and so A, B , I , > 0, where we specifically require R B = 0 (otherwise this reduces to an ODE problem). We again create a linear/quadratic control problem, where in this case, the state takes values in L2 (L). The above PDE is essentially the "Riccati" equation for this virtual control problem. We again apply semiconvex duality, and max-plus vector space concepts. This leads to an extension of the operator to this infinite-dimensional context, and finally, a fundamental solution for this class of PDEs. (1.2) 2. The linear-quadratic control problem and semigroups The proofs of the results in the sections on the DRE may be found in [9]. As indicated above, the fundamental solution to the DRE will be obtained through an associated optimal control problem. Recall that we are considering the DRE given by (1.1). Since we will be employing semiconvex duality (see below and [4, 10]), we will choose some (dualityparametrizing) symmetric matrix, Q, such that F (Q) > 0, where we note that, for any square matrix D, we will use the notation D > 0 to indicate that matrix D is positive definite throughout. We will need to consider the specific solution of DRE (1.1) with initial condition (2.1) P0 = Q.


Max-plus convolution and fundamental solution

23

We assume: There exists a solution of DRE (1.1), Pt , with initial condition (2.1), satisfying Pt > Q (i.e., Pt - Q positive-definite) for (A.e) t (0, T ) with T > 0, and we note specifical ly, that we may have T = +. We will be obtaining the fundamental solution t for solutions with initial conditions, P0 = p0 > Q. Note that we do not assume stability of the DRE, and finite-time blow-up is possible. We will let T = T (p0 ) = sup{t . 0 | Pt exists, and Pt > Q}, and we let T = T (p0 ) = T T where indicates the minimum operation. Remark 2.1. Note that with 0 and at least one positive eigenvalue, we may take Q = -k I for arbitrarily large k , so that one can ensure F (Q) > 0 (as well as for any p0 > Q). We will be using a control value function to motivate and develop the fundamental solution. Consider the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) problems on [0, T ) Ч I n , indexed by R z I n , given by R (2.2) (2.3) Vtz = H (x, V z ) = (Ax)
1 V z + 2 x Cx + ( V z) V z 1 V z (0, x) = (z , x) = 2 (x - z ) Q(x - z ).

Theorem 2.2. For any z I n , there exists a solution to (2.2),(2.3) R in C ([0, T ) Ч I n ) C ([0, T ) Ч I n ), and this is given by R R (2.4)
1 V z = 2 (x - t z ) Pt (x - t z ) + z Rt z

where P satisfies (1.1),(2.1), and , r satisfy 0 = I , R0 = 0, (2.5) = P
-1

C -A

and

R = C .

1 For : I n I given by (x) = 2 (x - z ) p0 (x - z ) (and actually for R R a much larger set of functions), we define the max-plus linear semigroup, S , by

(2.6)

S [](x) = V z ( , x) = 1 (x - z ) P (x - z ) + z R z . 2

We let , denote the max-plus addition and multiplication operations. We say that is uniformly semiconvex with (symmetric matrix) 1 constant K if (x) + 2 x K x is convex on I n , and we denote this space as R K n S (I ). Recall that S K is a max-plus vector space. R We will use the quadratic given in (2.3) to define our semiconvex duality. The main duality result (c.f., [10], [4], where proofs may be found) is


24

William M. McEneaney

Theorem 2.3. Let S K (I n ) where -K > Q. Then, for al l x, z R I, R
n

(2.7)

. (x) = max [ (x, z ) + a(z )] = n
z I R



(x, z ) a(z ) dz
I R
n

. = (x, ћ)


.- a(ћ) = D 1 [a]

(2.8)

a(z ) = -
I R
n

(x, z ) [-(x)] dx . [-(ћ)]} = D [].

= - { (ћ, z )

Using Theorem 2.3 and some technical arguments, for all t (0, T ) and all x, z I n R


(2.9)

St [ (ћ, z )](x) =
I R
n

(x, y ) Bt (y , z ) dy Bt (ћ, z ),

= (x, ћ) where for all y I R (2.10)
n

Bt (y , z ) = -
I R
n

(x, y ) -St [ (ћ, z )](x) dx [St [ (ћ, z )](ћ)]-
-

= (ћ, y )

.

We define the time-indexed max-plus linear operators Bt by (2.11) . Bt [a](z ) = Bt (ћ, z ) a(ћ) =
I R
n

Bt (y , z ) a(y ) dy ,

and one easily sees that these satisfy the semigroup property. (We may use a space of uniformly semiconcave functions as the domain.) We say that Bt is the kernel of max-plus integral operator Bt . .1 Theorem 2.4. Let (x) = 2 x p0 x and a(z ) = D []. Then, for t (0, T ), x I n , R (2.12) St [](x) = (x, ћ)
- - Bt [a](ћ) = D 1 Bt [a](x) = D 1 Bt D [](x).

Now, note that by (2.6) and (2.10), (2.13) Bt (x, y ) = - max n
xI R 1 2

(x-y ) Q(x-y )-

1 2

(x - t z ) Pt (x - z ) + 1 z Rt z 2

where t < T guarantees strict concavity of the argument of the maximum.


Max-plus convolution and fundamental solution

25

Lemma 2.5. Let and be 2n Ч 2n matrices with block structure given by (2.14) Let . F (x, z ) = max n
z I R 1 2

=

1,1 1,2



1,2 2,2

and

=

1,1 1,2 z y



1,2 2,2

,

x z



x + z

1 2



z y

.

Then, x x y y where has identical block structure to and , and is given by = where the operation is defined as F (x, y ) =
1 2



. and S =
2,2

1,1 2,1

= =
1,1

1,1 1,2

- ,

1,2

S

-1 1,2



,
2,2

-

1,2

= - S

1,2

S

-1 1,2



,

2,2

=

1,2

-1 1,2

,

+

.

Combining (2.13) and Lemma 2.5, one obtains the following. Theorem 2.6. (2.15) Bt (x, y ) =
1 2

x y



t

x y

where t has the same block structure as above. 3. The DRE fundamental solution semigroup Now we will use the semigroup nature of the St operators to obtain the semigroup nature of the Bt operators, and consequently the propagation of the Bt and t . The propagation of t = (1 ) t will be the dynamics of the fundamental solution of the DRE. Lemma 3.1. Let a(z ) = - = D 1 a. Then, (3.1)
1 2

(z - z ) qa (z - z ) + ra with qa < -Q, and
-1

1 (x) = 2 (x - z ) QU

q

a

(x - z ) + ra

where U = Q + qa . Alternatively, let (x) = 1 (x - x) qp (x - x) + rp with .2 qp > Q, and let a = D . Then, with = Q - qp (3.2)
1 a(z ) = 2 (z - x) Q -1

qp (z - x) + rp .


26

William M. McEneaney

Based on this lemma, it is natural to make the following definitions, - which inherit notation from D and D 1 . For symmetric qp > Q, define . - D [qp ] = Q(Q - qp )-1 qp , and for symmetric qa < -Q, define D 1 [qa ] = Q(Q + qa )-1 qa . One may show (see [9]): Theorem 3.2. For al l t1 , t2 0 such that t1 + t2 < T ,


B

t1 +t

2

( , x ) =
I R
n

Bt1 ( , z ) Bt2 (z , x) dz

x, I n . R

Theorem 3.3. The forward propagation of semigroup t is given by (3.3) where the
t1 +t
2

=

t1



t

2

operation is given in Lemma 2.5.

To summarize, suppose one wishes to obtain the solution of (1.1) at time t with initial condition P0 = p0 . Then, one performs the following steps: ћ Obtain q0 from p0 via q0 = D p0 = Q(Q - p0 )-1 p0 . -1 . 1 1 2 1 ћ Obtain qt from t and q0 via qt = t ,1 -t ,2 t ,2 + q0 t ,2 = t q0 . - ћ Obtain Pt from from qt via Pt = D 1 qt = Q(Q + qt )-1 qt . 4. Exp onentiation and a Semiring Recall that for a standard-algebra linear system, one views the fundamental solution as eAt = (eA )t . We would like some similar exponential-type representation here. Naturally, we define -exponentiation for positive integer powers through 2 = , 3 = [ 2 ] , et cetera. Using Theorem 3.3, this immediately yields nt = t n . However, this only works for integer powers. We will extend this to positive real powers so that we may simply write t = (1 ) t for any t > 0. . Let Q denote the set of rationals. Given any t (0, ), let et = {s (0, ) | p Q such that s = pt}. As is well-known, the collection of such et forms an uncountable set of equivalence classes covering (0, ). Suppose s et . Then, there exists p = m/n with m, n N such that s = pt. Let = t/n. Then, t = n and s = m . Consequently, by Theorem 3.3, s = m and t = n . With this in mind, we make the following extension of -exponentiation.
p t

Definition 4.1. Let s = pt with p = m/n, m, n N . We define . = m where = t/n.


Max-plus convolution and fundamental solution

27

We need to demonstrate that the definition is independent of the choice of m, n N . That is, suppose p = m0 /n0 = m1 /n1 . Let 0 = t/n0 and 1 = t/n1 . We must show 0m0 = 1m1 . We will use the following, trivially-verified result. Lemma 4.2.
n t m

=

(nm) t

.

With this lemma, the above independence is easily proven. Lastly, one extends the -exponentiation definition to exponents which may not be rational using continuity. There are underlying semirings with the , operations, and this seems to be quite interesting. These semirings are related to the convolution semiring of [5]. We only touch on the matter here. Let a, b . . [0, +) {+} = W + . Then define a b = ab/(a + b) which defines the + + operation on W . Also, define on W by a b = max{a, b}. Theorem 4.3. W + , , is a commutative idempotent semiring.

5. First-Order Quasilinear PDE The same approach, which was used above in the case of the DRE, can be applied to a first-order, quasilinear PDE with a quadratic nonlinearity. This PDE will take the form of a Riccati equation, and we will refer to it as the fully-first-order Riccati PDE (the FFOR PDE). The FFOR PDE will be (5.1) 0 = -Pt + AP - B P + 1 P P 2 . where the domain will be [0, T ] Ч L where L = [0, L]. For simplicity, we consider the scalar case, and so A, B , I , > 0, where we specifically R require B = 0 (otherwise this reduces to an ODE problem). We let
B EP =

(0, T ] Ч {0} (0, T ] Ч {L}

if B > 0 if B < 0.

The initial and boundary conditions will be (5.2) (5.3) P (0, ) ^ P (t, ) = p0 () =0 L B ^ (t, ) EP .

We will obtain a fundamental solution for (5.1)-(5.3) using technology analogous to that used for the DRE. In order to do so, we must devise a virtual control problem for which the time-reversed version of (5.1) is the associated Riccati equation.


28

William M. McEneaney

We begin by defining the dynamics of the virtual control problem. The state will take values in X = L2 (L; I ). The control will take values R in W = L2 (L; I ). In particular, we consider the control space W s = R L2 ([-T , 0], W ). The domain for the dynamics will be [-T , 0] Ч L, and for t [-T , 0], we will have state, (t, ћ) X . Let E
B X

=

(-T , 0] Ч {L} (-T , 0] Ч {0}

if B > 0 if B < 0.

The virtual control problem dynamics is given by first-order PDE, initial condition and boundary condition (5.4) (5.5) (5.6) t (t, ) = A (t, ) + B (t, ) + w(t, ), (-T , ћ) = x0 (ћ) X B ^ ^ (t, ) = 0 (t, ) EX .

Let the inner product and norm on X , W be denoted by x, y and x , respectively. Let C, Q > 0. The payoff and value are given by
0

(5.7) J z (-T , x, w) =
-T wW
s

1 2

(t, ћ), C (t, ћ) -

1 2

w(t, ћ)

2

dt + ( (0, ћ), z )

(5.8) W z (-T , x) = sup J z (-T , x, w) .1 where (x, z ) = 2 (0, ћ) - z , Q( (0, ћ) - z ) . The first step is to obtain the verification result. Theorem 5.1. Suppose V z C ([0, T ] Ч X ) C 1 ((0, T ] Ч X ) satisfies (5.9) (5.10) (5.11) 0 = -Vtz + ( +
z 1 2 x

V z , Ax - (
x

x

v ) , B x +



V

z2

+

1 2

x, C x ,

V (0, x) = (x, z ),
x

^^ V z (t, , x()) = 0

B ^ (t, ) EP , x H 1 (L).

Then, V z (T , x) J z (-T , x, w) for al l w W , for al l x H 1 (L). Further, if there exists a solution, to (5.4)-(5.6) with w (t, , (t, )), then letting w (t, ) = w (t, , (t, )), one has V z (T , x) = J z (-T , x, w ), and ~ ~ z z consequently V (T , x) = W (-T , x). The next step is to note that the solution for this problem has a simple form.


Max-plus convolution and fundamental solution

29

Theorem 5.2. For any z X , there is a solution to (5.9)-(5.11) in V z C ([0, T ] Ч X ) C 1 ((0, T ] Ч X ) of the form (5.12) 1 V z (t, x) = 1 (x - Z (t, ћ)), P (t, ћ)(x - Z (t, ћ)) + 2 Z (t, ћ), R(t)Z (t, ћ) , 2 where P satisfies (5.1)-(5.3) with (5.13) p0 () = Q L,

Z C 1 ([0, T ] Ч L) satisfies (5.14) (5.15) (5.16) 0 = P Zt + [AP + C ]Z + B P Z , Z (0, ) = z () ^ Z (t, ) = 0 L,
B ^ (t, ) EX ,

and R C 1 ([0, T ]) satisfies (5.17) Rt = C , R(0) = 0.

Note that in the case where C = 0, the Z PDE takes the simpler form (5.18) 0 = Zt + AZ + B Z . . Z (t, ћ) = M(t)[z ](ћ) =
L

Note also that Z (t, ћ) is given by a linear operator acting on z , denoted as (5.19) M (t; ћ, )z ( ) d .

Using this in (5.12), one has (5.20) 1 V z (t, x) = 2 (x - M(t)z ), P (t, ћ)(x - M(t)z ) +

1 2

M(t)z , R(t)M(t)z .

We may think of V z (t, ћ) as given by the max-plus linear semigroup V z (t, x) = St [ (ћ, z )](x). Introducing the semiconvex dual, one may propagate instead in the dual space. The dual-space semigroup operator is naturally found in the form of a max-plus integral operator with some kernel, which we denote by B (t; x, z ) for t [0, T ] and x, z X . One obtains B (t; x, z ) from B (t; y , z ) = - max
xX 1 2

(y - x), Q(y - x)
1 2

-

1 2

(x - M(t)[z ], P (x - M(t)[z ] +

M(t)z , R(t)M(t)z

.

Further details will appear in the full paper.


30 Bibliography

Vladimir F. Molchanov

[1] M. Akian, S. Gaub ert and A. Lakhoua, The max-plus finite element method for optimal control problems: further approximation results, Proc. joint 44th IEEE Conf. on Decision and Control and Europ ean Control Conf. (2005). [2] F.L. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadrat, Synchronization and Linearity, John Wiley, New York, 1992. [3] G. Cohen, S. Gaubert and J.-P. Quadrat, Duality and Separation Theorems in Idempotent Semimodules, Linear Algebra and Applications, 379 (2004), pp. 395- 422. [4] W.H. Fleming and W.M. McEneaney, A max-plus based algorithm for an HJB equation of nonlinear filtering, SIAM J. Control and Optim., 38 (2000), pp. 683- 710. [5] V.N. Kolokoltsov and V.P. Maslov, Idempotent Analysis and Its Applications, Kluwer, 1997. [6] G.L. Litvinov, and V.P. Maslov (eds.), Idempotent Mathematics and Mathematical Physics, Contemp orary Math. 377, Amer. Math. Society, Providence, 2005. [7] G.L. Litvinov, V.P. Maslov and G.B. Shpiz, Idempotent Functional Analysis: An Algebraic Approach, Mathematical Notes, 69 (2001), pp. 696-729. [8] V.P. Maslov, On a new principle of superposition for optimization problems, Russian Math. Surveys, 42 (1987), pp. 43-54. [9] W.M. McEneaney, A New Fundamental Solution for Differential Riccati Equations Arising in Control, Submitted to Automatica. [10] W.M. McEneaney, Max-Plus Methods for Nonlinear Control and Estimation, Birkhauser, Boston, 2006.

Polynomial quantization on para-hermitian symmetric spaces from the viewp oint of overgroups: an example1 Vladimir F. Molchanov
Quantization in the spirit of Berezin on para-Hermitian symmetric spaces G/H was constructed by the author in [2]. One of the variants of quantization is the so-called polynomial quantization (here for the initial algebra of operators, one has to take a representation of the universal enveloping algebra). A construction of polynomial quantization on para-Hermitian symmetric spaces G/H was presented in [4]. For rank one, explicit formulas were given in [3]. In this paper we consider a new approach to the polynomial quantization using the notion of an "overgroup". This
05-0100074a, No. 05-01-00001a and 07-01-91209 YaF a, the Netherlands Organization for Scientific Research (NWO): grant 047-017-015, the Scientific Program "Devel. Sci. Potent. High. Scho ol": pro ject RNP.2.1.1.351 and Templan No. 1.2.02.
1Supp orted by the Russian Foundation for Basic Research: grants No.


Polynomial quantization on para-hermitian spaces

31

approach gives the Berezin covariant and contravariant symbols and the Berezin transform in a highly natural and transparent way. In the paper we restrict ourselves to a simple but crucial example: G = SL(2, R) with the diagonal subgroup H and G = G Ч G. 1. Groups, subgroups, a cone, sections The group G = SL(2, R) consists of real matrices g= , - = 1.

Its subgroups H , Z , N of G consist of matrices h= 0 0 -
1

, z =

1

0 1

, n =

1 0

1

,

respectively. The Gauss and "anti-Gauss" decompositions of G are defined by G = N H Z and G = Z H N . The group G acts on Z and N by fractional linear transformations: + + ћg = , g = . + + These actions are obtained when we decompose z g "by Gauss" and n g "by anti-Gauss". We can reduce the second action to the first one: g = ћ g where . g= We assume that the groups act from the right, in accordance with this we will write vectors in the row form. Let us take the following bilinear form in the space R4 : [x, y ] = -x1 y1 - x2 y2 + x3 y3 + x4 y4 . Realize R as the space Mat(2, R) of real 2 Ч 2 matrices: x= 1 2 x1 - x x2 + x
4 3 4

-x2 + x3 x1 + x4

.

Denote the matrix corresponding to the vector xJ , J = diag{1, -1, -1, -1}, by x . Then the form [x, y ] can be written in terms of matrices: [x, y ] = -2 tr x y . As an overgroup for G, we take the direct product G = G Ч G. It acts on Mat(2, R) as follows: to a pair (g1 , g2 ) G we assign the transformation (1.1)
- x g1 1 xg2 .


32

Vladimir F. Molchanov

This action preserves det x = -(1/4)[x, x]. Therefore G covers the group SO0 (2, 2) with multiplicity 2, and the kernel of the homomorphism consists of two pairs: (e, e) and (-e, -e), e being the unit matrix in G. Let C be the cone in R4 defined by [x, x] = 0, x = 0 (or det x = 0, x = 0). Let us take the following two points in C : s- = (1, 0, 0, -1) = 1 0 0 0 , s+ = (1, 0, 0, 1) = 0 0 0 1 ,

and two parabolic sections - = {[x, s+ ] = -2} and + = {[x, s- ] = -2} containing s- and s+ respectively. Consider in G two unipotent subgroups Q- and Q+ consisting of pairs (z , n ) and (n , z ) respectively. They act simply transitively on sections - and + respectively and transfer points s- and s+ to the points u = u( , ) = (1 - , - - , - + , -1 - ), v (1.2) where N ( , ) = 1 - . In terms of matrices, the vectors u and v are written as follows: u= 1 - - = 1 - (1 ), v = - - 1 = - 1 ( 1), = v ( , ) = (1 - , + , - , 1 + ), [u, v ] = -2N (1 , 2 )N (2 , 1 ), respectively. Let u = u(1 , 1 ) and v = v (2 , 2 ), then

The relation between u and v is given by: Let X be the section of the cone C by It is a hyperboloid of one sheet: -x2 + x2 2 3 Using maps of points along generating G on sections. Let (g1 , g2 ) G. For X we (1.3) xx=

u = v J or u = v . the plane x1 = 1 (or tr x = 1). + x2 = 1, in R3 . 4 lines in C , we obtain actions of have:

- g1 1 xg2 . - tr(g1 1 xg2 )

For - and + , these actions are given by fractional linear transformations of and : u( , ) u( ћ g1 , g2 ),
- +

v ( , ) v ( ћ g2 , g1 ).

Each of the sections and is mapped on X along the generating lines (almost everywhere): ux= u v u( , ) v ( , ) = , vy= = . u1 N ( , ) v1 N ( , )


Polynomial quantization on para-hermitian spaces These maps give the following two systems of coordinates , on X : - + ,- N ( , ) N ( , - + , , y = 1, N ( , ) N ( , ) x = 1, - 1 + ,- , ) N ( , ) 1 + . N ( , )

33

Let us call these coordinates the horospherical coordinates corresponding to - and + respectively. The relation between these two systems is given by: x = y J or x = y . Let us take the following measures on the sections X , - and + : dx = |x4 |
-1

dx2 dx3 ,

du = d d ,

dv = d d .

Under the maps mentioned above, the measures are related as follows: dx = dy = 2N ( , )-2 d d . 2. Representations of G = SL(2, R) The representations T, , C, = 0, 1, of G act on functions ( ) on R by: (T, (g ))( ) = ( ћ g )( + )2, , where we use the notation: t
,

= |t| sgn t, C, = 0, 1, t R \ {0}.

Together with these representations, we consider the representations T, (g ) = T, (g ), so that (T, (g ) )( ) = ( g )( + )
2, ,

(notice that T, and T, are equivalent). The operator A (A
,

defined by

)( ) =
-

(1 - )-

2 -2,

( ) d ,
-1,

intertwines T, with T--1, and also T, with T- A--1, A, is a scalar operator: A where (, ) =
- -1,

. The product

A

,

= (, ) ћ id,

2 2 - tan . 2 + 1 2


34 Notice that (2.1) (- - 1, ) = (, ).

Vladimir F. Molchanov

3. Representations of G = G Ч G asso ciated with a cone For C, = 0, 1, let D, (C ) denote the space of functions f C (C ) satisfying the condition: f (tx) = t Let R
, ,

f (x), x C , t R \ {0}.

be the representation of G on D, (C ) by translations: (R
, - (g1 , g2 )f )(x) = f (g1 1 xg2 ).

In fact, it is the representation of the group SO0 (2, 2) associated with a cone [1]. This representation can be realized on functions on sections of the cone C , see  1. In the realization on X , the representation R, is given by (see (1.3)): (R On
- , - (g1 , g2 )f )(x) = f (x) tr(g1 1 xg2 ) ,

, x X.
,

and on + we have respectively:
,

(3.1) (R (3.2) (R

(g1 , g2 )f )( , ) = f ( ћ g1 , g2 ) (1 + 1 )(2 + 2 )
,

, .

,

(g1 , g2 )f )( , ) = f ( ћ g2 , g1 ) (2 + 2 )(1 + 1 )

Formulas (3.1) and (3.2) show that R, (g1 , g2 ) is the tensor product T, (g1 ) T, (g2 ) and T, (g2 ) T, (g1 ) respectively with = /2. Define the operator B, in the X -realization by (3.3) (B, f )(x) =
X ,

[x, y ]-

-2,

f (y ) dy , x X .
-

It intertwines R (B

with R

--2,

. It acts from
--2,

to + by

,

f )(u) = 2
+

[u, v ]

f (v ) dv , u - ,

and similarly from - to + . By (1.2) it can be written as (B
,

f )(1 , 1 ) = (-1) 2-
-1 + --2,

N (1 , 2 )N (2 , 1 )

f (2 , 2 ) d2 d2 .

It shows that B, = (-1) 2-
-1

A

,

A

,

, = /2.


Polynomial quantization on para-hermitian spaces Therefore (3.4) B, B-
-2,

35

= (/2, )

2

ћ id.

Let us go back to the X -realization and use the both horospherical coordinate systems. Then (3.5) (B
,

f )(x) = (-1) 2-

-2 X

N (1 , 2 )N (2 , 1 ) N (1 , 1 )N (2 , 2 )

--2,

f (y ) dy ,

where x and y have coordinates 1 , 1 and 2 , 2 in the horospherical coordinate systems corresponding to - and + , respectively. 4. The Berezin symb ols and the Berezin transform The group G contains three subgroups isomorphic to G. The first one is the diagonal consisting of (g , g ), g G. It preserves X under the action (1.1), hence X = G/H . The measure dx is invariant. The representation R, is the representation by translations: R
,

(g , g )f (x) = f (g

-1

xg ).

Other two subgroups G1 and G2 consist of pairs (g , e) and (e, g ), g G, respectively. By (3.2) we have on + : (R, (e, g )f )( , ) = f ( ћ g , )( + ), . Therefore, in the horospherical coordinates on X corresponding to + , we have that , 1 (4.1) (R, (e, g )f )( , ) = f ( ћ g , )N ( ћ g , ), ( + ), . N ( , ) This equation can be rewritten as follows. Denote
,

( , ) = N ( , ), .

It is the kernel of the intertwining operator for G (see Sect. 1) and it is an analogue of the Berezin supercomplete system. Then (4.1) is 1 (T/2, (g ) 1) f ( , ), ( , ) . (R, (e, g )f )( , ) = , ( , ) Similarly, in the horospherical coordinates on X corresponding to - , we obtain that 1 (1 T/2, (g )) f ( , ), ( , ) . (R, (e, g )f )( , ) = , ( , ) (and similar formulas for (g , e)). Let us go from the group G to its universal enveloping algebra Env(g) and retain symbols for representations. Then


36

Vladimir F. Molchanov

the dependence of representations on disappears and we omit for them. Now take for f the function f0 equal to 1 identically, then for X Env(g) we obtain that 1 (T/2 (X ) 1), ( , ), (R (0, X )f0 )( , ) = , ( , ) 1 (R--2 (0, X )f0 )( , ) = (1 T-/2-1 (X ))--2, ( , ). --2, ( , ) The right hand sides of these formulas are just the covariant and contravariant symbols of the operator T/2 (X ) in the polynomial quantization. We can normalize the operator B ator Q, will satisfy the condition Q, Q- Namely, (4.2) (Q, f )(x) = c(, )
X --2,

so that the normalized oper-

-2,

= id.
,

(1 - 1 2 )(1 - 2 1 ) (1 - 1 1 )(1 - 2 2 ) = 2 (/2, ),

f (y ) dy ,

where x and y have coordinates 1 , 1 and 2 , 2 as in (3.5) and (4.3) c(, )
-1

see (2.1), (3.4), (3.5). The kernel in (4.2) (with the factor c(, )) is nothing but the Berezin kernel. Therefore, the operator Q, is the Berezin transform. It transfers contravariant symbols to covariant ones. Note that if we want to write the Berezin transform using only one coordinate system, then we will have to change the operator (3.3), namely, we will have to write [x, y J ] instead of [x, y ]. Bibliography
[1] V. F. Molchanov, Representations of the pseudo-orthogonal group associated with a cone. Mat. Sb., 1970, vol. 81, No. 3, 358-375 (in Russian). Engl. transl.: Math. USSR Sb., 1970, vol. 10, 333-347. [2] V. F. Molchanov, Quantization on para-Hermitian symmetric spaces. Amer. Math. Soc. Transl., Ser. 2, 1996, vol. 175 (Adv. in Math. Sci.-31), 81-95. [3] V. F. Molchanov, N. B. Volotova. Polynomial quantization on rank one paraHermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, Nos. 1-3, 215-222. [4] V. F. Molchanov, N. B. Volotova. Polynomial quantization on para-Hermitian symmetric spaces. Vestnik Tamb ov Univ., 2005, vol. 10, No. 4, 412-424.


The structure of max-plus hyperplanes

37
1

The structure of max-plus hyp erplanes V. Nitica and I. Singer

A max-plus hyperplane (briefly, a hyperplane) is the set of all points x = (x1 , ..., xn ) in Rn satisfying an equation of the form max a1 x1 ... an xn a that is, max(a1 + x1 , ..., an + xn , an
+1 n+1

= b1 x1 ... bn xn bn

+1

, ),

) = max(b1 + x1 , ..., bn + xn , bn

+1

with ai , bi Rmax (i = 1, ...n + 1), where each side contains at least one term, and where ai = bi for at least one index i. We show that the complements of (max-plus) semispaces at finite points z Rn are "building blocks" for the hyperplanes in Rn max (recall that a semispace at z is a maximal -with respect to inclusion- max-plus convex subset of Rn \{z }). max Namely, observing that, up to a permutation of indices, we may write the equation of any hyperplane H in one of the following two forms: a1 x1 ... ap xp ap
+1 xp+1

... aq x

q n+1

= a1 x1 ... ap xp aq a1 x1 ... ap xp ap

+1 xq +1

... am xm a

,

where 0 p q m n and all ai (i = 1, ..., m, n + 1) are finite, or,
+1 xp+1

... aq xq an

+1 +1

= a1 x1 ... ap xp aq

+1 xq +1

... am xm an

,

where 0 p q m n, and all ai (i = 1, ..., m) are finite (and an+1 is either finite or -), we give a formula that expresses a nondegenerate strictly affine hyperplane (i.e., with m = n and an+1 > -) as a union of complements of semispaces at a point z Rn , called the "center" of H, with the boundary of a union of complements of other semispaces at z . Using this formula, we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior. We give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center, and we characterize the cases in which the boundary bd H of a nondegenerate strictly affine hyperplane H is also a hyperplane. Next, we give the relations between nondegenerate strictly affine hyperplanes H , their centers z , and their coefficients ai . In the converse direction we show that any union of complements of semispaces at a point z Rn with the boundary of any union of complements
1Partially supp orted by NSF grant DMS-0500832 and by grant nr. 2-CEx06-11-

34/2006.


38

V. Nitica and I. Singer

of some other semispaces at that point z , is a nondegenerate strictly affine hyperplane. We obtain a formula for the total number of strictly affine hyperplanes. We give complete lists of all strictly affine hyperplanes for the cases n = 1 and n = 2. We show that each linear hyperplane H in Rn max (i. e., with an+1 = -) can be decomposed as the union of four parts, where each part is easy to describe in terms of complements of semispaces, some of them in a lower dimensional space. The paper in extenso will appear in Linear Algebra and its Applications.

Image pro cessing based on a partial differential equation satisfying the pseudo-linear sup erp osition principle1 E. Pap and M. Strboja
We consider a general form of PDE-based methods for image restoration, and give a short overview of the underlying models. In these models, the original image is transformed through a process that can be represented by a second-order partial differential equation. Typically, this role is played by some nonlinear generalization of the heat equation, and it is possible to analyse the solutions from the viewpoint of pseudo-linear (idempotent) analysis. Our main result is that the generalization of the heat equation proposed by Perona and Malik satisfies the pseudo-linear superposition principle. 1. Intro duction The approach based on partial differential equations is well-known in image processing ([A, C, T]). In this approach, a restored image can be seen as a version of the initial image at a special scale. Image u is an instance of an evolution process, denoted by u (t, ћ). The original image is taken at time t = 0, u (0, ћ) = u0 (ћ) . The original image is then transformed, and this process can be described by the equation u 2 u (t, x) = 0, where x . Some pos t (t, x) + F x, u(t, x), u (t, x) , sibilities for F to restore an image are considered in [A].
1Partially supp orted by the Pro ject MNZZSS 144012, grant of MTA HTMT, French-Serbian pro ject "Pavle Savi? and by the pro ject "Mathematical Models for c", Decision Making under Uncertain Conditions and Their Applications" of Academy of Sciences and Arts of Vo jvodina supp orted by Provincial Secretariat for Science and Technological Development of Vo jvo dina.


Image processing based on PDE

39

Pseudo-linear superposition principle means the following. Instead of the field of real numbers, think of a semiring defined on a real interval [a, b] [-, ]. This is a structure equipped with pseudo-addition , which is typically idempotent (x x = x), and with pseudo-multiplication . The pseudo-linear superposition principle says that some nonlinear equations (ODE, PDE, difference equations, etc.) turn out to be linear over such structures, meaning that if u1 and u2 are two solutions of the considered equation, then a1 u1 a2 u2 is also a solution for any constants a1 and a2 from [a, b]. By pseudo-analysis we mean analysis over such semirings, in the framework of [L, M, N, O, P, Q, R]. One of its key ideas is the pseudolinear superposition principle stated above. This (pseudo-) linear intuition leads to the concepts of -measure, pseudo-integral, pseudo-convolution, pseudo-Laplace transform, etc. Similar ideas were developed independently by Maslov and his collaborators in the framework of idempotent analysis and idempotent mathematics, with some applications [G, H, J, K]. In particular, idempotent analysis is fundamental for the theory of weak solutions to HamiltonJacobi equations with non-smooth Hamiltonians, see [G, H, K] and also [Q, R] (which use the language of pseudo-analysis). In some cases, this theory enables one to obtain exact solutions in the similar form as for the linear equations. Some further developments relate more general pseudooperations with applications to nonlinear partial differential equations, see [S]. Recently, these applications have become important in the field of image processing [Q, R]. Our report is organized as follows. In Sect. 2 we consider a general form of PDE for image restoration. The starting PDE in image restoration is the heat equation. Because of its oversmoothing property (edges get smeared), it is necessary to introduce some nonlinearity. We consider then the following model ([A, T]) u 2 = div c | u| t u,

where we choose the function c such that the equation remains to be of the parabolic type. We take c(s) 1/ s as s , because we want to preserve the discontinuities [A]. Because of this behavior, it is not possible to apply general results from parabolic equations theory. An appropriate framework to study this equation is nonlinear semigroup theory ([A, B, D]). In Sect. 3 we show that Perona and Malik equation satisfies the pseudo-linear superposition principle.


40 2. PDE-based metho d in image pro cessing

E. Pap, M. Strbo ja

PDE-methods for restoration can be written in the following general form: (t, x) + F x, u(t, x), u (t, x) , 2 u (t, x) = 0 in (0, T ) Ч , (t, x) = 0 on (0, T ) Ч (Neumann boundary condition), u (0, x) = u0 (x) (initial condition),
u t u N

where u (t, x) is the restored version of the initial degraded image u0 (x). The idea is to construct a family of functions {u (t, x)}t>0 representing successive versions of u0 (x). As t increases, the image u (t, x) becomes more and more simplified. We would like to attain two goals. The first is that u (t, x) should represent a smooth version of u0 (x), where the noise has been removed. The second, u (t, x) should be able to preserve some features such as edges, corners, which may be viewed as singularities. The heat equation is the basic PDE for image restoration:
u t

(t, x) - u (t, x) = 0, t 0, x R2 , u (0, x) = u0 (x) .

The heat equation has been successfully applied in image processing but it has some drawback. It is too smoothing and because of that edges can be lost or severely blurred. In [A] authors consider models that are generalizations of the heat equation. Suppose that the domain image is a bounded open set of R2 . The following equation was initially proposed by Perona and Malik [T]:
u t u N

= div c | u|

2

u

in (0, T ) Ч ,

(2.1)

= 0 on (0, T ) Ч , u (0, x) = u0 (x) in

where c : [0, ) (0, ) . If we choose c 1, then it is reduced to the heat equation. If we assume that c (s) is a decreasing function satisfying c (0) = 1 and lim c (s) = 0, then inside the regions where the magnitude
s

of the gradient of u is weak, the edges are preserved. For each point x where and T with TћN = 0, |T| = u we use the usual notation

equation (2.1) acts like the heat equation and | u| = 0 we can define the vectors N = | u| u 1. For the first and second partial derivatives of ux1 , ux2 , ux1 x1 ,... We denote by uNN and uTT


Image processing based on PDE

41

the second derivatives of u in the N-direction and T-direction, respectively: uNN uTT =N
t 2

uN=

1 | u| 1 | u|
2

(u2 uxx + u2 uyy + 2ux uy uxy ), x y u2 u x
yy

= Tt

2

uT=

2

+ u2 uxx - 2ux uy u y

xy

.

The first equation in (2.1) can be written as u 2 2 (t, x) = c | u (t, x)| uTT + b | u (t, x)| uNN , t where b(s) = c(s) + 2sc (s). Therefore, (2.2) is a sum of a diffusion in the T-direction and a diffusion in the N-direction. The functions c and b act as weighting coefficients. Since N is normal to the edges, it would be preferable to smooth more in the tangential direction T than in the normal direction. Because of that we impose (2.2) (2.3)
s

lim

b(s) sc (s) 1 = 0 or lim =- s c(s) c(s) 2

If c(s) > 0 with power growth, then (2.3) implies that c(s) 1/ s as s . The equation (2.1) is parabolic if b(s) > 0. The assumptions imposed on c (s) are c : [0, ) (0, ) decreasing, 1 c(0) = 1, c(s) s as s , (2.4) b(s) = c(s) + 2sc (s) > 0.
1 Consider c(s) = 1+s , an often used function satisfying (2.4). Because of the behavior c(s) 1/ s as s , it is not possible to apply general results from parabolic equations theory. An appropriate framework to study this equation is nonlinear semigroup theory (see [A, B, D]).

3. Pseudo-linear sup erp osition principle for Perona and Malik equation Let [a, b] be a closed (in some cases semiclosed) subinterval of [-, +]. We consider here the total order on [a, b]. The operation (pseudoaddition) is a commutative, non-decreasing, associative function : [a, b] Ч [a, b] [a, b] with a zero (neutral) element denoted by 0. Denote [a, b]+ = {x : x [a, b] , x 0}. The operation (pseudo-multiplication) is a function : [a, b] Ч [a, b] [a, b] which is commutative, positively nondecreasing, i.e., x y implies x z y z , z [a, b]+ ,associative and for


42

E. Pap, M. Strbo ja

which there exist a unit element 1 [a, b] , i.e., for each x [a, b] , 1 x = x. We assume 0 x = 0 and that is distributive over , i.e., x (y z ) = (x y ) (x z)

The structure ([a, b] , , ) is called a semiring (see [I, O]). In this paper we shall consider only the min-plus (or tropical) semiring. It is defined on the interval (-, +] and has the following continuous operations: x y = min {x, y } , x y = x + y . Note that the pseudo-addition is idempotent, while the pseudo-multiplication is not. We have 0 = - and 1 = 0. We show that the pseudo-linear superposition principle holds for Perona and Malik equation. Theorem 3.1. If u1 = u1 (t, x) and u2 = u2 (t, x) are solutions of the equation (3.1) then (1 u1 ) (
2

u 2 - div c | u| u = 0, t u2 ) is also a solution of (3.1) on the set = +.

D = {(t, x) |t (0, T ) , x R2 , u1 (t, x) = u2 (t, x)}, with respect to the operations = min and

The obtained results will serve for further investigation of the weak solutions of the equation (3.1) in the sense of Maslov [G, H] and Gondran [E, F], as well as some important applications. Bibliography
[A] G. Aub ert, P. Kornprobst, Mathematical Problems in Image Processing, SpringerVerlag, 2002. [B] H. Breyis, Op? ateurs Maximaux Monotones et Semi-Groupes de Contractions dans er les Espaces de Hilbert, North-Holland Publishing Comp, Amsterdam-London, 1973. [C] F. Catte, P.L. Lions, J.M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal of Numerical Analysis, 29(1):182193, 1992. [D] T. Cazenave, A. Haraux, Introduction aux Problemes d'Evolution Semi-Lin?ares, e (Intro duction to Semilinear Evolution Problems), Math? ematiques & Apllications, Ellipses, 1990. [E] M. Gondran, Analyse MINPLUS, Analyse fonctionnelle/Functional Analysis, C. R. Acad. Sci. Paris, t. 323, S? erie I, p. 371-375, 1996. [F] M. Gondran, M. Minoux, Graphes, dioides et semi-anneaux, Editions TEC & DOC, Londres- Paris- New York, 2001. [G] V. N. Kolokoltsov, V. P. Maslov, Idempotent calculus as the apparatus of optimization theory. I, Functional. Anal. i Prilozhen 23, no. 1, (1989), 1-14.


Image processing based on PDE

43

[H] V. N. Kolokoltsov, V. P. Maslov, Idempotent Analysis and Its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1997. [I] W. Kuich: Semirings, Automata, Languages, Berlin, Springer-Verlag, 1986. [J] G. L. Litvinov, The Maslov Dequantization, Idempotent and Tropical Mathematics: a very Brief Introduction, Cont. Mathematics 377, AMS, (2005), 1-17. [K] V. P. Maslov, S.N. Samb orskij (eds.), Idempotent Analysis, Advances in Soviet Mathematics 13, Providence, Rhode Island, Amer. Math. So c.., 1992. [L] E. Pap, An integral generated by decomposable measure, Univ. u Novom Sadu Zb. Rad. Priro d.-Mat. Fak. Ser. Mat. 20 (1) (1990), 135-144. [M] E. Pap, g-calculus, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23 (1) (1993), 145-156. [N] E. Pap, Applications of decomposable measures, in Handbo ok Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory (Ed. U. Hohle, R.S. Rodabaugh), Kluwer Е Academic Publishers, 1999, 675-700. [O] E. Pap, Nul l-Additive Set Functions, Kluwer Academic Publishers, DordrechtBoston-London, 1995. [P] E. Pap, Decomposable measures and nonlinear equations, Fuzzy Sets and Systems 92 (1997) 205-222. [Q] E. Pap, Pseudo-Additive Measures and Their Aplications, Handbo ok of Measure Theory (Ed. E. Pap), Elsevier, Amsterdam, 2002, 1403-1465, [R] E. Pap, N. Ralevi? Pseudo-Laplace transform, Nonlinear Analysis 33 (1998) 553c, 560. [S] E. Pap, D. Vivona, Non-commutative and associative pseudo-analysis and its applications on nonlinear partial differential equations, J. Math. Anal. Appl. 246 (2) (2000) 390-408. [T] P. Peron, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7): 629-639, 1990.

Tropical analysis on plurisubharmonic singularities Alexander Rashkovskii

1. Plurisubharmonic singularities Recall that an upper semicontinuous, real-valued function on an in Cn is called plurisubharmonic (psh) if its restriction to every line is a subharmonic function. A basic example is log |f | for an function f . Moreover, by Bremermann's theorem [1], every psh u can be written as u(z ) = lim sup lim sup
y z m

open set complex analytic function

1 log |fm (y )|. m


44

Alexander Rashkovskii

Let 0 denote the ring of germs of analytic functions f at 0 Cn , and let m0 = {f 0 : f (0) = 0} be its maximal ideal. The log-transformation f log |f | maps 0 into the collection of germs of psh functions at 0. We will say that a psh germ u has singularity at 0 if u(0) = -. For functions u = log |f | this means f m0 ; asymptotic behaviour of arbitrary psh functions can be much more complicated. By P S H G0 we denote the collections of all psh germs singular at 0. The operations on 0 induce a natural tropical structure on P S H G0 with the addition u v := max{u, v } (which is based on Maslov's dequan1 tization: f + g N log |f N + g N | log |f | log |g | as N ) and multiplication u v := u + v (simply by f g log |f g | = log |f | log |g |). Thus P S H G0 becomes a tropical semiring, closed under (usual) multiplication by positive constants. A partial order on P S H G0 is given as follows: u v if u(z ) v (z ) + O(1) as z 0, which leads to the equivalence relation u v if u(z ) = v (z ) + O(1). The equivalence class cl(u) of u is called the plurisubharmonic singularity of the germ u. The collection of psh singularities P S H S0 = P S H G0 / has the same tropical structure {, } and the partial order: cl(u) cl(v ) if u v . It is endowed with the following topology: cl(uj ) cl(u) if there exists a neighbourhood of 0 and psh functions vj cl(uj ), v cl(u) in such that vj v in L1 ( ). By abusing the notation, we will right occasionally u for cl(u). 2. Characteristics of singularities The main characteristic of an analytic germ f m0 is its multiplicity (vanishing order) mf : if f = Pj is the Taylor expansion of f in homogeneous polynomials, Pj (tz ) = tj P (z ), then mf = min{j : Pj 0}. The basic characteristic of singularity of u P S H G0 is its Lelong number 1 u(z ) (u) = lim M (u, t) = lim inf = ddc u (ddc log |z |)n-1 (0); t- t z 0 log |z | ? here M (u, t) is the mean value of u over the sphere {|z | = et }, d = + , c ?)/2 i. If f m0 , then (log |f |) = mf . This characteristic of d = ( - singularity gives an important information on the asymptotic behaviour of u at 0: u(z ) (u) log |z | + O(1). Since (v ) = (u) for all v cl(u), Lelong number can be considered as a functional on P S H S0 with values in the tropical semiring R+ (min, +) ? of non-negative real numbers with the operations xy = min{x, y } and x y = x + y . As such, it is


Tropical analysis on plurisubharmonic singularities

45

(i) positive homogeneous: (cu) = c (u) for all c > 0, ? (ii) additive: (u v ) = (u) (v ), (iii) multiplicative: (u v ) = (u) (v ), and (iv) upper semicontinuous: (u) lim sup (uj ) if uj u. Lelong numbers are independent of the choice of coordinates. Let us now fix a coordinate system (centered at 0). The directional Lelong number of u in the direction a Rn (introduced by C. Kiselman [5]) is + (2.1) (u, a) = lim
t-

1 u(z ) M (u, ta) = lim inf , z 0 a (z ) t

where M (u, ta) is the mean value of u over the distinguished boundary of the polydisk {|zk | < exp(tak )} and a (z ) = k a-1 log |zk |. It has k the same properties (i)-(iv), and the collection { (u, a)}a gives a refined information on the singularity u. In particular, (u) = (u, (1, . . . , 1)). A general notion of Lelong number with respect to a plurisubharmonic weight was introduced by J.-P. Demailly [2]. Let P S H G0 be continuous and locally bounded outside 0. Then the mixed Monge- Amp` current ddc u (ddc )n-1 is well defined for any psh function u ere and is equivalent to a positive Borel measure. Its mass at 0, (u, ) = ddc u (ddc )n-1 ({0}), is called the generalized Lelong number, or the Lelong-Demail ly number, of u with respect to the weight . Since it is constant on cl(u), we have a different kind of functional on P S H S0 . It still has the above properties (i), (iii), and (iv), however in general is only ? subadditive: (u v , ) (u, ) (v , ). Note that (u, a) = a1 . . . an (u, a ). 3. Additive functionals Another generalization of the notion of Lelong number was introduced [12]. Let P S H G0 be locally bounded and maximal outside 0 (that satisfies (ddc )n = 0 on a punctured neighbourhood of 0); the collection all such germs (weights) will be denoted by M W0 . The type of u P S H relative to M W0 , (u, ) = lim inf
z 0

in is, of S0

u(z ) , (z )

gives the bound u (u, ). This functional is positive homogeneous, additive, supermultiplicative, and upper semicontinuous. Actually, relative types give a general form for all "reasonable" additive functionals on P S H S0 :


46

Alexander Rashkovskii Theorem 3.1. ([12]) Let a functional : P S H S0 [0, ] be such

that 1) (cu) = c (u) for al l c > 0; ? 2) (uk ) = (uk ), k = 1, 2; 3) if uj u, then lim sup (uj ) (u); 4) (log |z |) > 0; 5) (u) < if u -. Then there exists a weight M W0 such that (u) = (u, ) for every u P S H S0 . The representation is essential ly unique: if two maximal weights and represent , then cl() = cl( ). 4. Relative typ es and valuations Recall that a valuation on the analytic ring ч : 0 [0, +] such that ч(f1 f2 ) = ч(f1 ) + ч(f2 ),
0

is a nonconstant function ч(1) = 0;

ч(f1 + f2 ) min {ч(f1 ), ч(f2 )},

a valuation ч is centered if ч(f ) > 0 for every f m0 , and normalized if min {ч(f ) : f m0 } = 1. Every weight M W0 generates a functional on 0 , (f ) = (log |f |, ), with the properties (f1 f2 ) (f1 ) + (f2 ), (f1 + f2 ) min { (f1 ), (f2 )}, (1) = 0. It is a valuation, provided (u, ) is tropically multiplicative; is centered iff (log |z |, ) > 0, and normalized iff (log |z |, ) = 1. One can thus consider linear (both additive and multiplicative) functionals on P S H S0 as tropicalizations of certain valuations on 0 . For example, the (usual) Lelong number is the tropicalization of the multiplicity valuation mf . The types relative to the directional weights a are multiplicative functionals on P S H S0 , and a are monomial valuations on 0 ; they are normalized if mink ak = 1. It was shown in [4] that an important class of valuations (quasi-monomial valuations, or Abhyankar valuations of rank 1) can be realized as with certain weights M W0 ; when n = 2, all other centered valuations are limits of increasing sequences of the quasi-monomial ones [3]. 5. Lo cal indicators as Maslov's dequantizations Consideration of psh germs is the first step of Maslov's dequantization of analytic functions 0 f log |f | P S H G0 . One can perform the next step - namely, passage to the logarithmic scale in the arguments z .


Tropical analysis on plurisubharmonic singularities

47

For a fixed coordinate system at 0, let (u, a) be the directional Lelong numbers of u P S H S0 in the directions a Rn (2.1). Then the + function u (t) = - (u, -t), t Rn , is convex and increasing in each tk , - so u (log |z1 |, . . . , |zn |) can be extended (in a unique way) to a function u (z ) plurisubharmonic in the unit polydisk Dn = {z Cn : |zk | < 1, 1 k n}. This function is called the local indicator of u at 0 [7]. It is easy to see that it has the homogeneity property (5.1) u (z1 , . . . , zn ) = u (|z1 |, . . . , |zn |) = c-1 u (|z1 |c , . . . , |zn |c ) c > 0. It was shown in [9] that u (z ) can be represented as the (unique) weak m m limit of the functions m-1 u(z1 , . . . , zn ) as m , so the indicator can be viewed as the tangent (in the logarithmic coordinates) for the function u at 0. This means that for u = log |f |, f m0 , the sublinear function u (t) on Rn is just a Maslov's dequantization of f . - The indicator is a psh characteristic of asymptotic behaviour near 0. Namely, if u is psh in the unit polydisk Dn , then u(z ) u (z ) + supD u. When u has isolated singularity at 0, this implies the following relation between the residual Monge-Amp` masses: (ddc u)n (0) (ddc u )n (0). ere Since u is much simpler than the original function u, one can compute explicitly the value of its residual mass. The first equation in (5.1) suggests us to pass from plurisubharmonic functions to convex ones and from the complex Monge-Amp` operator to the real one, while the secere ond equation allows us to calculate the real Monge-Amp` measure in ere terms of volumes of gradient images. Denote u,x = {b Rn : + sup [ (u, a) - b, a ] 0}.
ak =1

The convex image u (t), t Rn , of the indicator is just the support - function to the convex set u = Rn \ u,x : u (t) = sup { t, a : a u }. + This gives Theorem 5.1. ([9]) The residual Monge-Amp` e mass of u P S H G0 er with isolated singularity at 0 has the lower bound (ddc u)n (0) (ddc u )n (0)) = n! Vol(
u,x

).

If F = (f1 , . . . , fn ) is a holomorphic mapping with isolated zero at 0, then its multiplicity at 0 equals (ddc log |F |)n (0) and the set log |F | is the convex hull of the union of the Newton polyhedra conv{ + Rn : + D() fj (0) = 0} of fj at 0, 1 j n. In this case, Theorem 2 gives us Kushnirenko's theorem on multiplicity of holomorphic mappings [6].


48

Alexander Rashkovskii

The results on local indicators have global counterparts concerning psh functions of logarithmic growth in Cn (i.e., u(z ) A log(1 + |z |) + B everywhere in Cn , a basic example being logarithm of modulus of a polynomial), see [10] and [11]. Similar notions concerning Maslov's dequantization in Cn and generalized Newton polytops were also introduced and studied in [8]. Bibliography
[1] H. Bremermann, On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions, Math. Ann. 131 (1956), 76-86. [2] J.-P. Demailly, Nombres de Lelong g? ? alis?s, th? ` s d'int? alit? et ener e eoreme egr e d'analycit?, Acta Math. 159 (1987), 153-169. e [3] C. Favre and M. Jonsson, Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), 271-311. [4] C. Favre and M. Jonsson, Valuations and plurisubharmonic singularities, http://arXiv.org/math/0702487. [5] C.O. Kiselman, Un nombre de Lelong raffin?, In: S? e eminaire d'Analyse Complexe et G? ? eometrie 1985-87, Fac. Sci. Monastir Tunisie 1987, 61-70. [6] A.G. Kouchnirenko, Poly` es de Newton et nombres de Milnor, Invent. Math. edr 32 (1976), 1-31. [7] P. Lelong and A. Rashkovskii, Local indicators for plurisubharmonic functions, J. Math. Pures Appl. 78 (1999), 233-247. [8] G. L. Litvinov and G. B. Shpiz, The dequantization transform and generalized Newton polytopes, in: "Idemp otent Mathematics and Mathematical Physics", G. L. Litvinov, V. P. Maslov (eds.), AMS, Providence, 2005, p. 181-186. [9] A. Rashkovskii, Newton numbers and residual measures of plurisubharmonic functions, Ann. Polon. Math. 75 (2000), no. 3, 213-231. [10] A. Rashkovskii, Indicators for plurisubharmonic functions of logarithmic growth, Indiana Univ. Math. J. 50 (2001), no. 3, 1433-1446. [11] A. Rashkovskii, Total masses of mixed Monge-Amp` e currents, Michigan Math. er J. 51 (2003), no. 1, 169-186. [12] A. Rashkovskii, Relative types and extremal problems for plurisubharmonic functions, Int. Math. Res. Not., 2006, Art. ID 76283, 26 pp.


Minimal elements and cellular closures

49

Minimal elements and cellular closures over the max-plus semiring1 Serge Sergeev i
This report is based on the publications [2] (part 1) and [7] (part 2). In part 1, I outline some simple consequences of the observation that extremals are minimal elements with respect to the certain preorder relation. Part 2 is occupied with some extensions of algebraic closure operation, which arise from the cellular decomposition considered in [3]. The common feature of these works is a bit of interplay between maxalgebra [1] and tropical convexity [3], [5]. All results are obtained in the setting of Rn ,m , the n-dimensional free semimodule over the semiring max Rmax,m = (R+ , = max, = ). 1. Extremals as minima An element u of a (sub)semimodule K Rn ,m is an extremal, if u = max x y , x, y K implies that u = x or u = y . The preorder relation j is defined by (1.1) u j v uj = 0, vj = 0, u/uj v /vj .

The role of j is explained in the following. Proposition 1.1. [5, 2] The fol lowing are equivalent: (1) y is a (max-)linear combination of x1 , . . . , xm Rn ,m ; max (2) for any j supp(y ), there exists some xl from x1 , . . . , x that xl j y .

m

such

Proposition 1.2. [2] Let a semimodule K be generated by a subset S of Rn ,m . The fol lowing are equivalent. max (1) y is an extremal of K ; (2) for some j , this y is a minimal element of S (and, equivalently, of K ) with respect to j . Proposition 1.2 enables to treat idempotent extremals as minima. The problem of finding partial maxima (and minima) in n-dimensional real space was investigated by F. Preparata et al. [6]. The following estimate is derived from their results.
1Supp orted by the RFBR grant 05-01-00824 and the joint RFBR/CNRS grant

05-01-02807.


50

Serge Sergeev i

Theorem 1.3. [5](for n = 3),[2] Let K be a semimodule in Rn ,m max generated by k elements. The problem of finding al l extremals of K requires not more than O(k log2 k ) operations, if n = 3, and not more than O(k (log2 k )n-3 operations, if n > 3 (with n fixed). Propositions 1.1 and 1.2 imply a number of statements for generators of idempotent semimodules in Rn ,m , see [2] for details. Here I mention max two of them. Theorem 1.4. [8, 2] Let K be a semimodule in Rn ,m generated by max S , and let E be the set of extremals of K such that ||u|| = 12 for al l u E . Then S = E F , where F is redundant in the sense that S - {u} generates K for any u F . As a corollary, the weak basis of a semimodule is essentially unique whenever it exists. The following is a tropical version of Minkowski's theorem. Theorem 1.5. [4, 2] A closedsemimodule in R its extremals. 2. Cellular closures Algebraic closure of a square matrix A is the series I A A2 . . ., where I is the identity matrix. This series converges iff (A) 1, where (A) is the maximal cycle mean of A. This (A) is also the maximal eigenvalue of the problem Ax = x. The corresponding eigenspace will be denoted by eig(A). The following theorem and its corollary are the "ground stone" of this section. Theorem 2.1. [7] Let A and B be two square matrices such that (A) 1 and (B ) 1. Then A = B iff the spaces generated by columns of A and B coincide. A square matrix A is definite, if (A) = 1 and all the diagonal entries equal 1. Corollary 2.2. Let A and B be two definite matrices. Then A = B iff eig(A) = eig(B ). R
n max,m

is generated by

I consider now the concepts of [3]. Let A be an n Ч m matrix over max,m and y an n-comp onent vector. Denote the collection S = {Sj : j
2the choice of norm do es not matter


Minimal elements and cellular closures

51

supp(y )}, where Sj = {i : y j Aћi }, by type(y | A) and call it the combinatorial type of y with respect to A. Combinatorial types can be formally defined as arbitrary collections of not more than n possibly empty subsets of {1, . . . , m}. Denote the set of indices i, whose Si are present in the type, by supp(S ). If S = type(y | A) for some y , then supp(S ) = supp(y ). The types are partially ordered by the rule S S if supp(S ) supp(S ) and Si Si for all i supp(S ). The set X S = {z : S type(z | A)} is the region of S . If Aik = 0 for all i Sk , then S is compatible and we introduce the matrix AS by kSi Aћk /Aik , if i supp(S ) and Si = ; S Aћ i = e i , if i supp(S ) and Si = ; 0, if i supp(S ). / If the region X S is not empty, then X S = eig(AS ). Hence any region is (essentially) the eigenspace of a definite matrix, and we use Corollary 2.2. Theorem 2.3. If S and T are (compatible) types such that X X are not empty and X S = X T , then (AS ) = (AT ) .
T S

and

Theorem 2.3 enables to define various cel lular closures of A to be (AS ) . This operation is correctly defined for every region, being independent of the type. Consider now the case when A is a square n Ч n matrix with a permutation whose weight n Ai(i) is nonzero. A permutation with maximal weight i=1 is called maximal. We define D to be the matrix such that Dij = Aij if j = (i) and Dij = 0 otherwise. If is maximal, then (D )-1 A is definite and is called the definite form of A [7]. Different maximal permutations lead to different definite forms. But we have that eigenspaces of all definite forms coincide (see [7]), and by Corollary 2.2 closures of all definite forms are equal. Thus, for any square matrix A with nonzero permutations, we can define its definite closure to be (D )-1 A) , where is a maximal permutation. Definite closure is a cellular closure, since eig((D )-1 A) is the same as X S with S = ({ (1)}, . . . , { (n)}), where is any maximal permutation. Bibliography
[1] P. Butkovi Max-algebra: linear algebra of combinatorics? Linear Algebra Appl. c, 367 (2003) 313-335.


52

Oleg Yu. Shvedov

[2] P. Butkovi H. Schneider, and S. Sergeev, Generators, extremals and bases of max c, cones. Linear Algebra Appl. 421 (2007) 394-406. Also arXiv:math.RA/0604454. [3] M. Develin and B. Sturmfels, Tropical convexity Documenta Math. 9 (2004) 1-27. Also arXiv:math.MG/0308254. [4] S. Gaubert and R. Katz, The Minkowski theorem for max-plus convex sets. Linear Algebra Appl. 421 (2007) 356-369 . Also arXiv:math.GM/0605078. [5] M. Joswig, Tropical halfspaces. - In Discrete and Computational Geometry (E. Goo dman, J. Pach and E. Welzl, eds.), MSRI Publications, Cambridge Univ. Press, 2005. Also arXiv:math.CO/0312068. [6] F.P. Preparata and M. Shamos, Computational Geometry. An Introduction. Springer, New York, 1985. [7] S. Sergeev, Max-plus definite matrix closures and their eigenspaces. Linear Algebra Appl. 421 (2007) 182-201. Also arXiv:math.MG/0506177. [8] E. Wagneur, Finitely generated moduloЕ ids: The existence and unicity problem for bases. - In J.L. Lions and A. Bensoussan (Eds.), Analysis and Optimization of Systems, Springer, Lecture Notes in Contr. and Inform. Sci. 111, pp. 966-976, 1988.

Semiclassical quantization of field theories1 Oleg Yu. Shvedov
1. It is well-known that equations of quantum field theory (QFT) are ill-defined [1]. One usually investigates the perturbative QFT instead of "exact" QFT: all quantities are presented as formal series in a small perturbation parameter; the QFT divergences are eleminated within a perturbation framework only. Semiclassical approximation [2] may be also viewed as an expansion in a small parameter. Since the well-defined results are obtained within the perturbation theory, it seems to be more reasonable to talk about semiclassical quantization rather than semiclassical approximation. Consider the field theory model with the Lagrangian L depending on the small parameter h ("Planck constant") as follows [3] (the scalar case is considered for the simplictiy): 1 1 (1) L = ч ч - V ( h). 2 h with V () being a scalar potential.
1Supp orted by the RFBR grant 05-01-00824 and the joint RFBR/CNRS grant

05-01-02807.


Semiclassical quantization of field theories

53

In the formal quantum theory, field (x) and momentum (x) are ^ ^ viewed as operators satisfying the canonical commutation relations. Semiclassical states depend on the small parameter h as: (2) (t) e
i h

S (t)

e

i h

dx[(x,t)(x)-(x,t) (x)] ^ ^

f (t).

Here S (t) is a real c-number finction of t, (x, t) and (x, t) are classical fields and canonucally conjugated momenta, (x) and (x) are quantum ^ ^ field and momentum operators, f (t) is a regular as h 0 state vector. Superpositions of states (2) are also viewed as semiclassical states. Presentation of semiclassical form in the form (2) is not manifestly covariant. There are space and time coordinates. It happens that the manifestly covariant form of the state (2) is the following: i i i h (3) e h S T exp{ dxJ (x)h (x)}f e h S TJ f . ^ h Here S is a real number, J (x) is a real function (classical Schwinger source), h (x) is a Heisenberg field operator, f is a state vector being regular as ^ h 0. The Schwinger source J (x) should be rapidly damping at space and time infinity [4]. 2. Investigate properties of the semiclassical state (3). First of all, h h note that the state TJ +hJ f can be expressed via the operator TJ . To do this, it is necessary to investigate the operator (4)
h R (x|J ) -ih(TJ )+ h TJ . J (x)

It happens to coincide with the well-known LSZ R-function [5]. Notice that R (x|J ) is expanded in h; one writes (1) (5) R (x|J ) = R (x|J ) + hR (x|J ) + ... The c-number function R (x|J ) is called as a retarded classical field generated by the Schwinger source J . It is shown in [6] that for the model (1) R (x|J ) is a solution of the equation (6) ч ч R (x|J ) + V (R (x|J )) = J (x), R |
x
<

suppJ

= 0.

which vanishes as x0 -. The following properties are corollaries of (4). 1. The Hermitian property (7) (8) U + (x|J ) = R (x|J ). R 2. The Poincare invariance property
g
-1

R (x|ug J )U g = R (wg x|J ).


54

Oleg Yu. Shvedov

3. The Bogoliubov causality property [1]: R-function R (x|J ) depends only on the source J at the preceeding time moments. Making use of the standard notations x > y iff x0 - y 0 |x - y|, x < y iff x0 - y 0 |x - y|, x y iff |x0 - y 0 | < |x - y|, one rewrites the Bogoliubov condition as R (x|J ) > (9) = 0, y x. J (y ) 4. Commutation relation (10) [R (x|J ); R (y |J )] = -ih R (x|J ) R (y |J ) - J (y ) J (x) .

5. Boundary condition at -. If x < y for all y suppJ , the LSZ R-functiion does not depend on the source: < ^ (11) R (x|J ) = h (x) h, x suppJ. In particular, the classical retarded field vanishes as x0 -. Making use of the operator (4), one can construst the semiclassical field: h h (x|J ) = (TJ )+ h (x)TJ ^ coincides with R (x|J ) at x0 +: (12) > suppJ. 3. Another interesting feature of semiclassical states is that some of them are approximately equal each other. We say that J 0 iff (x|J ) = R (x|J ), x (13)
h TJ f

e

i h

I

J

WJf

for some number I J and operator W J presented as a formal asymptotic series. It is shown in [6] for the model (1) that the source J is equivalent to zero iff the retarded field generated by J vanishes at +. Analogously to [6], one derives the following properties. 1. Poincare invariance. (14) 2. Unitarity (15) W
+ J

U gW J U

g

-1

=W

ug J

,
-1 J

I

ug J

= IJ ;

=W

;

3. Bogoliubov causality: as J + J2 0, J + J1 + J2 0 and suppJ2 > suppJ1 , the operator


Semiclassical quantization of field theories

55

(W and number

J +J2

)+ W +I

J +J1 +J

2

-I

J +J

2

J +J1 +J

2

do not depend on J2 . 4. Variational property: (16) I J - ihW + W J = J dxR (x|J ) J (x),

which is valid as J 0 and J + J 0. 5. Boundary condition at +: (17) R (x|J ) = W + (x) hW J , J^

> suppJ. It follows from (17) that the retarded classical field generated by the source J 0 will vanish at +. For the model (1), an inverse statement is also valid: for any field configuration c (x) with the compact support one can uniquely choose a source J 0 (denoted as J = Jc = J (x|c ); for example (1), it is found from the relation (6)) generating c (x) as a retarded classical field: c (x) = R (x|J ); it satisfies the locality condition J (x|c ) c (y ) = 0 as x = y . It is possible to treat this statement as a basic postulate of semiclassical field theory. Then the theory may be developed without additional postulating classical stationary action principle and canonical commutation relation. Namely, it follows from eq.(16) in the leading order in h that the functional x (18) I [c ] = I
J
c

-

dxJc (x)c (x)

satisfies the "classical equation of motion" (19) Jc (x) = - I [c ] . c ( x )

The functional I [] should satisfy the locality condition (20) 2 I = 0. c (x) c (y ) x=y

This means that it is presented as an integral of a local Lagrangian. Relation (19) allows us to reconstruct the classical retarded field, making use of known source J 0, since the boundary condition at - is


56

Oleg Yu. Shvedov

known. It follows from the Bogoliubov causality condition that the retarded field depends only on J at the preceeding time moments. If the sourse J (x) is not equivalent to zero, it can be modified at + and transformed to the sourse equivalent to zero. Therefore, the relation I (21) [R (ћ|J )] = -J (x), R |x suppJ , the property to the classical field equation I [(ћ|J )] = 0. (22) c (x) This is a classical stationary action principle. It is viewed a other general principles of semiclassical field theory. Thus, we see that classical action I [c ] in field theory is h the phase of the state TJ f as J 0 according to eq.(18). Let us rewrite the properties of the operator W J via Denote W [c ] W Jc . 1. Poincare invariance. (23) 2. Unitarity. (24) c ( y ) W + [c ] = (W [c ])-1 . 3. Bogoliubov causality. (25) W + [c ] W [c ] c (x) = 0, y > x; U g W [c ]U
g
-1

(21) is taken

coroollory of related with the field c .

= W [ug c ].

4. Yang-Feldman relation [7]. 2 I W [c ] [ (y |J ) - c (y |J )] = ihW + [c ] . c (x) c (y ) R c (x) 5. Boundary condition. > (27) W + [c ]h (x) hW [c ] = R (x|Jc ), x suppc , ^ Here h (x) = R (x|0) is the field operator without source. ^ 4. The covariant axioms of semiclassical field theory are as follows. C1. A Hilbert state space F is given. C2. An unitary represatation of the Poincare group is given. The operators of the representation U g : F F are asymptoitc series in h. C3. To each classical source J (x) with compact support one assignes a retarded field (LSZ R-function). It is an operator-valued distribution (26) dy


Convex analysis, transportation and galaxies

57

R (x|J ) expanded in h according to (5). It satisfies the properties (7), (8), (9), (10). C4. To each classical field configuration c (x) with compact support one assigns a c-number. It is a classical action I [c ] satisfying the locality condition (20). The property c (x) = R (x|J ) is valid iff (28) J (x) = - I [c ] . c ( x )

C5. To each classical field configuration c (x) with compact support one assigns the operator W [c ] expanded in h. It satisfies the relations (23), (24), (25), (26), (27). It is possible to develop a semiclassical perturbation theory, making use of these properties. Bibliography
[1] N.N.Bogoliubov, D.V.Shirkov. Intoduction to the Theory of Quantized Fields. N.Y., Interscience Publishers, 1959. [2] V.P.Maslov. Perturbation Theory and Asymptotic Methods. Moscow, Moscow University Press, 1965. V.P.Maslov. Op erational Methods. Moscow, Mir publishers, 1976. V.P.Maslov. The Complex-WKB Metho d for Nonlinear Equations. Moscow, Nauka, 1977. V.P.Maslov, O.Yu.Shvedov. The Complex Germ Metho d for Many-Particle and Quantum Field Theory Problems. Moscow, Editorial URSS, 2000. [3] R.Jackiw, Rev.Mod.Phys. 49 (1977), 681. [4] J.Schwinger. Particles, Sources and Fields. Addison-Wesley. 1970. [5] H.Lehmann, K.Symanzik, W.Zimmermann. Nuovo Cim. 6 (1957) 319. [6] O.Yu.Shevdov. Teor. Mat. Fiz 144 (2005) 492. [7] O.I.Zavialov. Renormalized Feynmann Graphs. Moscow, Nauka, 1979.

Convex analysis, transp ortation and reconstruction of p eculiar velo cities of galaxies1 Andre Sobolevski i i
We show how the problem of reconstruction of peculiar velocities of galaxies starting from redshift-space catalogues can be rendered as a convex quadratic optimization problem, invoking optimal transport techniques
1Supp orted by the joint RFBR/CNRS grant 05-01-02807.


58

Alexander V. Stoyanovsky

for efficient large-scale astrophysical data processing. Connection with tropical algebra is briefly discussed.

The Weyl algebra and quantization of fields Alexander V. Stoyanovsky
In this talk we present a logically self-consistent procedure of quantization of fields. In more detail our approach is exposed in the papers [1,2] and in the book [3]. As a basic example we use the 4 model in 4-dimensional space-time. 1. Difficulties of traditional approaches to quantum field theory Let us briefly discuss the logical contradictions in the known procedure of quantization of fields. Usually one starts with the action functional (1.1) J= F (x0 , . . . , xn , u1 , . . . , um , u1 0 , . . . , um ) dx0 . . . dxn , xn x

where x0 = t, x1 , . . . , xn are the independent variables, u1 , . . . , um are the ui dependent variables, and ui j = xj . For the 4 model the action has the x form 3 1 1 u2 j - m2 u2 - g u4 dtdx1 dx2 dx3 . (1.2) J = u2 - t x 2 4! j =1 One writes down the quantum field theory Schroedinger equation (1.3) ih = t H t, x, ui (x), ui , -ih i x u (x) dx,

where x = (x1 , . . . , xn ); the density of the Hamiltonian H is the Legendre transform of the Lagrangian F with respect to the variables ui ; ui(x) is t the variational derivative operator. Note that the Schroedinger equation is not well defined in quantum field theory even for the free scalar field (g = 0 in (2)). For example, if we consider mathematical equation (1.3) literally, then it is not difficult to check that this equation has no nonzero four times differentiable solutions 2 (the expression for the derivative t has no sense). The traditional ap2 proach is to "subtract infinity" from the RHS of equation (1.3) and to solve it in the Fock space of functionals. However, this approach contradicts physical as well as mathematical considerations. Physically, if states were


The Weyl algebra and quantization of fields

59

functionals and energy were finite, then, in principle, we could measure some quantities related with these functionals (such as energy). However, it is known that quantum mechanical quantities like energy and momentum are theoretically non-measurable in relativistic quantum dynamics, and the only measurable quantities are the scattering sections. Mathematically, equation (1.3) in the Fock space does not admit a relativistically invariant generalization (usually called the Tomonaga-Schwinger equation [4]), as shown in the important paper [5]. In this paper it is shown that the evolution operators of the Klein-Gordon equation from one space-like surface to another, which are symplectic transformations of the phase space of the field, do not belong to the version of the infinite dimensional symplectic group which acts on the Fock space. So quantization of free fields, for example, following the lines of the book [4], meets difficulties of the logical kind and is therefore not completely satisfactory. Due to this fact, the renormalization procedure for interacting fields, defined using the Bogolyubov-Parasyuk theorem, gives us a model in which it is difficult to say how quantum field theory turns into the classical one as h 0. 2. The infinite dimensional Weyl algebra The proposed way to overcome these difficulties is to replace the algebra of variational differential operators by the infinite dimensional Weyl algebra defined below, which admits an explicit action of the infinite dimensional group of continuous symplectic transformations of the phase space of a field. This phase space is the Schwartz space of functions (ui (s),pi (s)), where pi (s) are the variables conjugate to ui (s), and s = (s1 , . . . , sn ) are parameters on a spacelike surface, with the Poisson bracket (2.1) {1 , 2 } =
i

1 2 1 2 - ui (s) pi (s) pi (s) ui (s)

ds

of two functionals l (ui (ћ), pi (ћ)), l = 1, 2. Let us write this bracket in the form (2.2) {1 , 2 } =
i,j



ij

1 2 ds, y i (s) y j (s)

where y i = ui for 1 i m and y i = pi-m for m + 1 i 2m, and ij = i,j -m - i-m,j . The Weyl algebra is defined as the algebra of weakly infinite differentiable functionals (ui (ћ), pi (ћ)) with respect to the Moyal


60 -product (2.3) (1 2 )(y i (ћ)) ih = exp - 2

Alexander V. Stoyanovsky

i,j

ij i ds 1 (y i (ћ))2 (z i (ћ)) y (s) z j (s)
z (ћ)=y (ћ) i
i i

.

This product is not everywhere defined: for example, u (s) pi (s) is undefined. Note only that if all necessary series and integrals are absolutely convergent, then the -product is associative. Below we will be interested only in some concrete computations in the Weyl algebra. Let us replace the Schrodinger equation (1.3) by the Heisenberg equation in the Weyl algebra (2.4) ih = t H (t, x, ui (x), ui i , p (x))dx, x

and by its relativistically invariant generalization, where (2.5) [1 , 2 ] = 1 2 - 2
1

is the commutator in the Weyl algebra. The classical limits of equation (2.4) are the Hamilton equations = {, H dx} t equivalent to the Euler-Lagrange equations. (2.6) 3. Quantization of free scalar field Put g = 0 in (1.2). Since the obtained Hamiltonian 1 (3.1) H0 = (p(x)2 + (grad u(x))2 + m2 u(x)2 )dx 2 is quadratic, we have 1 (3.2) [H0 , ] = {, H0 }, ih hence, (t1 ; u(ћ), p(ћ)), sub ject to the Heisenberg equation, is obtained from (t0 ; u(ћ), p(ћ)) by the linear symplectic change of variables (3.3) (u(t0 , x), p(t0 , x) = ut (t0 , x)) (u(t1 , x), p(t1 , x) = ut (t1 , x)), given by the evolution operator of the Hamilton equations, i. e., of the Klein-Gordon equation, from the Cauchy surface t = t0 to the Cauchy surface t = t1 . Hence we can identify the Weyl algebras of various Cauchy


The Weyl algebra and quantization of fields

61

surfaces by means of evolution operators of the Klein-Gordon equation. In other words, we can consider the Weyl algebra W0 of the symplectic vector space of solutions u(t, x) of the Klein-Gordon equation on the whole space-time. The symplectic form on this vector space is obtained by taking Cauchy data on any spacelike surface (for example, on the surface t = const). Below we will fix this identification of the Weyl algebras of various spacelike surfaces. Define the vacuum average linear functional (3.4) 0||0

on the Weyl algebra W0 as the unique (not everywhere defined) functional with the properties (3.5) 0| u- (t, x)|0 = 0|u+ (t, x) |0 = 0, 0|1|0 = 1.

Here u = u+ + u- is the decomposition of a solution u(t, x) of the Klein- Gordon equation into the positive and negative frequency parts (we assume m > 0 so that this decomposition is unique). For W0 , define an operator in the standard Fock space with the matrix elements (3.6) 0|u- (-p(1) ) . . . u- (-p(N ) ) u+ (p ~ ~ ~
(1)

) . . . u+ (p(N ) )|0 . ~

Here u+ (p) is the Fourier transform (the momentum representation) of ~ u+ , p = (p0 , . . . , pn ). One can check the following two properties of this correspondence: i) -product of functionals goes to composition of operators in the Fock space, so that this correspondence is a (not everywhere defined) homomorphism from the algebra W0 to the algebra of operators in the Fock space; ii) complex conjugation of functionals goes to Hermitian conjugation of operators in Hilbert space. 4. Quantization of interacting fields Statement. There exists a map from the set of smooth functions g = g (t, x) with compact support to the set of functionals P (g ) W0 with the fol lowing properties. 1) P (g ) is a formal series in g with the first two terms (4.1) P (g ) = 1 + 1 ih g (t, x)u(t, x)4 /4! dtdx + . . . .

2) Classical limit: P (g ) = a(g , h) exp(iR(g )/h) where a(g , h) is a formal series in h, and conjugation by exp(iR(g )/h) in the Weyl algebra W0


62

Alexander V. Stoyanovsky

up to O(h) yields the perturbation series for the evolution operator of the nonlinear classical field equation (4.2) 2u(x) - m2 u(x) = g (x)u3 (x)/3! from t = - to t = . 3) The Lorentz invariance condition: (4.3) LP (L
-1

g ) = P (g )

for a Lorentz transformation L. 4) The unitarity condition: (4.4) P (g ) P (g ) = 1,

where P (g ) is complex conjugate to P (g ). 5) The causality condition: for two functions g1 , g2 equal for t t0 , the product P (g1 ) P (g2 )-1 does not depend on the behavior of the functions g1 , g2 for t < t0 . 6) The quasiclassical dynamical evolution (cf. with the Maslov-Shvedov quantum field theory complex germ [6]): for any spacelike surfaces C1 , C2 there exists a limit PC1 ,C2 of P (g ) modulo o(h) as the function g (x) tends to 1 if x belongs to the strip between the spacelike surfaces and to 0 otherwise. This limit possesses the property (4.5) P
C1 ,C3

=P

C2 ,C3

P

C1 ,C2

+ o(h).

7) The S -matrix: there exists a limit P of P (g ) as g (x) tends to the function g = const. This P is a formal power series in g . Any other choice of P (g ) with the properties 1-7 above is equivalent to some change of parameters m, g (x). This statement is completely similar to the Bogolyubov-Parasyuk theorem. Moreover, if we denote by S (g ) the operator in the Fock space corresponding to P (g ) and by S the operator corresponding to P , then S (g ) is exactly the Bogolyubov S -matrix and S is the physical S -matrix. The elements P (g ) are constructed in the same way as S (g ) in [4], using the renormalization procedure, the main difference being that composition of operators is replaced by -product of functionals, and the normally ordered product of operators is replaced by the usual (commutative) product of functionals. Note that the conditions on P (g ) (in particular, the causality condition) are the natural analogs of conditions on dependence of the evolution operator of a partial differential equation on the coefficient functions of this equation. Therefore the above apparatus is similar to the scattering theory in the theory of partial differential equations.


Polynomial quantization with pseudo-orthogonal group

63

Note also that the presence of interaction cutoff function g (x) is necessary from the physical point of view, since the scattering particles are considered as non-interacting at infinity (which means that g = 0 at infinity). Bibliography
[1] A. V. Stoyanovsky, Maslov's complex germ and the Weyl-Moyal algebra in quantum mechanics and in quantum field theory, math-ph/0702094. [2] A. V. Stoyanovsky, The Weyl-Moyal algebra and Maslov's complex germ in quantum field theory, to appear. [3] A. V. Stoyanovsky, Intro duction to the mathematical principles of quantum field theory, URSS, Moscow, 2007 (in Russian). [4] N. N. Bogolyubov, D. V. Shirkov, Introduction to the theory of quantized fields, GITTL, Moscow, 1957 (in Russian). [5] C. G. Torre, M. Varadara jan, Functional evolution of free quantum fields, Class. Quant. Grav. 16 (1999) 2651-2668, hep-th/9811222. [6] V. P. Maslov, O. Yu. Shvedov, Metho d of complex germ in the many particle problem and in quantum field theory, URSS, Moscow, 2000 (in Russian).

Polynomial quantization on para-hermitian spaces with pseudo-orthogonal group of translations1 Svetlana V. Tsykina
We construct polynomial quantization, which is a variant of quantization in spirit of Berezin, on para-Hermitian symmetric spaces G/H with the pseudo-orthogonal group G = SO0 (p, q ). For all these spaces, the connected component He of the subgroup H containing the identity of G is the direct product SO0 (p - 1, q - 1) Ч SO0 (1, 1), so that G/H is covered by G/He (with multiplicity 1, 2 or 4). The dimension of G/H is equal to 2n - 4, where n = p + q . We restrict ourselves to the spaces G/H that are G-orbits in the adjoint representation of G. A construction of quantization on arbitrary para-Hermitian symmetric spaces was given in [2]. The term "polynomial quantization" means in particular that both covariant and contravariant symbols are polynomials on G/H . Following the general scheme of [2], we introduce multiplication
05-0100074a), the Netherlands Organization for Scientific Research (NWO) (grant No. 047017-015), the Scientific Programs "Devel. Sci. Potent. High. School" (pro ject RNP 2.1.1.351 and Templan, No. 1.2.02).
1Supp orted by the Russian Foundation for Basic Research (grant No.


64

Svetlana V. Tsykina

of covariant symbols, establish the correspondence principle, and study the Berezin transform. The polynomial quantization on rank one para-Hermitian symmetric spaces has been constructed in [3]. In this paper, we consider the spaces G/H with G = SO0 (p, q ). Note that these spaces have rank 2. 1. The pseudo-orthogonal group and its Lie algebra Consider the space Rn equipped with the following bilinear form:
n

[x, y ] =
i=1

i xi yi ,

where 1 = . . . = p = -1, p+1 = . . . = n = 1, and x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) are vectors in Rn . Let G denote the group SO0 (p, q ). This group is the connected component of the identity, in the group of linear transformations of Rn that preserve [x, y ] and have determinant equal to 1. We assume that G acts linearly on Rn from the right: x xg . In accordance with that, we write vectors in the row form. We also assume that p > 1, q > 1. Let us write matrices g G in the block form corresponding to the partition n = 1 + (n - 2) + 1. Denote by H the subgroup of G consisting of matrices 0 (1.1) h = 0 v 0 , 0 where 2 - 2 = 1, v SO(p - 1, q - 1). The subgroup H consists of two connected components. The connected component He , containing the unit matrix E of G consists of matrices (1.1), where = cht, = sht. Thus, it is SO0 (p - 1, q - 1) Ч SO0 (1, 1). The second connected component of H (which does not contain E ) contains the matrix diag {-1, -1, 1, . . . , 1, -1, -1}, as a representative. The Lie algebra g of G consists of real matrices X of order n satisfying the condition X I + I X = 0, where I = diag {1 , . . . , n }, the prime denotes matrix transposition. Let 001 (1.2) Z0 = 0 0 0 . 100


Polynomial quantization with pseudo-orthogonal group

65

The stabilizer of Z0 in the adjoint representation is exactly the group H , therefore, the manifold G/H is just the G-orbit of the matrix Z0 in g. The operator ad Z0 has three eigenvalues: -1, 0, +1. Respectively, the Lie algebra g is decomposed into the direct sum of eigenspaces g = q- + h + q+ , where h is the Lie algebra of H . The subspaces q- 0 0 0 Y : X : 0 , 0 0 - 0 , q+ consist of matrices 0 0 - 0

respectively, where , are rows in Rn-2 . Both spaces q+ are Abelian subalgebras of g, they have dimension n - 2. The subgroup H preserves both subspaces q- and q+ in the adjoint action: (1.3) Z h-1 Z h, h H. Let h H have the form (1.1). For simplicity, we identify matrices X and Y with vectors and , respectively. Under the action (1.3) vectors q- and q+ are transformed as follows: (1.4) = ( + ) v , = ( - ) v .
n-2

Consider the space R I1 = diag {2 , . . . , n-1 }:

with bilinear form defined by the matrix
n-1

, =
i=2

i i i .

2. Representations of G asso ciated with a cone The group G = SO0 (p, q ) preserves manifolds [x, x] = c, c R, in Rn . Let C be the cone [x, x] = 0, x = 0, in Rn . Let us fix two points in the cone: s+ = (1, 0, . . . , 0, 1), s- = (1, 0, . . . , 0, -1). Consider the following two sections of the cone:
+

= =

{x1 + xn = 2} = {[x, s- ] = -2}, {x1 - xn = 2} = {[x, s+ ] = -2}.

-

The points s+ , s- belong to + , - respectively. They are eigenvectors of the maximal parabolic subgroups P + = Q+ H and P - = Q- H respectively, with eigenvalues - and + , where , are parameters of h H , see (1.1). Here Q- = exp q- , Q+ = exp q+ .


66

Svetlana V. Tsykina

The section + meets almost all generatrices of the cone C . The linear action of G on the cone induces the following actions of G on - and + respectively: (2.1) (2.2) x - x = - 2 ћ xg , x - , [xg , s+ ] 2 x - x = - ћ xg , x + , [xg , s- ]

defined almost everywhere on + . For the subgroups Q- and Q+ respectively, these actions turn out to be linear: x xg . Moreover, the subgroups Q+ act on + simply transitively. This allows to define the coordinates = (2 , . . . , n-1 ) on - and = (2 , . . . , n-1 ) on + transferring them from q- on q+ respectively, namely, for u - and v + we set: (2.3) (2.4) u= v = u( ) = s- e v ( ) = s e
X


= (1 + , , 2 , -1 + , ), = (1 + , , 2 , 1 - , ). - and s+ + under the actions = Q+ H and P - = Q- H respectively. ) be the space of C functions f on parity , i.e.

+Y

The stabilisers in G of the points s- (2.1) and (2.2) are the subgroups P + Let C, = 0, 1. Let D, (C the cone C with homogeneity and

f (tx) = t, f (x), x C , t R = R \ {0}, where we denote t, = |t| sgn t. Denote by T, the representation of G which acts on D, (C ) by translations: (T, (g )f ) (x) = f (xg ). Consider now the restrictions of functions from D, (C ) to the sections + . Such restrictions form a space D, (+ ) of functions f on + . This space is contained in C (+ ) and contains D(+ ). In the coordinates , , the representation T, of the group G acts on the space of restrictions D, (C ) by (2.5) (2.6) (T, (g )f ) ( ) (T, (g )f ) ( ) 1 = f ( ) - [ug , s+ ] 2 1 = f ( ) - [v g , s- ] 2
,

,
,

,

where u = u( ), v = v ( ) are defined by (2.3), (2.4), actions and are defined by (2.1), (2.2).


Polynomial quantization with pseudo-orthogonal group Define the operator A (2.7) where (A
, ,

67

on D, (+ ) by: N ( , )
Rn-2 2-n-,

f )( ) =

f ( )d ,

1 N ( , ) = - [u( ), v ( )] = 1 - 2 , + , , . 2 The function N ( , ) is a polynomial in , . The operator A, intertwines the representations T, and T2-n-, . These representations act on functions on different sections. We can change the position of and in (2.7). The product A2-n-, A, is a scalar operator: A where 0 (, ) = 23 Ч ( + 1)(3 - n - ) Ч (2 + n - 2) sin + n 2 -+p ++q ++n - ћ sin ћ sin ћ sin . sin 2 2 2 2
n-3 2-n-,

A

,

= 0 (, )E ,

3. The space G/H Consider the following realization of the space G/H . Let be the set of matrices: y x (3.1) z= , [x, y ] where x, y C , y = I y . For these matrices, rank and trace are equal to 1. The adjoint action z g -1 z g preserves . The stabilizer of the matrix z 0 , corresponding to the pair x = s- , y = s+ , is the subgroup H , so that is just G/H . Take vectors u = u( ) and v = v ( ) in the sections - and + of the cone C , respectively, for x and y in (3.1) We obtain an embedding - Ч + given by (3.2) The map that N ( with the action of each g z = z ( , ) = v u , [u, v ] u = u( ), v = v ( ),

(u, v ) - z given by formula (3.2) is defined for , Rn-2 such , ) = 0, since [u, v ] = -2N ( , ). Therefore, vectors , Rn-2 condition N ( , ) = 0 are local coordinates on . The adjoint the group G on is generated by its actions on and . For G, this action is defined on a dense set of .


68

Svetlana V. Tsykina

We can identify the tangent space of G/H at the initial point z 0 with the space q = q- + q+ in the Lie algebra g. Let S (q) denote the algebra of polynomials on q. The action (1.3) of the group H on q induces an action of H on S (q). Let S (q)H denote the algebra of polynomials invariant with respect to H . This algebra is generated by two polynomials , and , , . Let D(G/H ) denote the algebra of differential operators on G/H invariant with respect to G. This algebra is in the one-to-one correspondence with the algebra S (q)H . Let 2 and 4 denote operators in D(G/H ) corresponding to generators , and , , of S (q)H respectively. Let us call these operators 2 and 4 the Laplace operators on G/H . The operator 2 is the Laplace-Beltrami operator. These operators are differential operators of the second and the fourth order respectively, they are generators in D(G/H ). Explicit expressions of them are very cumber0 0

some. We write explicit expressions horospherical coordinates. These coordinates are defined as the Cartan subspace a, consisting of 00 0 0 At = 0 0 t1 0 0 t2

for their radial parts 2 and 4 in follows. Let us take in q = q+ + q- matrices 0 t1 0 0 0 t2 0 0 0 , 0 0 0 000

where t = (t1 , t2 ) R2 . Introduce in a the lexicographical order in coordinates. Let n denote the subalgebra of g formed by the corresponding positive root spaces. Let A = exp a, N = exp n. Consider the set of points z in obtained from z 0 via the translation by a = a(t1 , t2 ) A and then by n N , i.e. z = n-1 a-1 z 0 an. It is a neighbourhood U of the point z 0 . Parameters t1 , t2 of the subgroup A and also parameters of the subgroup N are coordinates in this neighbourhood (horospherical coordinates). Let f be a function defined on U that does not depend on n N . Then it is a function of t = (t1 , t2 ): f (z ) = F (t). Let D be a differential operator in D(G/H ). Then Df also does not depend on n N :
0

D f =D F ,
0

where D is a differential operator in t1 , t2 , the radial part of D with respect to N . It turns out to be a differential operator with constant coefficients.


Polynomial quantization with pseudo-orthogonal group Introduce operators D1 D2 = = + +n-3 t1 t2 - +1 t1 t2
2 2

69

- (2n - 7)

- (2n - 7).

Theorem 3.1. We have that 0 1 {D1 + D2 - (n - 4)(n - 6)} , 2 = 2
0 4

=

D1 D2 + 2(n - 4)3 .

4. Polynomial quantization on G/H We follow the scheme from [2]. The role of supercomplete system is played by the kernel ( , ) = , ( , ) = N ( , ), of the intertwining operator A2-n-, . As an analogue of the Fock space, we take the space of functions ( ). We start from the algebra of operators D = T, (X ), where X belongs to the universal enveloping algebra Env(g) for g. The covariant symbol F ( , ) of the operator D is defined by: 1 D ( , ), F ( , ) = ( , ) where D means that the operator D acts on ( , ) as on a function of . These covariant symbols are independent of . They are functions on G/H . Moreover, they are polynomials on G/H (i.e. restrictions on G/H of polynomials on the space of matrices z , see (3.1)). For generic the space A of covariant symbols is the space S (G/H ) of all polynomials on G/H . The map D F , which assigns to an operator its covariant symbol, is g-equivariant. For an arbitrary the operator D is reconstructed from its covariant symbol F : (4.1) (D)( ) = c(, ) F ( , v ) ( , v ) (u) dx(u, v ), (u, v )

where c(, ) = 0 (, )-1 . The multiplication of operators gives rise to a multiplication (denote it by ) of covariant symbols. Let F1 , F2 be the covariant symbols of operators D1 , D2 respectively. We have that 1 F1 F2 = (D1 ) (F2 ).


70 This multiplication is given by (F1 F2 )( , ) =

Svetlana V. Tsykina

F1 ( , v )F2 (u, )B ( , ; u, v ) dx(u, v ),

where dx(u, v ) is an invariant measure on G/H , and B ( , ; u, v ) = c ( , v )(u, ) . ( , )(u, v )

Let us call this kernel B the Berezin kernel. Thus, the spaces A turn out to be associative algebras with unit (with respect to ). On the other hand, we can define contravariant symbols of the operators. A function F ( , ) can be viewed as the contravariant symbol for the following operator A (acting on functions ( )): (A)( ) = c(, ) F (u, v ) ( , v ) (u) dx(u, v ). (u, v )

Notice that this expression differs from (4.1) only by the first argument of function F . A contravariant symbol can be reconstructed from the corresponding operator. Thus we obtain two maps D F ("co") and F A ("contra"), connecting operators D and A with polynomials F on G/H . The passage from the contravariant symbol of an operator to its covariant symbol is an integral operator with the Berezin kernal. Let us call B the Berezin transform. Theorem 4.1. The Berezin transform can be expressed in terms of Laplace operators: B= ( + n - 2 + a+b )( + 1 - a+b )( + n + a-b )( + n - 1 - 2 2 2 2 2 ( + n - 2)( + 1)( + n )( + n - 1) 2 2 D1 = (a + b)2 + 2(n - 3)(a + b) + (n - 4)2 , D2 = (a - b)2 + 2(a - b) - 2(n - 4). Note that on finite-dimensional subspaces in S (G/H ) the Berezin transform is a differential operator. Now let -. The first two terms of the asymptotic expansion of B are given by: (4.2) B 1- 1 2 .
a-b 2

)

where a, b are some variables and one has to consider


Polynomial quantization with pseudo-orthogonal group

71

The relation (4.2) implies the following correspondence principle (as the "Planck constant" one has to take h = -1/ ): (4.3) (4.4) F1 F2 - F1 F2 , - (F1 F2 - F2 F1 ) - {F1 , F2 },

as -, In (4.3) and (4.4), F1 F2 denotes the pointwise multiplication of F1 and F2 , and {F1 , F2 } stands for the Poisson bracket of F1 and F2 . Bibliography
[1] Berezin F.A. Quantization in complex symmetric spaces. Izv. Akad. Nauk. SSSR. Ser. mat., 1975, vol. 39, No. 2, 363-402. Eng. transl.: Math. USSR Izv., 1975, vol. 9, 341-379. [2] Molchanov V.F. Quantization on para-Hermitian symmetric spaces. Amer. Math. Soc. Transl., Ser. 2 (Adv. Math. Sci.-31), 1996, vol. 175, 81-95. [3] Molchanov V.F., Volotova N.B. Polynomial quantization on rank one para-Hermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, Nos. 1-3, 215-232.

The horofunction b oundary1 Cormac Walsh
The horofunction boundary (also known as the `metric' or `Busemann' boundary) is a means of compactifying metric spaces. Its definition goes back to Gromov [10] in the 1970s but it seems not to have received much study until recently, when it has appeared in several different domains [11, 16, 15, 1, 13]. To define this boundary for a metric space (X, d), one assigns to each point z X the function z : X R, z (x) := d(x, z ) - d(b, z ), where b is some basepoint. If X is proper, then the map : X C (X ), z z defines an embedding of X into C (X ), the space of continuous real-valued functions on X endowed with the topology of uniform convergence on compacts. The horofunction boundary is defined to be X () := cl{z | z X }\{z | z X }, and its elements are called horofunctions. This boundary is not the same as the better known Gromov boundary of a -hyperbolic space. For these spaces, it has been shown [5, 22, 17]
1This work was funded in part by grant RFBR/CNRS 05-01-02807.


72

Cormac Walsh

that the horoboundary is finer than the Gromov boundary in the sense that there exists a continuous surjection from the former to the latter. Of particular interest are those horofunctions that are the limits of almost-geodesics. An almost-geodesic, as defined by Rieffel [16], is a map from an unbounded set T R+ containing 0 to X , such that for any > 0, |d( (t), (s)) + d( (s), (0)) - t| < for all t T and s T large enough with t s. Rieffel called the limits of such paths Busemann points. See [1] for a slightly different definition of almost-geodesic which nevertheless gives rise to the same set of Busemann points. As noted by Ballmann [2], the construction above is an additive analogue of the way the Martin boundary is constructed in Probabilistic Potential Theory. One may pursue the analogy further in the framework of max-plus algebra, where one replaces the usual operations of addition and multiplication by those of maximum and addition. Indeed, this approach has already provided inspiration for many results about the horofunction boundary [1, 18]. We mention, for example, the characterisation of Busemann points as the functions in the horoboundary that are extremal generators in the max-plus sense of the set of 1-Lipschitz functions. So the set of Busemann points is seen to be an analogue of the minimal Martin boundary. There is also a representation of 1-Lipschitz functions in terms of horofunctions analogous to the Martin representation theorem. There are few examples of metric spaces where the horofunction boundary or Busemann points are explicitly known. The first cases to be investigated were those of Hadamard manifolds [3] and Hadamard spaces [2], where the horofunction boundary turns out to be homeomorphic to the ray boundary and all horofunctions are Busemann points. The case of finite-dimensional normed spaces has also received attention. Karlsson et. al. determined the horofunction boundary in the case when the norm is polyhedral [12]. Other examples of metric spaces where the horofunction boundary has been studied include the Cayley graphs of finitelygenerated abelian groups, studied by Develin [6], and Finsler p-metrics on GL(n, C)/Un , where explicit expressions for the horofunctions were found by Friedland and Freitas [8, 9]. Webster and Winchester have some general results on when all horofunctions are Busemann points [24], [23]. In the following sections, we describe our recent work elucidating the horoboundary of some particular metric spaces.


The horofunction boundary 1. Normed spaces

73

Rieffel comments that it is an interesting question as to when all boundary points of a metric space are Busemann points and asks whether this is the case for general finite-dimensional normed spaces. In [19], we answer this question in the negative and give a necessary and sufficient criterion for it to be the case. Let V be an arbitrary finite-dimensional normed space with unit ball B . Recall that a convex subset E of a convex set D is said to be an extreme set if the endpoints of any line segment in D are contained in E whenever any interior point of the line segment is. For any extreme set E of the dual unit ball B and point p of V , define the function fE ,p from the dual space V to [0, ] by f
E ,p

(q ) := IE (q ) + q |p - inf y |p
y E

for all q V .

Here IE is the indicator function, taking value 0 on E and + everywhere else. Our first theorem characterises the Busemann points of V as the Legendre-Fenchel transforms of these functions. Theorem 1.1. The set of Busemann points of a finite-dimensional normed space (V , || ћ ||) is
{fE ,p

| E is a proper extreme set of B and p V }.

We use this knowledge to characterise those norms for which all horofunctions are Busemann points. Theorem 1.2. A necessary and sufficient condition for every horofunction of a finite-dimensional normed space to be a Busemann point is that the set of extreme sets of the dual unit bal l be closed in the Painlev? e- Kuratowski topology. 2. The Hilb ert metric Let x and y be distinct points in a bounded open convex subset D of RN , with N 1. Define w and z to be the points in the Euclidean boundary of D such that w, x, y , and z are collinear and arranged in this order along the line in which they lie. The Hilbert distance between x and y is defined to be the logarithm of the cross ratio of these four points: Hil (x, y ) := log |z x| |wy | . |z y | |wx|


74

Cormac Walsh

If D is the open unit disk, then the Hilbert metric is exactly the Klein model of the hyperbolic plane. As pointed out by Busemann [4, p105], the Hilbert geometry is related to hyperbolic geometry in much the same way that normed space geometry is related to Euclidean geometry. It will not be surprising therefore that there are similarities between the horofunction boundaries of Hilbert geometries and of normed spaces. Define the function |z x| , for all x and y in D. (2.1) Funk (x, y ) := log |z y | This function satisfies the usual metric space axioms, apart from that of symmetry. Hilbert's metric can now be written Hil (x, y ) := Funk (x, y ) + Funk (y , x), for all x and y in D.

This expression of the Hilbert metric as the symmetrisation of the Funk metric plays a crucial role. It turns out that every Hilbert horofunction is the sum of a horofunction in the Funk geometry and a horofunction in the reverse Funk geometry, where the metric in the latter is given by rev (x, y ) := Funk (y , x). This allows us to simplify the problem by investigating separately the horofunction boundaries of these two geometries and then combining the results. Determining the boundary of the Funk geometry turns out to be very similar to determining that of a normed space, which was done in [19]. In [20], we characterise those Hilbert geometries for which all horofunctions are Busemann points. Theorem 2.1. A necessary and sufficient condition for every horofunction on a bounded convex open subset of RN containing the origin to be a Busemann point in the Hilbert geometry is that the set of extreme sets of its polar be closed in the Painlev? e-Kuratowski topology. It had previously been shown [12] that all horofunctions of the Hilbert geometry on a polytope are Busemann points. Theorem 2.2. Let D be a bounded convex open subset of RN . If a sequence in D converges to a point in the horofunction boundary of the Hilbert geometry, then the sequence converges in the usual sense to a point in the Euclidean boundary D.


The horofunction boundary 3. Finitely generated groups

75

An interesting class of metric spaces are the Cayley graphs of finitely generated groups with their word metric. Here one may hope to have a combinatorial description of the horoboundary. The first to consider the horoboundary in this setting was Rieffel [16] who studied the horoboundary of Zn with an arbitrary finite generating set in connection with his work on non-commutative geometry. In [21], we investigate the horofunction boundary of Artin groups of dihedral type. Let prod(s, t; n) := ststs ћ ћ ћ , with n factors in the product. The Artin groups of dihedral type have the following presentation: Ak = a, b | prod(a, b; k ) = prod(b, a; k ) , with k 3. Observe that A3 is the braid group on three strands. The generators traditionally considered are the Artin generators S := {a, b, a-1 , b-1 }. In what follows, we will have need of the Garside normal form for elements of Ak . The element := prod(a, b; k ) = prod(b, a; k ) is called the Garside element. Let M
+

:= {a, b, ab, ba, . . . , prod(a, b; k - 1), prod(b, a; k - 1)}. w = w1 ћ ћ ћ wn
r

It can be shown [7] that w Ak can be written for some r Z and w1 , . . . , wn M + . This decomposition is unique if n is required to be minimal. We call it the right normal form of w. The factors w1 , . . . , wn are called the canonical factors of w. An algorithm was given in [14] for finding a geodesic word representing any given element of Ak ; k 3. We use this algorithm to find a simple formula for the word length metric. Proposition 3.1. Let x = z1 ћ ћ ћ zm r be an element of Ak written in right normal form. Let (p0 , . . . , pk-1 ) Nk be such that p0 := r and, for each i {1, . . . , k - 1}, pi - pi-1 = mk-i , where mi is the number of canonical factors of x of length i. Then the distance from the identity e to x in the Artin-generator word-length metric is
k -1

d(e, x) =
i=0

|pi |.

Since d is invariant under left multiplication, that is, d(y , x) = d(e, y -1 x), we can use this formula to calculate the distance between any pair of elements y and x of Ak . With this knowledge we can find the following description of the horofunction compactification.


76

Cormac Walsh

Let Z be the set of possibly infinite words of positive generators having no product of consecutive letters equal to . We can write each element z of Z as a concatenation of substrings in such a way that the products of the letters in every substring equals an element of M + and the combined product of letters in each consecutive pair of substrings is not in M + . Because z does not contain , this decomposition is unique. Let mi (z ) denote the number of substrings of length i. Note that if z is an infinite word, then this number will be infinite for some i. Let denote the set of (p, z ) in (Z {-, +})k Ч Z satisfying the following: ћ pi - pi-1 mk-i (z ) for all i {1, . . . , k - 1} such that pi and pi-1 are not both - nor both +; ћ if z is finite, then pi - pi-1 = mk-i (z ) for all i {1, . . . , k - 1} such that pi and pi-1 are not both - nor both +. We take the product topology on . We now define to be the quotient topological space of where the elements of (+, . . . , +) Ч Z are considered equivalent and so also are those in (-, . . . , -) Ч Z . We denote these two equivalence classes by + and -, respectively. ^ ^ We let M denote the horofunction compactification of Ak with the Artin-generator word metric. The basepoint is taken to be the identity. Theorem 3.2. The sets and M are homeomorphic. Let Z0 be the set of elements of Z that are finite words. Let 0 denote the set of (p, z ) in Zk Ч Z0 such that pi - pi-1 = mk-i (z ) for all i {1, . . . , k - 1}. One can show that the elements of 0 are exactly the elements of corresponding to functions of the form d(ћ, z ) - d(e, z ) in M. In the present context, since the metric takes only integer values, the Busemann points are exactly the limits of geodesics (see [24]). Develin [6], investigated the horoboundary of finitely generated abelian groups with their word metrics and showed that all their horofunctions are Busemann. We have the following characterisation of the Busemann points of Ak . Theorem 3.3. A function in M is a Busemann point if and only if the corresponding element (p, z ) of is in \0 and satisfies the fol lowing: pi - pi-1 = mk-i (z ) for every i {1, . . . , k - 1} such that pi and pi-1 are not both - nor both +. The group Ak also has a dual presentation: Ak = 1 , . . . , k | 1 2 = 2 3 = ћ ћ ћ = k 1 , with k 3.


The horofunction boundary

77

- - ~ The set of dual generators is S := {1 , . . . , k , 1 1 , . . . , k 1 }. Again, one can find a formula for the word length metric and use it to determine the horoboundary. This time however, it turns out that there are no non-Busemann points.

Theorem 3.4. In the horoboundary of Ak with the dual-generator word metric, al l horofunctions are Busemann points. In general, one would expect the properties of the horoboundary of a group with its word length metric to depend strongly on the generating set. It would be interesting to know for which groups and for which properties there is not this dependence. As already mentioned, all boundary points of abelian groups are Busemann no matter what the generating set [6]. On the other hand, the above results show that for Artin groups of dihedral type the existence of non-Busemann points depends on the generating set.

Bibliography
[1] Marianne Akian, St? ephane Gaub ert, and Cormac Walsh. The max-plus Martin b oundary. Preprint. arXiv:math.MG/0412408, 2004. [2] Werner Ballmann. Lectures on spaces of nonpositive curvature, volume 25 of DMV Seminar. BirkhЕ auser Verlag, Basel, 1995. With an app endix by Misha Brin. [3] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Manifolds of nonpositive curvature, volume 61 of Progress in Mathematics. BirkhЕ auser Boston Inc., Boston, MA, 1985. [4] Herb ert Busemann. The geometry of geodesics. Academic Press Inc., New York, N. Y., 1955. [5] Michel Coornaert and Athanase Papadopoulos. Horofunctions and symb olic dynamics on Gromov hyp erbolic groups. Glasg. Math. J., 43(3):425-456, 2001. [6] Mike Develin. Cayley compactifications of ab elian groups. Ann. Comb., 6(3-4):295- 312, 2002. [7] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. [8] Shmuel Friedland and Pedro J. Freitas. p-metrics on GL(n, C)/Un and their Busemann compactifications. Linear Algebra Appl., 376:1-18, 2004. [9] Shmuel Friedland and Pedro J. Freitas. Revisiting the Siegel upp er half plane. I. Linear Algebra Appl., 376:19-44, 2004. [10] M. Gromov. Hyperb olic manifolds, groups and actions. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 183-213, Princeton, N.J., 1981. Princeton Univ. Press. [11] Hitoshi Ishii and Hiroyoshi Mitake. Representation formulas for solutions of Hamilton-Jacobi equations, 2006. Preprint.


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ivgeny wF feniminov

[12] A. Karlsson, V. Metz, and G. Noskov. Horoballs in simplices and Minkowski spaces. Int. J. Math. Math. Sci., 2006. Art. ID 23656, 20 pages. [13] Anders Karlsson. Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems, 21(5):1447-1457, 2001. [14] Jean Mairesse and Fr? ? ederic Math? eus. Growth series for Artin groups of dihedral typ e. Internat. J. Algebra Comput., 16(6):1087-1107, 2006. [15] Charles M. Newman. A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), pages Е 1017-1023, Basel, 1995. BirkhЕ auser. [16] Marc A. Rieffel. Group C -algebras as compact quantum metric spaces. Doc. Math., 7:605-651 (electronic), 2002. [17] Peter A. Storm. The barycenter method on singular spaces. Preprint. arXiv:math.GT/0301087, 2003. [18] Cormac Walsh. Minimum representing measures in idemp otent analysis. Preprint, 2005. [19] Cormac Walsh. The horofunction boundary of finite-dimensional normed spaces. To app ear Math. Pro c. Camb. Phil. Soc. arXiv:math.GT/0510105, 2005. [20] Cormac Walsh. The horofunction b oundary of the Hilbert geometry. To app ear Adv. Geom. arXiv:math.MG/0611920, 2006. [21] Cormac Walsh. Busemann points of Artin groups of dihedral typ e, 2007. Preprint. [22] Corran Webster and Adam Winchester. Boundaries of hyp erbolic metric spaces. Pacific J. Math., 221(1):147-158, 2005. [23] Corran Webster and Adam Winchester. Busemann Points of Metric Spaces, 2005. Preprint. [24] Corran Webster and Adam Winchester. Busemann points of infinite graphs. Trans. Amer. Math. Soc., 358(9):4209-4224 (electronic), 2006.

Квантование как приближенное описание некоторого диффузионного процесса
Е.М. Бениаминов

1. Описание и некоторые свойства модели

Рассматривается некоторая математическая модель процессаD состояE ние которого в каждый момент времени задается волновой функцией ! комплекснозначной функцией (x, p)D где (x, p) R2n , и n " размерE ность конфигурационного пространстваF В отличие от квантовой мехаE никиD где волновая функция зависит только от координат или только от импульсовD в нашем случае волновая функция зависит и от коордиE нат и и от импульсовF Так жеD как в квантовой механикеD предполагаетE сяD что для волновых функций выполняется принцип суперпозицииD и


untiztion s pproximte desription @in ussinA

UW

плотность вероятности (x, p) на фазовом пространствеD соответствуE ющая волновой функции (x, p), задается стандартной формулой @IFIA

(x, p) = (x, p)(x, p) = |(x, p)|2 .

В работе рассматривается классическая модель диффузионного процесса для волновой функции (x, p) на фазовом пространствеF ПредполагаетсяD что каждый комплексный вектор волновой функции одновременно находится в REх движенияхX точка приложения вектора движется по классической траекторииD заданной функцией Гамильтона H (x, p)Y точка приложения вектора перемещается случайно по координаE там и импульсамD находясь в диффузионном процессе с постоянными коэффициентами диффузий a2 и b2 по координатам и импульсамD соE ответственноY точка приложения каждого вектора движется по случайной траекE тории в результате движенийD описанных в двух предыдущих пунктахD а сам вектор вращается с постоянной угловой скоростью = mc2 / в системе координатD связанной с этой точкойD где m ! масса частицыD c ! скорость светаD ! постоянная ПланкаY длина всех комплексных векторов волновой функции в момент времени t умножается на exp(abnt/ ) @это чисто техническое требоE ваниеD которое не сказывается на относительных вероятностях нахожE дения частицы в фазовом пространствеAF ПредполагаетсяD что волновой вектор (x, p, t) в точке (x, p) в моE мент времени t по принципу суперпозиции равен сумме волновых векE торовD заданных распределением векторов 0 (x, p) в начальный моE мент времени и попавших в результате описанных выше движений в точку (x, p) в момент времени tF Процесс описывается дифференциальным уравнением диффузиE онного типаF Анализ дифференциального уравнения модели показыE ваетD что движение в модели раскладывается на быстрое и медленноеF В результате быстрого движения системаD начиная с произвольной волновой функции на фазовом пространствеD переходит к функцииD принадлежащей некоторому особому подпространствуF Элементы этоE го подпространства параметризуются волновыми функциямиD зависяE щими только от координатF Медленное движение по подпространству описывается уравнением ШредингераF Исходя из предположений о тепловой причине диффузий и соE ответствии следствий модели известным физическим экспериментам Лэмба E Резерфорда 2 @сдвиг Лэмба в спектре атома водородаAD в


VH

ivgeny wF feniminov

работе делается оценка коэффициентов диффузий и времени переходE ного процесса от классического описания процессаD в котором принцип неопределенности Гейзенберга может не выполнятьсяD к квантовомуD в котором принцип Гейзенберга уже выполняетсяF Время переходного процесса имеет порядок 1/T ћ 10-11 сD где T " температура средыF
2. Основные результаты

Рассмотрим диффузионный процесс на фазовом пространствеD в коE тором волновая функция (x, p, t) в момент времени t удовлетворяет дифференциальному уравнению @PFIA

= t

n

k=1

H H - xk pk pk xk
n

-

i

n

H-
k=1 2

H pk + a,b , pk
n

@PFPA

где



a,b

= a2
k=1

ip - xk

k

+ b2
k=1

2 abn + , p2 k

где H (x, p) " функция ГамильтонаY a2 и b2 " коэффициенты диффуE зий по координатам и импульсамD соответственноF Если в уравнении @PFIA отбросить последнее слагаемоеD то получим дифференциальное уравнение в частных производных первого порядE каF Эта часть уравнения @PFIA описывает детерминированную составE ляющую движения комплексных векторов (x, p, t)F Согласно уравнеE ниюD в этом движении точка приложения каждого вектора движется по классической траекторииD заданной гамильтонианом H (x, p)D а сам вектор при этом вращается в каждой точке траектории с угловой скоE ростью @PFQA

=

1

n

H-
k=1

H pk . pk

ЗаметимD что в случаеD когда конфигурационное пространство трехE 2 2 2 мерно и H = c m2 c2 + p2 D то dt = mc mc dt = mc d , где " H собственное время в системе координатD связанной с частицейD движуE щейся с импульсом pF То есть в этом случаеD векторD точка приложения которого движется по классической траекторииD вращается с постоянE ной угловой скоростью = mc2 / в системе координатD связанной с этой точкойF


untiztion s pproximte desription @in ussinA

VI

НаоборотD если в правой части уравнения @PFIA оставить только последнее слагаемое вида @PFPAD то получим уравнение
n 2 k=1 2 k n

@PFRA

=a t

ip - xk

+b

2 k=1

2 abn . + p2 k

Это уравнение описывает диффузионную составляющую движения векторов (x, p, t) на фазовом пространствеF В этом движении точки приложения векторов перемещаются в соответствии с классическим однородным диффузионным процессом с коэффициентами диффузий по координатам и импульсам равными a2 и b2 D соответственноF При этом сам вектор при малых случайных перемещениях из точки (x, p) в точку (x + dx, p + dp) переносится параллельноD а его длина в момент времени t умножается на exp(abnt/ )F ЗаметимD что параллельный пеE ренос векторов на фазовом пространстве задается связностьюD котоE рая выражается формулойX L(dx,dp) (x, p) - (x, p) -(i/ )(x, p)pdq , где L(dx,dp) (x, p) " параллельный перенос вектора (x, p) из точки (x, p) по бесконечно малому вектору (dx, dp)F В частном случаеD коE гда конфигурационное пространство трехмерноD такая связность на фазовом пространстве вызвана синхронизацией движущихся часов в точках фазового пространстваF Правая часть уравненния @PFRA " самосопряженный операторF ЗаE дача на собственные значения для этого оператора преобразованием Фурье по координатам сводится к стационарному уравнению ШрединE гера для гармонических колебанийF Отсюда показываетсяD что собE ственные значения оператора уравнения @PFRA неположительныD и верE на следующая теоремаF PFI Пусть (x, p, 0) произвольная функция, преобразование Фурье которой по p стремится к нулю при x . Тогда решение (x, p, t) диффузионного уравнения (2.4) экспоненциально по времени (с показателем равным -abt/ ) стремится к стационарному решению вида:
Теорема .

@PFSA

(x, p) = lim (x, p, t) =
t

1 (2 )n/

2 R
n

(y )(x, y )e

-i(y -x)p/

dy ,


VP

ivgeny wF feniminov

где
@PFTA @PFUA

(y ) =

1 (2 )n/ b a

2
2n

(x, p, 0)e

i(y -x)p/

(x, y )dpdx,

R n/4

(x, y ) =

e

-b(x-y )2 /(2a )

.

ЗаметимD что 2 (x, y ) представляет собой плотность вероятностей нормального распределения по x с математическим ожиданием y D и дисперсией a /(2b)F Если величина a /(2b) малаD то функция 2 (x, y ) близка к дельтаEфункции от x - y F Композиция выражений @PFTA и @PFSA строит проектор из пространE ства всех волновых функцийD заданных на фазовом пространствеD на некоторое подпространствоF Элементы этого подпространства параE метризуются функциями вида (y ), где y Rn D тF еF волновыми функE циями на конфигурационном пространствеF Если же предполагатьD что диффузия вызывается тепловыми возE действиями на электронD то коэффициенты диффузий по координатам и импульсам выражаются в статистической физике @смFD например 4D глFUD R и WA через температуру T по формуламX a2 = k T /(m ) и b2 = k T m, где k " постоянная БольцманаD m " масса электронаD " коE эффициент трения среды на единицу массыF ОтсюдаD a/b = ( m)-1 и ab = k T F То естьD в этом случаеD величина a/bD которая входит в выражение @PFUAD не зависит от температурыF С другой стороныD t " время переходного процессаD определенное в теореме ID имет видX t /(ab) = /(k T ) = T -1 ћ 7.638 ћ 10-12 с. С учетом этой оценкиD будем считать в уравнении @PFIA величину /(ab) малым параметром и предполагатьD что координаты и импульсы мало меняются за это время при классическом движенииD определенE ном гамильтонианом H (x, p)F PFP Движение, описываемое уравнением (2.1), асимптотически распадается при /(ab) 0 на быстрое движение и медленное движение. В результате быстрого движения произвольная волновая функция (x, p, 0) переходит за время порядка /(ab) к виду (2.5). Волновые функции вида (2.5) образуют линейное подпространство. Элементы этого подпространства параметризуются волновыми функциями (y), зависящими только от координат y Rn . Медленное движение, начинающееся с ненулевой волновой функции из этого подпространства, происходит по подпространству и параметризуется волновой функцией (y, t), зависящей от
Теорема .


untiztion s pproximte desription @in ussinA

VQ

^ i / t = H

времени. Функция , где
^ H =

(y , t)

удовлетворяет уравнению Шредингера вида
n

1 (2 )n
R
3n

H (x, p) -
k=1
i

H ib H + ( xk - yk ) Ч xk a pk

Ч(x, y )(x, y )e

(y -y )p

(y , t)dy dxdp,

и

(x, y )

задается формулой (2.7).
PFQ. ^ H

Если ab малая величина и то оператор с точностью до членов порядка
Теорема

H (x, p) = a /b

имеет вид:

p2 2m

+ V (x),

@PFVA

^ H-

2

n

2m

k=1

2 2 yk

a + V (y ) - 4b

n

k=1

2V 3nb 2 + 4ma . yk

Первые два слагаемые в формуле @PFVA дают стандартный операE тор ГамильтонаF Последнее слагаемое " константаD и ею можно преE небречьF Предпоследнее слагаемое рассмотрим @ввиду малости a /bA как возмущение к оператору ГамильтонаF СчитаяD что отклонения в спектре атома водорода @сдвиг ЛэмE баAD наблюдаемые в экспериментах ЛэмбаEРезерфода 2D вызываются предпоследним слагаемым в формуле @PFVAD можно оценить величину a/bF Расчеты стандартным методом возмущенийD аналогичные расчеE тамD выполненным в 5D дают следующую оценкуX a/b = 3.41 ћ 104 сGгF ОтсюдаD стандартное отклонение для нормального распределения 2 D по которому производится сглаживание волновых функцийD имеет вид a /(2b) = 4.24 ћ 10-12 см. Эта величина существенно меньше радиуE са атома водорода и близка к комптоновской длине волны электрона /(mc) = 3.86 ћ 10-11 смF Таким образомD расчеты показываютD что предложенная модель в виде дифференциального уравнения @PFIA достаточно адекватно опиE сывает физические процессы в стандартных случаях для стандартноE го гамильтонианаF Но эту модель можно применить и для расчетов процессов с нестандартным гамильтонианом или с гамильтонианомD быстро меняющимся во времениD как при внезапных возмущениях или для периодически меняющегося потенциала с частотой порядка ab/ D и сравнить с экспериментальными даннымиF


VR
Литература

eFwF qel9fndD fFuhF uirshteyn

I P Q

Маслов В.П. Уравнения Колмогорова E Феллера и вероятностная модель кванE товой механики GG Итоги науки и техникиF ТеорF верFD матF статF и кибернетF IWVPF ТF IWF СF SSEVSF Lamb W.E., Retherford R.C. Fine Structure of the Hydrogen Atom by a Microwave Method in

GG hysF evF IWRUF F UPF F PRIEPRQF
E.M. A Method and for Mechanics Probability

Beniaminov

Justication

of

the

View in

of

Observables Space

httpXGGrxivForgGsGquntEphGHIHTIIPF PHHIF R Исихара А Статистическая физикаF МFX МирD IWUQF RUP сF S Welton T.A. Some Observable Eects of The Quantum-Mechanical the Electromagnetic Field GG hysF evF IWRVF F URF F IISUEIITUF

Quantum

Distributions

Phase

Fluctuations of

Идемпотентные системы нелинейных уравнений и задачи расчета электроэнергетических сетей
А.М. Гельфанд и Б.Х. Кирштейн

1

Электроэнергетическую сеть можно рассматривать как граф с n верE шинамиD каждой вершине @узлуA которого сопоставлены два вещеE ственных числа E активная Pk и реактивная Qk составляющие инъекE ции мощности в узле k (k = 1, ..., n)D и каждому ребру @линий электроE a передачAD соединяющих k Eый и j Eый узлы E активная Ykj и реактивная r Ykj составляющие проводимостиF Установившиеся режимы электроэнергетических сетей характеризуE a r ются значениями активных Uk и реактивных Uk составляющих наE пряжений в узлахD которые должны удовлетворять системе 2n алгебE раических уравнений 1 E узловых уравнений балансов активной и реативных мощностейF Такие уравнения можно рассматривать как вещественную и комплексE ную составляющую системы n комплексных уравнений вида @IA

Wk = E k (
j (k)

akj E j ),

a r a r где Wk = Pk + iQk D Ek = Uk + iUk D E k = Uk - iUk D суммирование идет по всем узлам j связанных линией с узлом k D akj комплексные числаD a r которые определяется по комплексным проводимостям Zj = Uj + iUj F

HIEHPVHUEНЦНИЛаF

1Работа выполнена при частичной финансовой поддержке гранта РФФИ HSE


sdempotent equtions in eletroenergetis @in ussinA

VS

Эти уравнения дают 2n вещественных алгебраических уравнений отE a r носительно 2n неизвестных Uk , Uk , k = (1, ..., n)F Важной задачей анализа электроэнергетических сетей является анаE лиз устойчивости @определение запаса устойчивостиA установившегося режимаF На практикеD такой анализ сводится к проверке существоваE нии вещественной деформации решения при деформациях коэффициE ентов уравненийD входящих в системуF Такие деформации отвечают изменениям инъекций мощности в узлах или измению структуры граE фа сети @отключение линийAF При этом в качестве критерия потери устойчивости при деформациях уравнений системы обычно принимается либо расходимость итерациE онного процесса метода Ньютона для нахождении решенияD либо выE рождение матрицы Якоби системы уравненийF Такой подход не всегда математически корректно отвечает поставленной залачеD ноD главноеD не позволяет быстро и наглядно получить границу области устойчиE вости установившегося режимаF Мы рассматриваем некоторую процедуру простого макетирования заE дачи анализа устойчивости электроэнергетических систем с помощью анализа решений идемпотентной системы уравненийD полученной в реE зультате деквантования по Маслову 2 исходной системы @IAF Запишем систему уравнений установившегося режима более симметE рично в виде системы 2n комплексных алгебраических уравнений 3D относительно 2n комплексных переменныхF Для этого введем новые комплексные переменные Sk вместо E k и новые уравненияD полученE ные с помощью комплексного сопряженния уравнений @IA и последуE ющей аналогичной заменной переменныхF Получим систему из 2n комплексных уравнений @PA

Wk = Ek (
j (k)

akj Sj ),

W k = Sk (
j ( k )

akj Ej )

относительно 2n комплексных переменных Ek , Sk , k = 1, ..., nF Решение системы @PA удовлятворяет системе @IA тогда и только тогда когда @QA

E k = Sk

для всех k = 1, ..., nF В 4 для всякой алгебраической поверхности f (z ) = 0 в C2n определеE на ее амеба Af в R2n D образ пересечения этой поверхности с комплексE ным тором относительно отображения

Log : (z1 , . . . , z2n ) (C\{0})2n (log |z1 |, . . . , log |z2n |) R2n .


VT

eFwF qel9fndD fFuhF uirshteyn

Условие @QA в этих терминах означаетD что пересечение амеб всех уравE нений системы @PA содержит точкиD удовлетворяющие условиям @RA

log|Ek | = log|Sk |

для всех k = 1, ..., nF Известно 4D что амеба Af совпадает с дополненнием конечного числа открытых выпуклых подмножеств E в R2n X

R2n \Af = {E }.
В 5 определены наборы линейных функций на этих выпуклых мноE жествахD нижняя грань которых определяет кусочноEлинейное подмноE жество R2n D которое называется спайном амебыD лежит внутри Af и является ее гомотопическим ретрактомF Чтобы определить идемпотентную систему уравнений определим троE пические аналоги операций сложения и умножения в R обычным обE разомX тропическое сложение как x y = max{x, y } и тропическое умножение как x y = x + y F Рассмотрим идемпотентную систему уравненийD полученную из уравE нений @PA заменой обычных операций на их тропические аналоги и комплексных коэффициентов на логарифмы их модулейX @SA и @TA

log|Wk | = log|Ek | (
j (k)

log|akj | log|Sj |),

log|Wk | = log|Sk | (
j (k)

log|akj | log|Ej |),

ЗаметимD что многогранники Ньютона для уравнений @PA совпадают с выпуклой обoлочкой подмножества вершин единичного куба в R2n и не содержат поэтому внутри себя точек целочисленной решеткиF Как доказано в 6D в этом случае уравнение спайна амебы совпадают с уравнениями идемпотентной системыD результата деквантования f F Отсюда легко получается следующая
Теорема. Решение идемпотентной системы уравнений

@SA, @TA

сов-

падает с множеством точек пересечения пределов амеб уравнений при ретракции их на свои спайны.

@PA

Используя эту теорему можно следующим образом получить проE стую модель для анализа запаса устойчивости электроэнергетической


sdempotent equtions in eletroenergetis @in ussinA системыF Пусть

VU

E = (E1 , ..., En ) (C\{0})n решение системы @IAF Найдем решение идемпотентной системы уравE нений @TAD @UAD ближайшее к вектору Log((E , E )) в R2n F Будем гоE воритьD что область параметров деформации является областью приE тяжения @отталкиванияAD если при соответствующей деформации реE шение идемпотентной системы приближается @удаляетсяA от подпроE странстваD определяемого уравнениями @RA в R2n F Задача нахождения границыD разделяющих области притяжения и отE талкивания в случае идемпотентой системы может рассматриватьсяD как естественный модельный аналог задачи нахождения запаса устойE чивостиF Такая задачаD по сравнению с анализом зависимости от паE раметров решений вещественных многомерных систем алгебраических уравненийD решается существенно проще в идемпотентном анализеD где онаD по существуD сводится к анализу систем линейных уравнений заE висящих от параметровF
Литература

I Евдокунин ГFАF Электрические системы и сетиF СПбD PHHIF P Литвинов ГFЛF Деквантование Маслова, идемпотентная и тропическая математика: краткое введениеF Записки научных семинаров ПОМИD 326 @PHHSAD IRS!IVPF Q wontes eF Algebraic solution of the load-ow problem for a 4-nodes electrical networkF wthemtis nd gomputers in imultionF 45D @IWWVA F ITQ!IURF R qelfnd sFD uprnov wFD elevinsky eF Discriminants, resultants and multidimensional determinantsF firkh? userX fostonD IWWRF S ssre wFD ullgrd rF Amoebas, Monge - Ampere measures and triangulations of the Newton polytope F eserh ep orts in wthemtisFD to kholm niversityD10DPHHPF T ssre wFD sikh eF Amoebas: their spines and their contours gontemporry wthF 377 @PHHSAD F PUS ! PVVF


VV

yksn F nmensky
Классические и неархимедовы амебы в вопросах расширения полей
О.В. Знаменская

Поле P называется алгебраическим расширениемD или расширением Галуа поля KD если существует алгебраическое уравнение @IA

c0 + c1 x + c2 x2 + ћ ћ ћ + cn xn = 0

с коэффициентами в поле KD такоеD что поле P получается присоедиE нением к K всех корней этого уравненияF ИзвестноD что все такие расE ширения конечномерныF Таким образомD в классической теории конечные расширения строE ятся при помощи присоединения к исходному полю нулей полиномов от одного переменногоF Наша цель " изучение бесконечных аналоE гов этих расширенийD определяемых полиномами от нескольких переE менныхD для случая неархимедовых полейF Более точноD наша задача состоит в описании бесконечных расширений подполей неархимедова поля K рядов ПюизоF НапомнимD что в поле K с неархимедовым нормированием норма элемента a K может быть определена через показатель нормироE вания vl(a) поля K при помощи соотношения |a| = e-vl(a) F Здесь vl(a) есть отображение K R {}D определенное на элементах K и удовлетворяющее следующим условиям 1X A vl(a) = тогда и только тогдаD когда a = 0Y A vl(ab) = vl(a) + vl(b)Y A vl(a + b) min(vl(a), vl(b))F Пусть K " поле рядов Пюизо с коэффициентами в произвольном поле k D тFеF рядов a(t) вида

a(t) =
qj Aa

j t

qj

по дробным степеням qj переменного tD где Aa Q " вполне упорядоE ченное множествоF Показатель неархимедова нормирования vl в этом случае полагается равным min Aa F Бесконечные расширения поля K будем строить следующим обраE зомF Рассмотрим полином @PA

f=
A

a (t)z




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из K[z1 , . . . , zn ] с коэффициентами a (t) L KD где L " подполе KF Определим расширение P/L как множество всевозможных значений полиномов b z (t) ,
B

где z (t) = z1 (t), . . . , zn (t) является решением уравнения f = 0 для полинома f вида @PAF ОтметимD что самый простой случай расширений Галуа " цикE лические расширенияD получаются присоединением к исходному полю всех корней из единицыD тFеF решений двучленного уравнения xm - a = 0F ОчевидноD все корни полинома Галуа f (x) = xm - aX
ћ лежат на окружностиY ћ на окружности они равномерно распределеныF

Многомерный аналог первой из указанных геометрических характериE стик решений полиномов Галуа может быть сформулирован на языке амебF Амебой Af комплексной гиперповерхности V (\{0})n D задаваеE мой полиномом f @смF 2AD называется ее образ при отображении vog : (C \ {0})n - Rn , действующем по правилу

(z1 , . . . , zn ) - (log |z1 |, . . . , log |zn |).
АмебуD определяемую таким образомD назовем классическойF
Определение I @смF 3A. Классическая амеба Af называется соE лиднойD если число связных компонент дополнения к ней минимальноF

Многомерное обобщение того фактаD что все корни f (x) = xm - a лежат на единичной окружности на языке амеб выражается в томD что Af f F ЗаметимD что если n = 1D то амеба произвольного полинома от одного переменного есть конечное множество точек в R1 F Солидными в этом случае будут только амебыD состоящие из одной точки и имеющие лишь две связные компоненты в дополненииD а это и есть в точности амебы полиномов ГалуаF НапомнимD что многогранником Ньютона Nf полинома f от n пеE ременных называется выпуклая оболочка показателей его мономов в Rn F Конусом рецессии выпуклого множества E Rn называется макE симальный конус среди техD которые сдвигом можно поместить в E F

амеба

полинома солидна


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yksn F nmensky

Согласно результатам МF ФорсбергаD МF Пассаре и АFКF Циха 4D справедлива
Теорема .

I Существует естественная инъективная функция порядка на множестве {E } связных компонент дополнения Rn \ Af амебы гиперповерхности f = 0, сопоставляющая каждой компоненте E некоторую целочисленную точку (E ) из многогранника Ньютона Nf . Конус рецессии компоненты E совпадает с конусом, двойственным к Nf в точке (E ). Таким образомD многогранник Ньютона отражает структуру класE сической амебыF В частностиD число связных компонент дополнения Rn \ Aa не меньше числа вершин и не больше числа всех целых точек многогранника Ньютона Nf F ЯсноD что классическая амеба солиднаD если число компонент дополнения строго равно числу вершин Nf F С точки зрения многогранников НьютонаD многомерным аналогом полиномов Галуа являютсяD так называемыеD 3D тFеF полиномы видаX

ные полиномы

максимально разрежен-

f=
A

a z ,

где с ненулевыми коэффициентами входят только мономыD соответE ствующие вершинам Nf F МF Ниссе был заявлен результатD что классиE ческая амеба любого максимально разреженного полинома солиднаF Далее нас будет интересовать вопрос солидности неархимедовых амеб нулевого множества максимально разреженных полиномов @PAD при помощи которых строятся бесконечные расширения P/L неархиE медова поля рядов ПюизоF Определим по аналогии свойство солидности для неархимедовых амебF Пусть K " произвольное неархимедово поле и vl(a) " его поE казатель нормированияF
Определение P @смF 5, 6A. Амебой A(V ) алгебраической гиперE n поверхности V (K ) называется замыкание образа V при отобраE жении vog : (z1 , . . . , zn ) (-vl(z1 ), . . . , -vl(zn )).

Из теоремы I следуетD что конусы рецессии всех компонент доE полнения классической солидной амебы полномерныF СоответственноD для неархимедова случая можно дать следующее
Определение Q. Неархимедова амеба A(V ) называется солиднойD если любая связная компонента дополнения к ней имеет полномерный конус рецессииF


glssil nd nonrhimeden moes @in ussinA Пусть K " поле рядов ПюизоF
.

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P Неархимедова амеба максимально разреженного полинома, определяющего многомерное расширение Галуа поля K, солидна.
Теорема

Q В случае n = 2 если неархимедова амеба солидна, то, как граф, она не имеет циклов.
Теорема .

Согласно 6 существует двойственность между неархимедовой амеE бой A(V ) и подразбиением многогранника Ньютона полиномаD опреE деляющего V F С учетом этого справедлива

Пусть f (z ) =
A

c z D z = (z1 , . . . , zn ) Cn " полиномD все коE

эффициенты которого имеют рациональные модулиD тFеF все |c | QF Определим полином F следующим образомX

F=
A

c (t) ,
-|c |

где c (t) таковыD что |c (t)| = e справедлива

F В указанных предположениях

медова амеба

Теорема

R. A(V )

Если классическая амеба солидна.

Af

солидна, то и неархи-

В доказательстве теоремы используются понятия хребта амебы и тропического многообразияD определяемого при помощи тропикализаE ции полинома F F
Литература

I Боревич ЗFИFD Шафаревич ИFРF Теория чиселF МFX НаукаF ГлF редF физFEматемF литEрыF ! IWVSF SHR сD QEе издFдопF P qelfnd sFwFD uprnov wFwFD elevinsky eFF Discriminants, Resultants, and Multidimensional DeterminantsD firkh? userD fostonD IWWRF Q ssre wFD sikh eF Amoebas: their spines and their contours GG gontemporry wthFD PHHSD olF QUUD F PUSEPVVF R wF porsergD wF ssre nd eF sikhD Laurent Determinants and Arrangements of Hyperplane Amoebas, edvnes in wthF 151 @PHHHA RSF S uprnov wFwF Amoebas over non-Archimedean eldsD reprint PHHHF T iinsiedler wFD uprnov wFD vind hF Non-archimedean amoebas and tropical varietiesD httpXGGrxivForgGsGmthFeqGHRHVQIIF


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Обобщение ультравторичного квантования для фермионов при ненулевой температуре
Г.В. Коваль и В.П. Маслов

1

В работах ВFПF Маслова развит метод ультравторичного квантоE вания и концепция истинного символа 1, 2, 3F Этот метод позвоE ляет находить асимптотические серии систем большого числа частиц при нулевой температуреF В частностиD некоторые серии определяютE ся периодическими решениями системы уравнений ГамильтонаD соотE ветствующей истинному символу 2, 3 рассматриваемой физической системыF В данной работе найдено соответствие между уравненияE ми метода ультравторичного квантования по парам для фермионов и уравнениями вариационного метода БоголюбоваF Так как вариациE онный метод Боголюбова применим для случая ненулевой температуE рыD из принципа соответствия получено обобщение уравнений метода ультравторичного квантования фермионов на температурный случайF В статье 3 показаноD что асимптотика серий собственных знаE чений системы N тождественных фермионов в пределе при N определяется решениями следующей системы уравнений @IA
2

(x, y ) = +2 + (x, y ) = +2

-

2m

(x + y ) + U (x) + U (y ) + V (x, y ) (x, y )+

dz dw (V (x, y ) + V (z , w)) + (z , w)(x, z )(w, y ),
2

-

2m

(x + y ) + U (x) + U (y ) + V (x, y ) + (x, y )+

dz dw (V (x, z ) + V (y , w)) + (x, z )+ (w, y )(z , w),

где x, y M " координаты частицD пространство M определяется заE дачейD напримерD это может быть R3 D или трехмерный торD x D y " операторы ЛапласаD действующий по соответствующей переменной x или y D U (x) " потенциал внешнего поляD V (x, y ) " потенциал взаиE модействияD симметричный относительно перестановки переменных x и y D m " масса частицD " постоянная ПланкаD " действительное
1Работа выполнена при поддержке грантов РФФИ HSEHIEHHVPR и HSEHIEHPVHUE

НЦНИЛаF


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числоF Функции (x, y ), + (x, y ) L2 (M) " антисимметричны отноE сительно перестановок переменных x и y и удовлетворяют условию N @PA dxdy + (x, y )(x, y ) = . 2 Уравнения @IA получаются при ультравторичном квантовании по парамF В таком квантовании рассматриваемой системе фермионов отE вечает истинный символ вида @QA A[+ , ] =
2

dxdy + (x, y ) - +2

2m

(x + y ) + U (x) + U (y ) + V (x, y ) (x, y )+

dxdy dz dwV (x, y )+ (x, y )+ (z , w)(x, z )(w, y ),

который является функционалом от двух антисимметричных функций + (x, y ) и (x, y ) из L2 (M2 )F Этому символу соответствует система уравнений Гамильтона @RA

i

A (x, y , t) = , t + (x, y , t)

-i

+ A (x, y , t) = , t (x, y , t)

где в правой части уравнений стоит вариационная производная функE ционала @QAF Система уравнений @RA имеет интеграл движенияD вид которого совпадает с выражением в левой части равенства @PAF УравE нения @IA получаются из @RA в частном случаеD когда @SA

(x, y , t) = (x, y )e

-it

,

+ (x, y , t) = + (x, y )e

it

.

Запишем @IA в другом видеF Введем функции G(x, y )D R(x, y ) и R(x, y )

R(x, y ) = + (x, y ),
@TA

G(x, y ) = 2

dz + (x, z )(y , z ),

R(x, y ) = 2 (x, y ) -

dz (x, z )G(z , y ) .

В силу антисимметрии функций + (x, y )D (x, y ) для функций @TA выE полняются равенства @UA @VA

R(x, y ) = -R(y , x), G(x, y ) =

R(x, y ) = -R(y , x), dz R(z , x)R(z , y ).

dz G(x, z )G(z , y ) +


WR Из @PA следует

qFF uovlD FF wslov

@WA

dx G(x, x) = N .

I Функции ют системе уравнений
Предложение .

G(x, y ) R(x, y ) R(x, y )

,

,

удовлетворя-

@IHA
2

- - - - - -

2m

(x - y ) + U (x) - U (y ) G(x, y )-

dz (V (x, z ) - V (y , z )) R(x, z )R(z , y ) = 0,
2

2m

(x + y ) + U (x) + U (y ) + V (x, y ) R(x, y )-

dz (V (x, z )G(z , y )R(x, z ) + V (y , z )G(z , x)R(z , y )) = R(x, y ),
2

2m dz

(x + y ) + U (x) + U (y ) + V (x, y ) R(x, y )- V (x, z )G(y , z )R(x, z ) + V (y , z )G(x, z )R(z , y ) = R(x, y ).

ДействительноD непосредственной проверкой удостоверяетсяD что функции @TA удовлетворяют уравнениям @IHAD если функции + (x, y )D (x, y ) удовлетворяют @IAF Чтобы обобщить уравнения @IA на случай ненулевой температурыD применим принцип соответствия между этими уравнениями и уравE нениями вариационного принципа Боголюбова при ненулевой темпеE ратуреF То есть исходя из температурных уравнений вариационного принципа БоголюбоваD по принципу соответствия найдем обобщение уравнений @IA на температурный случайF


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Рассмотрим уравнения вариационного метода БоголюбоваF При температуре 0 для рассматриваемой системы фермионов из ваE риационного метода Боголюбова 4 получаются уравнения @IIA
2 dy V (x, y )RB (x, y )v (y )+

u (x) = + - v (x) = +

-

2m

+ U (x) - ч u (x) +

dy V (x, y ) (GB (y , y )u (x) - GB (y , x)u (y )) ,
2

-

2m

+ U (x) - ч v (x) +

dy V (x, y )RB (x, y )u (y )+

dy V (x, y ) (GB (y , y )v (x) - GB (y , x)v (y )) ,

где = 1, 2, . . . D функции u (x) и u (x)D а также v (x) и v (x)D комE плексно сопряжены друг другу и удовлетворяют условиям

@IPA

dx u (x)v (x) + v (x)u (x) =
dx u (x)u (x) + v (x)v (x) =

dx u (x)v (x) + v (x)u (x) = 0,



, , = 1, 2, . . . ,

где " символ КронекераF Кроме того в уравнениях @IPA функции RB (x, y ) и GB (x, y ) имеют вид


RB (x, y ) =
@IQA
=1

1 -n 2



(v (x)u (y ) - v (y )u (x)) ,

GB (x, y ) =
=1

(v (x)v (y ) (1 - n ) + u (x)u (y )n ) ,

где @IRA

n =

1 , exp( /) + 1

а ч определяется из условияD что функция GB (x, y ) @IQA удовлетворяет равенству @ISA

dx GB (x, x) = N .

Функции @IQA и комплексно сопряженная к RB (x, y ) функция RB (x, y ) при любой температуре удовлетворяют равенствам


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@ITA @IUA

RB (x, y ) = -RB (y , x), GB (x, y ) = G (y , x), B

RB (x, y ) = -RB (y , x),

а из уравнений @IIA следуетD что эти функции также удовлетворяют системе уравнений @IVA
2

- + + - - + +

2m

(x - y ) + U (x) - U (y ) GB (x, y )+

dz (V (x, z ) - V (y , z )) RB (x, z )RB (z , y )+

dz (V (x, z ) - V (y , z )) (GB (z , z )GB (x, y ) - GB (x, z )GB (z , y )) = 0,
2

2m

(x + y ) + U (x) + U (y ) + V (x, y ) RB (x, y )-

dz (V (x, z )GB (z , y )RB (x, z ) + V (y , z )GB (z , x)RB (z , y )) + dz V (x, z ) (GB (z , z )RB (x, y ) - GB (z , x)RB (z , y )) + dz V (y , z ) (GB (z , z )RB (x, y ) - GB (z , y )RB (x, z )) = 2чRB (x, y ),

где дополнительное уравнение получается комплексным сопряжением второго уравнения формулы @IVAF Если = 0D то из @IRA следуетD что n принимает значение 0 или 1 для всех = 1, 2, . . . F Тогда из @IPA следуетD что функции @IQA при нулевой температуре удовлетворяют условию @IWA

GB (x, y ) =

dz GB (x, z )GB (z , y ) +

dz RB (z , x)RB (z , y ).

Равенства @IWAD @ITA и @ISA совпадают соответственно с равенстваE ми @VAD @UA и @WAD если по принципу соответствия заменить RB (x, y ) на R(x, y )D RB (x, y ) на R(x, y )D GB (x, y ) на G(x, y )F Уравнения @IVA при такой замене не переходят в уравнения @IHAD однакоD если 2ч в @IVA заE менить на D то очевидно соответствие между одной системой и друE гойD соответствующие друг другу уравнения отличаются несколькими слагаемыми в левой частиF Кроме тогоD для частного вида взаимодейE ствияD напримерD такого как в модели БКШ 5D уравнения @IVA после замены совпадают с @IHAF


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ОтметимD что функции GB (x, y )D RB (x, y )D RB (x, y ) удовлетворяют большему числу условийD чем функции G(x, y )D R(x, y )D R(x, y )F ФункE ции (x, y )D + (x, y )D удовлетворяющие уравнениям @IAD в общем слуE чае не являются комплексно сопряженными друг другу 2F Поэтому из формул @TA следуетD что функция R(x, y ) не должна быть комплексE но сопряженной к R(x, y )D а функция G(x, y ) не должна удовлетворять условию @IUAF Из соответствия между функциями @TA и @IQAD получимD что темE пературным аналогом системы уравнений @IA является следующая сиE стема уравненийX

@PHA
2

u (x) = - v (x) = u (x) = - v (x) =

- - - -

2m
2

+ U (x) - ч u (x) + + U (x) - ч v (x) + + U (x) - ч u (x) + + U (x) - ч v (x) +

dy V (x, y )R(x, y )v (y ), dy V (x, y )R(x, y )u (y ), dy V (x, y )R(x, y )v (y ), dy V (x, y )R(x, y )u (y ),

2m
2

2m
2

2m

где G(x, y )D R(x, y ) и R(x, y ) выражаются следующим образомX


G(x, y ) =
=1

(v (x)v (y ) (1 - n ) + u (x)u (y )n ) , 1 -n 2 1 -n 2 (v (x)u (y ) - v (y )u (x)) , (v (x)u (y ) - v (y )u (x)) ,

@PIA

R(x, y ) =
=1



R(x, y ) =
=1



n выражается через и формулой @IRAD а функции u (x)D v (x)D u (x)D v (x)D = 1, 2, . . . кроме уравнений @PHA еще удовлетворяют условиям
@PPA

dx (u (x)v (x) + v (x)u (x)) = dx (u (x)u (x) + v (x)v (x)) =

dx (u (x)v (x) + v (x)u (x)) = 0,


,

, = 1, 2, . . . .


WV

qFF uovlD FF wslov

Параметр ч в уравнениях @PHA определяется из условияD что функция G(x, y ) из @PIA удовлетворяет условию @WAF P Если функции u (x), u (x), v (x), v (x), = удовлетворяют системе уравнений (20) и условиям (22), то функции (21) при любом 0 удовлетворяют системе уравнений (10) с = 2ч и условиям (7,9), а при = 0 еще удовлетворяют условию (8).
Предложение .

1, 2, . . .

В силу уравнений @PHA и условий @PPA это утверждение доказываE ется прямой подстановкой функций @PIA в формулы @UEIHAF Множество решений уравнений @IVA ширеD чем множество функE ций @IQAD выраженных через решения системы уравнений @IIAF В 2 показаноD что система уравнений @IVA может быть записана в виде паE рыX @PQA

A, L = 0,

а множеству решений температурных уравнений @IIA соответствует такое решение уравнения @PQAD для которого @PRA где @PSA

A = f (L), 1 1 -. exp( /) + 1 2

f ( ) =

Для уравнений @PHA и @PHA справедливо аналогичное утверждениеF РасE смотрим матрицы A и L вида @PTA

A=

G- R

1 2

1 2

-R - Gt

,

L=

T B

-B -T

,

где GD RD R " операторы в пространстве L2 (M)D задаваемые интеE гральными ядрами G(x, y )D R(x, y )D R(x, y ) соответственноD Gt " опеE раторD задаваемый в L2 (M) ядром Gt (x, y ) = G(y , x)D B и B " ядрами B (x, y ) = V (x, y )R(x, y ) и B (x, y ) = V (x, y )R(x, y ) соответственноD а оператор T " оператор Гамильтона для одной частицыD то есть опеE ратор вида
2

+ U (x). 2m Подстановка @PTA в @PQA приводит к четырем уравнениямD из коE торых два совпадаютD а три независимых приводятся к виду @IHAF ПоE этому справедливо следующее утверждениеF

T =-


gontt lssi(tion of wongeEemp equtions @in ussinA ere
Предложение .

WW

Q Система уравнений (4) может быть записана в виде (23), где A и L имеют вид (26), а 2ч = . Кроме тогоD решениям системы уравнений @IHAD вида @PIAD котоE рые получены из решений уравнений @PHDPPAD соответствуют такие A и LD что для них справедливо равенство @PRAF Это является следствием уравнений @PHDPPAF В заключение отметимD что уравнения @IHA в температурном слуE чаеD полученные здесь из принципа соответствияD могут быть строго получены из истинного символа 2, 3 для ультравторично квантованE ного уравненияD отвечающего матрице плотности 1F
Литература

I ВFПF МасловF Квантование термодинамики и ультравторичное квантованиеF МоскваD Институт компьютерных исследованийD PHHIF P ВFПF МасловF Ультратретичное квантование термодинамикиF GG ТМФD тF IQPD QD PHHPD сFQVVEQWVF Q FFwslovF he notions of entropyD rmiltoninD tempertureD nd thermodynmil limit in the theory of proilities used for solving model prolems in eonophysisF GG ussin tournl of wthF hysisD PHHPD vFWD nFRD pFRQUERRSF R НFНF БоголюбовF Избранные трудыF тFPF КиевD Наукова думкаD IWUHF S ДжF ШрифферF Теория сверхпроводимостиF МFD НаукаD ФизматгизD IWUHF

Контактная классификация уравнений Монжа-Ампера
А.Г. Кушнер

1. Геометрические структуры, ассоциированные с уравнениями Монжа-Ампера

Класс уравнений МонжаEАмпера выделяется из многообразия уравнеE ний второго порядка темD что он замкнут относительно контактных преобразованийF Это обстоятельство было известно еще Софусу ЛиD изучавшему уравнения МонжаEАмпера методами созданной им конE тактной геометрииF В IVUHEх и IVVHEх он поставил проблемы классиE фикации уравнений МонжаEАмпера относительно @псевдоAгруппы конE тактных преобразованийD в частностиD о приведении уравнений МонE жаEАмпера к квазилинейной форме и наиболее простом координатном представлении таких уравнений 5F


IHH

elexey qF uushner

В IWUW гF в работе 9 ВF ВF Лычагин показалDчто уравнения МонжаE Ампера допускают эффективное описание в терминах дифференциE альных форм на многообразии 1Eджетов гладких функцийF Отправной точкой является следующее наблюдениеF Пусть M " nEмерное гладкое многообразиеD J 1 M " многообразие 1Eджетов гладких функций на M F На J 1 M естественным образом опреE делена контактная структура " распределение Картана C D в каноничеE ских локальных координатах Дарбу (q , u, p) = (q1 , . . . , qn , u, p1 , . . . , pn ) задаваемое дифференциальной 1Eформой Картана U = du - pdq F ОграE ничение дифференциала формы Картана на подпространство КартаE на не вырождено на нем и определяет симплектическую структуру a = dU |C (a) 2 (C (a))F Со всякой дифференциальной nEформой n (J 1 M ) свяжем нелинейный дифференциальный оператор : C (M ) n (M ), действующий на гладкую функцию v следующим образомX @IFIA

(v ) = j1 (v ) ( ).

Здесь j1 (v ) : M J 1 M " 1Eджет функции v C (M )F Оператры называются операторами D а уравнеE ние E = { (v ) = 0} J 2 M " F СледуюE щее обстоятельство оправдывает эти названияX будучи записанным в локальных канонических координатах на J 1 M D оператор имеет тот же самый тип нелинейности по производным второго порядкаD что и классические опреаторы МонжаEАмпераD а именноD нелинейности типа определителя матрицы Гессе и ее миноровF При n = 2 мы получаем классическое уравнение МонжаEАмпераX

Монжа-Ампера уравнением Монжа-Ампера

@IFPA

Av

xx

+ 2B v

xy

+ C vyy + D(vxx v

yy

2 - vxy ) + E = 0,

где A, B , C, D, E " функции от независимых переменных x, y D функции v = v (x, y ) и ее первых производных vx , vy F Преимуществом такого подхода перед классическим является реE дукция порядка пространства джетовX мы используем более простое пространство 1Eджетов J 1 M вместо пространства 2Eджетов J 2 M D в коE торомD будучи уравнениями второго порядкаD должны лежать уравнения МонжаEАмпераF В случаеD когда коэффициенты уравнения @IFPA не зависят явно от функции v ситуация еще более упрощаетсяX в определении оператора @IFIA вместо пространства 1Eджетов можно рассматривать кокасательE ное расслоение T M многообразия M D а вместо контактной геометрии " симплектическуюF Такие уравнения МонжаEАмпера будем называть F

ad hoc

симплектическими


gontt lssi(tion of wongeEemp equtions @in ussinA ere

IHI

ЗаметимD что соответствие между дифференциальными nEформаE ми на J 1 M и операторами МонжаEАмпера не является взаимноEодноE значнымF Дифференциальные формыD аннулирующимся на любом инE тегральном многообразии распределении КартанаD образуют идеал C во внешней алгебре (J 1 M )D который называется F Им отвечает нулевой дифференциальный операторF Элементы факE торEалгебры (J 1 M )/C по этому идеалу называются формамиF Далее мы будем рассматиривать случай когда M " двумерное гладкое многообразиеF В терминах эффективных форм можно опредеE лить тип уравнения " эллиптическийD параболическийD гиперболичеE ский или переменныйF Функция Pf ( ) C J 1 M D определяемая поE точечно равенством Pf (a )a a = a a D называется формы F Уравнение E называется или в точке a J 1 M D если пфаффиан Pf ( ) отрицаE тельныйD нулевой или положительный в этой точкеF Если пфаффиан не равен нулю в точкеD то уравнение называется F ОчеE видным образом понятие типа распространяется на областьF В силу невырожденности симплектической структуры на подпроE странстве Картана C (a)D формула Xa a = Aa Xa a , определяет на C (a) ассоциированный с эффективной дифференциальной 2Eформой линейный оператор Aa F Здесь Xa C (a)F Этот оператор симметричен относительно симплектической структурыD а его квадрат скалярен и A2 + Pf ( ) = 0F ЗаметимD что операторы Aa не образуют поля эндоE морфизмов на J 1 M D ибо они определены только на подпространствах КартанаF Если в точке a J 1 M пфаффиан формы не обращается в нульD то в некоторой ее окрестости этой точки форму можно нормироE вать так чтобы Pf ( ) = +1F В этом случае на подпространсве Картана определена либо структура почти произведения @для гиперболических уравненийAD либо комплексная структура @для эллиптических уравнеE нийAF В первом случае мы получаем два вещественныхD а во втором " два комплексных 2Eмерных распределения на J 1 M D которые будем называть и обозначить через C+ и C- F Эти расE пределения косоортогональны друг другу и на каждой из плоскостей C+ (a) 2Eформа a не вырожденаF Характеристические распределения порождают еще одно распреE деление " вещественное одномерное распределение

идеалом Картана эффективными

эллиптическим

пфаффианом гиперболическим, параболическим невырожденным

характеристическими

l = [C+ , C+ ]

[C- , C- ],


IHP трансверсальное распределению Картана 6F

elexey qF uushner

2. Невырожденные уравнения и инварианты Лапласа

Пусть " невырожденная нормированная эффективная дифференциE альная 2Eформа на J 1 M F В каждой точке a J 1 M комплексификация касательного пространства Ta (J 1 M ) распадается в прямую сумму @PFIA

Ta (J 1 M )C = C+ (a) l(a) C- (a).

Обозначим распределения C+ , l, C- через P1 , P2 и P3 соответственноF Формула @PFIA порождает разложение в прямую сумму комплекса де Рама многообразия J 1 M D что позволяет найти дифференциальные инE варианты уравнения 3F s Определим тензорные поля qj,k : D(J 1 M )C ЧD(J 1 M )C D(J 1 M )C 1 на J M X @PFPA

q

s j,k

(X, Y ) = -Ps [Pj X, Pk Y ] .

Здесь Pj " проектор на распределение Pj D D(J 1 M ) " модуль векторE ных полей на J 1 M F 3 1 Мы получаем всего 4 нетривиальных тензорных поляX q1,2 D q2,3 , 2 2 q1,1 , q3,3 F Остальные тензоры @PFPA равны нулюF Определим две дифE ференциальные 2Eформы как свертки тензоровX

+ = q

2 1,1

1 , q3,2 ,

Эти формы мы будем называть D поскольку они явE ляются обобщением инвариантов Лапласа для случая линейных уравE нений 2F ЗаметимD что классические инварианты Лапласа определены только для гиперболических уравненийF Формы Лапласа играют важную роль при решении вопроса о конE тактной линеаризации уравнений МонжаEАмпераF ТакD напримерD если обе формы Лапласа нулевыеD то уравнение МонжаEАмпера локально контактно эквивалентно либо волновому уравнению vxx - vyy = 0D либо уравнению Лапласа vxx + vyy = 0 @смF также 10AF В терминах форм Лапласа формулируется решение проблемы экE вивалентности уравнений МонжаEАмпера линейным уравнениям вида

формами Лапласа

- = q

2 3,3

3 , q1,2 .

avxx + 2bv

xy

+ cv

yy

+ rvx + svy + k v + w = 0,

где a, b, c, r, s, k , w " функции только от независимых переменных x, y 4F В частностиD для таких уравнений формы Лапласа замкнутыF НаE примерD для уравнения ХантораEСакстона

v

tx

= v vxx + u2 , x


gontt lssi(tion of wongeEemp equtions @in ussinA ere

IHQ

возникающего в теории жидких кристалловD формы Лапласа имеют видX 1 = -dq2 dp1 , 2 = 2 (1 - ) dq2 dp1 . Это уравнение контактно эквивалентно линейному уравнению ЭйлераE Пуассона 8

v

tx

=

2 (1 - ) 2 (1 - ) 1 vt + vx - 2 u. (t + x) (t + x) ( (t + x))

Если выполняется условие - - = + + = 0D то формы Лапласа разложимыX + = + для некоторых дифференциальных 1Eформ + , + 1 (C+ )F Рассмотрим следующие 1Eмерные подраспределения распределения КартанаX X+ = C+ ker + и X+ = C+ ker + F Для уравнений общего положения эти распределения различныF Это позволяет построить eEструктуру для таких уравнений и найти полную систему их скалярных дифференциальных инвариантовF ЗаметимD что для уравнений МонжаEАмпераD контактно эквиваE лентных уравнениюD линейному относительно первых производных @тFеF уравнению вида

avxx + 2bvxy + cvyy + rvx + svy + w = 0,
где a, b, c, r, s, w " функции от x, y , v AD посторенные 1Eмерные распреE деления попарно совпадаютX X+ = X+ и X- = X- F Подробное изложение геометрии уравнений МонжаEАмпера @и не только двумерных3A можно найти в 4F
Литература

I runter tFuFD xton F Dynamics of Director Fields GG sew tF epplF wthF SI @TAD ppF IRWV ! ISPI @IWWIA P porsyth eFF Theory of dierential equationsD prt R GG rtil di'erentil equtionsD olFTD gmridge niversity ressD SWT pFD @IWHTA Q uushner eF Almost Product Structures and Monge-Ampere Equations GG vohevskii tournl of wthemtisD httpXGGljmFksuFruD olF PQD PHHTD pp ISI! IVIF R uushner eFD vyhgin FD utsov FD Contact Geometry and Nonlinear Dierential Equations GG gmridge niversity ressD RWT ppFD @PHHUA S vie ophusD Ueber einige partiel le Dierential-Gleichungen zweiter Orduung GG wthF ennFD SD ppF PHW!PST @IVUPA T vyhgin F Lectures on Geometry of Dierential EquationsD prt P GG 4v pienz ome @IWWQA U vyhgin F F nd outsov F xF yn ophus vie theorems for wongeEempre equtions GG hoklF ekdF xuk f PU @SAD ppF QWT!QWV @IWVQA


IHR
V worozov yFsF

ivgeny xF wikhlkin
Contact Equivalence of the Generalized Hunter-Saxton Equation

GG УМНFE IWUWFE ТFQRD выпFIFE СFIQUEITSF IH Туницкий ДFВF О контактной линеаризации уравнений Монжа-Ампера GG Известия РАНD Серия 4Математика TH @PAD стрF IWS!PPH @IWWTA
уравнения второго порядка

@PHHRA W Лычагин ВFВF

and the Euler-Poisson Equation

GG reprint rivX mthEph G HRHTHITD ppF I!Q

Контактная геометрия и нелинейные дифференциальные

Об амебе дискриминанта алгебраического уравнения
Е.Н. Михалкин

Рассмотрим общее алгебраическое уравнение nEой степени @IA

zn + x

n-1

z

n-1

+ . . . + x1 z - 1 = 0.

Нас будет интересовать дискриминантное множество = { = 0} уравнения @IA @здесь ! дискриминант этого уравненияAF Запишем D используя параметризацию ПассареEЦиха 1X @PA где @QA

xk (s) =

nsk , s

-

, s , s

k n

, k = 1, . . . , n - 1; s CP

n-2

,

= (n - 1, . . . , 2, 1), = (1, 2, . . . , n - 1)

! целочисленные векторыD , ! знак скалярного произведенияF В случаеD когда уравнение @IA содержит лишь один параметр xi @в этом случае рассматриваемое уравнение называется триномиальнымAD дискриминантное множество представляет собой некоторое подмноE жество точек комплексной плоскости C @смF 1AF Но если уравнение @IA содержит два параметра и болееD то дискриминантное множество удобE но исследоватьD используя амебу дискриминанта @определение амебы было дано ГельфандомEКапрановымEЗелевинским 2AF ОтметимD что когда интересующее нас уравнение содержит лишь два параметра xi D то амеба дискриминанта A лежит в R2 F В статье 3 из интегральной формулы Меллина 4 было получено интегральное представление для одного из решений уравнения @IA с интегрированием по компактуF Была найдена и область сходимости полученного интегралаF А именноD доказана следующая


yn moe of disriminnt @in ussinA
Теорема .

IHS

I Ветвь алгебраической функции z0 (x) решения уравнения (1) с условием z(0) = 1, допускает представление в виде интеграла

z0 (x) =
1

1 1+ 2 in
0

t

1-n n

(1 - t)

-

1+n n

n-1

e

i n

ln 1 -
k=1 n-1

e

k n

i

xk t n (1 - t)
k

k

n-k n

-

-e

-

i n

ln 1 -
k=1

e

k - n i

xk t n (1 - t)

n-k n

dt,

где ветви логарифма определены в области пространства Cn-1 переменного x = (x1 , . . . , xn-1 ), полученной удалением из Cn-1 двух семейств комплексных гиперплоскостей
n-1

- =
t[0;1] k=1 n-1 k=1

xk t n (1 - t) xk t n (1 - t)
t[0;1]
k

k

n-k n

e e

k - n i

=1 ,

+ =

n-k n

k n

i

=1 ,

и выбираются условием

ln 1 = 0.

Поставим задачу исследовать взаимное расположение дискримиE нантного множества уравнения @IA и семейства гиперплоскостей + @- AF Решение задачи будет более нагляднымD если перейти к логаE рифмической шкале

Log : (x1 , x2 , . . . , x
Обозначим через

n-1

) - (log |x1 |, log |x2 |, . . . , log |x

n-1

|).

n-1

F+ (x; t) =
k=0

xk t n (1 - t)

k

n-k n

e

k + n i

-1

! пару функцийD линейных относительно xF Использовав параметризацию @PA дискриминантного множества уравнения @IAD а также параметризацию нулевого множества функций F+ ( x ; t ) l , l C, xl ( ) = + an-1 n-1 + . . . + a+ 1 1 при n-l l l a+ = t n (1 - t) n e+ n i , l = 1, . . . , n - 1, l


IHT

ivgeny xF wikhlkin

Контур амебы дискриминанта уравнения (1) при является огибающей для семейства амеб гиперплоскостей l при n . Более того, в случае n = 3 для указанного семейства гиперплоскостей является огибающей контур амебы дискриминанта уравнения (1) и при ,s > 0. Значения , s , , s находятся ,s из равенств (3).
Теорема P. s R+-1 n + arg l =

можно показать справедливость следующего утвержденияF

В дополнение к вышеизложенномуD в докладе будет приведена геоE метрическая иллюстрация Теоремы P для дискриминанта

(x) = 27 + 4x3 - 4x3 + 18x1 x2 - x2 x 1 2 1
кубического уравнения @RA

2 2

z 3 + x2 z 2 + x1 z - 1 = 0.

В дополнение к этомуD для уравнения @RAD в логарифмической шкале

Log : (x1 , x2 ) - (log |x1 |, log |x2 |)
будет найдено уравнение кривойD которая соответствует пересечению дискриминантного множества с комплексными прямыми + F Отметим некоторые ее свойстваX это петляD проходящая вокруг каспидальной точкиD симметричная относительно прямой

u = v , где u = log |x1 |, v = log |x2 |.
Литература

I P Q R

Passare M., Tsikh A. Algebraic equations and hypergeometric series sn the o ok 4he legy of xiels renrik eel4F pringerX ferlinEreidelergExew orkD PHHRF F TSQ ! TUPF Gelfand

multidimensional determinantsF

I.,

Kapranov

M.,

firkh? userX fostonD IWWRF

Zelevinsky

A. Discriminants, resultants and

Михалкин Е.Н. О решении общих алгебраических уравнений с помощью ин-

QTS ! QUIF

тегралов от элементарных функцийGG

СибF матемF журнF PHHTF ТF RUD PF СF

В кнFX Комплексный анализ и дифференциальные операторы @к ISHEлетию СFВF КовалевскойAD КрасГУD PHHHD СF IQR ! IRTF S Passare M., Tsikh A. Amoebas: their spines and their contoursGG gontemporry wthF 377 @PHHSAD F PUS ! PVVF
алгебраических уравнений.

Семушева А.Ю., Цих А.К. Продолжение исследований Меллина о решении


niversl lgorithms @in ussinA
Программа для демонстрации универсальных

IHU

алгоритмов решения дискретного уравнения Беллмана в различных полукольцах
С.Н. Сергеев и А.В. Чуркин

1

Программа предназначена для демонстраE ции некоторых универсальных алгоритмов обращения матрицы и реE шения уравнения Беллмана в различных полукольцахF В зависимости от выбора полукольцаD программа может либо найти обратную матриE цу и решить уравнение Ax = B D где A и B E пользовательская матриE ца и векторEстолбец соответственноD либо найти матрицу A и решить уравнения Беллмана x = A x B Перед запуском программы польE зователь выбирает одно из полуколецD требуемую задачу и алгоритм расчетаF Затем исходные данные заносятся в матрицу @для наглядE ности максимальный размер ограничен величиной IHхIHAF Результат расчета выводится либо в виде матрицыD либо в виде вектора столбцаD в зависимости от поставленной задачиF В процессе разработки проE граммы использовался объектно E ориентированный подходD позволяE ющий в полной мере использовать универсальность предложенных алE горитмовD подключая в качестве объектов полукольца с определенной арифметикойD актуальной для решения конкретной задачиF Использование идемпотентных операций и позволяет записать ряд важных алгоритмов обращения матриE цы в универсальном видеF Выбор пользователем требуемого полукольE ца определяет тип данныхD с которыми будет работать универсальE ный вычислительный алгоритмF В программе реализована возможE ность выбора из следующих полуколецX IA ="+" и = " Ч " E обычная арифметикаF PA = "max" и = " + " E арифметика mxEplusD в которой опеE рация взятия максимума используется вместо сложенияD а сложение E вместо умноженияF Такая арифметика часто используется в задачах максимизацииD системах автоматического управления и дрF QA = "min" и = " + " E арифметика minEplusD в которой операция взятия минимума используется вместо сложенияD а сложение E вместо умноженияF Используется в задачах нахождения кратчайшего путиD задачах оптимизацииF

Назначение программы.

Примеры полуколец.

НЦНИЛаF

1Работа выполнена при поддержке грантов РФФИ HSEHIEHHVPR и HSEHIEHPVHUE


IHV

eFF ghurkinD FxF ergeev

RA = "max" и = " Ч " E арифметикаD в которой вместо слоE жения используется операция взятия максимумаF SA = "max" и = "min" E арифметика mxEminD в которой вместо сложения используется операция взятия максимумаD вместо умножения E операция взятия минимумаF Используется в задачах мноE гокритериальной оптимизацииF TA = "or" и = "and" E логическая арифметика над булевскиE ми переменнымиF Каждому полукольцу в программе соответствует пользовательE ский тип данныхD определяемый отдельным классомF Кроме описаний правил сложения и умноженияD внутри класса определяются также вид нуляD единицы и правило взятия операции 4B4F Такой подход даE ет возможность дополнять программу новыми типами полуколецD не меняя структуру основной программы и не внося никаких изменений в ту ее частьD которая занимается вычислительными алгоритмамиF В последнее время большое количеE ство работ @напримерD 1E5A посвящено разработке универсальных версий алгоритмов линейной алгебры и численного анализаF РассматE риваемая программа использует универсальные версии ряда классиE ческих алгоритмов обращения матриц и решения систем линейных уравненийF На выбор пользователю предлагаются ряд алгоритмовD как точныхD так и с использованием метода последовательных приближеE нийX IAМетод исключения по схеме ГауссаY PAМетод окаймленияY QAИтерационный метод ЯкобиY RAИтерационный метод ГауссаEЗейделяY SAАлгоритмы для матриц специального видаX симметричныхD треE угольныхD теплицевых и дрFD в том числе предложенные в 4 и 5F В случае выбора обычной арифметики эти алгоритмы ведут сеE бя классическим образомD однако для случая идемпотентного полуE кольца они позволяют найти матрицу A или решить соответствующее уравнение БеллманаF Для полуколец mxEplus или minEplus уравнение Беллмана представляет собой основное функциональное уравнение диE намического программирования и выражает принцип оптимальности БеллманаX управление на каждом шаге должно быть оптимальным с точки зрения процесса в целомF Для наглядного представления инE формации в программе заложена возможность визуализации исходE ной матрицы в виде графа с соответствующими весамиF На отдельной

Универсальные алгоритмы.

Возможности визуализации.


niversl lgorithms @in ussinA

IHW

вкладке диалогового окна программы отображается введенная польE зователем информация об исходной матрице и результат соответствуE ющего расчетаF В таком режиме работы программыD напримерD задача о решении уравнения Беллмана в полукольце minEplus будет отобраE жаться как кратчайший путь между узлами заданного пользователем графаF

Применение при разработке программы объектно ориентированного подхода позволяет не только независимо менять полукольца и алгоE ритмы вычисленияD но и управлять базовым типом числовых данных для контроля за точностью вычисленийF В следующей версии програмE мы предполагается реализовать механизм выбора одной из числовых арифметикF Среди них арифметика целых чиселD арифметика чисел с плавающей точкойD дробноEрациональная арифметика с использоваE нием цепных дробейD в том числе с контролируемой точностьюF Это позволит сравнить ошибку округленияD накопленную в ходе примеE нения того или иного вычислительного алгоритма с ошибкой самого метода @для итерационных алгоритмовAD что позволит судить об их итоговой эффективностиF
Литература

Использование различных арифметик для контроля точности.

I fFeF grreF An algebra for network routing problems. tF of the snstF of whsF nd epplisF IWUID xUD pFPUQEPWWF P FgF fkhouseD fFeF grreF Regular algebra applied to path-nding problems. tF of the snstF of wthsF nd pplisF IWUSF xISF pFITIEIVTF Q qFvF vitvinovD FF wslovF Correspondence principle for idempotent calculus and some computer applications. snX sdemp oteny @tF qunwrdenD editorAD ulF of the sF xewton snstFD gmridge nivF ressF IWWV pFRPHERRQF R qF vitvinovD iF wslovF Universal numerical algorithms and their software implementation. rogrmming nd omputer softwreF PHHHD FPT xSD pFPUSEQVHF S F ergeevF Universal algorithms for generalized discrete matrix Bel lman equations with symmetric Toeplitz matrix. iEprint rivXmthFeGHTIPQHWF


IIH
Кривые в

omn lvert

C

2 , амебы которых определяют

фундаментальную группу дополнения
Роман Ульверт

В классической работе ван Кампена 1D продолжающей исследоE вания многих математиковD начиная от ЗарисскогоD был предъявлен метод вычисления фундаментальной группы дополнения к плоской комплексной кривойF Результат ван Кампена впоследствии был переE формулирован БF Мойшезоном и МF Тайхер с использованием понятия брэйдEмонодромииD рассмотренном в 2F Дадим краткое описание гомоморфизма брэйдEмонодромииF Пусть алгебраическая кривая C задана множеством нулей полинома f (x, y ) C[x, y ]F Будем смотреть на f как на полином ВейерштрассаX

f = 0 (y )xd + 1 (y )x

d-1

+ ћ ћ ћ + d (y ).

Обозначим B := C \ D где = {y1 , . . . , ys } " дискриминант полинома f F Ограничение проекции (x, y ) y на множество E := C Ч B \ C определяет локально тривиальное расслоение p : E B F Вычисление фундаментальной группы 1 (C2 \ C ) сводится к вычислению группы 1 (E )F Выберем (x0 , y0 ) E и обозначим через F слой над точкой y0 F База и слой расслоения p представляют собой дополнения к конечным наборам точек в CD поэтому группы 1 (B , y0 ) и 1 (F, x0 ) являются свободными группами и точная последовательность расслоения

1 - 1 (F, x0 ) - 1 (E , (x0 , y0 )) - 1 (B , y0 ) - 1.
расщепляетсяD так что группа 1 (E , (x0 , y0 )) есть полупрямое произвеE дение групп 1 (B , y0 ) и 1 (F, x0 )F Действие группы 1 (B , y0 ) на 1 (F, x0 ) может быть описано в терминах групп косF Для этого слой F отождеE ствим с диском D2 D из которого выброшено множество K = {x1 , . . . , xd } различных точекF Каждый элемент группы 1 (B , y0 ) определяет биекE цию множества K D а следовательно и элемент группы Brd = Br(D, K ) кос из d нитейF Таким образом определен гомоморфизм 1 (B , y0 ) Brd D носящий название гомоморфизма брэйдEмонодромииF Этот гомоE морфизм позволяет выписать соотношения между образующими групE пы 1 (E , (x0 , y0 ))F Переход к рассмотрению амебы AC кривой C D то есть образа криE вой под действием отображения логарифмической проекции (x, y ) (ln |x|, ln |y |)D ставит вопрос о томD каким образом знание амебы моE жет помочь в вычислении фундаментальной группы 1 (C2 \ C )F ОбоE значим через E связную компоненту R2 \ AC порядка = (x , y )F


gurves in C2 @in ussinA

III

Определение и свойства порядка можно найти в статье ФорсбергаE ПассареEЦиха 3F Для нас существенноD что целые числа x и y выE ражают коэффициенты зацепления петель = {x = eu+it , y = ev } и x = {x = eu , y = ev+it } с кривой C D причем x и y не зависят от выE y бора точки (u, v ) E F Пусть (x0 , y0 ) = (eu , ev )D где (u, v ) E F Тогда классы петель [ ], [ ] 1 (C2 \ C, (x0 , y0 )) коммутируютF ВозникаE x y ет вопросD можно ли описать фундаментальную группу дополнения к плоской кривой C D используя в качестве образующих только классы петельD получающихся из петель , D где пробегает порядки всех y x связных компонент дополнения к амебе кривойc Чтобы ответить на этот вопросD мы должны вернуться к описаE нию группы 1 (C2 \ C ) с использованием брэйдEмонодромииF Так как старшая степеньD с которой переменная x входит в полином f D равна dD то найдется хотя бы одна связная компонента E из R2 \ AC поE рядка = (d, y )F Выберем (u, v ) E такD чтобы v = ln |yi | для всех yi из дискриминанта полинома f F Тогда для всех t [0, 2 ] диск {ln |x| u, y = ev+it } содержит ровно d корней полинома f (x, ev+it )F Следовательно определена коса из d нитейD которую можно сопостаE вить петле = {x = eu , y = ev+it }F Отсюда видноD что петли , y x y естественным образом связаны с брэйдEмонодромией кривойD то есть могут претендовать не только на роль образующих группы 1 (C2 \ C )D но и давать соотношения между образующимиF В качестве простого примера кривойD фундаментальная группа дополнения которой определяется амебойD рассмотрим дискриминант [2, 3] кубического уравнения z 3 + z 2 + xz + y = 0F Он задается полиноE мом 27y 2 + 4x3 + 4y - 18xy - x2 и представляет собой каспидальную криE вую с невырожденной амебойF Фундаментальная группа 1 (C2 \ [2, 3]) (3,0) (0,2) описывается с помощью петель x и y F БрэйдEмонодромия дает 2 Br3 F изоморфизм 1 (C \ [2, 3]) =
Литература

I vn umpen iFF On the fundamental group of an algebraic plane curve GGemerF tF wthF SS @IWQQAD F PSS E PTHF P woishezon fF Stable branch curves and braid monodromies GGelgF qeometry @ghigoD IWVHAD vetF xotes in wthF VTPD pringerD reidelergD IWVID F IHU E IWPF Q porserg wFD ssre wFD sikh eF Laurent determinants and arrangements of hyperplane amoebas GGedvF in wthF ISI @PHHHAD F RS E UHF


LIST OF PARTICIPANTS AND AUTHORS

sxseD homine de olueuD fFF IHSD UVISQ ve ghesny gedexD Marianne.Akian@inria.fr

eusex wrinneD

France.

feuvys eliD

heprtment of wthemtisD pulty of ienes t fxD oute de oukrD QHQVD fxD Tunisia. Ali.Baklouti@fss.rnu.tn

fiveusx iheslv FD
Kingdom.

hool of wthemtil ienesD xottinghm niversityD xqU PhD Viacheslav_Belavkin@nottingham.ac.uk
(Бениаминов Евгений Михайлович),

United

fixsewsxy ivgeny wF gryusx endrey F

ussin tte niversity for the rumnitiesD pulty of wthemtisD ulF ghynov ISD wosowD Russia. ebeniamin@yandex.ru wFF vomonosov wosow tte niversityD pulty of hysisD ghir of unE tum ttistis nd pield heoryD IIWWWPD veninskie qoryD wosowD Russia. churandr@mail.ru
(Чуркин Андрей Валерьевич),

giws eD xhimovski prospF RUD IIURIVD wosowD vdanilov43@mail.ru

hexsvy ldimir sF

(Данилов Владимир Иванович),
Russia.

sxseD homine de olueuD fFF IHSD UVISQ ve ghesny gedexD Nadir.Farhi@inria.fr

pers xdirD

France.

pei prD

heprtment of wthemtis 8 gomputer ieneD gehD hkrD lamffaye@yahoo.fr

Senegal.

pygu ldimir F

siD fF gheremushkinsky PSD wosowD

(Фок Владимир Владимирович),
Russia.

fock@itep.ru
France.

sxseD homine de olueuD fFF IHSD UVISQ ve ghesny gedexD Stephane.Gaubert@inria.fr hel(nEsnformtik goFD ulF kky ID wosowD gelfand_a@oaoesp.ru

qefi t? ephneD

qiv9pexh elexnder wF

(Гельфанд Александр Маркович),
Russia.

qyxgrey elexnder fFD

frown niversityD wthF heprtmentD ISI hyer stFD rovideneD sD Alexander_Goncharov@brown.edu

USA.


? qfsg tjn

niversity of xovi dD pulty of ehnil ienesD rg hositej yrdovi TD PIHHH xovi dD Serbia. tatjana@uns.ns.ac.yu

(Грбич Татjана),

qvsxu yleg F

ss eD fF uretny ID wosowD

(Гулинский Олег Викторович),
Russia.

bedelbaeva_aigul@mail.ru

qisgr hmitryD

veweD niversite de leniennesD ve wont rouy SWQIQ leniennesD France. d.gurevich@free.fr wFF vomonosov wosow tte niversityD heprtment of wehnis nd wthE emtisD pulty of righer elgerD IIWWWP veninskie qoryD wosowD Russia. guterman@list.ru

qiwex elexnder iF

(Гутерман Александр Эмильевич),

sqyxsx ergeyD

treht niversityD xvEQSHV e trehtD The Netherlands. igonin@mccme.ru

sixfiq sliD

sweD niversite vouis steurD U rue ene hesrtesD TUHVR trsourg gedexD France. itenberg@math.u-strasbg.fr

ueexy elexnder F

se eD prospF THEletiy yktyry TD IIUQIPD wosowD sasha@cs.isa.ru

(Карзанов Александр Викторович),
Russia.

urevewy yheslvD usrix foris uhF

sweD niversite vouis steurD U rue ene hesrtesD TUHVR trsourg gedexD France. kharlam@math.u-strasbg.fr hel(nEsnformtik goFD ulF kky ID wosowD bkirsh@aha.ru
(Кирштейн Борис Хаймович),
Russia.

uyvyuyvy ssili xF

wosow snstitute of ionomisD ehersky ulF TGID IPWQRRD wosowD nd the niversity of rwikD heptF of ttistisD goventry gR UevD Russia and United Kingdom. vkolok@fsmail.ru giws eD xhimovski prospF RUD IIURIVD wosowD gleb_koshevoy@mail.ru

(Колокольцов Василий Никитич),

uyriy qle eF

(Кошевой Глеб Алексеевич),
Russia.


wFF vomonosov wosow tte niversityD pulty of hysisD ghir of unE tum ttistis nd pield heoryD IIWWWPD veninskie qoryD wosowD Russia. kovgen@mail.ru

uyev qenndy F

(Коваль Геннадий Васильевич),

urxi elexey qF

estrkhn tte niversityD ulF tishev PHD RIRHSTD estrkhnD kushnera@mail.ru

(Кушнер Алексей Гурьевич),

Russia.

ueivehi imon F vssxy qrigory vF wevy itor F

FvF oolev wthemtil snstituteD prospF uoptyug RD TQHHWHD xovosiirskD Russia. sskut@math.nsc.ru sndependent niversity of wosowD IIWHHPD fF lsyevsky perF IID wosowD Russia. glitvinov@gmail.com
(Маслов Виктор Павлович), (Литвинов Григорий Лазаревич),

(Кутателадзе Семен Самсонович),

wFF vomonosov wosow tte niversityD pulty of hysisD ghir of unE tum ttistis nd pield heoryD IIWWWPD veninskie qoryD wosowD Russia. v.p.maslov@mail.ru

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(Молчанов Владимир Федорович),

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niversity of tvngerD RHQT tvngerD alexander.rashkovskii@uis.no

(Рашковский Александр Юрьевич),
Norway.

wFF vomonosov wosow tte niversityD pulty of hysisD ghir of unE tum ttistis nd pield heoryD IIWWWPD veninskie qoryD wosowD Russia. sergiej@gmail.com

(Сергеев Сергей Николаевич),

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(Шведов Олег Юрьевич),

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(Штрбоjа Мирjана),

(Стояновский Александр Васильевич),


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(Ульверт Роман Викторович),

(Цыкина Светлана Викторовна),

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? iole olytehnique de wontr? gFF THUW uF gentreEvilleD wontr? u? e elD el e rQg QeUD nd qiehD wontrelD Canada. Edouard.Wagneur@gerad.ca sxseD homine de olueuD fFF IHSD UVISQ ve ghesny gedexD cormac.walsh@inria.fr
France.

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ierin pederl niversityD heprtment of wthemtis nd snformtisD TTHHRID prF voodnyi UWD ursnoyrskD Russia. znamensk@lan.krasu.ru

(Знаменская Оксана Витальевна),