Volume 66, Issue 3

We prove that the Cayley graph and the coset geometry of the von Dyck group D ( a , b , c ) are linked by a vertex-to-edge duality.

A poset P is called reversible if every order preserving bijective self map of P is an order automorphism. P is called hereditarily reversible if every subposet of P is reversible. We give a complete characterization of hereditarily reversible posets in terms of forbidden subsets. A similar result is stated also for preordered sets. As a corollary we extend the list of known examples of hereditarily reversible topological spaces.

We give necessary and sufficient conditions for a family M${\cal M}$ of subsets of a set X which completely separates X to be a sublattice of the lattice of C(X,≤)$\mathscr{C}(X,\leq)$ of all convex subsets of X , with respect a suitable linear ordering ≤ of X . As an application we give a characterization of Hausdorff topological spaces which are orderable or suborderable.

In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation X 4 + Y 4 = 2( U 4 + V 4 ).

Given a cyclic field extension K /ℚ of degree d and a nonzero rational integer m , we show that the equation mp 2 = x 4 - y 2 has no nontrivial solutions in K/Q${K/{\mathbb Q}}$ when p belongs to a subset of rational prime numbers of relative density at least φ ( d )/(2 d ).

In this paper, we find all solutions of the Diophantine equation x 2 + C = y n in integers x , y ≥ 1, a , b , c ≥ 0, n ≥ 3, with gcd( x , y ) = 1, when C = 2 a 3 b 17 c and C = 2 a 13 b 17 c .

In this paper, we shall derive representations of the Chowla-Walum type formula for the error term in a divisor problem related to the derivatives of the Riemann zeta-function. As a direct corollary of this formula, we shall consider estimations of this error term.

PEANO, G.: Sur une courbe, qui remplit toute une aire plane , Math. Ann. 36 (1890), 157–160] invented a surjective continuous mapping f of the unit interval I = [0, 1] onto I × I . A wellknown geometric construction was given by [HILBERT, D.: Über die stetige Abbildung einer Linie auf ein Flächenstück , Math. Ann. 38 (1891), 459–460]. In general Peano curves are constructed as limit of a series of piecewise linear functions. We here present another type of proof. A detailed survey on Peano curves is given in [WUNDERLICH, W.: Über Peano-Kurven , Elem. Math. 28 (1973), 1–10].

In the paper some properties of compositions of ϱ -upper continuous functions are presented. We will show conditions for a function f : I → I under which g ∘ f is ϱ -upper continuous for each ϱ -upper continuous function g : I → ℝ. A small generalization of Luzin-Menchoff Theorem and Zahorski Theorem will be discussed.

The aim of this paper is to characterize the products of Świa̧tkowski functions.

The object of this paper is to investigate a series of inclusion relationships of two new subclassses of meromorphically p -valent functions, defined by means of a linear operator. We also study some integral preserving properties and convolution properties of the considered classes.

We obtain some oscillation criteria for all solutions to a second-order mixed neutral differential equation with distributed deviating arguments. The results presented improve those reported in the literature.

In this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equation Rn△(t,x(t))+b(t)|Rn−1△(t,x(t))|γ−1Rn−1△(t,x(t))+q(t)f(|x(τ(t))|γ−1x(τ(t)))=0$$R_n^\triangle (t, x(t))+b(t)|R_{n-1}^\triangle(t, x(t))|^{\gamma-1} R_{n-1}^\triangle(t, x(t))+q(t)f(|x(\tau(t))|^{\gamma-1}x(\tau(t)))=0$$ on an arbitrary time scale T with sup T = ∞, where n ≥ 2, γ > 0 is a constant, τ : T → T with τ ( t ) ≤ t and limt→∞⁡τ(t)=∞$\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $ and Rk(t,x(t))=x(t),ifk=0,rk(t)Rk−1△(t,x(t)),if1≤k≤n−1,rn(t)|Rn−1△(t,x(t))|γ−1Rn−1△(t,x(t)),ifk=n,$$R_k(t, x(t))= \begin{cases} x(t),&\text{if}\ \;\;k=0, \\ r_k(t)R^\triangle_{k-1}(t, x(t)),&\text{if}\ \;\;1\leq k\leq n-1, \\ r_n(t)|R^\triangle_{n-1}(t, x(t))|^{\gamma-1} R^\triangle_{n-1}(t, x(t)),&\text{if}\ \;\;k= n, \end{cases}$$ with r k ( t ) (1 ≤ k ≤ n ) are positive rd-continuous functions. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

In this paper, we study the existence and uniqueness of global weak solution, the regularity of the solutions and the existence of global attractor for a fourth order parabolic equation modeling epitaxial thin film growth with Neumann boundary conditions in two space dimensions.

In this paper, we consider the growth of meromorphic solutions of some generalized q -shift difference equation A n ( z ) f ( qz + n ) + … + A 1 ( z ) f ( qz + 1) + A 0 ( z ) f ( qz ) = 0. Moreover, we also investigate the existence of transcendental meromorphic solutions of Fermat type of q -shift difference equations.

In this paper, oscillation criteria are obtained for higher-order half-linear delay difference equations involving generalized difference operator of the form Δb(pn(Δbm−1xn)α)+qnxn−σβ=0,  n≥n0,$${\Delta _b}({p_n}{(\Delta _b^{m - 1}{x_n})^\alpha }) + {q_n}x_{n - \sigma }^\beta = 0,\qquad n \geq {n_0},$$ where ∆ b is defined by ∆ b y n = y n +1 - by n , b ∈ ℝ - {0}, p : ℕ → ℝ + , α , β are the ratio of odd positive integers with β ≤ α ; m , n , n 0 , σ are non-negative integers, q : ℕ → ℝ. The cases of b negative and positive and q n ≥ 0, which has important role for oscillation of this equation, are considered. Also we provide some examples to illustrate our main results.

If { x j } is a sequence in a normed space X , the space of bounded multipliers for the series ∑jxj$\sum\limits_j {{x_j}} $ is defined to be M∞(∑xj)={{tj}∈l∞:∑j=1∞tjxjconverges}${M^\infty }\big({\sum {x_j}}\big) = \Big\{ {\{ {t_j}\} \in {l^\infty }:\;\sum\limits_{j = 1}^\infty {{t_j}} {x_j}\; \text{converges}} \Big\}$ and the summing operator S:M∞∑xj→X$S:{M^\infty }\left( {{\mkern 1mu} \sum {{x_j}} } \right) \to X$ is defined to be S({tj})=∑j=1∞tjxj$S(\{ {t_j}\} ) = \sum\limits_{j = 1}^\infty {{t_j}} {x_j}$. We show that if X is complete, the series ∑jxj$\sum\limits_j {{x_j}} $ is subseries convergent iff the operator S is compact and the series is absolutely convergent iff the operator is absolutely summing. Other related results are established.

In this paper we introduce Kantorovich variant of the Mittag-Leffler operators including the modified Kantorovich-Szász-Mirakjan operators. We give αβ -statistical approximation theorems for these operators in various function spaces. The results include the statistical, lacunary statistical and λ-statistical cases. Moreover, we compute the rate of convergence in different Lipschitz type spaces.

We consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

Let ϕ be an injective, continuous, Lie product preserving map on M n (ℝ), n > 3. In the paper we show that then there exist an invertible matrix T ∈ M n (ℝ) and a continuous function ψ : M n (ℝ)→ ℝ, where ψ ( A ) = 0 for all matrices of trace zero, such that either ϕ ( A ) = TAT −1 + ψ ( A ) I for all A ∈ M n (ℝ), or ϕ ( A ) = −TA t T −1 + ψ ( A ) I for all A ∈ M n (ℝ). We determine that a similar proposition holds true for the set M n (ℂ), n > 3.

The aim of this paper is to enhance certain optimality conditions for constrained vector programming problem with ℓ -stable data published by [GINCHEV, I.: On scalar and vector ℓ-stable functions, Nonlinear Anal. 74 (2011), 182–194], and by [BEDNAŘIK, D.—PASTOR, K.: On second-order condition in constrained vector optimization , Nonlinear Anal. 74 (2011), 1372–1382].

There are several ways how to redefine the Bates and Watts curvatures in models with constraints. One of possible approaches is based on a reparametrization of models. It enables us to construct linearization regions for the bias of parameter estimators, for the confidence regions, etc., in an easy way.

Kernel estimates are an integral part of non-parametric statistics. Convolutions of the kernels can be found in some expressions describing properties of kernel estimates. This paper presents the exact construction of convolutions of polynomial kernels and their properties.

We give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable q , and then to use generating functions and Rothe's identity from classical q -calculus.