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.     

Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

The famous irreducibility criteria of Schönemann–Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann–Eisenstein–Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.     

On a mild generalization of the Schönemann irreducibility criterion

We state a mild generalization of the classical Schönemann irreducibility criterion in ?[x] and provide an elementary proof.     

On the Rate of Overconvergence of the Generalized Eneström-Kakeya Functional for Polynomials

The classical Eneström-Kakeya Theorem, which provides an upper bound for the moduli of zeros of any polynomial with positive coefficients, has been recently extended by Anderson, Saff and Varga to the case of any complex polynomial having no zeros on the ray [0,$+∞$). Their extension is sharp in the sense that, given such a complex polynomials $p_n(z)$ of degree $n≥1$, a sequence of multiplier polynomial can be found for which the Eneström-Kakeya upper bound, applied to the products $Q_{mi}(z)$ · $p_n(z)$, converges, in the limit as $i$ tends to $∞$, to the maximum of the moduli of the zeros of $p_n(z)$. Here, the rate of convergence of these upper bounds is studied. It is shown that the obtained rate of convergence is best possible.     

On a Generalized Chu–Vandermonde Identity

In the present paper we introduce a generalization of the well–known Chu–Vandermonde identity. In particular, by inductive reasoning, the identity is extended to a multivariate setup in terms of the fourth Lauricella function. The main interest in such generalizations derives from the species diversity estimation and, in particular, prediction problems in Genomics and Ecology within a Bayesian nonparametric framework.     

On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

We consider the generalized Hurwitz equation \({a_1x_1^2 + \cdots + a_nx_n^2 = dx_1 \cdots x_n - k}\) and the Baragar–Umeda equation \({ax^2 + by^2 + cz^2 = dxyz + e}\) for solvability in integers.     

On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications

Ukrainian Mathematical Journal - We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms $$ left(hunderset{F_s,K}{ast }fright)(x) $$ in a two-parameter...     

On Convergence of Chlodovsky and Chlodovsky–Kantorovich Polynomials in the Variation Seminorm

The aim of this paper is to study the variation detracting property and rate of approximation of the Chlodovsky and Chlodovsky–Kantorovich polynomials in the space of functions of bounded variation with respect to the variation seminorm.     

On the Centralizer Dimension and Lattice of Generalized Baumslag–Solitar Groups

A generalized Baumslag–Solitar group (a GBS group) is a finitely generated group G acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups. Each GBS group is the fundamental group π₁(A) of some labeled graph A. We describe the centralizers of elements and the centralizer lattice. Also, we find the centralizer dimension for GBS groups if A is a labeled tree.     

On Constrained Markov–Nikolskii type Inequalities for k-Absolutely Monotone Polynomials

We consider a classical problem of estimating norms of higher order derivatives of an algebraic polynomial via the norms of the polynomial itself. The corresponding extremal problem for general polynomials in the uniform norm was solved by V. A. Markov. In 1926, Bernstein found the exact constant in the Markov inequality for monotone polynomials. It was shown in [3] that the order of the constants in constrained Markov–Nikolskii inequality for k-absolutely monotone polynomials is the same as in the classical one in case \({0 < p \leqq q \leqq \infty}\) . In this paper, we find the exact order for all values of \({0 < p, q \leqq \infty}\) . It turnes out that for the case q < p the constrained Markov–Nikolskii inequality is significantly better than the unconstrained one.     

On the Generalized Szász–Mirakyan Operators

In this paper, we construct sequences of Szász–Mirakyan operators which are based on a function ρ. This function not only characterizes the operators but also characterizes the Korovkin set ${\left \{ 1,\rho ,\rho ^{2} \right \}}$ in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function ρ and which are subspaces of the space of continuous functions on ${\mathbb{R} ^{+}}$ . We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function ρ. Further, we prove some shape-preserving properties of the operators such as the ρ-convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szász operators.     

Hyperelliptic Sigma Functions and Adler–Moser Polynomials

Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in “Functional Analysis and Its Applications,” for each $$g &gt; 0$$ , a...     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.