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In this paper, we prove some extensions and generalizations of the classical Enestrom-Kakeya theorem.

Recently, Lebesgue-Radon-Nikodym theorem with respect to fermionic 𝑝 -adic invariant measure on ℤ 𝑝 was studied in Kim. In this paper we will give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to 𝑝 -adic 𝑞 -measure on ℤ 𝑝 . In special case, 𝑞 = 1 , we can derive the same results in Kim.

This paper is a revised version of my previous paper [4]. Main purpose is to give a right proof for non-archimedean Hartogs theorem(Stawski’s theorem) in case when the value group of the field K is discrete.

In this paper we obtain results concerning the bound for the number of zeros for the polynomial �� (�� ) which generalizes well known result due to A.Ebadian ,M.Bidkham and M.Eshaghi Gordji (Number of zeros of a polynomial in a given domain, Tamkang Jour, of Mathematics, Vol 42, No.4,(2011), 531-536) and also improves upon some well-known results.

In this paper we give an effective version of the Bombieri–Vinogradov theorem of Bombieri, Friedlander, and Iwaniec. As applications, we refine Chen’s theorem on primes with difference h for h ≤ x 1/56, with a consequence on integers of the form p + 2 k .

In the present paper, we will derive the Liouville theorem and the finite dimension theorem for polynomial growth harmonic functions defined on Alexandrov spaces with nonnegative Ricci curvature in the sense of Kuwae-Shioya and Sturm-Lott-Villani.

In 1947 Henry Scheffe published a result which afterwards became known as Scheffe’s theorem, stating that the distributions of a sequence (f n ) of densities, which converge almost everywhere to a density f, converge uniformly to the distribution of f. But almost 20 years earlier Frigyes Riesz proved a sufficient condition for convergence in the p-th mean (p ≥ 1), wherefrom the Scheffe theorem is just a special case.

Villena's theorem ensure the automatic continuity of a derivation on a non-associative H*-algebra with zero annihilator. In this paper, we formulate the Jewell-Sinclair theorem in a non-associative content, and we show that the Villena's theorem may be also followed from our formulation of such theorem.

Our object is to present an independent proof of the extension of V.A. Markov's theorem to Gâteaux derivatives of arbitrary order for continuous polynomials on any real normed linear space. The statement of this theorem differs little from the classical case for the real line except that absolute values are replaced by norms. Our proof depends only on elementary computations and explicit formulas and gives a new proof of the classical theorem as a special case. Our approach makes no use of the classical polynomial inequalities usually associated with Markov's theorem. Instead, the essential ingredients are a Lagrange interpolation formula for the Chebyshev nodes and a Christoffel–Darboux identity for the corresponding bivariate Lagrange polynomials. We use these tools to extend a single variable inequality of Rogosinski to the case of two real variables. The general Markov theorem is an easy consequence of this.

We recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

Recently Totik and Varju [17] presented an estimate of the uniform norm of a monic polynomial with prescribed zeros on the unit circle. In this paper we improve their estimate and extend it to the case of polynomials with some zeros on an arbitrary analytic curve in the complex plane.

The classical Julia–Wolff–Carathéodory Theorem is one of the main tools to study the boundary behavior of holomorphic self-maps of the unit disc of ℂ. In this paper we prove a Julia–Wolff–Carathéodory type theorem in the case of the polydisc of . The Busemann functions are used to define a class of “generalized horospheres” for the polydisc and to extend the notion of non-tangential limit. With these new tools we give a generalization of the classical Julia Lemma and of the Lindelöf Theorem, which the new Julia–Wolff–Carathéodory Theorem relies upon.

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.

The paper is concerned with endomorphism algebras for weak Doi-Hopf modules. Under the condition “weak Hopf-Galois extensions”, we present the structure theorem of endomorphism algebras for weak Doi-Hopf modules, which extends Theorem 3.2 given by Schneider in [1]. As applications of the structure theorem, we obtain the Kreimer-Takeuchi theorem (see Theorem 1.7 in [2]) and the Nikshych duality theorem (see Theorem 3.3 in [3]) in the case of weak Hopf algebras, respectively.

We prove that there exists an absolute constant c>0 such that if A is a set of n monic polynomials, and if the product set A.A has at most n 1+c elements, then |A+A|≫n2. This can be thought of as step towards proving the Erdős–Szemeredi sum-product conjecture for polynomial rings. We also show that under a suitable generalization of Fermat’s Last Theorem, the same result holds for the integers. The methods we use to prove are a mixture of algebraic (e.g. Mason’s theorem) and combinatorial (e.g. the Ruzsa–Plunnecke inequality) techniques.

In the paper an elementary and simple proof of the Fundamental Theorem of Algebra is given.

Let Fq[t] denote the polynomial ring over the finite field Fq, and let SN denote the subset of Fq[t] containing all polynomials of degree strictly less than N . For a matrix Y = ` ai,j ∈ FR×S q satisfying ai,1 + · · ·+ai,S = 0 (1 ≤ i ≤ R), let DY (SN ) denote the maximal cardinality of a set A ⊆ SN for which the equations ai,1x1 + · · ·+ai,SxS = 0 (1 ≤ i ≤ R) are never satisfied simultaneously by distinct elements x1, . . . , xS ∈ A. Under certain assumptions on Y , we prove an upper bound of the form DY (SN ) ≤ q (C/N) for positive constants C and γ.

We prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables with finite second moment.

The following theorem is proved. Theorem. Let P (x) = P 2n k=0 akx k be a polynomial with positive coefficients. If the inequalities a 2 2k+1 a2ka2k+2 0 for every x 2 R . We show that the constant 1 cos2( � n+2 ) in this theorem could not be increased. We also present some corollaries of this theorem. 2000 Mathematics Subject Classification 30C15; 26C10.

A variant of Cauchy mean-value theorem is presented. Applying a counterpart of the Lagrange mean-value theorem, we introduce new type of means. The functions generating the homogeneous means of this type are determined. In some special cases the effective formulas for these means are presented. Mean-value-theorems which do not lead to any mean are also mentioned.

Let P be a non empty poset. Observe that there exists a chain of P which is non empty. Let I1 be a relational structure. We say that I1 is chain-complete if and only if: (Def. 1) I1 is lower-bounded and for every chain L of I1 such that L is non empty holds sup L exists in I1. One can prove the following proposition (1) Let P1, P2 be non empty posets, K be a non empty chain of P1, and h be a monotone function from P1 into P2. Then h◦K is a non empty chain of P2. Let us note that there exists a poset which is strict, chain-complete, and non empty. Let us mention that every relational structure which is chain-complete is also lower-bounded.

In this paper, we prove some generalizations of results concerning the Enestrom-Kakeya theorem. The results obtained considerably improve the bounds by relaxing the hypothesis in some cases.

The paper deals with absolute summability factors for infinite series. The main result obtained in this paper generalizes a recent paper of Rhoades and Savas [B. Rhoades, E. Savas, A summability factor theorem for generalized absolute summability, Real Anal. Exchange 31 (2) (2005-2006) 355-363].

We give a brief introduction to re-publication of Chapter 2 of the author's 1964 Dissertation. The result has come to be known as the Neimark-Sacker bifurcation theorem. Some of the applications of the theorem are cited and a brief overview of the reduction of the mapping to normal form is given in 3-dimensions. By introducing weighted monomials the reduction method is carried out without the use of the Center Manifold theorem which was not known to the author and was published the same year as the dissertation. The first detailed proof of the theorem is carried out in the above cited Chapter 2 and is contained in the companion article to follow this one.

We generalize a theorem of Birch on solving systems of homogeneous equations of odd degree to the case of solving systems of homogeneous equations of degree not divisible by a fixed integer s ≥ 2. Birch's theorem is the case s = 2. In addition, we give a new exposition of the proof of Brauer's theorem on solving systems of homogeneous equations.

For complex square matrices, the Levy–Desplanques theorem asserts that a strictly diagonally dominant matrix is invertible. The well-known Gersgorin theorem on the location of eigenvalues is equivalent to this. In this article, we extend the Levy–Desplanques theorem to an object in a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. As a consequence, we prove a Gersgorin type theorem for the spectral eigenvalues of an object in a Euclidean Jordan algebra.

This paper deals with comparing the Apostolatos-Kulisch theorem and the Mayer-Warnke theorem, which form a basis for the so-called formal (algebraic) approach to outer interval estimation of solution sets for interval linear systems of equations. It is shown that despite the greater generality of the Mayer-Warnke theorem, it does not essentially extend the scope of applicability of the formal approach, and the practical significance of this extension is very small.

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)^2-1 values of the coefficient. We more generally handle the situation where several specified coefficients vary., 21 pages

Let $$ P(z) = a_n z^n + a_n - 1z^{n - 1} + \cdots + a_0 $$ be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coefficients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).

We extend the free version8 of the Laha–Lukacs theorem to probability spaces with two states. We then use this result to generalize the noncommutative central limit theorem of Kargin23 to the two-state setting.

We prove a version of the Hilbert Lemniscate Theorem in C n . More precisely, any polynomially convex compact subset K of C n can be approximated externally by special polynomial polyhedra P defined by proper polynomial mappings from C n to C n with “almost” all their zeros in P . We precise this version when K is balanced. And we also give several applications of these results: approximation of the pluricomplex Green function for a compact set with pole at infinity, a Lelong–Bremermann Theorem for functions in L + and uniform polynomial approximation of holomorphic functions.

In an earlier paper Savas and Rhoades [Ekrem Savaş, B.E. Rhoades, A note on | A |k summability factors, Nonlinear Anal. 66 (2007) 1879-1883. [1]] obtained a summability factor theorem for absolute summability of order k ? 1. In this paper we extend that result to doubly infinite matrices. © 2008 Elsevier Ltd. All rights reserved.

The Perron-Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles.

In this paper, we extend the Perron-Frobenius theorem for positive polynomial operators in Banach lattices. The result obtained is applied to derive necessary and sufficient conditions for the stability of positive polynomial operators. Then we study stability radii: complex, real and positive radii of positive polynomial operators and show that in this case the three radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.

We prove the following generalization of the classical Shephard–Todd–Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring \(A:=k_{p_{ij}}[x_1,\cdots,x_n]\). Then the fixed subring AG has finite global dimension if and only if G is generated by quasi-reflections. In this case the fixed subring AG is isomorphic to a skew polynomial ring with possibly different pij’s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

We prove a quantitative result on the existence of linearly independent polynomial configurations in the difference set of sparse subsets of the integers. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy and then applying a simple lifting argument.

For a polynomial of degree n, we have obtained an upper bound involving coefficients of the polynomial, for moduli of its p zeros of smallest moduli, and then a refinement of the well-known Enestrom-Kakeya theorem (under certain conditions).

The joint source-channel coding problem is considered. The joint source-channel coding theorem reveals the existence of a code for the pair of the source and the channel under the condition that the error probability is smaller than or equal to ∈ asymptotically. The separation theorem guarantees that we can achieve the optimal coding performance by using the two-stage coding. In the case that ∈ = 0, Han showed the joint source-channel coding theorem and the separation theorem for general sources and channels. Furthermore the ∈-coding theorem (0 ≤ ∈ < 1) in the similar setting was studied. However, the separation theorem was not revealed since it is difficult in general. So we consider the separation theorem in this setting.

We prove the following simple uniqueness theorem: Let A and B be two integral symmetric matrices with the same irreducible characteristic polynomial. If there exist rational orthogonal matrices Q such that B=QTAQ, then Q is unique up to a sign. An application of the theorem to the reconstruction conjecture is provided.

The aim of the present paper is to define a class of nonlinear functions on the set of non-negative real numbers and study the dynamics and periodic points of these functions. We show that this class of functions contains chaotic functions and exhibits the phenomenon of bifurcation. We also apply the results on dynamics of these functions to obtain general tests of divisibility of numbers and conditions of divisibility of polynomials. We not only obtain generalizations of the well-known remainder theorem and the factor theorem of algebra but also compute an explicit value of the quotient.

The Fifteen Theorem proved by Conway and Schneeberger is a criterion for quadratic forms over the rational integer ring to be universal. In this article, we give a proof of an analogy of the Fifteen Theorem for quadratic forms over polynomial rings, which is known as the Four Conjecture proposed by Gerstein.

Formulation of the sampling theorem for deterministic signals with a limited energy is presented. Basic primary sources of the sampling theorem and the role of the authors of these sources in the development of signal theory are analyzed. A conclusion about the correctness of referring to the signal-theory sampling theorem as the Kotel’nikov-Shannon theorem is drawn.

Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p ⩾ 0 . Assume that p is good for G. Pommerening's theorem [K. Pommerening, Uber die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Uber die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra g of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening's theorem, which is a further simplification of Premet's first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet's proof of the existence theorem for good transverse slices to the nilpotent Ad ( G ) -orbits in g .

The theorem in question gives a sufficient condition for a quadratic space over a local ring R to contain a hyperbolic plane over R.

We prove a Hadamard-type theorem that associates the generalized order of growth $\rho_{f}^*({\alpha}, {\beta})$ of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate $\rho_{f}^*({\alpha}, {\beta})$ with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G.

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer�Sobczyk type decomposition theorem for modular measures on D-lattices.