Peters's tables

Johann Theodor Peters (1869[START_REF] Roegel | A reconstruction of Peters's table of 7-place trigonometrical values for the new division[END_REF] was a German astronomer and computer of mathematical and astronomical tables. In 1910 and 1911, together with Julius Bauschinger, he published the first widely available 8-place table of logarithms [16]. This work was the basis of many later tables, most of which have been reconstructed by us. 1Peters also published a number of purely trigonometrical tables. His first large such table was the 7-place table published in 1918 [START_REF] Theodor | Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Grades[END_REF], and it was followed by 6-place tables published in 1929 [START_REF] Theodor | Sechsstellige Tafel der trigonometrischen Funktionen : enthaltend die Werte der sechs trigonometrischen Funktionen von zehn zu zehn Bogensekunden des in 90 • geteilten Quadranten u. d[END_REF] and 1930 [START_REF] Theodor | Sechsstellige trigonometrische Tafel für neue Teilung[END_REF].

The present table belongs to these purely trigonometrical tables and was published in 1937 [START_REF] Theodor | Sechsstellige Werte der Kreis-und Evolventen-Funktionen von Hundertstel zu Hundertstel des Grades nebst einigen Hilfstafeln für die Zahnradtechnik[END_REF].

Tables of involutes

Peters's table gives 6-place values of the six usual trigonometric functions, at intervals of 0 • .01, together with their differences. In addition, Peters gives values related to the construction of involute curves (Evolventen in German, abreviated ev ). Involute curves are used in various domains, for instance for the shape of teeth in gears.

Peters's table has its origin in the work of Werner F. Vogel (1893[START_REF] Theodor | Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades[END_REF] who introduced a new computation of the involute function in 1936 [103, pp. 163-167]. Vogel approached Peters and they worked out the present table.

Figure 1 shows how such a curve is computed, and how the involute function is defined. Given a circle of center O and radius OA, the involute curve (red) is obtained as the path of point P where GP is a tangent to the circle and the length of arc GA is equal to GP . In other words, P is the path of a point of a string which is unwound from the circle, and constantly kept taut. The length of the arc AB (blue) is the value of the involute function associated with the angle α. The length of the arc depends on the radius of the base circle, which is taken equal to 1.

Taking α = POG as the parameter, we have therefore tan α = GP OG = GP = [AG] = [AB ]+ [BG], where [x] refers to the length of the arc x centered on O. We have [AB ] = θ r , where θ r is the angle θ expressed in radians. Similarly, [AB ] = α r . Consequently,

inv(α) = θ r = [AB] = [AG] -[BG] = tan α -α r (1)
Peters introduced the function arc(α • ) = α r = α • × π 180 and therefore

inv(α) = arc(θ • ) = tan α -arc(α • ) (2)
θ • = inv(α) × 180 π (3)
For instance, for α = 20 • , we obtain arc(α) = 0.349066, inv(α) = 0.014904, and finally θ • = 0.8540.

Peters's table gives the six standard trigonometrical functions, together with the values of inv(α), θ • , and arc(α), on two facing pages, for a range of half a degree, at 0 • .01 intervals, all to six places. Three additional columns give the values of these three functions for 90 -α, so that all the values are given for the entire quadrant.

Peters does not give the sources of his values, but they are presumably based on his other trigonometrical tables.

The main table is supplemented by a number of appendices, in particular tables of polygons and more specialized tables for use in gearing applications, and conversion tables. These tables have not been reproduced here.

As an example of application, we can consider drawing an involute using the parameter α. Since OP = sec α, the rectangular coordinates of the point of the involute determined by α are x P = sec α • cos θ (4) y P = sec α • sin θ

(5)

Peters's table gives the secants as well as θ, and also cos θ and sin θ, so that it is relatively easy to obtain the rectangular coordinates. During WWII, there was a Japanese edition of Peters's table [START_REF] Clare | Japanese World War II editions of mathematical tables[END_REF]. Most of Peters's table was also reprinted by Vogel in English in 1945, but to seven places [102,[START_REF] Noah Lowan | Involutometry and trigonometry[END_REF]. Finally, Peters's table was also reprinted in 1963 (see [START_REF] Strubecker | Review of Peters, Sechstellige Werte der Kreis-und Evolventen-Funktionen[END_REF] for a review).

Figure 1 :

 1 Figure 1: The construction of the involute curve.

Figure 2 :

 2 Figure 2: Excerpt of Peters's table.

Figure 3 :

 3 Figure 3: Excerpt of Peters's table.

Figure 5 :

 5 Figure 5: Excerpt of Peters's table.

For more information on Peters's tables, we refer the reader to our summary[START_REF] Roegel | The genealogy of Johann Theodor Peters's great mathematical tables[END_REF].

Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.


Denis Roegel

The genealogy of Johann Theodor Peters's great mathematical tables

Johann Theodor (Jean) Peters (1869Peters ( -1941) ) is the author of numerous mathematical tables, in particular the well-known 8-place tables of logarithms coauthored with Julius Bauschinger and published in 1910-1911.

In 2016, we published reconstructions of almost all of the large mathematical tables authored by Johann Theodor Peters (1869Peters ( -1941)), after having earlier already reconstructed a table of factors coauthored by Peters [START_REF] Peters | Factor table giving the complete decomposition of all numbers less than 100,000[END_REF]. This represents a total of 22 volumes for which we have each time written an introduction and tried to analyze as best as we could the methods employed by Peters. Working on all these tables, or even on only some of them, may give rise to some confusion and numbness, as one becomes easily drowned by the various tables, sometimes with almost identical titles. And yet, all these tables are actually different.

The purpose of this article is to sort all these tables out and to give a global picture. It does also serve as a general introduction to all the tables which have been reconstructed.

A brief biography

Peters was born Johann Theodor Peters in Köln in 1869 [33,[START_REF] Gigas | Peters und sein Werk. Nachrichten aus dem Reichsvermessungsdienst[END_REF]32,[START_REF] Wilhelm | [END_REF][START_REF] Holland | Biographical notes on Johann Theodor Peters[END_REF], but throughout his life he was mostly known as "Jean Peters." He studied mathematics and astronomy at the University of Bonn. He became interested in calculations at the beginning of the 1890s. In his dissertation, he used eight In 1899, Peters came to the Astronomisches Rechen-Institut (ARI) in Berlin. In 1901 he was named "observator," and in 1910 he obtained the position of professor. 1 He soon started to work on new general tables of logarithms and trigonometry, which would in particular be useful for astronomical applications.

Peters's main work at the Institute was to compute the Berliner Astronomisches Jahrbuch in which he was involved for almost 30 years. He also worked on the Nautisches Jahrbuch and the Prussian Grundkalender .

From 1922 to 1924, Peters was the director of the ARI. He retired in 1934 and devoted himself to the sole computation of tables. In 1938, he was honored by the silver Leibniz Medal from the Prussian Academy of Sciences. He died in 1941.

Peters's tables

Although Peters has been working on mathematical and more specialized astronomical tables, we will concern ourselves only with his largest tables, which are general in nature, but particularly aimed at astronomical and geodetical applications. The range of Peters's tables displays both his interests and desire to provide a number of useful tables for different audiences, but also is a testimony to the changes in technology and usage. Tables of trigonometrical values started to gain importance in the 1910s and at the same time tables of logarithms were less needed, merely because calculating machines could be used for multiplications. Decimalization was another drive and at the end of the 1910s, Peters published tables where the arguments were given at intervals of thousandths of sexagesimal degrees, and not at intervals of seconds or tens of seconds. A few years later, total decimalization was reached with the centesimal division of the quadrant. 2 These changes have not taken over the old sexagesimal division, perhaps because in turn digital computers made it again easier to stick with the old division. This is the background of Peters's work, going from traditional hand-computed tables to more rational tables, and taking advantage of mechanical calculators, but without anticipating the death of table-computing.

Logarithms

Together with Julius Bauschinger, Peters published in 1910 and 1911 a table of the logarithms of numbers from 20000 to 20000 and the logarithms of trigonometrical functions for every second of the quadrant, all to 8 places [15]. For this table, the computations were done to 12 places, and in some cases to 20 places. The basis for this table were Briggs's tables published in 1624 [START_REF] Briggs | Arithmetica logarithmica[END_REF] and 1633 [START_REF] Briggs | Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633[END_REF].

Bauschinger and Peters's table spawned a number of other tables, as summarized in the genealogical chart below (figure 2).

From the 12-place table used in preparing the 8-place table, a 7-place table was derived in 1911 [38] and a 5-place table of the logarithms of the trigonometrical functions followed in 1912 [39], the latter mostly derived from the 8-place table.

Peters then prepared a 12-place manuscript table giving the logarithms of the trigonometrical functions at 0 • .001 intervals. This table was used to construct the 10-place table of the logarithms of trigonometrical functions, together with a volume of auxiliary tables [44,43].

In 1919, Peters also published a fundamental table of 52-place logarithms [START_REF] Dümmler | Zweiundfünfzigstellige Logarithmen[END_REF] 3 containing 52-place values of log a, log(1 + a • 10 -n ) and log(1a • 10 -n ) for 1 ≤ a ≤ 9 and 1 ≤ n ≤ 26. Using a number of multiplications by numbers 1 ± a • 10 -n , a given number can be expressed as a product of such factors, and consequently its logarithm can be found. This is the well-known "radix method" already used by Briggs [START_REF] Briggs | Arithmetica logarithmica[END_REF].

In 1921, Peters derived 6-and 7-place tables of the logarithms of trigonometrical functions [START_REF] Theodor | Sechsstellige Logarithmen der trigonometrischen Funktionen von 0 • bis 90 • für jedes Tausendstel des Grades[END_REF][START_REF] Theodor | Siebenstellige Logarithmen der trigonometrischen Funktionen 0 • bis 90 • für jedes Tausendstel des Grades[END_REF] from the above mentioned 12-place manuscript.

Then in 1922, Peters published a 10-place table of the logarithms of the numbers [START_REF] Theodor | Zehnstellige Logarithmentafel volume 1 : Zehnstellige Logarithmen von 1 bis 100000 nebst einem Anhang mathematischer Tafeln[END_REF], being the first volume of the tables published in 1919. This first volume also contained an extensive appendix with numerous auxiliary tables, written with Johannes Stein.

After a long hiatus devoted to other types of tables, Peters returned to logarithms in 1940, when he published a 2-volume 7-place table of logarithms of numbers, of antilogarithms, of addition and subtraction logarithms and of logarithms of trigonometrical functions for the new division [START_REF] Theodor | Logarithmen der Zahlen, Antilogarithmen, etc., 2: Logarithmen der trigonometrischen Funktionen für jede 10. Sekunde d. Neugrades, etc[END_REF].

Trigonometrical functions

In 1911, Peters computed a fundamental table of 21-place values of sines and cosines at 10 intervals in the semi-quadrant, as well as at 1 intervals between 0 • and 10 [37]. 4 These values can therefore be used to compute 20-place values for other angles in the quadrant. Peters also gave the first, second and third differences of the values for arguments below 10 , and they could be used to perform interpolations. But Peters's first large table of trigonometrical functions for every day purpose was his 7-place table published in 1918 [42]. It gave the values of the trigonometrical functions at intervals of 0 • .001 and was derived from Briggs's table published in 1633 [START_REF] Briggs | Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633[END_REF].

In 1929, Peters published a table of 6-place values of all six trigonometrical functions [START_REF] Theodor | Sechsstellige Tafel der trigonometrischen Funktionen : enthaltend die Werte der sechs trigonometrischen Funktionen von zehn zu zehn Bogensekunden des in 90 • geteilten Quadranten u. d[END_REF], this time based on Andoyer's great tables published in 1915-1918 [3].

A 6-place table published in 1930 and giving the trigonometrical values for the new centesimal division [START_REF] Peters ; Wichmann | Sechsstellige trigonometrische Tafel für neue Teilung[END_REF] was actually derived from an 8place manuscript table, and so may have been the 6-place table published in 1938 [START_REF] Theodor | Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades[END_REF].

In 1937, Peters published a sexagesimal 6-place table of trigonometri-cal and involute functions, with auxiliary tables useful for gear computations [START_REF] Theodor | Sechsstellige Werte der Kreis-und Evolventen-Funktionen von Hundertstel zu Hundertstel des Grades nebst einigen Hilfstafeln für die Zahnradtechnik[END_REF]. The 8-place table published in 1939 with Leslie Comrie and giving the sines, cosines, tangents and cotangents for every sexagesimal second [START_REF] Theodor | Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten[END_REF] was ultimately based on Andoyer's tables [3]. The secants and cosecants had also been computed, but could not be published.

Peters's final publication was a 7-place table for the new centesimal division and published in 1941 [START_REF] Theodor | Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades[END_REF].

Other tables

Apart from tables of logarithms and trigonometrical functions, Peters prepared, and sometimes published, several other general mathematical tables.

Peters's first large table was a multiplication table, published in 1909 [START_REF] Theodor | Neue Rechentafeln für Multiplikation und Division mit allen ein-bis vierstelligen Zahlen[END_REF]. It gave the products of all integers from 0 to 9999 by all integers from 1 to 99, and spanned 500 pages.

In 1922, Peters revised the Hütte tables [START_REF] Peters | Hütte Hilfstafeln : zur I. Verwandlung von echten Brüchen in Dezimalbrüche ; II. Zerlegung der Zahlen bis 10000[END_REF].

Peters also published several more specialized tables for astronomical purposes, in particular in the 1930s [40, [START_REF] Theodor | Tafeln zur Verwandlung von rechtwinkligen Platten-Koordinaten und sphärischen Koordinaten ineinander[END_REF][START_REF] Peters | Tafeln zur Berechnung der Präzessionen zwischen den Äquinoktien 1875.0 und 1950[END_REF][START_REF] Theodor | Präzessionstafeln für das Äquinoktium 1950[END_REF][START_REF] Theodor | Tafeln zur Berechnung der jährlichen Präzession in Rektaszension für das Äquinoktium 1950[END_REF][START_REF] Theodor | Hilfstafeln zur Verwandlung von Tangentialkoordinaten in Rektaszension und Deklination[END_REF].

Around 1929 [START_REF] Holland | Biographical notes on Johann Theodor Peters[END_REF], Peters was working on a table of factors up to 100000. This table was collated with two other tables computed independently by Lodge, Ternouth and Gifford and published in 1935 [START_REF] Peters | Factor table giving the complete decomposition of all numbers less than 100,000[END_REF].

Peters left unpublished manuscripts for a 8-place table of the trigonometrical functions, presumably to the 1000th of the degree of the old division (and used in the table published in 1930 [START_REF] Peters ; Wichmann | Sechsstellige trigonometrische Tafel für neue Teilung[END_REF], although the latter is for the new division), for a 8-place table of reciprocals (computed around 1924 [START_REF] Holland | Biographical notes on Johann Theodor Peters[END_REF]) and for 8-place tables of antilogarithms, and 8-place tables of addition and subtraction logarithms (computed between 1924 and 1926). Whether any of these manuscripts do still exist is unknown.

In his obituary [33], Kopf mentioned that a traverse table ("Strichtafel") was due to be printed, but such a table never seems to have been published. 5

Chronology of Peters's great tables

We give here a chronological list of Peters's main mathematical tables, and table 1 shows a summary organized by the main features of the tables. Table 1: Summary of the features of Peters's great tables. The lines correspond to the types of tables (for instance of logarithms of trigonometrical functions), the columns give the number of places, and at the intersections of the lines and columns we give the ranges or the intervals, and the years the tables were published.

• 1909: Multiplication table, with products of 1 to 99 by 0 to 9999 [START_REF] Theodor | Neue Rechentafeln für Multiplikation und Division mit allen ein-bis vierstelligen Zahlen[END_REF];

• • 1918: Trigonometrical functions to seven places, at intervals of 0.001 • [42];

• 1919: Logarithms of trigonometrical functions to ten places, at intervals of 0.001 • [44];

• 1922: Logarithms of numbers 10000 to 99999 to ten places [START_REF] Theodor | Zehnstellige Logarithmentafel volume 1 : Zehnstellige Logarithmen von 1 bis 100000 nebst einem Anhang mathematischer Tafeln[END_REF];

• 1921: Logarithms of trigonometrical functions to six places, at intervals of 0.001 • [START_REF] Theodor | Sechsstellige Logarithmen der trigonometrischen Funktionen von 0 • bis 90 • für jedes Tausendstel des Grades[END_REF];

• 1921: Logarithms of trigonometrical functions to seven places, at intervals of 0.001 • [START_REF] Theodor | Siebenstellige Logarithmen der trigonometrischen Funktionen 0 • bis 90 • für jedes Tausendstel des Grades[END_REF];

• 1929: Trigonometrical functions to six places, at intervals of 10 [48];

• 1930: Trigonometrical functions to six places, at intervals of 0.01 g (new division) [START_REF] Peters ; Wichmann | Sechsstellige trigonometrische Tafel für neue Teilung[END_REF];

• 1937: Trigonometrical functions and involute functions to six places, at intervals of 0.01 • [55];

• 1938: Trigonometrical functions to six places, at intervals of 0.001 g (new division) [START_REF] Theodor | Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades[END_REF];

• 1939: Trigonometrical functions to eight places, at 1 intervals [START_REF] Theodor | Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten[END_REF];

• 1940: Logarithms of numbers (10000 to 100000), antilogarithms, addition and subtraction logarithms, all to seven places [START_REF] Theodor | Logarithmen der Zahlen, Antilogarithmen, etc., 2: Logarithmen der trigonometrischen Funktionen für jede 10. Sekunde d. Neugrades, etc[END_REF] • 1940: Logarithms of trigonometrical functions to seven places, at intervals of 0.001 g (new division) [START_REF] Theodor | Logarithmen der Zahlen, Antilogarithmen, etc., 2: Logarithmen der trigonometrischen Funktionen für jede 10. Sekunde d. Neugrades, etc[END_REF];

• 1941: Trigonometrical functions to seven places, at intervals of 0.001 g (new division) [START_REF] Theodor | Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades[END_REF].

Conclusion

It is very easy to be impressed by the considerable volume of tables produced by Peters, and it is therefore not surprising that Archibald called Peters "perhaps the greatest mathematical table maker of all time" [12, p. 68].

Having worked through most of Peters's tables as well as many other great tables, we think that Archibald's statement is somewhat exaggerated. Peters has indeed produced a large number of tables, and these tables were all remarkable for their quality. But Peters's main tables do not make use of very advanced concepts, and they lack the groundbreaking features found in Napier's [START_REF] Roegel | Napier's ideal construction of the logarithms[END_REF], Briggs's [START_REF] Briggs | Arithmetica logarithmica[END_REF][START_REF] Briggs | Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633[END_REF], Prony's [START_REF] Roegel | The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation[END_REF], Sang's [START_REF] Alex | The logarithmic tables of Edward Sang and his daughters[END_REF] and some other tables. Instead, the greatness of Peters lies in his ability to organize computations, to check them, to use simple interpolations with first and second differences, in particular with calculating machines, to anticipate which tables would be needed, and to have been able to produce remarkable tables using earlier large tables. None of the large tables constructed by Peters were actually Peters's work first highlights the greatness of Briggs's work, as well as the importance and usefulness of such fundamental tables as constructed by Andoyer, and second stresses the importance of an efficient management in the construction of tables, and that great tables can be produced without computing everything from scratch. dropped in the last editions. Reconstructions are given in [START_REF] Roegel | A reconstruction of Bauschinger and Peters's eight-place table of logarithms[END_REF] and [START_REF] Roegel | A reconstruction of Bauschinger and Peters's eight-place table of logarithms[END_REF].]. numbers of from one to four places." We have only had the 1919 edition in hands, and we reconstructed it in [START_REF] Roegel | A reconstruction of Peters's table of products[END_REF].].

[37] Johann Theodor Peters. Einundzwanzigstellige Werte der Funktionen Sinus und Cosinus : zur genauen Berechnung von zwanzigstelligen Werten sämtlicher trigonometrischen Funktionen eines beliebigen Arguments sowie ihrer Logarithmen. Berlin: Verlag der Königl. Akademie der Wissenschaften, 1911. [54 pages, Appendix 1 to the "Abhandlungen der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse.", not seen, but reprinted at the end of the English edition of [START_REF] Theodor | Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten[END_REF]].

[38] Johann Theodor Peters. Siebenstellige Logarithmentafel der trigonometrischen Funktionen für jede Bogensekunde des Quadranten. Leipzig: Wilhelm Engelmann, 1911. [reconstructed in [START_REF] Roegel | A reconstruction of Peters's seven-place table of logarithms of trigonometric functions[END_REF]].

[39] Johann Theodor Peters. Fünfstellige Logarithmentafel der trigonometrischen Funktionen für jede Zeitsekunde des Quadranten. Berlin: Reimer, 1912. [reconstructed in [START_REF] Roegel | A reconstruction of Peters's five-place table of logarithms of trigonometric functions[END_REF]].

[40] Johann Theodor Peters. Tafeln zur Berechnung der Mittelpunktsgleichung und des Radiusvektors in elliptischen Bahnen für Excentrizitätswinkel von 0 • bis 24 [41] Johann Theodor Peters. Dreistellige Tafeln für logarithmisches und numerisches Rechnen. Berlin: P. Stankiewicz, 1913. [not seen, second edition in 1948 (seen), reconstructed in [START_REF] Roegel | A reconstruction of Peters's 3-place tables[END_REF]].

[42] Johann Theodor Peters. Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Grades. Berlin-Friedenau: Verlag der Optischen Anst. Goerz, 1918. [Reprinted in 1938and 1941, as well as in 1942 in English with the title "Seven-place Values of trigonometric functions for every thousandth of a degree.", all four editions seen. Reconstructed in [START_REF] Roegel | A reconstruction of Peters's seven-place table of trigonometric functions[END_REF].].

[43] Johann Theodor Peters. Zehnstellige Logarithmentafel : Hilfstafeln zur zehnstelligen Logarithmentafel. Berlin: Preuß. Landesaufnahme, 1919.

[not seen, second edition in 1957 (seen), reconstructed in [START_REF] Roegel | A reconstruction of Peters's auxiliary tables to his ten-place logarithms[END_REF]].

[44] Johann Theodor Peters. Zehnstellige Logarithmentafel, volume 2 : Zehnstellige Logarithmen der trigonometrischen Funktionen von 0 • bis 90 • für jedes Tausendstel des Grades. Berlin: Reichsamt f.

Figure 1 :

 1 Figure 1: From left to right, the astronomer Karl Heinrich Willy Kruse (1889-1945), Leslie John Comrie (1893-1950) and Peters, 1930. (from [12])

  places

  1910: Logarithms of the numbers 20000 to 200000 to eight places (with Julius Bauschinger) [15]; • 1911: Logarithms of trigonometrical functions to eight places and at intervals of 1 (with Julius Bauschinger) [15]; • 1911: Logarithms of trigonometrical functions to seven places at intervals of 1 [38]; • 1912: Logarithms of trigonometrical functions to five places and at intervals of 15 seconds of the arc [39]; (log sin and log tan every 1.5 below 2 • ) • 1913: Trigonometrical functions and of logarithms of trigonometrical functions, all to three places, and at intervals of 6 [41];

  The genealogy of Peters's major numerical tables. Dashed circles indicate manuscript tables. The vertical line separates the tables of logarithms (left) from the (non logarithmic) tables of trigonometrical functions (right). produced de novo, but they were based almost entirely on those of Briggs and Andoyer.

												?	Peters	1938 (6 p.)
												Peters	(8 p.)
							Andoyer	1915-1918		?		Peters	1930 (6 p.)	Peters	(12 p.)	?	Peters	1941 (7 p.)
									Peters	1918 (7 p.)		Peters	1929 (6 p.)	Peters	1939 (8 p.)
	Briggs 1633 Briggs 1633	Bauschinger & Peters	(12 p.)	Bauschinger & Peters	1910-1911 (8 p.)	Peters	1912 (5 p.)	Peters	(12 p.)		Peters Peters	1921 (6 p.) 1921 (7 p.)
	Briggs 1624									Peters	1919 (10 p.)	Peters	1940/2 (7 p.)
					Peters	1911 (7 p.)						Peters	1922 (10 p.)	Peters	1940/1 (7 p.)
	Prytz	1886										Andoyer	1922	Figure 2:

For more particulars on Peters's activities at the ARI, see Holland's summary[START_REF] Holland | Biographical notes on Johann Theodor Peters[END_REF]. Holland has gone through the ARI annual research accounts in search of mentions of Peters. The ARI also holds a file on Peters with a few items, see Wielen's summary of its contents[START_REF] Wielen | Die Reglements und Statuten des Astronomischen Rechen-Instituts und zugehörige Schriftstücke im Archiv des Instituts[END_REF].

In 1937, decimalization was made mandatory in Germany for surveying, see[26, p.

349].

Thistablehas not yet been reconstructed, but might be reconstructed in the future.

This table has also not yet been reconstructed, but might be reconstructed in the future.

An example of an early table of that kind can be found in the Handbuch der Schiffahrtskunde published in 1819[START_REF]Hamburgische Gesellschaft zur Verbreitung der Mathematischen Wissenschaften. Handbuch der Schiffahrtskunde[END_REF]. For another example, see Shortrede's table published in 1864[START_REF] Shortrede | Traverse tables to five places for every 2 of angle up to 100 of distance[END_REF].


T M Michelitsch

B A Collet

A P Riascos

A F Nowakowski

F C G A Nicolleau

Fractional random walk lattice dynamics

We analyze time-discrete and continuous 'fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in n = 1, 2, 3, .. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices L α 2 where α = 2 recovers the normal walk. First we demonstrate that the interval 0 < α ≤ 2 is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain the fundamental matrix Z (α) , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix Z (α) relates fractional random walks with normal random walks. We show that the fractional transition matrix elements exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic t -n α -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

Introduction

The study of dynamical processes on networks has become a huge interdisciplinary research area during the last decade [START_REF] Newmann | Complex Systems: A Survey[END_REF]. The characteristic features of many natural systems such as biological, social and computer systems (the world wide web) [START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF], and last but not least the molecular structure of materials can be described as complex networks [START_REF] Noh | Random Walks on Complex Networks[END_REF]. There is a huge amount of works addressing the random motions on networks and many models have been developed to capture the structural features of random motions on networks [START_REF] Dorogovtsev | Critical phenomena in complex networks[END_REF]37,[START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF]. In order to analyze dynamical processes on networks, it has been demonstrated that random walks are a highly useful concept to describe problems such as of exploration, search, navigation and propagation of information on networks [START_REF] Bénichou | Intermittent search strategies[END_REF]. Whereas normal random walks with emerging Brownian motions have been widely used to describe the dynamics on networks, e.g. [START_REF] Noh | Random Walks on Complex Networks[END_REF][START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF][START_REF] Doyle | Random Walks and Electric Networks[END_REF], and many others, it has turned out that many processes associated with long-range jumps on the network, cannot be properly described by normal random walks allowing in a time-step only moves to connected nodes. Many complex phenomena such as anomalous transport phenomena with long-range jumps in diffusion processes indeed cannot be described by Brownian motions. However, their power-law long-range jump characteristics can be described by Lévy motions [START_REF] Metzler | Lévy statistics and anomalous statistics and anomalous transport: transport: Lévy flights and flights and subdiffusion[END_REF][START_REF] Chechkin | Introduction to the Theory of Lévy Flights[END_REF] in the framework of Lévy statistics where such stochastic processes include long-range moves and are in many cases described by continuous space fractional operators [START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF][START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF]. Further diffusion processes have been analyzed governed by a time-fractional Fokker Planck equation [START_REF] Barkai | Fractional Fokker-Planck equation, solution, and application[END_REF].

The fractional calculus in the continuous space has become a standard tool to describe a large ensemble of interdisciplinary problems of complex behavior involving effects due to spatial long-range interactions. A prominent example is for instance the development of fractional quantum mechanics by Laskin [START_REF] Laskin | Fractional Schrödinger equation[END_REF]. Depending on the functional spaces where these fractional operators are defined we find various definitions of fractional operators in the continuous space in the literature [START_REF] Hilfer | Threefold Introduction to Fractional Derivatives[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF][START_REF] Samko | Fractional Integrals and Derivatives: Theory and Applications[END_REF][START_REF] Samko | Fractional Weyl-Riesz Integrodifferentiation of Periodic Functions of Two Variables via the Periodization of the Riesz Kernel[END_REF][START_REF] Podlubny | Fractional Differential Equations[END_REF]. Whereas these fractional operators are obtained as convolutional integrals in a continuous space, the fractional calculus on discrete networks and lattices is less developed. An approach to define fractional differential operators on lattices is suggested by Tarasov [START_REF] Tarasov | Lattice fractional calculus[END_REF]. Beside the applications on diffusion problems on the lattice, the importance of fractional lattice models appears also for a description of fractional lattice vibrational phenomena, a generalization of crystal lattice dynamics. Some initial steps towards such a fractional generalization generalization of nonlinear classical lattice dynamics has been introduced by Laskin and Zaslavsky [START_REF] Laskin | Nonlinear fractional dynamics on a lattice with long-range interactions[END_REF]. In a lattice dynamics model which defines by Hamilton's variational principle the 'Laplacian matrix' which contains all constitutive information of the harmonic interparticle interactions, it is therefore desirable to develop a 'fractional generalization' of the existing lattice dynamics approach [START_REF] Michelitsch | Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels[END_REF][START_REF] Michelitsch | Nonlocal constitutive laws generated by matrix functions: Lattice dynamics models and their continuum limits[END_REF].

In the context of Markovian processes on networks, the concept of 'fractional random walk on undirected networks' generalizing the 'normal random walk' was recently introduced by Riascos and Mateos [START_REF] Riascos | Long-range navigation on complex networks using Lévy random walks[END_REF][START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF][START_REF] Riascos | Fractional quantum mechanics on networks: Long-range dynamics and quantum transport[END_REF]. In contrast to the normal random walk where the walker can reach in one time-step only connected nodes, the fractional random walker is allowed to reach in one time-step any node in a connected network introducing a small world property to the network [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF]. It was found for different type of complex networks numerically, and for the cyclic ring explicitly, that the efficiency to navigate on networks is increased when the fractional random walk is chosen instead of a normal walk. Moreover, the power law decay of the fractional Laplacian matrix on sufficiently large 1D cyclic rings (periodic one-dimensional lattices) indicates in the context of fractional diffusion the emergence of Lévy flights, [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF] and see also the analysis in [START_REF] Michelitsch | Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels[END_REF].

Inspired by the above findings where the fractional generalization of a network Laplacian matrix introduces a small world property to large world networks, the fractional random walk dynamics opens a huge new interdisciplinary field which merits further studies. It is now highly desirable to better understand the mechanism of 'fast fractional navigation' in a network. There is a huge potential of applications in interdisciplinary research areas as various as search strategies on computer networks such as the world wide web, the propagation of information in societies, the spread of pandemic diseases, of cancer cells in the body, etc.

The goal of the present paper is to derive a theoretical framework to describe fractional random walk dynamics on undirected regular networks with main focus on n-dimensional cubic periodic and infinite lattices. The advantage of these regular network systems is that they accessible to an explicit spectral analysis as the eigenspace is terms of Bloch-vectors is explicitly available. In the present paper the fractional random walk versus normal random walk is analyzed by means of a fractional generalization of master equation. For the sake of simplicity and analytical accessibility, we confine us in the present paper on rather simple species of regular undirected networks with constant degree for all nodes with special focus on n-dimensional periodic cubic lattices. In the n-dimensional cubic lattice we assume only next neighbor node connections, i.e. in any physical dimension j = 1, .., n each node has two connected neighbor nodes. Thus the degree of all nodes is identically equal to 2n. Further we assume that the cubic lattice is periodic in any spatial dimension j = 1, .., n where the spatial dimension n = 1, 2, 3, 4, .. ∈ N. Despite of these restrictions, many derivations of this paper can be easily generalized to more complex undirected networks.

The present paper is organized as follows. In subsequent section 2 we define a 'good', physically admissible fractional Laplacian matrix for undirected regular and n-dimensional cubic networks. The fractional Laplacian matrix is generated by a power law matrix function of a 'simple' Laplacian matrix involving only next neighbor node connections. In section 3 we evoke briefly some general basic features of time-discrete and time-continuous random walks on regular undirected networks in the framework of Markovian process which we will need for the analysis of fractional random walks.

To define the 'fractional random walk' a 'fractional master equation' involving a fractional Laplacian matrix L α 2 as generator matrix for the time evolution is introduced (section 4). The fractional random walk contains for α = 2 the normal random walk. We show explicitly (see appendix) that fractional random walk approach is admissible in the interval 0 < α ≤ 2. We derive expressions for the transition matrix and average return probabilities. We show that the fractional transition matrix elements exhibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times t by slowly relaxing long-wave modes modes leading to a characteristic t -n α -decay (sections 4.1 and 4.2). The asymptotic expressions further recover known results for normal and fractional random walk on the 1D cyclic ring [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF].

In order to obtain further characteristic information on the fractional random walk we introduce a Green's matrix in section 4.3 which yields a convenient representation of the fundamental matrix Z (α) , containing the fractional mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix Z (α) relates fractional random walks with normal random walks (section 4.3). The analysis of the mean relaxation time (Kemeny constant) demonstrates for the n-dimensional cubic lattice the increased efficiency of a fractional random walk strategy. This analytical result supports the numerical findings of recent works [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF].

Fractional Laplacian matrix on regular undirected networks and cubic lattices

For general derivations performed in this paper, we consider undirected regular networks of N nodes. Let us denote with p = 0, .., N -1 the nodes of this network. We assume for simplicity that all nodes of the network have constant degree K p = K. In order to obtain explicit expressions we specify the network to cubic lattices with only next neighbor node connections. We consider the cubic lattice in n = 1, 2, 3, .., dimensions with periodic boundary conditions in any dimension j = 1, .., n. Topologically such a lattice can be conceived as n-dimensional hypertorus ('n-torus'), for instance in 1D this is a cyclic ring, in 2D a conventional torus, and so forth. We assume the lattice contains N = N 1 × .. × N n lattice points where each lattice point can be identified with a node of the network. The lattice is assumed to be N j -periodic in any spatial dimension j = 1, .., n. For the cubic lattice the nodes are denoted by the vector p = (p 1 , p 2 , .., p n ) (p j = 0, ..N j -1). N × N -matrices M defined on the cubic lattice are denoted by M p q =: M p 1 ,..,pn|q 1 ,..,qn . In order to define the lattice (fractional) Laplacian matrices, it is sufficient to confine ourselves to scalar fields u p (one field degree of freedom) associated to each node of first of all unspecified physical nature. Before we define fractional Laplacian matrices, let us first define a 'good' i.e. physically admissible Laplacian matrix on a undirected network. To this end we introduce a potential energy in the form of a positive (semi-) definite quadratic form

V = 1 2 N -1 p=0 N -1 q=0 u * q L pq u p ≥ 0 (1)
where on an undirected network the N × N -Laplacian matrix L pq = L qp is symmetric. Assuming identical degree K = K p for all nodes, it can be represented as

L pq = Kδ pq -A pq (2)
where A pq denotes the adjacency matrix with A pq = 1 (A pq = 0) if node p is (not) connected with node q and the degree K p = K = N -1 q=0 A pq counts the number of connections of a node p. We assume further that A pp = 0 ∀p = 0, .., N -1. In the approach to be developed it is sufficient to consider a 'large world network' Laplacian matrix which we define as K N << 1, i.e. the degree (number of connections of a node) is much less than the total number of N nodes. It follows that translational symmetry N -1 q=0 L pq = 0 is fulfilled, which means that the constant vector with the components u p = 1 is eigenvector of the Laplacian matrix with the only vanishing eigenvalue λ 0 = 0 [which we denote throughout this paper by λ 0 ]. Further, the requirement of elastic stability of (1) requires that N -1 eigenvalues of the Laplacian matrix λ > 0 = 1, ..N -1 are positive. The sign convention of the Laplacian matrix is chosen here such that the Laplacian matrix is the matrix-analogue to -d 2 dx 2 , namely as a positive semi-definite operator. For a discussion we refer to [START_REF] Michelitsch | Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit[END_REF][START_REF] Michelitsch | Nonlocal constitutive laws generated by matrix functions: Lattice dynamics models and their continuum limits[END_REF] 1 The Laplacian matrix of a cubic lattice with only next neighbor node connections is defined as the form

L = 2n 1 -A n (3)
where 1 denotes the N × N unity matrix. The components of the Laplacian matrix can be represented as

L p q = 2nδ p q -A p q . (4)
This representation needs to be read by introducing the notations δ p q = n j=1 δ p j q j for the components of the unity matrix on the cubic lattice with p = (p 1 , .., p n ), q = (q 1 , .., q n ) where δ ij denotes the Kronecker symbol. The adjacency matrix for the cubic lattice with only next neighbor node connections can be represented by

A p q = n j=1 [D j + D † j ] p q , (5)
where D j and D † j = D -1 j denote the (on the periodic and infinite lattice unitary) next neighbor shift operators in the j = 1, .., n-directions defined by D j u p 1 ,..p j ,..,pn = u p 1 ,..p j +1,..,pn and its adjoint (inverse) shift D † j u p 1 ,..p j ,..,pn = u p 1 ,..p j -1,..,pn . On the periodic and infinite lattices the shift operators are unitary [START_REF] Michelitsch | Nonlocal constitutive laws generated by matrix functions: Lattice dynamics models and their continuum limits[END_REF]. D j shifts the field associated to lattice point p = (.., p j , ..) to the field associated with the adjacent lattice point in the positive j-direction (.., p j+1 , ..). The inverse (adjoint) shift operator D † j = D -1 j shifts to the adjacent 1 Contrarily to the convention in these works where the Laplacian matrix is defined negative semi-definite matrix-analogue to the Laplacian operator d 2 dx 2 .

lattice point in the negative j-direction (.., p j-1 , ..). For the n-dimensional cubic lattice the components of the shift operators can be represented by

[D j ] p q = δ p j+1 q j n s =j δ psqs , [D † j ] p q = δ p j-1 q j n s =j δ psqs (6)
and with [START_REF] Metzler | Lévy statistics and anomalous statistics and anomalous transport: transport: Lévy flights and flights and subdiffusion[END_REF], the components of the Laplacian (4) of the cubic lattice with next neighbor node connections are L p 1 ,..pn|q 1 ,..,qn = 2n n j=1 δ p j q j -n j=1 δ p j+1 q j + δ p j-1 q j n s =j δ psqs .

The Laplacian matrix (2) has a spectral representation of the form

L = N -1 =0 λ | | (8)
where we employed Dirac's bra-ket notation for the representation of the eigenvectors. The eigenvector |0 refers to the vanishing eigenvalue λ 0 = 0 and has constant components [|0 ] p = 1 √ N ∀p reflecting translational symmetry. Due to the symmetry (self-adjointness) of the Laplacian matrix L pq = L qp , we have always the property of ortho-normality of the eigenvectors 2 . With these preliminary comments we can define the fractional Laplacian matrix by its spectral representation as a power matrix function

L α 2 = N -1 =0 λ α 2 | |, α > 0 (9)
having the same good properties as the Laplacian ( 8). So we introduce the 'fractional potential' as

V α = 1 2 N -1 p=0 N -1 q=0 u * q [L α 2 ] pq u p ≥ 0 (10)
which is as well a positive semi-definite quadratic form. For the periodic cubic lattice the field u p fulfills the periodicity conditions u p 1 ,..p j +N j ,..pn = u p 1 ,..p j ,..pn , j = 1, .., n

which are also fulfilled by the shift operator matrices [START_REF] Metzler | Lévy statistics and anomalous statistics and anomalous transport: transport: Lévy flights and flights and subdiffusion[END_REF]. This allows cyclic index convention 0 ≤ p j ≤ N -1 ∀j = 1, .., n [START_REF] Michelitsch | Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit[END_REF]. As a consequence the spectral representation of the fractional Laplacian matrix L α 2

on the cubic lattice can be expressed by Bloch vectors as

[L α 2 ] p q = [L α 2 ] p-q = 1 N e i κ •( p-q) λ α 2 , λ =   2n -2 n j=1 cos (κ j )   , α > 0 ( 12
)
where we have introduced the more compact notation (..) = N 1 -1 1 =0 .. Nn-1 n=0 (..) and where the sum is performed over the entire set of N = n j=1 N j Bloch-vectors κ = (κ 1 , .., κ n ) having the components κ j = 2π N j j (and j = 0, .., N j -1, with j = 1, .., n). We see again that only the eigenvalue which corresponds to the zero wave vector with = (0, .., 0) is vanishing whereas all other N -1 eigenvalues λ α 2 are positive. For α = 2 (12) recovers the spectral representation of the Laplacian matrix with next neighbor node connections (7). 3 Some general remarks on random walks on regular undirected networks

Let us define the notion of a random walk on the network in the framework of Marvovian process. Let P q (t) be the probability that the random walker occupies node q = 0, .., N -1 at a time t. The normalization condition N -1 q=0 P q (t) = 1 indicates that the random walker is at time t for sure somewhere on the network. As we deal with probabilities we have 0 ≤ P q (t) ≤ 1 and the normalization condition is fulfilled at any time 0 ≤ t < ∞. It is further convenient to introduce the probability vector P (t) = [P 0 (t), ..P q (t), ..P N -1 (t)] having the occupation probabilities P q (t) as components.

The random walk is now defined by the time evolution of the occupation probabilities P q (t) of the nodes q. We assume the random walker is allowed to undertake one discrete move from one node to another connected node during a given time increment δt (time-discrete random walk). We consider each move of the random walk to be a Markovian process which means, when we decompose the time evolution of the occupation probabilities into discrete steps δt, that the probabilities at a time t + δt depend only on the probabilities at t. Assuming a linear dependence, this time evolution can be expressed by a discrete master equation, namely [where we employ alternatively both, index and matrix notations]

P p (t + δt) = N -1 q=0 W pq (δt)P q (t), P (t + δt) = W (δt) • P (t) (13)
where the N × N matrix W (δt) is referred to as transition matrix connecting the probabilities P (t + δt) with P (t). Since we are considering Markovian processes, transition matrix W (δt) in ( 13) depends only on the time increment δt but not on (previous times) t. The matrix elements W pq (δt) indicate the conditional probability that the random walker that occupies node q at time t has moved to node p when the time has increased by an increment t + δt. It follows from the normalization of the P q (t) that W pq (δt) has to fulfill the normalization condition N -1 p=0 W pq (δt) = 1, and since we deal with probabilities all matrix elements are within 0 ≤ W pq (δt) ≤ 1. Further we see that the transition matrix after m time-steps is obtained by

W (t = mδt) = W m (δt), m ∈ N W (0) = W 0 (δt) = 1. (14)
The W (t = mδt) = W m (δt) denotes the transition matrix connecting the probability vector at time t = mδt (after m time steps) P (t = mδt) = W (t = mδt) • P (0) with an initial probability vector P (0). We emphasize that normalization and positiveness of the matrix elements of the transition matrix W (t = mδt) ∀m ∈ N remain conserved for all times t = mδt

N -1 p=0 W pq (t) = 1, 0 ≤ W pq (t) ≤ 1 . ( 15
)
Note that the normalization condition is equivalent to the fact that the stationary (equal-) distribution P p = 1 N ∀p is an eigenvector referring to the (largest) eigenvalue 1 of transition matrix W (t). It is further clear that in order to maintain [START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF] for all times for m → ∞ time increments, that all other N -1 eigenvalues of the transition matrix must be positive and inferior to 1 with exponentially evanescent contributions for increasing number m of time steps. It follows that lim t→∞ P q

(t) = 1 N with lim t→∞ W pq (t) = W pq (∞) = 1 N
∀p, q = 0, .., N -1, i.e. all matrix elements of the stationary transition matrix take the value 1 N reflecting the Markov chain convergence theorem [START_REF] Norris | Markov chains[END_REF]. Let is now specify the transition matrix W (δt) and link it with the properties of the network. We can do so by assuming W (δt) to be generated for a small (in the limiting case infinitesimal) time-step δt as

W pq (δt) = δ pq -δtL pq . (16)
The infinitesimal time-step relation [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF] accounts for the fact that for (infinitesimally) small time-steps δt the occupation probabilities change only infinitesimally by δ P (δt) = -δtL • P (t) being of order δt. The Laplacian matrix L contains the information on the topological structure of the network, and acts as the generator matrix of the continuous-time random walk when δt → 0. For undirected networks the Laplacian matrix is symmetric L pq = L qp , a transition matrix defined by the generating transformation ( 16) maintains symmetry at all times W pq (t) = W qp (t). In view of the structure of the Laplacian matrix (2) we observe that W (δt) = δ pq (1 -δtK) + δtA pq . The adjacency matrix A pq determines the probability transition received by node p from a node q in a time-step δt, and δtL pp = Kδt is the amount of probability emitted by a node p to all connected nodes during one-time step δt. We see that only connected nodes p, q (with A pq = 1) can exchange information during one time-step δt. We will see subsequently how this limitation of large world networks K << N changes when instead a fractional power L α 2 of the large world network Laplacian matrix L is chosen. From ( 16) follows the interpretation of the Laplacian matrix elements, namely L pq indicates the probability rate that a node p receives information from node q during a tile-step δt and due to L pq = L qp in our model this holds vice versa. It is important to chose δt small enough [START_REF]additional information and can be achieved by simple rescaling of the time step δt → Kδt[END_REF] that (i) W pp (δt) = 1 -δtL pp ≥ 0 where L pp = K meets the requirements to be positive. A further requirement is that all non-vanishing non-diagonal elements of the Laplacian matrix (2) are negative (ii) L pq = -A pq < 0, p = q so that the elements of the transition matrix are non-negative 0 ≤ W pq ≤ 1∀p = q having interpretation as probabilities. It follows then from (i) with L pp =p =q L pq = p =q A pq that the non-diagonal elements additionally fulfill W pq = -δtL pq < 1 (p = q). We see that the Laplacian of type ( 2) is meeting this requirement. We come back to this issue in the context of the fractional random walk, as it defines a restriction of admissible range for the exponent α.

For networks where the random walk does not allow the walker to remain at a node p as time increases by an increment δt = 1, the Laplacian in [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF] has to be renormalized as L pq = 1

Lpp L pq where the diagonal element is the degree of the network L pp = K p [For networks with variable degrees K p this leads to non-symmetric transition matrices which is not considered in the present paper]. (For further discussions see e.g. [START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF][START_REF] Noh | Random Walks on Complex Networks[END_REF] and others). As we confine ourselves in the present paper to regular networks K p = K, this re-normalization of the Laplacian matrix does not change the dynamics as it only rescales the time (δt = 1 K in ( 16)). The time evolution due to ( 13) is then determined by P (t = mδt) = W m (δt) P (t = 0) =: W (t) P (0). The matrix element [W m (δt)] pq = [W (t = mδt)] pq indicates the probability that the walker starting at node q reaches after m steps the node p. Let now δt → 0 where t = mδt is kept finite then we get δt

lim m→∞ W m = ( 1 -t m L) m = e -Lt = W (t).
The transition matrix W (t) describes hence the time evolution for the time-continuous random walk

P p (t) = N -1 q=0 W pq (t)P q (t = 0), W pq (t) = [e -Lt ] pq . (17)
The matrix element W pq is the probability of the walker who occupied at t = 0 node q to reach at time t node p. As outlined above the transition matrix of ( 17) is symmetric W pq (t) = W qp (t) for all times 0 ≤ t < ∞. The transition matrix of the time continuous random walk fulfills hence the evolution equation

dW dt = -L • W (t), dP p dt = - N -1 q=0 L pq P q . ( 18
)
Employing Dirac's bra-ket notation the transition matrix has the spectral representation

W (t) = e -Lt = |0 0| + N -1 =1 e -λ t | | (19)
where the initial condition

W (t = 0) = 1 = N -1 l=0 | | and the stationary distribution (statistic equilibrium) W (∞) = |0 0| with W pq (∞) = 1
N ∀, p, q is rapidly taken. We refer as 'normal random walks' to the random walks described by a Laplacian of the form (3), in order to distinguish them from 'fractional random walks' which is the subject of the subsequent analysis.

Fractional random walks on regular undirected networks and cubic lattices

We define the random walk in the same way as in the previous section by performing the transition from timediscrete to time continuous walk. The only modification is that a fractional power of the network Laplacian occurs as generator matrix for the dynamics. Denoting W (α) (t) the corresponding 'fractional' transition matrix, the time evolution for the fractional continuous random walk is defined by

d dt W (α) (t) = -L α 2 • W (α) (t), 0 < α ≤ 2 (20)
where L indicates a Laplacian matrix of type [START_REF] Barkai | Fractional Fokker-Planck equation, solution, and application[END_REF]. Now we come back to the crucial conditions (i) and (ii) raised in the previous section on the sign of the fractional Laplacian matrix. It is shown in the appendix that the fractional Laplacian matrix in the interval 0 < α ≤ 2 has negative non-diagonal elements leading to non-negative non-diagonal elements of the fractional transition matrix. Hence 0 < α ≤ 2 is the admissible choice for fractional Laplacian matrix L α 2 to describe random walk dynamics [START_REF] Michelitsch | Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels[END_REF]. This statement on the sign of fractional Laplacian matrix elements is in agreement with previous explicit results for the periodic chain [START_REF] Michelitsch | Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit[END_REF]. So we restrict ourselves on the admissible interval 0 < α ≤ 2 for fractional random walks throughout our analysis. The fractional transition matrix defined by ( 20) is then again obtained as an exponential having the spectral representation

W (α) (t) = e -L α 2 t = |0 0| + N -1 l=1 e -λ α 2 t | |, 0 < α ≤ 2 (21)
where for α = 2 the normal random walk transition matrix ( 19) is recovered. The transition matrix [START_REF] Michelitsch | Nonlocal constitutive laws generated by matrix functions: Lattice dynamics models and their continuum limits[END_REF] has the same stationary transition matrix independent of α, namely the equal distribution

W (α) (∞) = |0 0| with W (α)
pq (∞) = 1 N ∀p, q. Now let us focus on cubic lattices. The fractional transition matrix ( 21) can be written as

W (α) p-q (t) = 1 N + 1 N = 0 e i κ •( p-q) e -λ α 2 t . (22)
For large cubic lattices where all N j → ∞∀j = 1, .., n the fractional transition matrix takes the representation 3

W (α) p-q (t) = 1 (2π) n π -π dϕ 1 .. π -π dϕ n e i n s=1 ϕs(ps-qs) e -t 2 n j=1 (1-cos ϕ j ) α 2 (23)
where the initial condition W (α) (t = 0) = 1 is fulfilled by [START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF], namely W

p-q (t = 0) = δ p q = n j=1 δ p j q j . In [START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF] we introduced ϕ = (ϕ 1 , .., ϕ n ) which denotes the quasi-continuous Bloch vector of the infinite lattice limit [ϕ s = 2π

Ns s → ϕ s and dϕ s ∼ 2π Ns as N s → ∞ where -π ≤ ϕ s ≤ π can be chosen for the integration limits]. To characterize fractional random walks an interesting question is how the transition matrix behaves asymptotically for large lattices. Let us first consider the behavior for large times t >> 1. From ( 21) and [START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF] we can see that for large times the dynamics is dominated by the ensemble of 'slowly relaxing' diffusional

modes 4 W (α) (t >> 1) ∼ { s(t)} | s s |e -tλ α 2
s for which tλ α 2 s is still not large. It is clear that, the more time t increases, the smaller the set { s (t)} becomes with non-vanishing 'surviving modes' corresponding to sufficiently 'small' eigenvalues λ α 2 s << 1. For the cubic lattice the small eigenvalues of the fractional Laplacian are obtained from (12) as λ α 2 ( ϕ) ≈ ϕ α for sufficiently small ϕ = | ϕ| << 1. For increasingly large times the 'surviving modes' are located within an n-dimensional sphere in the ϕ-wave-vector space (around the origin) where the radius R of this wave-space sphere is developing in time as tλ

α 2 s ≤ R α t ∼ 1, i.e. R α (t) ∼ t -1
α . The number D α,n (t) of slowly relaxing modes within this n-dimensional sphere 'selected to have survived at time t' is measured by the volume of this sphere as D α,t (t) ∼ R n α (t) = t -n α . This means in the large time limit the decay of the fractional transition matrix should behave as W (α) p-q (t) ∼ t -n α . We will confirm this argumentation by the subsequent derivation of asymptotic expressions for the fractional transition matrix (23). 3 In the limiting case Nj → ∞ the contribution of the stationary limit

W (α) p-q (∞) = 1 N = 1 n j=1 N j
→ 0 can be neglected.

Asymptotic behavior of Fractional transition matrix: Emergence of Lévy flights

Let us consider transition matrix [START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF] for large ( n s=1 (p s -q s ) 2 ) 1 2 = | p -q| >> 1 on large lattices and finite times. We then can evaluate [START_REF] Michelitsch | The Fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion[END_REF] by introducing the vector k with the components k j = ϕ j | p -q| where the integration limits can be put ±∞ to obtain the leading asymptotic contribution

W (α) p-q (t) ≈ 1 (2π| p -q|) n ∞ -∞ dk 1 .. ∞ -∞ dk n e i k• ( p-q) | p-q| e -t 2 n j=1 1-cos k j | p-q| α 2 . ( 24
)
Let us consider now this integral for large | p -q| α >> t where asymptotically this integral is dominated by

2 n j=1 1 -cos k j | p-q| α 2 ∼ | k| α
| p-q| α to arrive at the asymptotic expression

W (α) p-q (t) ≈ 1 (2π) n ∞ -∞ dk 1 .. ∞ -∞
dk n e i k•( p-q) e -t| k| α .

(

) 25
This asymptotic representation for the fractional transition matrix is within the admissible interval 0 < α ≤ 2, the well known Fourier representation of an α-stable Lévy distribution in the n-dimensional infinite space, e.g. [START_REF] Chechkin | Fundamentals of Lévy flight processes[END_REF]. The cases α = 1 and α = 2 indicate Cauchy-and normal distributions, respectively. Having arrived at the asymptotic relation [START_REF] Noh | Random Walks on Complex Networks[END_REF] for the fractional transition matrix, it is straight-forward to obtain the representation 5 [START_REF] Michelitsch | The Fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion[END_REF], and see also [START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF]

W (α) p-q (t) ≈ t -n α W | p -q| α t ( 26
)
where function W depends only on the ratio | p-q| α t and fulfills

W ( | p-q| α t → 0) = 1 (2π) n d n ξe -ξ α W (0) = 1 (2π) n 2π n 2 Γ( n 2 ) ∞ 0 e -ξ α ξ n-1 dξ = 2 (2 √ π) n 2 α Γ( n α ) Γ( n 2 ) (27)
and finally, the asymptotic expression for W (α)

p-q (t) for large lattices and t 1 is given by:

W (α) p-q (t) ≈ 2 (2 √ π) n 2 α Γ( n α ) Γ( n 2 ) t -n α . ( 28
)
The result in equation ( 28) establishes the probabilities of transition at time t from the stating position q to the final location determined by p for a random walker following the fractional dynamics determined by equation [START_REF] Michelitsch | Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels[END_REF]; this result is discussed in the following part in the context of probabilities of return.

On the other hand, the transition probabilities at each step are given by the transition matrix:

W (α) p-q (t) ≈ 1 (2π) n ∞ -∞ dk 1 .. ∞ -∞ dk n e i k•( p-q) e -| k| α t ( 29
)
and a similar approach allows to obtain for | p -q| α >> tC α,n :

W (α) p-q ∝ tC α,n | p -q| n+α (30)
where the constant C α,n =

2 α-1 αΓ( α+n 2 ) π n 2 Γ(1-α 2 )
is given in the literature [START_REF] Michelitsch | The Fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion[END_REF][START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF] (having here physical dimension [sec -1 ]). The result in Eq. ( 30) for n-dimensional lattices emerges from the fractional Laplacian L α 2 . This

5 Put in (25) k = ξt -1 α .
power-law decay | p -q| -n-α for the transition probabilities of a random walker is the footprint of a dynamics with Lévy flights in n-dimensions defined within admissible interval 0 < α ≤ 2.

The time derivative of (25) yields

d dt W (α) p-q (t = 0) = -L α 2 p-q ≈ C α,n | p -q| n+α = 1 | p -q| n+α (-1) (2π) n ∞ -∞ dk 1 .. ∞ -∞ dk n e i k• ( p-q) | p-q| | k| α = (-1) (2π) n ∞ -∞ dk 1 .. ∞ -∞ dk n e i k•( p-q) | k| α d dt W (α) p-q (t = 0) ≈ -(-∆ n ) α 2 δ n ( p -q). (31)
The last relation is nothing but the fractional diffusion equation of a Lévy flyer in the n-dimensional continuous space where the fractional Laplacian matrix takes the representation of a Riesz fractional derivative kernel (continuous fractional Laplacian kernel) [START_REF] Michelitsch | The Fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion[END_REF][START_REF] Ortiguera | Riesz Potential Operators and Inverses via Fractional Centered Derivatives[END_REF].

Fractional return probabilities

Now closely connected with the derivations of the previous paragraph, let us analyze the return probability of the fractional random walk which is defined as the probability that a walker starting at t = 0 at a node p returns at this node after time t. The return probability is obtained by the uniform value of the diagonal elements of the transition matrix W (α) (t), namely

p α (t) = W (α) pp (t) = 1 N trace(W (α) (t)) = 1 N + 1 N N -1 l=1 e -λ α 2 l t (32)
having identical values for all nodes p, decaying rapidly to the stationary equal distribution p(∞) = 1 N and obeying the initial condition p α (t = 0) = 1. Since all diagonal elements of the transition matrix have the same value, the return probability for any node p coincides with the global average return probability.

Let us now consider large cubic lattices N j → ∞, ∀j = 1, .., n. Then the return probability is obtained by the diagonal elements of ( 23) as [where p α (∞) = 1 N → 0 can be set to zero]

p α (t) -p α (∞) = W (α) 0 (t) -W (α) 0 (∞) = 1 (2π) n π -π dϕ 1 .. π -π dϕ n e -t 2 n j=1 (1-cos ϕ j ) α 2 . ( 33
)
The return probability of the normal random walk is recovered by [START_REF] Samko | Fractional Weyl-Riesz Integrodifferentiation of Periodic Functions of Two Variables via the Periodization of the Riesz Kernel[END_REF] for α = 2 and yields p 2 (t) = e -2nt I n 0 (2t) in accordance with the well known expression of the literature e -t I n 0 ( t n ) which is recovered [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF] when instead of the Laplacian matrix L pq , the modified Laplacian matrix

Lpq K = 1
2n L pq is employed. The large time asymptotics of the return probability can be obtained from expression by setting | p -q| = 0 in the asymptotic relation [START_REF] Redner | A guide to first passage processes[END_REF] and accounting for ( 27) we arrive at

p α (t) -p α (∞) ≈ 2 (2 √ π) n 2 α Γ( n α ) Γ( n 2 ) t -n α . ( 34
)
The fractional return probability decays for large cubic lattices in n-dimensions also with a characteristic t -n α -power law. We have to interpret this power law decay in the same way as for the fractional transition matrix, namely as a collective effect of such normal modes with Bloch-wave vectors κ located on a in the limit infinitesimal small sphere around the origin κ = 0 where in this region λ → 0. The large time power law behavior emerges only in sufficiently large lattices, i.e. when the infinitesimal small κ-sphere is densely populated with normal modes. The collective effect of these long wave diffusional modes act like a Lévy flyer in the continuous space where the discrete structure of the lattice is ignored. For the cyclic ring n = 1 the p α (t) -p α (∞) ∼ t -1 α given in [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF] is recovered as well as the p α (t) -p α (∞) ∼ t -n2 for normal random walks (α = 2) on the n-dimensional cubic lattice. We notice that in range 0 < α < 2, the relaxation of the fractional random walk p α (t) -p α (∞) ∼ t -n α is faster than those of the normal random walk α = 2 in the same lattice. In the extreme limit of vanishing (infinitesimally positive) α → 0, it can be seen that the return probability p α (t) starting at p α (t = 0) = 1 is instantaneously collapsing as t -n α versus the equilibrium value p α (∞) = 1 N . From this observation follows that the fractional walk on the n-dimensional becomes the faster, the smaller α.

Derivation of the fractional fundamental matrix

The goal of this paragraph is to obtain further characteristic features of the fractional random walk such as for instance mean relaxation times which also is referred to as 'global time' [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF] and closely related the Kemeny constant. We demonstrate that this information is contained in the fractional Green's matrix to be deduced in this paragraph, and closely related, the fractional fundamental matrix. To this end we note that L α 2 due to its eigenvalue λ 0 = 0 is not invertible in the N -dimensional space of all N eigenvectors | . However it is invertible within the N -1-dimensional subspace of eigenvectors | of positive eigenvalues, i.e. the basis of this subspace does not contain the eigenvector |0 with L|0 = 0. Denoting the representation of the Laplacian matrix in this subspace as L then we have

W (α) (t) -W (α) (∞) = e - L α 2 t = N -1 =1 e -λ α 2 t | | ( 35
)
where the spectral sum contains only relaxing terms, where

L α 2 is invertible in this N -1-dimensional subspace [with unity 1(N-1)×(N-1) = 1N×N -|0 0| = N -1 l=1 | |] L-α 2 = ∞ 0 (W (α) (t) -W (α) (∞))dt = N -1 =1 λ -α 2 | | (36)
We refer to as [START_REF] Tarasov | Lattice fractional calculus[END_REF] the fractional Green's matrix. We see that the set of eigenvalues of the (fractional) Green's matrix [START_REF] Tarasov | Lattice fractional calculus[END_REF] is the set of relaxation times τ α = λ -α 2 occuring in [START_REF] Riesz | L'intégrale de Riemann-Liouville et le problème de Cauchy[END_REF]. To link the Green's matrix with some common notions in the literature, let us evoke the definition of the so called 'fundamental matrix' Z which is defined in in the literature for time-discrete random walks as [START_REF] Kemeny | Finite Markov Chains[END_REF][START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF]

Z = δt ∞ m=0 [W (δt) -W (∞)] m = δt[ 1 -(W (δt) -W (∞))] -1 (37)
where W (δt = 1) is the transition matrix corresponding to one time-step [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF] which is converging since all

N -1 eigenvalues (1 -δtλ -α 2 ) of [W (δt) -W (∞)
] are inferior to 1. In contrast to [START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF] where the time-step is chosen δt = 1, we introduce here the transition matrix with a multiplyer of the time increment δt, so that Z has physical dimension of time. In this way we can account for time-discrete, and in the limit δt → 0, for time-continuous random walks.

For the time-discrete fractional random walk, we employ in ( 37)

W (α) (δt)-W (α) (∞) = 1-|0 0|-δtL α 2 = 1N-1×N-1 -δt L α 2 which is just the relaxing part having spectral representation W (α) (δt) -W (α) (∞) = N -1 =1 | |(1 -δtλ α the fractional fundamental matrix 6 Z (α) = δt ∞ m=0 [W (α) (δt) -W (∞)] m = δt[1 -(W (α) (δt) -W (∞))] -1 Z (α) = δt |0 0| + N -1 =1 | | ∞ m=0 [(1 -δtλ α 2 )] m Z (α) = δt|0 0| + N -1 =1 | |λ -α 2 = δt|0 0| + L-α 2 . (38)
We see that the fractional fundamental matrix Z (α) for a discrete time walk is related to our above introduced fractional Green's function for the time-continuous walk [START_REF] Tarasov | Lattice fractional calculus[END_REF] and coinciding in the limit δt → 0. It is clear that this reasoning remains true for any transition from time-discrete to time-continuous random walks, independent of the choice of Laplacian matrix. Relation ( 38) is consistent with spectral representations given in the literature for discrete time random walks [START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF] [where there δt = 1]. Indeed the fractional fundamental matrix Z (α) contains the entire statistical information needed to characterize the 'efficiency' of fractional random walk. Before we exploit this information more closely for the cubic lattice, let us derive some important analytical relations between Z (α) and normal random walk transition matrix W (2) . To this end account first for the following integral relation holding for scalar λ > 0 and α > 0 [where we confine our analysis nevertheless as emphasized above to the admissible range 0 < α ≤ 2]

λ -α 2 = 1 Γ( α 2 ) ∞ 0 e -λt t α 2 -1 dt. ( 39
)
By using this relation and the spectral representation [START_REF] Zoia | Fractional Laplacian in bounded domains[END_REF] we can represent Z (α) for the time-continuous walk by

Z (α) = L-α 2 = 1 Γ( α 2 ) ∞ 0 e -Lt t α 2 -1 dt = 1 Γ( α 2 ) ∞ 0 (W (2) (t) -W (2) (∞))t α 2 -1 dt = ∞ 0 e - L α 2 t dt = ∞ 0 (W (α) (t) -W (α) (∞))dt. (40)
The relations [START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF], (40) hold in the entire admissible range 0 < α ≤ 2 where W (2) (t) = e -Lt is the transition matrix of the normal random walk. The representation (40) is especially useful when the normal walk transition matrix is explicitly known. As we will see, this is the case for normal walks on cubic n-dimensional lattices. Let us now evaluate (40) to obtain the fractional fundamental matrix of cubic n-dimensional infinite lattice where all N j → ∞. Then in view of (12) the spectral sums take integral representations 1

N (..) → 1 (2π) n π -π ... π -π (..)dϕ 1 .
.dϕ n . The transition matrix for the normal walk which is obtained from [START_REF] Michelitsch | A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot-type fractal functions[END_REF] for α = 2 and yields

W (2) p-q (t) = 1 (2π) n π -π dϕ 1 .. π -π dϕ n e -t 2 n j=1 (1-cos ϕ j ) e i( p-q)• ϕ = e -2nt n j=1 I |p j -q j | (2t) (41)
where I p (τ ) = 1 π π 0 e τ cos ϕ cos (pϕ)dϕ denotes the modified Bessel function of the first kind of integer orders p [START_REF] Abramowitz | Handbook of Mathematical Function[END_REF]. Using (40) we can determine the fractional fundamental matrix for infinite cubic lattices as

Z (α) p 1 -q 1 ,..,p j -q j ,..,pn-qn = 1 Γ( α 2 ) ∞ 0 e -2nt t α 2 -1 n j=1 I |p j -q j | (2t)dt = 1 (2n) α 2 Γ( α 2 ) ∞ 0 τ α 2 -1 e -τ n j=1 I |p j -q j | τ n dt (42) 6 As |[(1 -δtλ α 2 )| < 1 this is a converging geometrical series ∞ m=0 [(1 -δtλ α 2 )] m = [1 -(1 -δtλ α 2 )] -1 = (δt) -1 λ -α 2 as |[(1 -δtλ α 2 )| < 1 ∀ = 1, .., N -1.
containing the normal random walk transition matrix elements W

(2) p 1 -q 1 ,..,p j -q j ,..,pn-qn τ 2n = e -τ n j=1 I |p j -q j | τ n (in rescaled time), in accordance with the known expression e -τ I |p-q| (τ ) for n = 1 [START_REF] Redner | A guide to first passage processes[END_REF].

Fractional mean relaxation time

Having arrived at the representation (42) for the fractional fundamental matrix of the cubic lattice, it is now not a big deal to determine the fractional mean relaxation time τα =

1 N trace(Z (α) ) = 1 N N -1 =1 λ -α
2 (for a further discussion see [START_REF] Doyle | Random Walks and Electric Networks[END_REF][START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF][START_REF] Zhang | Mean first passage time for random walks on undirected networks[END_REF]). The smaller mean relaxation time, the faster the stationary stationary distribution is reached, and the higher the efficiency for the walker to explore the network.

From the above considerations follows that the fractional global time τα is determined by the trace of the fractional fundamental matrix [START_REF] Zoia | Fractional Laplacian in bounded domains[END_REF], and for the infinite cubic lattice by (42). The fractional mean relaxation time of the cubic lattice is then the uniform value of the diagonal elements of (42

) τα = Z (α) 0,..,0 = 1 Γ( α 2 ) ∞ 0 W (2) 0 (t) -W (2) 0 (∞) t α 2 -1 dt = 1 Γ( α 2 ) ∞ 0 e -2nt (I 0 (2t)) n t α 2 -1 dt = 1 (2n) α 2 Γ( α 2 ) ∞ 0 e -τ I 0 τ n n τ α 2 -1 dτ. ( 43
)
Interesting is the limiting case when α → 0+ infinitesimally positive, then since 1

Γ( α 2 )
∞ 0 e -τ τ α 2 -1 dτ = 1 is a normalized integral which is for α → 0+ concentrated arround τ = 0, we can write for (43) in this limiting case τ0+ = lim

α→0+ τα = (I 0 (0)) n 1 Γ( α 2 ) ∞ 0 e -τ τ α 2 -1 dτ = 1 (44)
since I 0 (0) = 1. The limiting value τα→0+ is in accordance with the observation that yields the limit of the diagonal element of [START_REF] Tarasov | Lattice fractional calculus[END_REF], namely τα→0+ = lim α→0 2 as function of α for the cubic lattice with N = 50 n nodes. We depict the results obtained for n = 1, 2, 3.

1 N N -1 =1 λ -α 2 = N -1 N → 1 for large lattices.
Figure 1 shows the mean relaxation time τα = 1

N N -1 =1 λ -α
2 for the cubic lattice with N = 50 n nodes for dimensions n = 1, 2, 3, respectively. We see that the mean relaxation time in the fractional range 0 < α < 2 is in the one-dimensional case n = 1 reduced compared with the normal walk α = 2. For one-dimensional cyclic chains the relaxation to the stationary distribution is faster for the fractional walk as for the normal walk. We demonstrate in the subsequent subsection that a fractional random walk dynamics governed by a renormalized fractional Laplacian matrix enhances the speed of fractional random walk. In the following paragraph we compare the efficiency of fractional random walks as a function of α. The appropriate measure to do so is to consider the Kemeny constants of these walks. This is outlined in the subsequent part.

Efficiency of fractional random walk

In order to compare efficiency of fractional random walks for different α let us consider again the time-discrete fractional random walk which is generated by (where we put here without loss of generality δt = 1)

P (α) (t + 1) = ( 1 -L(α) ) • P (α) (t) = 1 k (α) A (α) • P α) (t). ( 45
)
In order to compare the dynamics for different α we need to consider fractional random walks for different α on the same time-scale. This is achieved when we normalize the fractional Laplacian matrices in (45) such that they all have the same diagonal elements L(α) pp = 1 in the order of magnitude of the time step, i.e. independent of α. Hence we employ in the master equation (45) the re-normalized fractional Laplacian matrix

L(α) pq = 1 k (α) L α 2 pq = 1 - 1 k (α) A (α) pq , (46)
where

k (α) = L α 2 pp = 1 N trace(L α 2 ) = 1 N N -1 =1 λ α 2
denotes the fractional degree and A pq for p = q) [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF]. In the appendix we have shown that 0 < α ≤ 2 is the admissible interval where the fractional Laplacian has required good properties that the fractional connectivity matrix has positive non-vanishing elements A (α) pq ≥ 0. In view of (45) the quantities 1 k (α) A (α) denote the probabilities that the walker located at node q moves to node p where because of A For the time-discrete fractional random walk characterized by (45), the eigenvalues of L (α) take the form

λ α 2 k (α)
[where λ denote as above the eigenvalues of (2)].

Fractional Kemeny constant

The Kemeny constant for this type of random walk is with [START_REF] Zoia | Fractional Laplacian in bounded domains[END_REF] determined by [START_REF] Kemeny | Finite Markov Chains[END_REF] 7

K (α) = trace   L α 2 k (α) -1   = k (α) N -1 =1 λ -α 2 (47) where L α 2 k (α) -1
denotes the modified fractional Green's function occuring in the fractional fundamental matrix [START_REF] Zoia | Fractional Laplacian in bounded domains[END_REF] when the renormalized fractional Laplacian (46) is employed. In the study of discrete time random walks, the Kemeny constant is a global time that, for regular networks, gives the average of the times needed by the random walker to reach any site of the structure; this value can be calculated in terms of the Laplacian eigenvalues [START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF]. [For further derivations and definitions of Kemeny constant we refer to [START_REF] Doyle | Random Walks and Electric Networks[END_REF][START_REF] Kemeny | Finite Markov Chains[END_REF]]. For the fractional dynamics on n-dimensional lattices, the Kemeny K (α) constant is given by:

K (α) ≡ 1 N N -1 m=1 λ α/2 m N -1 l=1 λ -α/2 l . (48)
The value k

(α) ≡ 1 N N -1 m=1 λ α/2 m = 1 N trace(L α/2
) is the fractional degree. For n-dimensional lattices with N 1 is k (α) ≈ (2n) α/2 . The Kemeny constant for lattices in n = 1, 2, 3 dimensions is depicted in Figure 2. We see how this characteristic time, in the fractional range 0 < α < 2 is reduced, compared with the normal random walk α = 2. The relaxation to the stationary distribution is faster for the fractional walk as for the normal walk. A fractional random walk search strategy to explore the lattice has thus due to long range moves and emerging a small-world property increasing the capacity of the random walker to reach any site of the lattice in comparison with the normal strategy. These results confirm the recent findings in [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF].

K (α) /N n =1 n =2 n =3 8
Fig. 2: Representation of the Kemeny constant K (α) as function of α for the cubic lattice with N = 50 n nodes. We depict the results obtained for n = 1, 2, 3.

In Figure 2 we plot K

(α) = 1 N N -1 m=1 λ α 2 m N -1 =1 λ -α
2 for the infinite cubic lattice for spatial dimensions n = 1, 2, 3. This figure shows that the random walk governed by (45) is always faster the smaller α in 0 < α < 2 for a fractional random walk, as compared to a normal random walk α = 2. A fractional random walk search strategy to explore the lattice has thus due to long range moves and emerging small world property increased efficiency compared to a normal walk strategy. These results confirm the recent findings in [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF].

Conclusion

We analyzed time-discrete and time-continuous 'fractional' random walks on undirected regular networks with special emphasis on cubic lattices in n = 1, 2, 3, .. dimensions. The random walk of our model is described by a master equation involving fractional powers of Laplacian matrices L α 2 . The fractional network Laplacian matrix represents the matrix analogue to the continuous fractional Laplacian (Riesz fractional derivative). We demonstrated that the fractional random walk is admissible in the same range 0 < α ≤ 2 of power law index as Lévy flights.

We derived analytical expressions for the fractional transition matrix, fractional fundamental matrix (Green's matrix) and obtained representations for the return probabilities, the mean relaxation time and closely related the Kemeny constant. We show for the cubic lattice explicitly the emergence of Lévy flights for sufficiently large lattices. For the return probabilities we obtain characteristic t -n α power law decay for fractional walks in n-dimensional cubic lattices. This scaling is the same as one obtains for fractional diffusion (Lévy flights) in the continuous n-dimensional space [START_REF] Michelitsch | The Fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion[END_REF]. The fractional random walk model can be conceived as a model for anomalous diffusion on regular networks such as cubic lattices. It can be concluded that, due to long range moves, the efficiency to explore the lattice is increased when instead of a normal random walk a fractional random walk is chosen.

The fractional random walk approach derived in this paper can be generalized for an analysis of anomalous transport phenomena on more complex undirected networks. The subject of fractional random walks on regular undirected networks deserves further investigation. For instance with the present framework further characteristic quantities such as the MPFT (mean first passage time) for regular undirected networks can be derived.

Appendix

In this appendix we demonstrate that 0 < α ≤ 2 is an admissible interval for fractional random walks. To this end let us consider the transition matrix

W (δt) = 1 -δtL (49)
which requires non-negative matrix elements 0 ≤ W pq ≤ 1 together with the normalization condition N -1 p=0 W pq (δt) = 1. This property is fulfilled if L is a 'good' Laplacian matrix where (i) its non-diagonal elements must be non-positive L pq = -A pq ≤ 0 (p = q) and as a consequence of translational symmetry (ii) its diagonal elements L pp = K = N -1 q=0 A pq > 0 must be be positive indicating the degree of the network. Now the goal is to determine for which α-range the fractional Laplacian matrix L α 2 has also the required good properties (i) and (ii) when the Laplacian matrix L has those good properties. Let us first consider (i): (a) Assume W (δt) is such a good transition matrix, then also all its powers W m (δt) = W (mδt) and hence also the exponential W (t) = e -Lt = lim m→∞ ( 1 -t m L) m is a good transition matrix, especially with W pq (δt) = -L pq δt = A pq δt ≥ 0 (p = q). It follows that the elements of transition matrix [e -Lt ] pq are all positive and normalization remains conserved in time which exists for 0 < α < 2 with a negative constant C α < 0, occuring since the integrand for λ > 0 is identically negative as e -λt -1 < 0. It can be seen by simple partial integration that C α = -2 α Γ(1-α 2 ) = Γ(-α 2 ) < 0 (0 < α < 2).

(c) The fractional Laplacian matrix can then be represented by

C α L α 2 = ∞ 0 t -α 2 -1 (e -Lt -1)dt (51)
which can be verified by employing the spectral representation of L and (50). The non-diagonal elements of the matrix (51) have the same sign as the non-vanishing elements of e -Lt = W (t), namely +1, and remaining positive by the integration. We can conclude that the sign of (non-vanishing) non-diagonal elements of the fractional Laplacian matrix L (e -λ t -1) dt, 0 < α < 2 (53)

and it follows again because of e -λ t -1 < 0 and C α < 0 that L α 2 pp > 0 in the interval of existence 0 < α < 2 of (53).

The positive in general non-integer number L α 2 pp = k (α) of the identical diagonal elements of the fractional Laplacian matrix can be conceived as the fractional degree in this interval. This interpretation was already suggested earlier [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF].

In conclusion it follows that the non-diagonal matrix elements of the fractional Laplacian matrix holds equally as for L the good property [L α 2 ] pq < 0 for p = q and L pp > 0 in the interval 0 < α ≤ 2 where α = 2 recovers the good properties of L . The fractional Laplacian thus can be represented 0 < α ≤ 2

L α 2 = k (α) 1 -A (α) , 0 < α ≤ 2 (54)
as suggested in [START_REF] Riascos | Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights[END_REF][START_REF] Riascos | Fractional diffusion on circulant networks: emergence of a dynamical small world[END_REF] generalizing normal random walks to fractional random walks. The fractional degree k (α) = [L α 2 ] pp and fractional analogue of adjacency matrix A (α) pq ≥ 0 was introduced there with k (α) =

q =p A (α)
pq , having the same good properties for the fractional case as for the normal walk case where α = 2.

In the context of discrete fractional random walk problem (45) the non-negative quantities A (α) pq k (α) have the interpretations of transition probabilities. This interpretation holds when 0 ≤ A (α) pq k (α) ≤ 1 which we showed to be fulfilled in the interval 0 < α ≤ 2. For α > 2 it has been shown earlier for the infinite cyclic chain, that alternating signs occur with oscillatoric behavior of the non-diagonal elements as a function of α [START_REF] Michelitsch | Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit[END_REF].

It follows that the fractional Laplacian matrix approach yields a good fractional transition matrix W (α) (t) = e -L α 2 t only within the range 0 < α ≤ 2. We emphasize that this is the same α-interval of existence of Lévy flights and α-stable Lévy distributions.

2

  I. e. |j = δ j and 1 =

Fig. 1 : 1 N

 11 Fig. 1: Representation of the fractional characteristic time (mean relaxation time) τα = 1 N

  pq the fractional adjacency matrix which is constructed by the negative non-diagonal elements of the fractional Laplacian matrix L

1 k

 1 walker must move to another node as time increases by a time-step δt = 1. This is reflected by the fact the normalization condition N -1 p=0 (α) A (α) pq = 1 and by 0 < A (α) pq < 1 within the admissible range 0 < α ≤ 2.

  8 . (b) Consider now for λ ≥ 0 the integral ∞ 0 t -α 2 -1 (e -λt -1)dt = λ

α 2 2 ] 1 p=0 2 pp can be directly seen from L α 2 pp = 1 N

 221221 is determined by the (negative) sign of the constant C α , namelysign([L α pq ) = sign(C α )sign([e -Lt ] pq ) = (-1) × 1 = -1, p = q, 0 < α < 2 Lps = 0.Let us now analyze property (ii) for the fractional Laplacian matrix. The positiveness of the in our case idential diagonal elements L α

The contribution of 'fast relaxing modes' is rapidly surpressed.

). Now in order to evaluate the geometrical series(37) we have to pay attention on the zero order [W (δt) -W (∞)] 0 = 1N×N = |0 0| + 1(N-1)×(N-1) being the complete N × N -unity matrix which containing 0 0| which is not in the N -1-dimensional subspace of relaxing modes, whereas all orders m > 0 have only contributions within in the N -1-dimensional subspace. We can hence write for the expansion of

where in the literature the definition of Kemeny constant often differs by the constant 1.

Acknowledgements

We are indepted to G.A. Maugin pour fruitful discussions.


Charles Mottes

David Makowski

Thierry A Doré

A model-based approach for

  L'archive ouverte pluridisciplinaire

Results

example of modifying a typical cropping system of French Burgundy Method A model-based approach for adapting cropping systems to climate change Mottes Charles 1 ,2 , Makowski David 1 ,2 and Doré Thierry 1 ,2 1 INRA -UMR 211 Agronomie -F-78850 Thiverval-Grignon, France 2 AgroParisTech -UMR 211 Agronomie -F-78850 Thiverval-Grignon, France

• Models are powerful tools to assess prototypes of cropping systems under different pedoclimatic conditions.

• Present a methodological framework to virtually assess adaptations of cropping systems to climate change (A1B):

• Assess the robustness of modelling results to parameter uncertainty and the capacity of the model to sort cropping systems according to performances.

• Assess whether a cropping system modification improves agronomic and environmental performances compared to an actual system under climate change. • The model CliMultiCS (1) can be used to sort cropping system modifications according to performances in spite of the uncertainty on the absolute values simulated.

Context and objectives

• Estimations of crop yields and nitrate leaching with CliMultiCS are highly uncertain (10% < Cv < 34% vs parameter Cv=5.8%).

• Returning and incorporating all crop residues into soil virtually better increase crop yields in future climate than in actual climate but emit more greenhouse gas and slighly increase nitrate leaching.

• Multiple conflicting performances should be taken into account to avoid misadaptations of cropping systems to climate change.

We expect that uncertainties on model results are compatible with our objective of sorting modifications of cropping systems according to performances in actual and future climate.

Conclusions

Actual climate

Future climate

Computation of 12 performances Computation of 12 performances

CliMultiCS (1) for each modified cropping system

(1) CliMultiCS is a model assessing the effects of cropping systems and climate change on 12 performances including crop yields, greenhouse gas emissions, nitrate leaching, phosporus conservation, erosion, water conservation, soil carbon and other environmental performances. CliMultiCS is build on modules from EPIC, APSIM and CERES. It also integrates indicator based modules to assess weed presure and soil compaction risk.

Computation of metrics to assess the

Estimation of uncertainty by:

Mean performance MODIFIED -Mean performance REFERENCE

Assessment of the significance of the differences with a non parametric Wilcoxon-Mann-Whitney test.

performance Cv

Fig. 1 :

 1 Fig. 1: Comparison of a subsample of performances obtained for the reference (REFERENCE) and the modified cropping system (MODIFIED). Performances are summed over 33 years. GHG: Greenhouse gas emissions, NO 3 -: nitrate leaching, Cv: coefficient of variation

  Differences of the mean performances of the MODIFIED cropping system and the REFERENCE cropping system under actual and future climate. Performances correspond to cumuls over 33 years. GHG: Greenhouse gas emissions, NO 3 -: nitrate leaching, ***: p-value<0.of means of performances between the modified and the reference cropping system under both actual and future climate better mean yields than REFERENCE system in actual and future climate. Yield increases are higher in future climate than in actual climate. MODIFIED emits more GHG and nitrate than REFERENCE in future climate. The increase of GHG emissions is higher in future climate than in actual climate while nitrate leaching increase is lower.