A Generalization of Lerch’s Formula

We give higher-power generalizations of the classical Lerch formula for the gamma function.     

Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

In 1996 Sabitov proved that the volume \(V\) of an arbitrary simplicial polyhedron \(P\) in the \(3\) -dimensional Euclidean space \(\mathbb {R}^3\) satisfies a monic (with respect to \(V\) ) polynomial relation \(F(V,\ell )=0\) , where \(\ell \) denotes the set of the squares of edge lengths of \(P\) . In 2011 the author proved the same assertion for polyhedra in \(\mathbb {R}^4\) . In this paper, we prove that the same result is true in arbitrary dimension \(n\ge 3\) . Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular \(2\) -faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If \(P_t, t\in [0,1]\) , is a continuous deformation of a polyhedron such that the combinatorial type of \(P_t\) does not change and every \(2\) -face of \(P_t\) remains congruent to the corresponding face of \(P_0\) , then the volume of \(P_t\) is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in \(\mathbb {C}^n\) from their orthogonal edge lengths.     

Chen’s Conjecture and Its Generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that （k ＋l1）^2, （k ＋l2）^2, ..., （k ＋lg）^2 can not be expressed as 2^n＋p^α is at least c（g）x, where p is an odd prime and the constant c（g） depends only on g.     

A Generalization of Stenger’s Lemma to Maximal Dissipative Operators

It is shown that for any maximal dissipative operator A in some Hilbert space \({\mathcal H}\) , which is the orthogonal sum \({\mathcal H=\mathcal F\oplus \mathcal G}\) of two Hilbert spaces \({\mathcal F,\, \mathcal G}\) with \({{\rm dim}\,\mathcal G < \infty}\) , the compression \({\left. T:=P_\mathcal F\,A\right|_{{\rm dom}\,A\cap\mathcal F}}\) of A to \({\mathcal F}\) is again a maximal dissipative operator in \({\mathcal F}\) .     

Generalization of Mirsky’s theorem on diagonals and eigenvalues of matrices

Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof. Moreover, we show that there is a unique companion-matrix-type solution for this problem.     

A Generalization of Bochner’s Extension Theorem to Rough Tubes

This paper generalizes Bochner’s extension theorem to tubes X+iℝ ^m where the set X⊂ℝ ^m is not necessarily a manifold.     

On a Generalization of Voronin’s Theorem

Mathematical Notes - Voronin’s theorem states that the Riemann zeta-function ζ(s) is universal in the sense that all analytic functions that are defined and have no zeros on the right...     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Generalization of Kummer’s second theorem with applications

The aim of this research paper is to obtain single series expression of

On a Projective Generalization of Alperin’s Conjecture

In this paper, we prove that a projective generalization of theKnörr–Robinson formulation of Alperins conjecture holds ifthe ordinary form holds for a certain quotient group.     

Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory

In this paper we study the arithmetic of the Rikunas generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that and k where is a primitive n-th root of unity and =+^–1. In particular, this result with k implies the classical Kummer theory.Mathematics Subject Classification (2000): 11R20, 12E10, 12G05     

Generalization of Popoviciu-Type Inequalities Via Fink’s Identity

We obtained useful identities via Fink’s identity, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the ?eby?ev functional. Some results relating to the Grüss- and Ostrowski-type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functional associated with the obtained inequalities.     

A Generalization of Takegoshi’s Relative Vanishing Theorem

We present a generalization of Takegoshi’s relative version of the Grauert–Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi’s vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.     

Generalization of Matsui’s Algorithm 1 to linear hull for key-alternating block ciphers

We consider linear approximations of an iterated block cipher in the presence of several strong linear approximation trails. While the effect of such trails in Matsui’s Algorithm 2, also called the linear hull effect, has been previously studied by a number of authors, their effect on Matsui’s Algorithm 1 has not been investigated until now. The goal of this paper is to fill this gap and examine how to generalize Matsui’s Algorithm 1 to work also on linear hulls. We restrict to key-alternating ciphers and develop a mathematical framework for this kind of attacks. The complexity of the attack increases with the number of linear trails that have significant contribution to the correlation. We show how to reduce the number of trails and thus the complexity using related keys. Further, we illustrate our theory by experimental results on a reduced round version of the block cipher PRESENT.     

A Uniqueness Theorem of Meromorphic Maps and a Generalization of the Borel’s Lemma

This paper gives a generalization of the classical Borel's Lemma. Then as an application of this generalized Borel's Lemma, a uniqueness theorem for two linearly non-degenerate meromorphic maps of C^m into Pⁿ(C) (n ≥ 2) sharing 2n + 2 hyperplanes in general position is proved.     

A Generalization of Strassen’s Theorem on Preordered Semirings

Order - Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is...     

Generalization of Golubov’s Result for the Haar System

Mathematical Notes -     

On a Generalization of Rédei’s Theorem

In 1970 Rédei and Megyesi proved that a set of p points in AG(2,p), p prime, is a line, or it determines at least directions. In 81 Lovász and Schrijver characterized the case of equality. Here we prove that the number of determined directions cannot be between and . The upper bound obtained is one less than the smallest known example.     

Generalization of Taylor’s formula and differential transform method for composite fractional q-derivative

The Ramanujan Journal - In the present paper, we first establish a generalized q-Taylor’s formula involving composite fractional q-derivative. Next, we define the generalized q-differential...     

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.