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A NEW EMPIRICAL STATISTICAL METHODOLOGY FOR DETECTION OF PARALLELISMS AND DATING OF DUPLICATES A. T. Fomenko

In this article we present in greater detail the results originally published in Dok!o Akad. Nauk SSSR [I, 2], which were obtained by the application of one of the procedures developed by the author and reported at the Third International Vil'nyus Conference on Probability Theory and Mathematical Statistics [3, 4]. These seven procedures are applicable to historical texts and may be used to date ancient events and to detect dependent texts, duplicates, or parallelisms. In particular, one of the procedures enabled us to process the rich material representing the sequences of lengths of rule in various ancient dynasties. In his seminal study [5], Morozov pointed to three pairs of dynasties for which the length of rule was represented by visually similar graphs. This could imply that in reality these dynasties were reflections of the same real dynasty "replicated" as a result of erroneous dating of various lists (texts) describing the same events. At the same time, my analysis in [6] revealed about 10 allegedly independent pairs of dynasties with such and even more strikingly similar graphs. It therefore became necessary to develop a formal methodology capable of distinguishing quantitatively (and not merely by visual likeness) between dependent and independent dynasties. Let us state this problem in rigorous terms. Consider n successive real rulers in the history of some region, where n is a fixed integer, e.g., n = 15. We will refer to this sequence as the real dynasty. Each chronicler describing this dynasty A uses his own method to compute the length of rule ai of ruler i and thus generates a sequence of numbers ~ = (~l,...,an), while we will call the numerical dynasty. It is conveniently represented by the vector ~ in the Euclidean space R n. Another chronicler describing the same real dynasty in general comes up with a different numerical dynasty b = (bl,...,bn). The difference is essentially attributable to errors and objective difficulties which prevent exact determination of the actual lengths of rule. The main types of errors actually made by chroniclers were identified and described in [I, 6]. Thus, each real dynasty in general may be "replicated" by various chroniclers into several different numerical dynasties. Each real dynasty A may be represented by some set V(A) of points (vectors) in R n. The "fuzziness" of this set shows how large were the errors of the various chroniclers. Large errors result in a set V(A) with widely scattered points. If the errors are small, V(A) has a small "diameter." How large were the errors that chroniclers made in the description of real dynasties? The author formulated and tested in [I] the principle of small distortions, which provides an answer to this question. Let us briefly review this principle. Two numerical dynasties are said to be dependent if they represent the same real historical dynasty, i.e., constitute two descriptive versions of the same dynasty. Conversely, two numerical dynasties are called independent if the two real dynasties that they describe are essentially different. Real dynasties are regarded as being essentially different if the number of rulers occurring in both dynasties simultaneously does not exceed n/2, i.e., half the number of rulers in the particular dynastyl Note that two arbitrarily chosen dynasties may intersect, i.e., some rulers may simultaneously occur in both dynasties. Thus, alongside dependent and independent pairs of numerical dynasties, we also have "intermediate" pairs of dynasties in which the number of common rulers is greater than ~ n/2. If the total number of dynasties is large, the number of intermediate pairs of dynasties is relatively small and we can mainly focus on dependent and independent pairs of dynasties. The principle of small distortions is stated as follows: If the difference between two numerical dynasties is "small," then they are dependent, i.e~, represent the same real dynasty; if, on the other hand, two numerical dynasties are independent (i.e., represent essentially different real dynasties), then they "significant~ly" differ from one another. This principle (conjecture) claims that real chroniclers on the whole make

Translated 154-177, 1984.

from Problemy Ustoichivosti

Stokhasticheskikh Modelei -- Trudy Seminara, pp.

0090-4104/86/3502-2417512.50

9 1986 Plenum Publishing Corporation

2417


vt i"
~-Fig. small errors, icles. ~ Fig. 2 small distortions i.e., they introduce

, ,,'Z-,'f X./
v(DI
Fig. 3 into real numerical

I /,c,7-7
data in their chron-

This principle can be visually represented in terms of the sets V(A) and V(B) in R n. For each real dynasty A, the set of corresponding numerical dynasties constitutes a "spherical cluster" V(A) (Fig. I). If our principle holds, the spherical clusters V(A) and V(B) corresponding to different real dynasties A and B do not overlap, i.e., they are sufficiently distant from one another. Our principle naturally requires empirical verification. Is there a natural numerical coefficient (measure) %(a, b) which can be evaluated for each pair of numerical dynasties a and b so that it is "small" for dependent dynasties and "large" for independent dynasties? In other words, this coefficient should clearly discriminate between dependent and independent dynasties. Such a coefficient (a measure on the set of dynasty pairs) was proposed by the author in [I]. The principle of small distortions was verified by using specific historial material. The existence of such a natural measure %(a, b) is not a priori self-evident: Its existence is nontrivial and it discloses an interesting law which characterizes all the numerical data of ancient chronicles surviving to this day. Let us briefly recall the definition of %(a, b). By processing some 300 historical chronicles, records, primary sources, tables, etc., the author compiled a list of all the known numerical dynasties from 4000 B.C. to 1800 A.D. in the history of Europe, the Mediterranean, Near East, and Egypt. Then a complete list of all 15-ruler dynasties was compiled, i.e., tables of all groups of 15 successive rulers [I, 6]. Each dynasty was then represented by a point (a vector) in R Is, and we thus obtained some set D. The most typical chronicling errors were simulated, such as a) transposition of two successive rulers, b) substitution of one ruler for two successive rulers, reporting his rule to be equal to the sum of the lengths of rule of the two actual rulers, c) errors in the computation of the length of rule: the longer the rule, the greater the computational error. Each dynasty from the set D was subjected to all the perturbations a) and b). Each point in D was thus replicated into several points, and the set D on the whole increased. We denote the resulting set of 15"I011 points by V(D). Then the set V(D) was used to derive a probability density function @. To this end, the space R Is was divided into sufficiently small standard cubes so that none of the points in V(D) was located on the boundary of any of the cubes. If x is an interior point of a cube, we set ~(x) = [the number of points from V(D) contained in the eube]/[total number of points in V(D)]. If x is on the boundary of a cube, we set ~q~(x)=0. This function cp has its maximum in the region where particularly numerous dynasties from V(D) are clustered and falls off to zero where no points from V(D) are observed (Fig. 2). The measure %(a, b) now can be defined as follows. Let a = (al,...,als) and b = (bl,..., bls) be two numerical dynasties from V(D). Construct the parallelepiped ~(a, b) centered at the point a whose diagonal is the vector b -- a (Fig. 3). Now allow for chronicling errors of type c), i.e., errors made by real chroniclers when computing the length of rule of different rulers. Expand the parallelepiped ~(a, b) by enclosing it in a larger parallelepiped ~(a, b), whose orthogonal projections on the coordinate axes are segments with end points ~q- la~--b~l+r(a~), where f is the chronicling error function- a nondecreasing positive function of a i [I, 6, 13]. For 0 ~< a i < 20, we have f(a i) = 2, for 20 ~< a i < 30, we have f(a i) = 3; for 30 <~ ai < lOO,:we have f(a i) -- 5.[ai/10] -- 5. Now h(a, b) is defined as the number ~r , In
llI(a,t,)

Fig. 4 the number %(a, b) is depicted by the volume of the prism whose base is the paralellepiped ~(a, b) and which is bounded from above by the graph of the function ~.

2418


1
-

n

w

I

,r,,

f7(%6)

~ ''~

Independent Dependent Independent dynasties dynasties d~p~sties Fig. 5

Fig. 4

The number %(~, b) may be interpreted as the probability that the distance from the point a of the random vector r distributed in the space R Is with the density function 9 does not exceed the distance between a and b. For further details, see [I, 6, 13]. An extensive numerical experiment subsequently showed that for known dependent numerical dynasties g and b, the coefficient %(~, b) varies from I0-12 to 10-8 . The experiment was conducted using known dependent dynasties from historical chronicles between 4000 B.C. and 19th century A.D. Conversely, for known independent numerical dynasties between 13th century A.D. and ~9th century A.D., the number %(~, b) was found to be no smaller than 10 -3 (Fig. 5). The principle of small distortions was thus fully corroborated using reliably dated historical material for the period between 13th and 19th century A.D. In particular, the application of our dating methodology to the period between 13th and 19th century A.D. failed to detect any discrepancies in the traditional dates of historical dynasties. Our algorithm is applied as follows to quences of all rulers in a given region). strand, i.e., a subsequence of rulers who to the given dynasty. Since real dynastic ferent strands (numerical dynasties) may be compare two real dynastic streams (i.e~ two seFrom each stream of rulers we select a dynastic cover the overall time interval corresponding streams often contain many corulers, several difselected from any given stream.

Moreover, for each ruler we consider all the possible beginning and ending dates Of his rule. Again recall that these dates are sometimes recorded differently by different chroniclers, and therefore all the possible versions should be taken into consideration when selecting the strands. This factor also increases the number of possible strands extracted for a given dynasty. In a comparison of two dynasties, it may turn out that only two strands are dependent from among all the possible pairs of strands, while the remaining strands are independent. It is significant that when comparing the dynasties (or strands) a and b, the lengths of rule are treated as rough approximations, since the error function f(a i) is built into the algorithm. In other words, if the length of rule ~i is less than 20 years, the difference ~i -- bi is considered only to within · years; if ~i varies between 20 and 30 years, then the difference is taken to within · years. If a i exceeds 30 years, the admissible error may be · years, increasing linearly thereafter with the increase of [ai/10] (the whole part)~ Therefore, the algorithm only requires roughly approximate lengths of rule, and not their exact values, which in most cases are unknown. The character of the graph plotting the length of rule, i.e., the shape of the polygonal curve representing this graph, is also of importance. The algorithm itself and the results obtained by this procedure are seen to be highly stable with respect to errors in the length of rule between the above limits. The application of this methodology to historical material traditionally dated before 13th century A.D. unexpectedly detected pairs of dynasties (strands) a and b which are regarded~as independent (in all senses) and yet their closeness coefficient %(a, b) is of the same order of magnitude as for known dependent dynasties, i.e., not greater than I0 -s, In this article, Tables 1-13 list the rulers (according to traditional dating) which are included in the most interesting of these special pairs of dynasties that we discovered. For all these pairs of dynasties, %(a, b) < 10 -8 , i.e., they are dependent, duplicates or parallels of one another. In graphical analysis, we compare the graphs of the length of rule in the given sequence of rulers and also examine the superposition of two dynasties on the time axis by rigidly shifting one dynasty until it overlaps the other. As a result of this rigid shift, the

2419


TABLE i (see Fig. 6 below) Carolingians, the empire of Charlemagne, 7th9th cent. A.D. Shift by 360 years. For variant rules see [14,15] i) Pepin of Herstal 681-71~ (33) 2) Charles Martel 721-724 (20) 3) Pepin the Short 754-768 (14) 4) Charlemagne 768-814 (46) 5) Carloman 768-771 or 772 (3 or 4). C~arlemagne's famous "gift" (774)- gives the lands of Italy to the Pope 6) Louis I the Pious (814-833) (abdication)

A strand from the Third Roman Empire, 3rd-4th cent. A.D. (mainly Eastern}. For variant rules see [14,16] i) Constantius II 324-361 (37) 2) Theodosius 1 379-395 (16) 3) Arcadius 395-408 (13) 4) Theodosius II 408-450 (42) 5) Constantine III 407-411 (4). The famous "gift" of Constantine I the Great (4th century A.D. ): gives Rome to the Pope 6) Leo I 457-474 (17). The Carolingians on the left "recreate antiquity" 7) Zeno 474-491 (17) 8) Theodoric 493-526 (33) 9) Anastasius 491-518 (27) i0) Odoacer 476-493 (17) Ii) Justin I 518-527 (9). Breakup of the official Third Roman Empire in the West. Gothic war of 4th cent. A.D.

(19)

7) 8) 9) i0) II)

Lothair the Western 840-855 (15) Charles the Bald 840-875 (35) Louis the Germmn 843-875 (32) Louis II the Western 855-875 (20) Charles the Fat 880-888 (8). The breakup of the Carolingian empire (the West). A shift of some 360 years. War

A mean shift by ends of rule is 359.6 years, which coincides with a rigid shift of 360 years for overlapping of left and right columns.

A strand from the Third Roman Empire
310 330 350 370 ~90 ~10 ~0 ~50 ~70 ~90 510 530 5S0

6o

7~

Carolingians Fig. 6 relative position (in time) of the individual rulers is preserved. For visual presentation, vertical segments are drawn joining the beginnings (and also the ends) of rule of the superimposed rulers. Here we only give some basic examples. We also computed the mean shift, equating the ends of rule of the superimposed rulers. The parallelism of all these dynasties (see Tables 1-13 and Figs. 6-14) is perfectly consistent with the author's discovery that the global chronological chart GCC (i.e., the modern "textbook" of ancient and medieval history; see definitions and description in [I-4, 6-13]) decomposes into four identical chronicle strata. Table 14 presents a more detailed description of the GCC than that in [2] (see [2, Fig. 3]). The line E (the left-hand column) represents the ancient and medieval history of Europe, the Mediterranean, and Near East using traditional dating. Line B represents the biblical chronology and history according to the books of the Old and the New Testaments, This history is presented with an upward shift, highlighting its superposition on the events in European history as discovered by the author. The letters K, T, ~, P, C, H in Table 14 and in Fig. 3 in [2] represent various historical epochs (periods). For brevity, we relabeled with the letter T the epochs that were marked in Fig. 3 in [2] by black triangles with the letters MT. The line Co in Table 14 is the original, i.e., the chronicle which in all probability describes the actual history of the regions and the true chronology (the first line from the bottom in Fig. 3 in [2]). The line CI (3rd line from the bottom in Fig. 3 in [2]) presents a distorted original Co containing several duplicates. The lines C2, C3, C4 in Table 14 (respectively, the 4th, 5th, and 6th lines from the bottom in Fig. 3 in [2]) represent the line Ci shifted downward in time by 333 years, 1053 years, and 1778 years, respectively. Thus, the events in Table 14 labeled by the same numbers and the same letters, i.e., located in the same horizontal positions in the

2420


TABLE

2 (see Fig.

7 below) Third Roman Empire 4th-6th cent. A.D. Variant rules see [14,16,17] I) Constantius I Chlorus 293-306 (13). A duplicate of Julius Caesar (a shift of 340 years) 2) 3) 4) 5) Diocletian 284-305,304 (21) Licinius 308-324 (16) Constantine I 306-337 (31) Basil the Great (?), 333-378 (45) 'Me shift from Henry to Basil is 728 years = 1106-378o The shift from the "birth" of Hildebrand to Basil 1053-333 = 720 years. The famous reform of Basil (basileus?). Contest of Basil with Valens (Herod?) Honorius 395-423 (28) Theodosius 1 379-395 (16) Arcadius 395-408 (13) Theodosius II 408-450 (42) Valentinian III 425-455 (30) Anarchy and Ricimer 456-472 (16). Contemporaries: Severus, Petronius; Recimir. See two close names on the loft. Anarchy and Odoacer 476-493 (17). Capture of Rome and corontion. Odoacer is the leader of German gerulians. Theodoric 497-526 (29). A variant from [17] Note closeness of names Theodoric and Frederick Or: Theodoric - Odoacer (co-ruler) 476-526 (50). Theodoric's death is the beginning of the war in Italy in 4th cent. A.D. (the Gothic). The Gothic dynasty 526-541 (15). Opponents: Belisaruis, Narses Totila 541-552 (ii> Teias (very young) 552-553 (I or 2). Died near Naples. Lost the battle against Narses near Troy and Naples (= Trojan War). End of Third Roman Empire in Italy. Destruction and fall of the Goths.

Roman GermanEmpire lOth-13thcent. A.D. Variant rules see [14,15,17] I) Otto III the Red (-Chlorus) 983-1002 (19). A duplicate of Julius Caesar Ist cent. B.C. 2) Henry II 1002-1024 (22) 3) Conrad II 1024-1039 (15) 4) Henry III 1028-1056 (28) 5) Henry IV 1053-1106 (53). During his rule, Hildebrand from 1049CbeRinning of activity in Rome) to 1085 (death) (36 years) = the original of Basil the Great. In 1053, the beginning of Hildebrand's famous reform; then his contest with Henry IV (Canossa) 6) Henry V 1098-1125 (27) 7) Lothair 1125-1138 (13) 8) Conrad III 1138-1152 (14) 9) FrederiCk I 1152-1190 (38) i0) Henry VI 1169-1197 (28) Ii) Anarchyand Philip of Ghibelline 11981208 (IO). Contemporaries: Suburus, Petrus~ Rainerius 12) Otto IV 1207-1217 (17 or 16) or 1197-1218 (21). Capture of Rome and coronation. Otto IV is a German 13) Frederick II as a Roman emperor 1220 (final coronation)-1250 (30). The execution of Vinea (= Boethius?) 14) Or: Frederick II 1196-1250 (54) and coruler Otto IV until 1218. Frederick's death is the beginning of the war in Italy in 13th cent. A.D. 15) Conrad IV 1237-1254 (17). Opponent Charles of Anjou 16) Manfred 1254-1266 (12) 17) Conradin (very young) 1266-1268(2). Died in Naples. Lost the battle against Charles of Anjou near Troy and Naples. End of the Empire lOth-13th cent. A.D.. Destruction and fall of the Hohenstaufens.

6) 7) 8) 9) I0) ii) 12) 13) 14)

15) 16) 17)

Mean shift by ends of rule is 723 years, which is close to the rigid shift of 720 years for overlapping the left and the right columns.

A strand from the Third Rom~m Empire, 3rd-6th century A.D.
250 2so ~ ~20 ~ ~60 3~ ~QO ~20 t+~ t~60 t~80 500 520 5~0

hL_\ ~g

F-1

F-~,5

Roman-German Empire, lOth-iBth century A. D. Fig. 7

2421


TABLE 3 (see Fig. 8 Below)
Roman German Empire lOth-13th century A.D. The beginning of the Saxon dynastry in 911. Variant rules in both co]urals, see [14,15,18] i) Conrad I 911-918 (7) 2) Henry I 919-936 (17) 3) Otto I the Great 936-973 (37). Superimposed on Alberic II = Albrecht I 4) Otto II from the death of Otto I in 973 to his death in 983 and Otto III 983-1002 (total of 29 years) 5) Henry II 1002-1024 (22) 6) Conrad II from coronation in Rome in 1027 to his death in -1039 (2) 7) Henry Ill the Black 1028-1056 (28). The great split of the Church under Hildebrand Habsburg Empire 13th-17th century A.D. The beginning of the Austrian line in 1273. A rigid shift of some 362 years ensures overlap with the column on the left. I) Adolf of Nassau 1291-1298 (7) 2) Rudolf of Habsburg 1273-1291 (18) 3) Henry VII 1309-1314 and Louis V 1314-1347 (total of 38) 4) Charles IV 1347-1378 (31). To No. 10: in the Habsburg Empire there were only three eruptions of Vesuvius, in 1306, 1500, and 1631. 5) Wenzel 1378-1400 (Z2) 6) Rupert of the Palatinate 1400-1410 (i0) 7) 8) 9) Sigismund 1410-1438 (28). The great split of the Church in 1378-1417, see the books of Ezra, Nehemiah, and Esther in the Bible. Frederick III 1440-1493 (53) Maximilian I Plus 1493-1519 (26). In his time, "Almagest" of Ptolemy from the time of Antoninus Pius (138-161). Overlap by a shift of i000 + 300. The eruption of Vesuvius in 1500. The beginning of the Italian wars of Germany 14941527. Brescian revolt in 1512 (see left

s) Henry Iv i053-1106 (53)
9) Henry V 1098-1125 or Henry V from the coronation in Rome in 1111 to his death in 1125 + Lot/lair II 1125-1137 (27 or 26) ~e eruption of Vesuvius in 1138-1139.

iO)

i0)

ii)

12)

War in Italy with the participation of Germany 1143-1155. The revolt of Arnold of Brescia. Frederick I Barbarossa 1152-1190. Famous emperor. Rome captured by Frederick in 1154. Difference in dates 373 years (shift). Pope Adrian IV. Establishment of Franciscan and Dominican orders (1223 and 1220) Henry VI 1191 (coronation in Rome) - 1197

column).
11)

(6)

12) 13) 14) 15) 16) 17) 18)

Charles V 1519-1556 (37). Famous emperor. In his time, Frederick the Wise and war with Barbarossa. Rome captured by Charles V in 1527. Pope Adrian VI. Establishment and official confirmation (1540) of the Jesuit order. Ferdinand 1556-1564 (6) Maximilian II 1564-1576 (12) Rudolf II 1576-1612 (36) Matthias 1612-1619 (7) (Matthew) Ferdinand II 1619-1637 (18) End of the Empire 1618-1619 Breakout of the Thirty-Year War in Germany in 1618.

13)
14) 15) 16) 17) 18)

Phinp 1198-12os (lO)
Frederick II 1211-1250 (39). Three coronations i196~1211, 1220 Wilhelm 1250-1256 (6) Conrad IV 1237-1254 (17) End of the Empire 1250-1254 War in Italy 1250-1268. Begir~nir~ of the 17-year Interregnum in Germany (1256)

Holy Roman Empire of theGerman Nation, 10th-13th century A. D.
900 9ZO g~O g60 980 1000 fOZOI0r
1 Z 5 6

1080 10~ flQO 1120 11~ 1160 11~ 1?20 IZZO 17u 1260 "1280 11 ~
io !

2

1

8

Roman German Empire - the Habsburgs, 13th-17th centuryoA.1?.

Fig. 8
table, are duplicates, i.e., events which are identified as (completely or partially) coinciding by the new dating methods. Events labeled simultaneously by several letters in the first and the second columns of the table are the sum (superposition) of the events entered in the same horizontal positions in the other columns of the table under the same label. For instance, the events No. 16, P/C in the first column of Table 14 are obtained by super-

2422


TABLE 4 (see Fig.

9 below) Second Roman Empire from Ist cent. B.C. to 3rd cent. A.D. in Italy. Duration 299 years, from 82 B.C. to 217 A.D. Overlap observed with. upwrd rigid shift of 1053 years The beginning of the Empire; the three great emperors of Ist cent. B.C. : Pompey the Great (anarchy and war) Sulla Lucius (transposed with No. i) Julius Caesar (- Chlorus in Third Empire) A) Octavian Augustus 23 (or 27) B.C. to 14 A.D. (37). Octavian, adopted son of Julius Caesar, accedes to power at the age of 19 (young). B) C) Tiberius 14-37 (23). Reason for the shift: llth century A.D. - 10th century of Jesus Christ (ist century). The emperors are caesars, often bearing the name Germanicus. ,Many gold coins are known from ancient Rome before the Second [Empire. The emperors are often called Augustus (Holy). Octavian Augustus (the Holy) 23 B.C.-14 A.D. (37) Germanicus 6-19 (13). In his time, Jesus Christ 0-33, a duplicate of Hildebrand under a shift of 1053 years. The betrayal by Judas, "Christ' s passion." Tiberius + Caligula 14-41 (27) Tiberius + Caligula + Claudius + Nero 1468 (54)? A doubtful join. Claudius + Nero 41-68 (27)? Full name includes "Black." Nero 54-68 (14). No joins in this variant Two Titus Vespasians 69-81 (12) Vesuvius erupts, destruction of Pompeii and Hereulaneium in 79 A.D. Domitian 81-96 (15) Trajan + Hadrian 98-138 (40). Both named Trajan (join by name) Antoninus Plus 138-161 (23) Lucius Ve~-ds 161d169 (8) Marcus Aurelius 161-180 (19). The equestrian statue of Marcus Aurelius is famous ancient Roman relic. Confusion in medieval chronicles [17]. Com~odus +Caracalla 180-217 (37), duplicate of Theerodic the Goth in 6th cent. Septimius Severus 193-211 (18) Anarchy, Julia Maesa and her favories 217235 (18). End of Second Roman Kmpire. War with the Goths. is close to the rigid shift of 1053 years

Holy Roman Empire of the German Nation in Italy lOth-13th century A.D. Duration 292 years (962 or 964 to 1254). Variant rules see [14, 16-19] (right and left) The beginning of the Empire; the three great emperors of 10th century: Otto I the Great (anarchy and war) Otto II the Wild Otto III the Red (Chlorus) A) Otto I as a German king: 936-973 (37). Octivien, son of Alberie (duplicate of Julius Caesar), accedes to power at the age of 16 (young). B) Otto II 960 (German coronation) 983

(23)
C) The emperors are German kaisers. There are virtually no gold imperial coins of lOth-13th century. A.D. Possibly attributable to the right column. Empire officially known as Holy Henry II the Holy + Conrad the Salian 1002-1039 (37) Conrad II the Salien 1024-1039 (15). In his time Pope Hildebrand in Rome 10531073-1085. The betrayal of Cencius, "Papal passion" [17] Henry III the Black 1028-1056 (28) Henry IV 1053-1106 (53). The names on the right are close: all contain a con%mon part Tiberius Claudius Nero and Germanicus Henry V the Black 1098-1125, German king (27) ? or No. 6 Henry V the Black 1111-1125, Roman emperor (14) Lothair 1125-1137 (12) Vesuvius erupts 1138-1139 (duplicate of the 1500 eruption?) Conrad IIl 1138-1152 (14) Frederick I Ba_rbarossa 1152-1190 (38). Chronicles confuse with Frederick II Henry VI 1169-1197 (28) Philip of 6~ibelline 1198-1208 (I0) Otto IV 1198-1218 (20). In his time the famous column for Marcus Aurelius was erected (see [17], vol, 4, p. 568, note 74) Frederick II 1211-1250 (39). His title is Gattin (the Goth?) Conrad IV 1237-1254 (17) Interregnum 1256-1273 (17). End of the Roman Empire of 10th-13th century A.D. War in Italy in mid-13th cent. Duplicate of Gothic (Trojan) war.

I) 2)

i) 2)

3) 4) 5) 6) 7) 8) 9) i0) ii) 12) 13)

3) 4) 5) 6) 7) 8) 9) i0) ii) 12) 13)

14) 15) 16)

14) 15) 16)

Mean shift by ends of rule 1039 years, overlapping left and right columns.

which

position (summing) of the events No. 16, C from the line C2 and No~ 16, P from the line CI~ Table 16 lists the duplicates of the events detected by the code questionnaire technique [4, 6]. The personalities listed in one column are duplicates or analogs. No events listed in the first column are duplicates. Their originals (preimages) are probably the events in 13th century Italy. Table 15 describes the parallelism detected by the author between the events in medieval and ancient Greece. They are overlapped by an upward rigid shift of the ancient events by some 1800 years. This table is also totally consistent with the decomposition of the GCC into the sum of four identical chronicle strata Cl, C2, C3, C4. See also [I-4, 6-13, 22]. The tables list the dates of rule, and in parentheses we indicate the length of rule, for example, Arcadius 395-408 (13). In addition to numerical data, the tables reproduce some fragments of the code questionnaires in order to bring out the parallelism of events. Complete tables of code questionnaires are too extensive to be reproduced in full.

2423


Holy Roman Empire of the German Nation, 10th-13th century A.D.
980 9SO 9?0 990 '/Ore 'tO30 10SO 'lO~ "tOgD~0 1~0 ~SO 'too 'rrgo 1~1o lz~o 12~ I770 ~zgo

o

,

Second Roman Empire, Ist century B.C. 3rd century A.D.
#

Fig. TABLE 5 (see Fig. i0 below)

9

Holy Roman Empire of lOth-13th century A.D. 911-1307. The Saxon dynasty begins in911. Duration of empire 396 years. German rulers identified. Overlap 911A.D. with 928 B.C. (on the right). For variant rules see [14, 15,17,18) (on the left) and [191 or the Bible (references to the Bible denoted by [B]) i) Henry I 919-936 (17) 2) LothaJr 947-950 (3) 3) Otto I the Great 936-973 (37) 4) Otto II 960-983 (23) 5) Otto III the Red 983-996 (13), from accession to the throne in 983 to Roman coronation in 996. For continuation see below 6) Otto III from Roman coronation in 996 to 1002, i.e., 996-1002 (6) 7) Henry II 1002-1024 - Conrad II 10241039 (total 37) 8) Henry III 1028-1056 (28) 9) Henry IF 1053-1106 (53). Contest with Pope Hildebrand. The excommunication of Henry (Canossa). Acceded to the throne at the age of 6. At the end of his life retired to his secluded castle. Betrayal and coronation of his son Conrad while Henry was still alive. Son rules in father's place (see [17], vol. 5, pp. 233-

The Kingdom of Judah lOth-15thcentury B.C? [19] ..... existed, according to the Bible, for 395 years. Overlaps with the right colunm after a rigid shift of approximately 1830 years, i.e., 1838 = 928 + 910. Dates in the right column reckoned from "zero," taking 928 B.C. as the "zero" point. For dates see [19,B] I) Rehoboam 0-17 (17)

3) 4)

2) Abijah 17-2o (3)
5)

Asa 20-55 (35) or 20-61 (41) Jehoshaphat 55-79 (24), 61-86 (25)

Jehoram of Judah (8) by [B] or (6) (9 by),
: [19] + Ahaziah of Judah (i), total

(7), i.e., 86-94 by [BI
6) 7) 8) 9) Athaliah 96-101 (6). See dates in 2 Chronicles, 1-2 Kings Jehoash of Judah 92-130 (38) by [19] or (40) by [B] Amaziah 130-159 (29) Uzziah from 159 by [19] to 211, ruling (52) by [B] or (43) by [191; 211-159-52 by [B]. Contest with Chief Priest Azariah. Excommunication of Uzziah. Came to the throne at the age of 16. At the end of his life was a "leper" and lived in a "separate house." His son ruled de facto in his place (2 Chronicles, 26:21-23)

235)
i0) ii) 12) Lothair II 1125-1138 (13) Conrad III 1138-1152 (14) Henr~'vI 1169-1197 (28). Frederick I attacks Rome in 1167. "Plague" in the German army; they withdraw. Overlap Germany --Assyria (see on the right) Frederick II 1196-1250 (54). Famous Roman emperor. Conrad IV 1250-1254 (4) Charles of Anjou 1254-1285 (31) Unrest and events of the period 12851307 (22) in Italy (2). End of Empire of lOth-13th century Adolf of Nassau 1291-1298 (7) Albretch I 1298-1308 (10) Avignon captivity of the Pope (and the papal throne) in France, 1305-Jan. 1376 (70)

10) Jotham 211-227 (16) [B], (7) [191 11) Ahaz 227-243 (16) [B], (20) [19] 12) Hezekiah 256-285 (29). ~ae Assyrian king
Sennacherib attacks Jerusalem in the reign of Hezekiah, "Plague" in Sennacherib's army; they withdraw, 2 King 19:35-36. Compare ~laeodoric (6th cent.)

13) 14) 15) 16) 17) 18) 19)

Menasseh 285-344 (55) [B] or (45) [19]. Famous king 14) Amon 340-342 (2) 15) Josiah 342-373 (31)
16) 17) 18) 19) Jehoahaz (less than I) + Jehoiakim (ii) + Jehoiachin (less than i) + Zedekiah (ii) 373397 (22) or (24). End of Kingdom Jehoiakim 374-385 (Ii) Zedekiah 386-397 (ii) Babylonian captivity, Babylon ruled by Persia 397-467. Persia-France (?)

13)

By [19], the Kingdom of Judah begins in 928 B.C. Since in our table the zero year of the Kingdom of Judah is 910 A.D., the shift is approximately 928 + 910 = 1838 years, which is close to the third fundamental shift of 1778 (or 1800) years On the GCC [I, 2, 6].

2424


Holy RomanEmpire of the German Nation, 10th-13th century A.D.
910 940 980 IOZO 1060 1100 "t?~O 1180 1~20 126G 1300 1~ 1~86'

The
2

i~-e 0f
g

e

rules
9 1 17

~,~a Fig.

The Biblical Kingdom of Judah i0

TABLE 6 (see Fig. ii below) Roman coronations of Holy Roman emperors of the German Nation 10th-13th century A.D. Rigid shift of 1840 years. Variant rules see in [14~ 15,17] I) Hugh of Axles 926-947 (21), king of Italy, the beginning of the Empire 2) l~othair 947-950 (3), king 3) Otto I the Great 936 German coron.960 beginning of Otto II; or 936 German coron.- 962 Roman coron. Thus (24) or (26). Pope John Xll - Octavian [17]. See Octavian Augutus. 4) 962 Roman coron.-973 German coron. (II)~ 973-death of Otto I and German coron, of Otto II 5) 973 German coron.-996 Roman coron. (23) 6) 7) 8) 9) I0) Ii) 12) 13) 996 Roman coron.-1014 Roman coron. (18). Complete table of all variants on the right is given in [6]. 1014 Roman coron.-1027 Roman coron. (13) 1014 Roman coron.-1046 Roman coron. (32) 1046 Roman coron.-1084 Roman coron. (38) 1084Roman coron.--l125 death of Henry V, end of Franconian dynasty and beginning of Saxon dynastry 1125-1134 Roman coron., (9) 1134 Roman coron.-l155 Roman coron. (21) Pope Alexander III from accession to invasion of Frederick I in 1167, i.e., 1159-1167 (8). In 1143-1155, Italian wars of Germany. In 1154, capture of Rome by Frederick I

Kingdom of Israel (reference to the Bible denoted by [B]) starts at 922 B.C. [19]. For simplicity, years reckoned from zero. Here 922 B.C. = 928 A~ i) Jeroboam 0-22 (22). The beginning of the Kingdom 2) Nadab 22-24 (2) (or Navotb) 3) Baasha 24-48 (24). See Table 5: ~Asa of Judah -duplicate of Otto I. Close names AsaBaasha (Jesus?) Compare Hildebrand in llth century A.D. Under Octavian, Jesus born in ist century A.D. 4) Omri 51-63 (12). Dates of all rules according to 2 Chronicles and 3-4 Kings 5) 6) 7) 8) 9) i0) 11) 12) 13) ~hab 63-85 (22). "Double counting" in [B 1 reveals lacunae, see [5]. Ahaziah (2) + Jehoram of Isarel (12), i.e~, 85-99 (14). This is the ist variant of Jehoram in [B]. Jehoram of Israel 94-106 (12). 2nd variant of Jehoram in [B]. Jehu (28) + lacuna (2) (unrest) 99-127-129 (total 30 years) Jehoahaz (17) + Jehoash (16)~ i.e., 127144-q60 (total 33 years) Jeroboam II 160-201 (41). Implied superpositions: Assyria-Gern~ny, Persia--France, Babylor:-Rome or Avignon, Hittites-Geths Menahe= 203-213 (10) Pekah 215-235 (20) Hoshea 235-243 (8). War with Assyria, invasion of Shalmaneser. Superposition of Assyrian on Germany (left). End of the Kingdom of Israel. Superposition of the "phazahos" from [B] on TRK and TRNK (Franks, Goths)

According to [19], the Kingdom of Israel begins in 922 B.C. Since the zero year of the Kingdom of Israel in the table corresponds to 920 A.D., the shift is approximately 920 + 922 = 1842 years, which is close to the third fundamental shift of 1778 (or 1800) years on the author's GCC [~,

2, 6].

2425


Holy Roman Empire of the German Nation, lOth-13th century A. D.
910
=

9,]0
=

950
-_

9"/0
-_
m

990
T

1010
9

1030
~

"1050 1070
: ._

1090
._

l'rlO
e

1130
e
m

1150
e

1170
-_

5

11 9

12

=m

?-I
i.
"1o

,io
7!

,;o
I 1o

,io l
13

m

The Biblical " Kingdom of Israel

Fig. TABLE 7

II

ist period of the Roman pontiphicate 141-314 A.D. Variant rules see [14,21] (also the right column) I) 2) 3) 4) 5) 6) 7) 8) 9) i0) ii) 12) 13) 14) 15) 16) 17) 18) 19) TABLE 8 St. Pius 141-157 (16) St. Anicetus 157-168 (ii) St. Soter (= the savior) 168-177 (9) St. Eleutherius 177-192 (15) St. Victor 192-201 (9) Zephyrinus 201-219 (18) Calixtus 219-224 (5) Urban 1 224-231 (7) Pontianus 231-236 (5) Fabian 236-251 (15) Unrest 251-259 (8) Dionysius 259-271 (12) Felix 1 275-284 (9) or Eutychianus? Eutychianus 271-275 (4), Felix I? Caius 283-296 (13) Marcellinus 296-304 (8) Marcellus 304-309 (5) Eusebius 309-312 (3) Melchiades 311-314 (3)

2nd period of the Roman pontiphicate 314-532 A.D. Consistent with superposition of Second and Third Roman Empires I) Sylvester 314-336 (22) 2) Julius 1 336-353 (17) 3) Liberius (= the liberator) 352-367 (15) 4) Damasus 385-398 (13) 5) Siricius 385-398 (13) 6) Anastasius, Innocent 398-417 (19) 7) Boniface 418-423 (5) 8) Celestine 423-432 (9) 9) Sixtus III 432-440 (8) I0) St. Leo (Leo I) 440-461 (21) Ii) Unrest and Hilarius 461-467 (6) 12) Simplicius 467-483 (16) 13) Felix II 483-492 (9) 14) Gelasius 492-496 (4) 15) Symmachus 498-514 (16) 16) Hormisdas 514-523 (9) 17) John I 523-526 (3) 18) Felix III 526-530 (4) 19) Boniface II 530-532

Royal Rome, First Empire 753-500 B.C. See T. Livius. A shift of 1050 years (approximately) i) 2) 3) 4) 5) 6) 7) 8) 9) i0) ii)

A strand of the Third Roman Empire 3rd-4th cent. A.D. Time intervals identified in [6]. Some representatives From each interval are identified I) 2) 3) 4) 5) 6) 7) 8) 9) i0) ii) 300-337 (37), Constantine I 337-380 (43), Basil the Great 380-423 (43), Honorius 423-444 (21), Aeolus 444-476 (42), Ricimer 476-526 (50), Odoacer, Theodoric 526-552 (26), the Gothic dynasty The fall of Third Roman Empire in the West, expulsion of the Goths. The war with the Goths in 6th cent. B.C. See below The clan of Goths, enemies of Rome in the war of 6th century. The allies of the Goths are Franks, phonetically TRK or TRNK (compare "F" and"T"). Be]isarius - Roman military leader opposing the Goths. Name without vowels VLZR (on the left, son of Volusius). Belisarius accused of treason The ~essage of Totila the Goth to the Roman Senate. The plot inRome and its exposure. These events occur in 543 A.D. The dates on the left are shifted by the formula X+300, where X is the year when I[ome was founded. The first battle for Rome in 545-547. The Goths lose. The second march on Rome in 548-549. Roman leader Narses. The Goths are crushed.

iZ) 13)

Romulus Quirinus (37) Numa Pompilius (43) Tulla Hostilus (32) Ancus Martius (24) Tarquinius Priscus (38) Servius Tu]lius (44) Tarquinius Superbus (25) The fall of Royal Rome, the expulsion of the kings, the war with the Tarquinians around 500 B.C. See below The clan of Tarquinians, phonetically TRKVN (without vowels), enemies of Rome. Their duplicates -- Trojans (see Troy). Valerius -- Roman military leader opposing the Tarquinians. Name without vowels VLZR. Valerius accused of treason The message of L. Tarquinius Superbus to th, Roman Senate. The plot in Rome and its exposure. Upward shift of 1050 years places these events at 543-544 A.D. The first battle for Rome in 545. The Tarquinians lose. The second march on Rome in 546. Roman leader Lars Porsena. The Tarquinians are crushed.

12) 13)

2426


TABLE

9

(see Fig.

12 below) A strand of the Third Roman Empire 3rd-6th cent. A.Do Variant rules see [14, 16, 5, 19]. Dates sometimes reckoned from the death of the co-ruler. Analysis of complete list see [6] i) 2) 3) 4) 5) 6) Lucius Aurelian Unrest 275-276 Probus 276-282 Unrest 282-284 Diocletian the Diocletian and 270-275 (5) (I) (6) (2) Great 284-305 (21) Constantius Chlorus jointly

Second Roman Empire from 82 B.C. to 3rd cent. A.D. The first 8 items in this column are approximate, without affecting the closeness of the streaks. Periods of unrest also indicated I) 2) 3) 4) 5) 6) Lucius Sulla 82-78 (5) Unrest 78-77 (I) Sertorius 78-72 (6) Unrest 72-71 (2) Pompey the Great 70-49 (21) Pompey and Caesar jointly
60-49

(11)

293-305 (13)
7) Unrest 305-309 (4) 8) Constantius Chlorus, the winner of the Ist Tetrarchy 305-306 (i), after No. 5 9) Tetrarchs and Constantine Augustus 306-324 (18) i0) Constantine Augustus 306-337 (31). Beginning of No. 12 from the death of Constantine II) The birth of Basil the Great in the 27th year of Constantine Augustus (27) 12) Constantius ii 337-361 (24) 13) Constantius II and Constans jointly, 337-350 (13) from the end of No. I0 14) Julian 361-363 (2) from No. 12 15) Unrest of 363 (i) 16) Valentinian ! 364-375 (ii) 17) Valentinian and Valens (Pallans?) jointly 367-375 (Ii) 18) Valens 364-378 (14) 19) Valens-Valentinian-Gratian jointly 364-375 (11) 20) Jovianus 363-364 (I), transposed 21) Unrest of 378 (I) 22) Gratianus and Valentinian II after Valens and unrest 379-392 (13) 23) Theodosius 1 379-395 (16) 24) Eugenius 392-394 (2) 25) Joint rule of Eugenius 392-394 (2) 26) Arcadius 395-408 (13) 27) Honorius 395-423 (28) 28) Aetius 423-444 or 423-438 (14) or (21) 29) Valentinisn III 437-455 or 444-455

7) Unrest 49-45 (4) 8) Julius Caesar, the winner of the ist Triumvirate 45-44 (i) 9) Triumvirs and Octavian Augustus

44-27 (17)
i0) Octavian Augustus 27 B.C.-14 A.D. (41) or (37) from 23 B.C. Ii) The birth of Jesus in the 27th year of Octavian Augustus (27) 12) Tiberius 14-37 (23) 13) Tiberius and Germanicus jointly 6-19

(13)
14) 15) 16) 17) Caligula Unrest of Claudius Claudius 37-41 (4) 41 (I) 41-54 (13) and Pallans jointly 41-54

(13)
18) Nero 54-68 (14) 19) Nero-Burrus-Seneca jointly 54-62

(8)
20) Galba 68-69 (i) 21) Unrest of 69 (i) 22) Two Titus Vespasians 69-81 (12). Identical names 23) Domitian 81-96 (15) 24) Nerva 96-98 (2) 25) Joint rule of Nerva 96-98 (2)

26) Trajan 98-117 (19)
27) Hadrian 117-138 (21) 28) Titus Antonius Pius 138-161 (23) 29) Marcus Aurelius 161-180 (19) 30) 31) 32) 33) 34) 35) 36) Lucius Commodus 176-192 (16) Pertinax 193 (i) Didius Julianus 193 (i) Claudius Albinus 193 (I) Pescennius Niger 193-194 (i) Septimius Severus 193-211 (18) Caracalla 193-217 (24). The famous reforms of the Second Empire 37) End of the Second Roman Empire. The crisis of mid-3rd cent. War with the Goths. A shift of 333 years (approximately)

(11)
30) 31) 32) 33) 34) 35) 36)

or

(18)

Ricimer 456-472 (16) Olybrius 472 (I) Glycerius 473, 474 (I) Julius Nepos 474 (i) Romulus Augustulus 475-476 (I) Odoacer 476-493 (17) Theodoric 493-526 or 497-526 (33) or (29). Famous reforms 37) End of ~ird Roman Empire in the West~ The famous war with the Goths in mid-6th century.

This parallelism is secondary, being a consequence of the main superpositions listed above. Both streaks include additional numerical data which are not taken into consideration when calculating the closeness coefficient of the streaks.

2427


TABLE

I0 Eastern Roman Empire 306-700 B.C., with the capital in New Rome i) 2) il Licinius 308-324 (16) Arius 330-333 (3) (5) (8) (variants) Basil the Great? 333-378 (45) Theodosius I 379-395 (16) Arcadius 395-408 (13). Separation of the Western Empire from the Eastern Empire Theodosius II 408-405 + Martian 450-457 (49), dispute in the Council of Ephesus The invasion of Attila and anarchy 451-453 (2) Leo 1 457-474 (17) Zeno 474-491 (17). The Germanic leader Odoacer attacks Rome. The Western ruler is Ricimer (Rezin?) 456-472. Zeno turns to Theodoric the Goth for help Anastasius 491-518 (27) Two Justins: Justin I 518-565 (a total of 47). Suppression of the Nika revolt in New Rome, a massacre 5 emperors: Justin II+ Tiberius II + Mauricius+ Phocas + Heraclius, total 565641 (76) Constant II 642-668 (26), Arabs Constantine III 641-642 (I) Constantine IV 668-685 (17) Heraclion 641-642 (i) Justinian II, first reign 685-695 (i0) . Imperial war, Arabs The famous crisis of the end of the 7th century A.D. The breakup of the Eastern Empire

The Biblical Kingdom of Judah with the capital in Jerusalem, iOth-7th cent. B.C. i) 2) 3) 4) 5) 6) 7) 8) 9) Rehoboam (17) Abijah (3), "the Father of God" - trans. Ass (Jesus?) (46) (41) Jehoshaphat (25) Jehoram of Judah (8). Separation of Edom. Then a gap of 76 years (see below) Uzziah (52). Involvedin a Temple dispute, cursed and struck with leprosy Interregnum (2). A lacuna in 2 Chronicles Jotham (16) Ahaz (16). Rezin king of Syria and Pekah king of Israel attack Jerusalem and Ahaz turns to Tiglath-pileser for help (a duplicate of Theodoric on the right?) Hezekiah (29) Manasseh (55) (50). A famous king, accused of a massacre in Jerusalem (suppressing a rebellion?) Superposition of the capital on New Rome A 76-yr insert: 4 kings + Amon (2 yrs) = 5 kings, a total of 78 years, i.e., 78 = 76 + 2 Josiah (31), invasion of the Pharaoh Jehoahaz (I) Jehoiakim (II) Jehoiachin (i) Zedekiah (Ii). Nebuchadnezzar king of Babylon "takes the people into captivity" The end of the Kingdom of Judah, Babylonian captivity

6) 7) 8) 9)

i0) ii)

lOj Ii)

12) 13) 14) 15) 16) 17) 18)

12) 13) 14) 15) 16) 17) 18)

This is a secondary parallelism which follows from the parallelisms listed above and in [6]. The shift of some 1300 years is the sum of two shifts 1000 + 300.

TABLE

11
A strand of the Third Roman Empire 4th-5th cent. A.D. On the left sum of shifts iO00 + 300 I) 2) 3) 4) 5) 6) 7) 8) 9) i0) ii) 12) 13) 14) 15) 16) 17) 18) 20) 21) 22) Constantine I after deposing Max!entius 313327 (24). Rift and war with co-ruler Licinius Constantine II 337-340(3) after No. 1 Constantius II after the death of Constantine II 340-361 (21), then No. 4 Julian (Elius?) 361-363 (2) Jovain 363 (I) Valentinian I 364-375 (ii) Valens (the Wicked) 364-378 (14), contest with the prophet Basil the Great. Killed in battle Gracianus and Valens and unrest 379-383 (4). Then No. 9 after No. 7 Valentinian II 379-392 (13) Alaric and John the Eloquent 378-403 (25?) (32?) Theodosius 1 379-395 (16) Arcadius 395-408 (13) Honorius 395-423 (28) Constantium III 421 (7 months) John 423 (2 months) Interregnum, regency 423-444 (21) Valentinian III after regency 444-455 (ii), invasion of Attila (TTL) Petronius Maximum 455-456 (i) Ricimer 456-472 (16), invasion of Genseric, deportation of nations Romulus Augustulus 475-476 (i). Invasion of Odoacer and capture of Romulus The end of the independent existence of the Third Western Roman Empire as a "pure Roman" state.

The Biblical Kingdom of Israel lOth-8th cent. B.C. A shift of 1300 years i) 2) 3) 4) 5) 6) 7) 8) 9) i0) ii) 12) 13) 14) 15) 16) 17) 18) 19) 21) 22) Jeroboam I, Jeroboam's "heresy," split and war with Rehoboam (22). On the right "heresy" - Arianism? Nadab (2) Baasha (24). On the right Basil the Great (Jesus-Asa-Baasha?) Elah (Elius?) (2) Zimri (i) Omri (12) Ahab (the Wicked) (22), contest with the great prophet Elijah. Mortally wounded in battle Ahaziah (2). Rules in Samaria (superimposed) on Rome on the right) Jehoram of Israel (12) Jehu and the prophet Elisha (revolution to seize power) (28) Jehoahaz (17) Jehoash of Israel (16) Jeroboam II (41) Zechariah (6 months) (I) Shallum (i month) (i) Interregnum (24) After the interregum - Menahem, invasion of king Pul or Tul (i0). Superposition of the names Tul (TL) and Attila Pekahiah (2) Pekah (20), invasion of Tiglathpileser (the deporter) Hoshea (until captivity) (i), invasion of Shalmaneser and capture of Hoshea The end of the independent existence of the Kingdom of Israel. Hoshea is the last independent king.

2428


TABLE 12 (see Fig. 13 below)
First Byzantine Empire 527-829 AoD. 302 years [14, 20] I) Duration of Second Byzantine Empire 829-1204 A.D. 375 years ~14, 20] i) Duration of

2) 3) 4) 5) 6) 7) 8)

Justinian I Basileus and Theodora 527-565 (38). This is ~e beginning of the Eastern Roman Empire proper. A rigid shift of approximately 340 years Justin II 565-578 + Tiberius II 578-582 (17) Mauritius 582-602 (20) Phocas 602-610 (8) Heraclius 610-641 (31). Then unrest in No. 6 on both left and right Constantine III 641, Heracleon = Heraclius II 641 (total around i). Beginning of unrest Constans II 642-668-Constantine IV 668-685 Justinian II 685-695 (53) Unrest: Leontius II 695-698 or 694-6977 Tiberius III Apsimar 697-704 oe 698-7057 Justinian II (second time) 705-711~ Philippicus Vardanus 711-713, Anastasius II 713-715 or 716, Theodosius III from 715 or 761to 717. Total (22) The war of Justinian II (see above) - a partial duplicate of the Gothic-TrojanTarquinian war, abbreviated as GTR-war Leo III the Isaurian 717-741 (24) Constantien V Kopronymus 741-775 (34) Leo IV775-780 + Constantine Vl 780-797 + Irene 797-802 + Nicephorus 802-811 (36) Unrest: Stauracius 811, MichaellRhangabe 811-813, Leo V 813-820 or 821, Michael II 820 or 821-829. Total (19). In both col~unns numerous periods of unrest, thus complicating the overall picture. Both Empires overlapped by a shift of 340 years.

Theophilus 829-842 + Michael III and Theodora 842-867 (38). Then the Macedonian dynasty begins. Compare with biography of Justinian I Basil I (Basileus I) 867-886 (19). Compare Justinian I Basileus Leo VI 886-912 (26) Alexander 912-913 (i) Constantine VII 910-912-959 (47) or (49) Romanus II 959-963, Nicephorus II Phocas 963969, John I Tzimisces 963-975 or 976 (total 16) Constantine X or VIII 975-1028 (53), different versions Unrest: Constantine VIII 1025-1028, Romanus III 1028-1034, Michael IV 1034-1041, Michael V 1041-1042, Constantine IX Monomachus 1042-1054, Theodora 1054-1056, Michael VI 1056-1057. Total (20). To overlap with the unrest on the left: a shift of 340 years. In 1047 the revolt of Tornika (Nika TR?) Compare with the Nika revolt in the time of Justinian Io The Nika revolt - a duplicate of the GTR-war by [6]. John II Comneus II18-i143 (25) Alexius I Comneus 1081-1118 (37), transposed with No. 10 (John) Manuel I Comneus 1143-1180 (37) Unrest: Alexius II Comneus 1180-1183, Andronicus I 1183-1185, Isaac II Angelus 11851195, Alexius III 1195-1203, Alexius IV 12031204, Isaac II Angelus (second time) 1203-1204, Alexius V 1204. Total (24). Fall of Constantinople in 1204.

2) 3) 4) 5) 6) 7) 8)

9) i0) II)

9)

i0) ii) 12) 13)

12) 13)

TABLE

13

(see

Fig.

13 below) Second Byzantine Empire, the strand from Basil I to John VIII, i.e.~ 867-1143 A.D. Duration 276 years. Variant rules see [14, 20] I) Basil I (Basileus !). Nika revolt in the time of Justinian I in the ist Byzantine Empire. 867-886 (19). Theodora was the wife of Justinian I Leo VI the Wise 886-912 (26) Alexander 912-913 (i) Romanus 1 919-945 (26) Constantine VII 910 or 912-959 (47) Romanus !I + Nicephorus ii Phocas + John I Tzimisces 959-976, 975 (16) or (17) Nicephorus II Phocas + John I Tzimisces 963-976 (13), 2nd variant Basil II Bulgaroktonus 975 or 976-1025 (50) Unrest 1025-1057. For the list of emperors in this period see Table 12 Alexius I 1081-1118 (37) John II 1118-1143 (25) (Nos. Comneni). Followed by Manuel of 1180-1204 (nor included in The fall of Constantinople in I0 and ii both I and the unrest the parallelism). 1204

Third Byzantine Empire 1204-1453 A.D. Duration 249 years, numerous periods of unrest. Both empires overlap after a shift of 340 years

i)

The beginning of the Nicaean Empire in 1204. See Nika inthe GTR-war of 6th century A.D. Theodore I Lascaris 1204-1222 (18) John!ll Dukas Vatatzes 1222-1254 or 1256(32). The GTR-war Theodore II Lascaris 1254 or 1256-1258 or 1259 (3) MichaelVIII 1259 or 1260-1282 or 1283 (25) Andronicus II Paleologus 1282 or 1283-1320 or 1328 (46) Andronicus III Paleologus 1320-1341(21) 2nd variant: Andronicus III Paleologus 1328-1341 (13) John V Paleologus 1341-1391 or 1376 (50) Unrest 1376-1391. Andronicus IV 1376-1379~ John V (second time) 1379-1391, John VII 1390 (15) Manuel II 1391-1424, 1425 (34) John VIII by [20] or VI by [14] 1424 or 1425-1448 (24). The fall of Constantinople in 1453. The end of the Byzantine Empire

2)
3)

2)
3) 4) 5) 6) 7) 8) 9) I0) ii)

4) 5)
6)

7)
8) 9)

i0)
Ii)

2429


TABLE

14a Biblical chronology. Chronicle B on the GCC, years B.C. A shift of 1800 years upward in time The dates of the corresponding events are shifted upward because of the parallelism discovered by the author. In addition to the shift, there is an identification with the left column which results in overall shortening.

Traditional chronology. Chronicle E on the GCC in [2], years B.C. Conventional "textbook" representation i) K 1460-1236. The Trojan Kingdom of the Seven Kings, Trojans, Greeks. 2) T 1236-1226. The Trojan War (Greece?). The expulsion of the Trojans, the fall of Troy 3) H 1226-850. The kings of Ancient Greece

4) T 850-830. 2nd version of the dating of the Trojan War by T. Livius. The apple of contention of Venus (Eve?) 5) T 760-753. The founding of Rome, Romulus and Remus, the kidnapping of the Sabines 6) K/P 753-522. Royal Rome of the Seven Kings according to T. Livius. Great Greek colonization 8th-6th century 7) T 522-509. The war with Tarquinians. Deposing of the kings, the beginning of republican Rome. 8) H/C 509-82. Ancient republican Rome. Greco-Persian wars. Cyrus, Darius, Xerxes. Peloponnesian Wars. Macedonians, Philip II. The fall of Byzantium. The Empire of Alexander the Great. A famous period in the history of classical Greece. Roman wars with Samnites. Punic wars. Hannibal. The end of clasical Greece. The beginning of Hellenism.

4) T 850-830. Genesis 1-3. Adam and Eve, the apple of contention, the expulsion from Paradise 5) T 760-753. Genesis 4:1-16. Cain and Abel, the murder of Cain 6) K 753-522. Genesis 4-5:31. Enoch, Irad, Mehujael, Methuselah, Lamech, Seth, Enosh, Kenand, Mahalelel 7) T 5Z2-509. Genesis 5:32-8. The legend of Noah and the Flood, the destruction of mankind, the ark, the covenant 8) H 509-82. Genesis 9, 10:1-32. The descendants of Noah, peopling of the earth. Noah's sons Shem, Ham, and Japheth. Japheth's sons. Comment to No.7: There is a parallelism between Noah and Moses, duplication of the same term "the ark of covenant" (Moses) and "the ark" (Noah), duplication of covenants. A legend that Noah founded a city near Rome (see [17], Vol. 3, p. 437).

9) T 82-83. The beginning of Imperial Rome. Sulla, Pompey, Caesar, Octavian Augustus, civil wars of ist century.

9) T 82-23. Genesis 11:1-9. Babylonian confusion, the scattering of the people, chaos.

2430


Chronicle C ~ on GCC, years A.D. A shift of 1778 (approximately 1800) years downward in time i) K 306-535. Third Roman Empire 4th-6th century, both East and West. 2) T 535-552. The Gothic war in Italy, the expulsion of the Goths, the fall of Naples and Rome 3) H 552-901. Medieval Papal Rome, Greece

Chronicle C 3 in GCC, years A.D. A shift of 1053 years downward in time The reason for the lO00-year shift is the following: some dates were originally recorded as X.I century, which stands for the ist century of Christ (X = Christus). Later the meaning of this X was forgotten, and the for~31a "X.! century" was formally decoded as "XI century," i.e., llth century. Similarly the year I.!00 stood for year i00 of Jesus (I = abbreviation of Jesus)~ but was eventually decoded as the year ii00, since the letter I also stood for r'one thousand." As a result, these dates were shifted by !000 years.

4) T 901-924. The war in Alberic I and Theodora legend on "the woman of tion." 5) T 931-954. The War in Alberic II and Theodora

Italy. I. The contenItaly. II 6) K 300-535. Roman Empire of 4th-6th century. In 325, the founding of New Rome. The Eastern campaigns of Rome 7) T 535-552. Gothic war in Italy. The fall of Naples and Rome. Justinian, Belisarius, Narses. Go~s, Franks --TRN. 8) H 552-901. Medieval Papal Rome. The ~ars of the Romans with Lombards 705, 711 and later until 765 and 769. The war in the south of Italy. The wars with Saracens. Frankisb wars in Italy. Comment to No. 13 in chronicles G and E on the left: Charlemagne Z Joshua the son of Nun, the defeat of Roland - the defeat of the army near the city of Ai; both Charlemagne and Joshua stopped ~le sun during their battle (~mnique episodes), the traitor Ganelon - the "traitor" Achan.

6) P 962-1250. Holy Roman Empire of the German Nation 10th-13th century. Crusades 7) T 1250-1268. The war in Italy. Manfred, Conrad, the fall of medieval Troy.

8) C 1300-1550. The Habsburg Empire. Medieval Greece and the battles of 1316 (the original of the battle of Marathon). Duke Walter II. The wars between the Franks and the Turks. Mohmmaed II, Mohammedans. The fall of Byzanti~n in 1453. The Ottoman Empire. The end of independent existence of medieval Greece

The end of chronicle (line) C 4, We described the skeletal events which moved down as a result of the shift of 1778-1800 years. The reason for this shift - the use of abbreviations to record dates. 9) T 931-954. The wars in Italy. Alberic II and Theodora II. Restoration of many ancient customs. The beginning of the Holy Roman Empire of the German Nation

2431


~o

~o

TABLE Chronicle C 3 on GCC, years A.D. after a shift of 1053 years Chronicle C 2 on GCC, years A.D. after a shift of 333 years i0) P 965-1250. Roman Empire of Genllan Nation. beginning of a era" under Pope debrand. Rift Holy the The "new Hil-

14b Chronicle C 1 on GCC, years A.D. Distorted original, no shift Chronicle C o on GCC, years A.D. Original, no shift. This coltmm contains a part of the modern "textbook." This part served as the original for the downward shifted chronicles C2, C3, C 4- There are virtually no historical data before !Oth century

Traditional chronology. Chronicle E on GCC, years A.D.

i0) P/K 23 B.C.-235 A.D. Second Roman Farpire ist-3rd century. The beginning of the "new era," religious reforms, Jesus Christ

Biblical chronology. Chronology B on GCC, years A.D. after a shift of 1800 years I0) K 23 B.C.-217 A.D. Genesis 11:10-32. Arpachshad, Shelah, Eber, Peleg, Reu, Serug, Nahor, Terah, Haran, Abram. Haran -- Arius?

i0) K

306-535. Roman This chronicle has not Empire 4th-6th century. yet been moved do~n, In the 6th century, but it already conBasil the Great and his tains a small number religious reforms. A of duplicates rift in the church, Arius

ii) T 235-251. Julia Maesa, anarchy, Gothic war. T 270-300. War

-

12) ~/K/P/C 300-535. Roman Empire of 4th6th century. Division into two kingdoms: East and West

ii) T 270-300. Abraham Ii) T 1250-1268. The and Sarah. The confamous war in Italy. frontation with Pharaoh The fall of Troy and TRN. Genesis 12 Naples. 12) K 306-535. Genesis 12) C i273-16i9. The 13-38. Abraham and Habsburg (Roman) EmHaran. Division into pire. Eastern Roman two kingdoms. Isaac, (Byzantine) Empire Esau, Jacob, Joseph

!I) T 535-552. The famous Gothic war in Italy. The fall of Naples, Rome 12) ~ 681-887. Carolingians, the empire of Charlemagne. Eastern Roman Empire 13) T 901-924. War in Italy. Alheric I, Theodora I.

12) P/K 300-535. Roman Empire of 4th-6th century. The division of the empire into two: East and West. 13) P/K 535-552. Gothic war in Italy. The expulsion of Goths from Ita]y

12) K 306-526. Few remnants of traditional data about the period of 4th-6thcentury. 13) Insignificant remnants of traditional data about the 6th century.

13) T 535-552. The 13) T 535-552. Genesis 13) End of chronicle Gothic war in Italy. 39-50. Exodus (Moses). (line) C 3 The end of the Roman Leviticus. Numbers. Empire. The wars of Deuteronomy. Joshua. Charlemagne - the wars Chanson de Roland. of Jnshua. 14) ~/H/P 566-901. Medieval g/H/P 556-901. Papal Rome. Carolingians, History of the judges. the empire of CharlemaKne

I!4 ~8.

Judgesi-

141) ~/H 552-901. CarolinI gians, the empire of CharI lemagne

14) ~/H 552-901. InsignifiCant remnants of data about 6th-9th centuries.


italy. I

15) Insignificant remnants of data about the Ist half of lOth century.

15) T 90i-914-924. War in Italy. i5) T 90i-924. Judges 19-21. 15) T 1250-1268. The fmnous 15) T 901-924. War in Alberic I~ Theodora I. T War with the sons of Benwar in Italy. The fall of Alberic I, Theodora 931-954. War, Alberic II, Theojamin. T 931-954. Ruth. Ithe Hohenstaufens, the fall dora II 2 Samuel. I Kings i-ii. 1-2 of Troy, Naples. Manfred, Chronicles 1-9. Saul~ Samuel Charles of Anjou, Conrad. David~ ~olomon Medieval legends about Troy in Italy. 16) P/C 962-1250. Holy Roman 16) P 962-1250 1 Kings 12-22, 16) C 1273-1619. The Habs16) P 962-1250. Roman Empire of the German Nation. burg Empire. Renaissance Empire. Rich data. 2 Kings 1-23, 2 Chronicles Emperors crowned with two 10-34. Kingdoms of Israel in Europe, revival of anble empire." crowns: Roman and German and Judah (duality) cient motifs. End of chron(duality) icle C 2 Kings 2435-36. The ~[ebuchadof the Possible reason for the shift of 333 years: the date "3r4 yearofM~imilian" could have been ~r as MCL.III, i.e., Maximus Caesar Leo. Decoding gives the date 1153 A.D., which differs from the true date of 1496 by 343 years. Recall that Maximilian I 1493-1519. 18) C 1273-1619. The Habsburg Empire. The chronologists Scaliger, Petavius 16th-17th century. Dionysius Petavius - the original of Dionysius ExigUus (6~ century). The beginning of real history. 17) T 1250-1268. War Italy. The fall of Hohenstaufens. The Troy, Naples. Rich 17) T in The the The fall of Rich data

German 16) P 962-1250. Roman Ge~nan Empire. Rich data. "Dou-

17) T war the of de

1250-1268. The famous 17) T 1250-1268. 2 in Italy. The fall of 25~ 2 Chronicles Hohenstaufens. The fall war with Pharaoh, Troy, Naples. Charles nezzar. The fall Anjou, Manfred. Kingdom of Judah

1250-1268. War in Italy. fall of the Hohenstaufens. fall of Troy, Naples. data

18) C 1273-1619. ~ne Habsburg Empire. Avignon captivity of the Pope in France 1305-1378, 70 years. ~len return to Rome in Italy 18) Ca 1273-i~00. The books of Ezra, Neh~niah, Esther. Babylonian captivity, under Persian rule, 70 years. Then return to Jerusalem.

18) C 1273-1619. The Habsburg Empire. Council of Trent creates a global chronology and a Biblical canon.

bo bo %o


A strand from the Third Ron~un Empire, 3rd-6th century A.D. 270 290 3?0 330 ~50 370 3'90 ~10 ~30 0,50 ~-70 #.90 510 5~0
7

Second Roman

18 \

\\

\\

\l ~

IIl \'0 \\'5 --

tury~.c.-~d
century A.D. Fig. 12

\'\~-~
~

\----U~

~..

6I

9

10

First Byzantine Empire
11

5O0

5

~

~0660 6eO ?o0 720

L/,l fl/l
1 2, 6

. ,."
8

,o,

.

.

.

.

.

"

/ lU/ /Byzantine F~)ire
Third Byzantine Empire

Fig.

13

2434


TABLE

15
Ancient chronology lOth-3rd century B.C. The history of Ancient Greece [14, 20] I) The great Greek colonization of 8th6th century B.C. 2) Kingdoms of Judah and Israel, 928-531 B.C. 3) Trojan War is dated by T. Livius at 850-830 B.C. This is the 2nd version of its dating 4) The war with the Tarquinians in Rome 522-509 B.C. The tyranny of Pisistratus (TRN) 56(I-527 B.C. The fall of the Pisistratus tyrapmy 510-514 B.C. The war of Zedekiah (in the Bible) with Pharaoh (TNg). Above TEN -- Franks.

Medieval chronology IOth-16th century A.D. Downward shift by 1810 years I) The Crusades 10th-13th century. Colonization of the Mediterranean 2) Holy Roman Empire of the Gern~ Nation 911-1305 A.D. 3) Two wars in Italy 10th century A.D.: 901-924 and 931-954. Alberic I, ~leodora I, Alberic II, Theodora II 4) The famous war in 13th century Italy: 1250-1268. The fall of the Hohenstaufens, the fall of Troy, Naples. Manfed, Charles of Anjou~ Conradin. The accession of the Anjou line. The fall of the pontiphicate in Rome 5) Avignon captivity 1305-1376, 70 years 6) Wars in medieval Greece 1314-1332, 18 years 7) War in Greece 1374-1387, 13 years The eclipse of Thucydides 8) The rise and fall of Navarre and the Mistra despotate 1400-1450 9) Ottoman Empire 1298-1451. Mohammadans and expansion iO) The falll of Constantinople and Byzantium in 1453. War with Mohanm~edans, then fall of Greece II) Ottoman Empire 15th-16th century.

5) Babylonian captivity 531-461 B.C.; 70 years 6) Greco-Persian wars 492-479 B.C., 13 years 7) Pelophonesian War 431-404 B.C., 27 years 8) The rise and fall of Sparta 400-360 B.C. 9) Macedonian Kingdom 540-359 B.C. 10) From the fall of Byzantium in 364 B.C. and 11)
the siege of Byzantium by Macedonians in 340 B.C. (Philip II) The empire of Alexander the Great 4th-3rd century B.C. Hellenism~ the spread of Greek culture through the Mediterranean. End of clasical Greece. Items Nos, 12-19 contain addenda Cyrus 1 560-530 B.C. (30). In 546 Cyrus captures Lydia. Without vowels LD or TL? Ruled for 14 years

12) Charles of Anjou 1254-1285 (31). In 1268 Charles captures Italy. Without vowels TL-LT 13) Manfred (Kaiser - KZR) 1254-1266. Ruled for 12 years 14) Charles II of Naples 1285-1289 (4). Bis the second 15) Frederick II of Sicily 1302-1337 (appr.) (35). In his time, Ferdinand, Margarita (MR - donna), Mathilda 16) Duke Walter II de Briehne (HRZG for Herzog = Duke) 1337-1356 (19) 17) The restoration of the Parthenon at the end of 14th century A.D. 18) The death of Pleton in 1450 19) Mohanmled II the Conqueror 14511480 A.D. (29)
=

12)

13) Croesus (KRZ) 560-546 B.C.

14) Cambyses (KM-bis, i.e., EM the second)
53o-522 B.C. (8)

15) Darius I Hystaspes 521-486 B.C. (35). 16) 17) 18) 19)

In his time, Artaphernus, Mardonius (MR-domna?, Miltiadus (Mathilda?) Xerxes the Great (XRXS -- Herzog?) 486-464 B.C. (22) The construction of the Parthenon in 447 B.C. The death of Plato in 347 B.C. Philip II the Conqueror 359-336 B.C. (23)

TABLE 16
I) Trojan war 13th cent. B.C. 2) Tarquinian War 6th cent. B.C. in Rome 3) Civil war ist cent. B.C. in Rome 4) Civil war 3rd cent. A.D. in Rome 5) Gothic war 6tn cent. A.D. in Rome 6) Civil war 901924 A.D. in Rome 7) Civil war 931954 A.D. in Rome 8) The beginning of the Roman Empire 10th-13th cent. A.D. 9) War in Italy 13th cent. A.D. Fall of Troy Odysseus (Ulysses) (= Achilles?) Larcius (- Marcius Coriolanus) Sulla and Cicero (NRCC) Aurelian Lucius Narses (Narcius) Alberic I? Marotia? IAgamemnon ~Tarquinius ISuperbus Pompey the Great iDiocletian ithe Great Justinian and Theodora Theophylactus !Theodora I Achilles Valerius Julius Caesar Constantius Chlorus Belisarius Alberic I Alberic II Octavian Augustus Patroclus Junius~ son of Marcus Brutus Marcus Brutus

John !I John X John XI

Hugh
Theodora I! Otto I, Otto II, Otto III, Alberic II Charles of Anjou (NRCC) Innocent IV

Anjou?

Charles of

i

Jotm XXI

2435


Holy Roman Empire of the German Nation
880 ~ 100o "lO~ ~120 11~0 IzqO 13oo 13~0 I~2o q~80 15~0

,

,,

Medieval

............

/ILl[ Jl_/l_

11

O3 l

~ l

~o

Fig. 14 LITERATURE CITED A. T. Fomenko, "A methodology for identification of duplicates and some applications," Dokl. Akad. Nauk SSSR, 258, No. 6, 1326-1330 (1981). 2. A. T. Fomenko, "A new empirical statistical methodology for ordering of texts and its applications to dating problems," Dokl. Akad. Nauk SSSR, 268, No. 6, 1322-1327 (1983). 3. A. T. Fomenko, "Information functions and related statisti--~al relationships," Abstracts of Papers at the Third International Vil'nyus Conference on Probability Theory and Mathematical Statistics [in Russian], Vol. 2, Inst. Mat. Kibern. Akad. Nauk Litov. SSR, Vii'nyus (1981), pp. 211-212. 4. A. T. Fomenko, "Some statistical relationships in the information distribution density " in texts with a scale," Semiot. Inf., No. 15, 99-124 (1980). 5. N. A. Morozov, Christ [in Russian], Vols. I-7, Moscow--Leningrad (1924-1932). 6. A. T. Fomenko, New Experimental Statistical Procedures for Dating of Ancient Events and Their Application to the Global Chronology of the Ancient and Medieval World [in Russian], Preprint, Gos. Komitet Telev. Radioveshch., Moscow (1981). 7. A. T. Fomenko, "The second derivative of the lunar elongation and statistical relationships in the distribution of some astronomical data," in: Operations Research and MIS [in Russian], No. 20, Kiev Univ. (1982), pp. 98-113. 8. A. T. Fomenko, "On the second derivative of the lunar elongation," in: Topics in the Mechanics of Controlled Motion [in Russian], Perm (1980), pp. 161-166. 9. A. T. Fomenko, "On the properties of the second derivative of the lunar elongation and related statistical relationships," Vopr. Vychisl. Prikl. Mat., Tashkent, No. 63, 136150 (1981). 10. A. T. Fomenko, "The jump of the second derivative of the Moon's elongation," Celestial Mech., 29, 33-40 (1981). 11. A. T. Fomenko, "On the subject of mystifications," Tekh. Nauka, No. 11, 26-29 (1982). 12. A. T. Fomenko, "An authors' invariant of Russian literary texts," in: Methods of Quantitative Analysis of Texts of Narrative Sources [in Russian], Inst. Istorii SSSRAN SSSR, Moscow (1983), pp. 86-109. 13. A. T. Fomenko, "On the geometry of distribution of integer points in hyperregions," Proc. of Seminar on Vector and Tensor Analysis [in Russian], No. 21, Mosk. Gos. Univ., Moscow (1983), pp. 106-152. 14. J. Blair, Chronological Tables [Russian translation], Vols. I-2, Moscow (1808-1809). 15. C. Bemont and G. Monod, The History of Europe in the Middle Ages [Russian translation], Petrograd (1915). 16. R. Cagnat, Cour d'epigraphie latine, 4th ed., Paris (1914). 17. F. Gregorovius, The History of the City of Rome in Middle Ages [Russian translation], Vols. I-5, St. Petersburg (1902-1912). 18. Kohlrausch, The History of Germany [Russian translation], Vols. I-2, Moscow (1860). 19. E. Bickerman, The Chronology of the Ancient World [Russian translation], Moscow (1975). 20. F. Grevorovius, The History of the City of Athens in Middle Ages [Russian translation], St. Petersburg (1900). 21. S. G. Lozinskii, Papal History [Russian translation], Moscow (1934). 2436 I.


22.

V. V. Fedorov and A. T. Fomenko, "Statistical evaluation of chronical closeness of historical texts," in: Problemy Ustoichivosti Stokhasticheskikh Modelei, VNIISI, Moscow

(1983),

pp.

101-107.

SEMIINVARIANTS

IN THE THEORY OF CHARACTERIZATION OF

PROBABILITY DISTRIBUTIONS* Huang Heu Ngy

I.

Introduction

In this article we consider semiinvariants, or cumulants -- a well-known concept in probability theory. In fact, not all the results are new. They are presented mainly in order to demonstrate the role of semiinvariants. In what follows, we are dealing with a real random variable (r.v.) ~ whose distribution function (d.f.) is denoted by F(x) = P(~ < x) and the characteristic function (c.fo) by f(t). We assume that moments of all orders exist for ~, i.e.,

~i~l~<

~,

~=1,2

.....

We recall that a semiinvariant of order k (k = I, 2, 3,o..)

is defined as

'%k=-~-T~=o.
Therefore with each r.v. ~ is associated a sequence of semiinvariants 2. Theorems of Characterization of Probability Distributions in (s.s.) (Xz, >(2, X3,...).

1 dklnf(t)

Terms of Semiinvariants We first give the characterization theorems for the best known distributions. The proof of the theorems is based on the uniqueness theorem for analytical functions and is therefore omitted. THEOREM I. The r.v. ~ is degenerate at the point a, i.e., P(~ = ~) = I, if and only if its s.s. has the form (a, 0, 0,...). THEOREM 2. The r.v. ~ is normal with the parameters the form (a, oz, O, 0,...). (a, o2) if and only if its SoSo has

THEOREM 3. The r.v. ~ is Poisson with the parameter X, i.e., P(~ = k) = %ke->~/k!, k >/ 0, if and only if its s.s. has the form (X, X, ~ .... ). THEOREM 4. The r.v. ~ is exponential with the parameter 0 > 0, i.e., P(~ < x) = max (0 r I -- exp (--x/@)), -~~ < x < ~, if and only if its s.s. is (G, e 2, 2@3,...,(k -- 1)!0k,...)o THEOREM 5. The r.v. ~ is binomial with the parameter p, i.e., P(g = 0) = p, P(~ = I) = q, p + q = I, p, q > 0, if and only if its semiinvariants are given by

Zk= X skrq~, k >1 I,
r=l

where

ski= 1,

k=l,2

.....

ski=(-- 1)J-~{(j - !)[s~-,, j-~ Skk= (-- 1)~-~ (k -- 1)i.

r+Ylsk-,, it}, 2Kj,~
*Presented by V. V. Kalashnikov. Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei -- Trudy Seminara, pp. 177-179, 1984.

0090-41.04/86/3502-2437512.50

9 1986 Plenum Publishing Corporation

2437