By Gerald A. Edgar

Fractals are a tremendous subject in such different branches of technology as arithmetic, desktop technology, and physics. Classics on Fractals collects for the 1st time the historical papers on fractal geometry, facing such themes as non-differentiable services, self-similarity, and fractional size. Of specific price are the twelve papers that experience by no means sooner than been translated into English. Commentaries by way of Professor Edgar are incorporated to assist the coed of arithmetic in interpreting the papers, and to put them of their ancient viewpoint. the quantity includes papers from the subsequent: Cantor, Weierstrass, von Koch, Hausdorff, Caratheodory, Menger, Bouligand, Pontrjagin and Schnirelmann, Besicovitch, Ursell, Levy, Moran, Marstrand, Taylor, de Rahm, Kolmogorov and Tihomirov, Kiesswetter, and naturally, Mandelbrot.

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Xn (mod 2k) ∈ C Λ(C) = √ 2k is an n-dimensional unimodular lattice with the minimum norm min{ dE2k(C) , 2k}. Moreover, if C is Type II, then Λ(C) is Type II. Proof Clearly Λ(C) is an n-dimensional lattice. Let a1 , a2 ∈ Λ(C). Then ai = √1 (ci + 2kzi ), where ci ∈ C and zi ∈ Zn for i = 1, 2. Then [a1 , a2 ] = 1 ([c1 , c2 ] + 2k 2k 2k[c1 , z2 ] + 2k[c2 , z1 ] + 4k 2 [z1 , z2 ]) ∈ Zn since [c1 , c2 ] is a multiple of 2k. Thus, Λ(C) is integral. √Note that 2kZn ⊂ √ 2kΛ(C) ⊂ Zn . It is (2kZn ) = (2k)n and [ 2kΛ(C) : 2kZn ] = √easy to see that Vn/2 n/2 =√1 = det Λ, that is, (2k) .

2 for more details. 6 More Problems Related to a Prize Problem In this section, we further describe one of the long-standing open problems in algebraic coding theory. This is about the existence of a binary self-dual [72, 36, 16] code. We refer to [Ki1]. Let C be a binary Type I code, and C0 the doubly even subcode C0 of C (that is, the subcode of C consisting of all codewords of weight ≡ 0 (mod 4)). 6 More Problems Related to a Prize Problem 45 Sloane defined the shadow S of C by S := C0⊥ \C [CS3].

Bn ) ∈ GF(q)n be regarded as a block, and p = (p1 , . . , pn ) ∈ GF(q)n be regarded as a point. We say that b covers p, or p is in b, provided that (a) supp(p) ⊂ supp(b) and (b) for all i ∈ supp(b), (at least) one of the following conditions holds: (i) pi = bi , (ii) pi = 0. More generally, a q-ary t-(v, k, λ) design D = (P , B) is a pair consisting of the set P ⊂ GF(q)n of elements (called points) of weight t and a collection B of weight k elements of GF(q)n (called blocks) such that every point p ∈ P is covered by exactly λ blocks.