کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5778704 | 1633781 | 2017 | 53 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
Distance and tube zeta functions of fractals and arbitrary compact sets
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
ریاضیات (عمومی)
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N-dimensional Euclidean space RN, for any integer Nâ¥1. It is defined by the Lebesgue integral ζA(s)=â«Aδd(x,A)sâNdx, for all sâC with Res sufficiently large, and we call it the distance zeta function of A. Here, d(x,A) denotes the Euclidean distance from x to A and Aδ is the δ-neighborhood of A, where δ is a fixed positive real number. We prove that the abscissa of absolute convergence of ζA is equal to dimâ¾BA, the upper box (or Minkowski) dimension of A. Particular attention is payed to the principal complex dimensions of A, defined as the set of poles of ζA located on the critical line {Res=dimâ¾BA}, provided ζA possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, ζËA(s)=â«0δtsâNâ1|At|dt, called the tube zeta function of A. Assuming that A is Minkowski measurable, we show that, under some mild conditions, the residue of ζËA computed at D=dimBâ¡A (the box dimension of A) is equal to the Minkowski content of A. More generally, without assuming that A is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of A. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Advances in Mathematics - Volume 307, 5 February 2017, Pages 1215-1267
Journal: Advances in Mathematics - Volume 307, 5 February 2017, Pages 1215-1267
نویسندگان
M.L. Lapidus, G. RadunoviÄ, D. ŽubriniÄ,