koch helge (13 Ergebnisse)

Zustand: VG. Stockholm 1899 first edition. Norstedt. Bihang till Svenka band 25, afd. 1 no 5. octavo. later wraps. 24p. largely unopened. VG plus.

Paperback. Zustand: Brand New. 520 pages. 9.75x6.25x1.18 inches. In Stock.

(28,5 x 22,5 cm). SS. (77)-104. Original-Broschur. (Sonderdruck aus: Acta mathematica). Erste Ausgabe. - Weiterführung des 1896 von E. Borel hergestellten Zusammenhangs über die analytische Fortsetzbarkeit einer Taylorreihe, mit Rücksicht auf die von G. Mittag-Leffler gewonnenen Ergebnisse.

Teknisk högskolelitteratur i Stockholm AB (THS), 1984. Andra reviderade upplagan. (7), 444 sidor. Häftad.

Zustand: VG. Article on pp. 255-266 in one complete issue of Rendiconti del Circolo Matematico di Palermo , tomo XXVIII, fasc II, 1909. Octavo wraps. Other mathematical papers in issue as well by Lichtenstein, Bagnera, Bucca, Broggi, Sellerio and others. VG. excellent condition, no owner marks; a slight bit of faint foxing on cover. Binding secure and text clean. Light spine wear.

hardcover. Zustand: Sehr gut. 320 Seiten; 9783141225631.2 Gewicht in Gramm: 1.

In-4, [274 x 215 mm] de (2), 12, (2) pp. Demi-percaline bleue. (Reliure de l'époque.). Relié avec: - KOCH, Helge von. Sur les déterminants infinis et les équations différentielles linéaires. Stockholm, Imprimerie centrale, 1892. (2), 34 pp. - KOCH, Helge von. Sur les intégrales régulières des équations différentielles linéaires. pp. 337 à 346. Tiré à part des "Acta mathematica", tome 18. Réunion de trois mémoires en éditions originale du grand mathématicien suédois. "The first step on the long road which eventually led to funtional analysis, since it provided Fredholm with the key for the solution of his integral equation." DSB 7, 436. Cachets de bibliothèque annulés sur le titre. In-4, [274 x 215 mm] de (2), 12, (2) pp. Demi-percaline bleue. (Reliure de l'époque.) - - - - - - - - - - - - - - - - - - - - -.

Hardcover. First edition. GEOMETRIC FRACTALS. First edition, extremely rare offprint issue, inscribed by the author to Fredholm, of von Koch's paper, which contains the first examples of geometric fractals, the famous 'Koch curve' and 'Koch snowflake'. The term 'fractal' was coined much later, by Benoit Mandelbrot in his 1975 book Les objets fractals, forme, hasard et dimension. A fractal is an object that displays 'self-similarity' on all scales; it need not exhibit exactly the same structure at all scales, but the same 'type' of structures must appear on all scales. A fractal should also have a 'dimension' that is not a whole number, unlike a line (dimension 1) or a plane (dimension 2). Today fractals have found a bewildering variety of applications in both the arts and sciences (fractal patterns have been found in the works of Jackson Pollock). The first example of a fractal was the 'Cantor set,' introduced by Georg Cantor (1845-1918) in 1883. This set is obtained by starting with a line segment and then removing the middle third, leaving two shorter line segments; then one removes the middle third of each of these two segments; iterating the process indefinitely results in the Cantor set, an infinite set of points that is 'self-similar,' in the sense that if one magnifies a section of the set, one obtains the whole set again. The Cantor set is not really a curve: it is an infinite set of points, but is 'countable', like the whole numbers 1,2,3,., but unlike the original line segment whose points cannot be 'counted' in this way (a fact that was also proved by Cantor). In this sense the Cantor set cannot be regarded as a geometric fractal. Koch modified Cantor's construction to obtain a genuine curve. The Koch curve is constructed by starting with a straight line segment, erecting an equilateral triangle with base the middle third of the original segment, and then erasing that middle third. This produces a shape with four line segments, each one-third the length of the original segment. The same construction is now applied to each of these four segments. The Koch curve is the result of iterating this process indefinitely. The Koch snowflake is the result of applying the process to an equilateral triangle instead of a line segment, so that a Koch snowflake is three Koch curves joined together. The Koch curve is self-similar, in the sense that any part of it, when magnified, looks the same as part of the original curve. The Koch curve has infinite length, because the total length increases by a factor 4/3 with each iteration, so that after n iterations the length is increased by a factor (4/3)n, which becomes larger without limit as n increases. The 'fractal dimension' of the Koch curve maybe be calculated by counting the number N(d) of circles of diameter d needed to cover the curve; as d get smaller N(d) gets larger, but the ratio - log N(d) / log d approaches a definite number as d approaches zero. In fact, it approaches log 4/log 3 = 1.26186 - the fractal dimension of the Koch curve. Koch's aim in this paper was not, of course, to give examples of fractals, whose definition was 70 years in the future. Rather, as the title of the paper states, he wished to give a geometric construction of a curve that was continuous but not differentiable, that is, it had no well-defined tangent at any point. The first example of such a curve had been given by Karl Weierstrass (1815-97) in a paper presented on 18 July 1872 to the Prussian Academy of Sciences (it was not published until 1895). Before Weierstrass most mathematicians believed that continuous curves were differentiable except at isolated points. But Koch was dissatisfied with Weierstrass's example: in the introduction to the present paper, he wrote, "it seems to me that his example is not satisfactory from the geometrical point of view since the function is defined by an analytic expression that hides the geometrical nature of the corresponding curve and so from this point of view one does not see why t.

Paperback. Zustand: Brand New. 7th revised updated edition. 344 pages. German language. 7.99x5.00x0.91 inches. In Stock.

Zustand: Good. Stockholm 1892 Centrale. These pour le doctorat. 4to., 79pp. Cloth spine, paper-covered boards. Title in ink on front cover. Good, some browning of text, water stains on front board.

[Berlin, Stockholm, Paris, Beijer, 1892-93]. 4to. Without wrappers as extracted from "Acta Mathematica. Hrdg. von G. Mittag-Leffler.", Bd. 16. Fine and clean. Pp. 217-295. First printing of Koch's last paper relating to his doctoral thesis. Von Koch is known principally for his work in the theory of infinitely many linear equations and the study of the matrices derived from such infinite systems. He also did work in differential equations and in the theory of numbers. (DSB).

[Berlin, Stockholm, Paris, Beijer, 1891 - 1892]. 4to. Without wrappers as extracted from "Acta Mathematica. Hrdg. von G. Mittag-Leffler.", Bd. 15 and 16. Fine and clean. Pp. 53-63" Pp. 217-295. First printing of Koch's two paper which together constitute his doctoral thesis. Von Koch is known principally for his work in the theory of infinitely many linear equations and the study of the matrices derived from such infinite systems. He also did work in differential equations and in the theory of numbers. (DSB).

First edition, offprint issue, of one of the earliest mathematical constructions later recognized as a fractal: the Koch curve, or snowflake. The front wrapper has the ownership signature of the mathematician's sister, the artist Ebbe von Koch (1866-1943). Koch (1870-1924) described a curve that was continuous but not differentiable at any single point, thereby making it impossible to draw a tangent line. Although Karl Weierstrass had in 1872 demonstrated the existence of a similar function through purely analytic means, Koch was dissatisfied with the lack of a clear geometric representation and sought an explicit visual construction. When almost a century later Benoit Mandelbrot coined the term "fractal", Koch's curve was recognized as a canonical example of such self-similar, geometrically representable constructions. Applications of the Koch curve can be found in antenna engineering, heat transfer technology, and acoustics. Octavo. Diagrams within text. Original blue printed wrappers, sewn as issued. Front wrapper numbered in pencil, a handful of small marks to same: a fine copy.