On a mild generalization of the Schönemann - Eisenstein - Dumas irreducibility criterion

We state a mild generalization of the classical Schönemann and Eisenstein- Dumas irreducibility criterion in ?[x] and provide an elementary proof. In the end of the paper, we also provide a concrete example of a polynomial which is irreducible by the main result of the paper but whose irreducibility does not follow from existing criteria.     

A generalization of Eisenstein�CSch?nemann irreducibility criterion

One of the results generalizing Eisenstein Irreducibility Criterion states that if ${\phi(x) = a_nx^n\,{+} \,a_{n-1}x^{n-1} \,{+} \,\cdots\,{+} \,a_0}$ is a polynomial with coefficients from the ring of integers such that a _s is not divisible by a prime p for some ${s \, \leqslant \, n}$ , each a _i is divisible by p for ${0 \, \leqslant \, i \, \leqslant \, s-1}$ and a ₀ is not divisible by p ², then ${\phi(x)}$ has an irreducible factor of degree at least s over the field of rational numbers. We have observed that if ${\phi(x)}$ is as above, then it has an irreducible factor g(x) of degree s over the ring of p-adic integers such that g(x) is an Eisenstein polynomial with respect to p. In this paper, we prove an analogue of the above result for a wider class of polynomials which will extend the classical Sch?nemann Irreducibility Criterion as well as Generalized Sch?nemann Irreducibility Criterion and yields irreducibility criteria by Akira et?al. (J Number Theory 25:107?C111, 1987).     

On a generalization of the Hadwiger–Nelson problem

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG _Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v ? w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger–Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG _Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].     

On a generalization of Gross’s formula

We prove a relation between Whittaker functionals of cusp forms on ${{\widetilde{SL}}_2}$ and the toric periods of forms on the quaternion algebra. As an application we prove a generalization of a formula of Gross.     

On a generalization of Deuringʼs results

Using the Dieudonné theory we will study a reduction of an abelian variety with complex multiplication at a prime. Our results may be regarded as generalization of the classical theorem due to Deuring for CM-elliptic curves. We will also discuss a sufficient condition for a prime at which the reduction of a CM-curve is maximal.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

On dominant contractions and a generalization of the zero–two law

Zaharopol proved the following result: let \({T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}\) be two positive contractions such that T ≤ S. If \({\|S-T\| <1 }\) then \({\left\|S^n-T^n\right\| <1 }\) for all \({n\in\mathbb{N}}\). In the present paper we generalize this result to multi-parameter contractions acting on L ¹. As an application of that result we prove a generalization of the “zero–two” law.     

On a generalization of a Greguš fixed point theorem

Let C be a closed convex subset of a complete convex metric space X. In this paper a class of selfmappings on C, which satisfy the nonexpansive type condition (2) below, is introduced and investigated. The main result is that such mappings have a unique fixed point.     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

A generalization of the Chung-Erdös inequality for the probability of a union of events

A generalization of the Chung-Erdös inequality for the probability of a union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least m events from a set of events A₁,..., A_n, where 1 ≤ m ≤ n. Bibliography: 6 titles.     

On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality

We show that a problem asked by Yau (Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319, 1992) cannot be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng (Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. 17, pp. 517–542, 2010).     

On a generalization of Kronecker’s limit formula

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.