Given a reduced group G, the class of groups A such that$A \cong \operatorname{Tor}(A, G)$is studied. A complete characterization is obtained when G is separable.
Let h be a positive integer, and A a set of nonnegative integers. A is called an exact asymptotic basis of order h if every sufficiently large positive integer can be written as a sum of h not ...necessarily distinct elements from A. The smallest such h is called the exact order of A, denoted by g(A). A subset A − F of an asymptotic basis of order h may not be an asymptotic basis of any order. When A − F is again an asymptotic basis, the exact order g(A − F) may increase. Nathanson 48 studied how much larger the exact order g(A − F) when finitely many elements are removed from an asymptotic basis of order h. Nathanson defines, for any given positive integers h and k, \documentclass12pt{minimal}
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$${G}_{k}(h) {=\max { }_{{ A \atop g(A)\leq h} }\max }_{F\in {I}_{k}(A)}g(A - F),$$
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$${I}_{k}(A) =\{ \vert F\vert \ \vert F\vert = k\text{ and }g(A - F) < \infty \}$$
\end{document}. Many results have been proved since Nathanson’s question was first asked in 1984. This function Gk(h) is also closely related to interconnection network designs in network theory. This paper is a brief survey on this and few other related problems. G. Grekos 11 has a recent survey on a related problem.
All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is ...given.
Tartalom ◊ Köszönetnyilvánítás ◊ Jelölések ◊ I. rész: A vizsgált objektum. Alaptulajdonságok és alaptételek ◊ II. rész: Reprezentációk ◊ III. rész: Nevezetes részcsoportok ◊ IV. rész: Az ...automorfizmusokról általában ◊ V. rész: Ch-csoportok ◊ VI. rész: Csoportok a Ch-n kívül ◊ VII. rész: Nyitott kérdések ◊ Függelék ◊ Irodalom ◊ Utóirat
javított változat
Tartalom ◊ Köszönetnyilvánítás ◊ Jelölések ◊ I. rész: A vizsgált objektum. Alaptulajdonságok és alaptételek ◊ II. rész: Reprezentációk ◊ III. rész: Nevezetes részcsoportok ◊ IV. rész: Az automorfizmusokról általában ◊ V. rész: Ch-csoportok ◊ VI. rész: Csoportok a Ch-n kívül ◊ VII. rész: Nyitott kérdések ◊ Függelék ◊ Irodalom ◊ Utóirat
javított változat
Asymptotic formulas for the number of subgroups of a given index of the free product of finitely many cyclic groups are given. The classical modular group $\Gamma$ is discussed in detail, and a table ...of the number of subgroups of $\Gamma$ of index $n$ is given for $1 \leqslant n \leqslant 100$.
Two experiments were performed to investigate stimulus determinants of pattern complexity and pattern goodness. Two hundred and ninety-six undergraduates rated complexity and goodness of ...two-dimensional patterns, which consisted of solid and/or open circles. The patterns were invariant under transformations of reflection or rotation, and they formed cyclic groups or dihedral ones. The results were summarized as follows. (1) Goodness of patterns increased with the order of cyclic and dihedral groups with different weights. (2) Complexity of patterns having line-segments decreased with the order of cyclic and dihedral groups with equal weights, whereas that of patterns having no line-segments was medium regardless of the order. (3) Simplicity and goodness of patterns with a vertical axis of reflection were higher than those with the other orientation axes. (4) Patterns consisting of solid circles were rated more complex than those of open ones. (5) Complexity increased as a positively accelerated function of the number of circles, whereas goodness increased as a negatively accelerated function. It was concluded that complexity and goodness were determined by compound factors, which are processed at different stages of human visual system.
A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a ...substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.
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