Some vanishing sums involving binomial coefficients in the denominator

Identities involving binomial coeffcients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace [1] using probability theory turn out to be special cases of the Chu-Vandermonde identities. Here, we obtain some generalizations of the identities observed by Horrace and give different types of proofs; these, in turn, give rise to some other new identities. In particular, we evaluate sums of the form Pm j=0 (1) j j d (mj) (n+jj ) and deduce that they vanish when d is even and m = n > d=2. It is well-known [2] that sums involving binomial coeffcients can usually be expressed in terms of the hypergeometric functions but it is more interesting if such a function can be evaluated explicitly at a given argument. Identities such as the ones we prove could perhaps be of some interest due to the explicit evaluation possible. The papers [3], [4] are among many which deal with identities for sums where the binomial coeffcients occur in the denominator and we use similar methods here

Sztuka i przestrzeń. Pawła Florenskiego interpretacja kategorii przestrzenności

This paper attempts to acquaint Polish readers with the Russian Orthodox philosopher Pavel Florensky’s (1882–1937) thinking on the category of space‑ness, which together with the categories of symbolism and light is one of the key‑words in his theory of aesthetics. The concept of space is, according to Florensky, a criterion for distinguishing art types, although for him it is only a model, which can be used to help form judgments about the world. It is also the basis of being for the artwork itself. Painting and printmaking, among other arts types mentioned by Florensky (poetry, music, theatre, and architecture), have their own particularities in their organization of space. Painting and printmaking are, for Florensky, the truest art forms — art par excellence. The presentation of a threefold interpretation of the category of space‑ness, together with the description and justification of the divisions which are made in art based on these spatial categories by the philosopher is the leitmotiv of this paper. The aim of the paper is to present and discuss the opinions and arguments of the Orthodox thinker from his work Анализ пространственности (и времени) в художественно‑изобразительных произведениях, which has not yet been translated into Polish or English

Tiresome paths, water gates and Euler’s formula

A hallmark of mathematics is its power to look at seemingly different problems with the same eyes and find a common idea which resolves both. It is not surprising that the two problems we discuss here, about routes to be taken with various constraints and about watering fields, can both be treated using ideas from graph theor

Some Observations on Khovanskii\u27s Matrix Methods for extracting Roots of Polynomials

In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii’s method for finding the m-th (m ≥ 4) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii’s, which also involves m×m matrices, for approximating the root of an arbitrary polynomial of degree m

An examination of home internet and mobile device use in the U.S

It is important to understand the determinants of the extent and types of activities performed by home Internet users and by mobile device users, as well as the interactions between these modes of usage. Few studies have looked at the interactions between these two channels of digital com-munication. While most prior research has been based upon surveys of attitudes toward and inten-tions to use a technology (and lack a final link to actual usage), in this study, the extent of actual home Internet use, mobile device use, and the simultaneity between these modes of usage are ex-amined. We find that, overall, mobile device use is enhanced by home Internet use, while the two act as substitutes in the case of advanced and sensitive applications