Minimal but inefficient presentations of the semi-direct products of some monoids

In recent papers of Ruskuc, Saito and J. Wang, the semi-direct product of two arbitrary monoids and a standard presentation, say P, for this product have received considerable attention. Wang defined a trivialiser set of the Squier complex associated with P and after that necessary and sufficient conditions for P to be efficient have been given by Cevik. As a main result of this paper, we give sufficient conditions for a presentation of the semi-direct product of a one-relator monoid by an infinite cyclic monoid to be minimal but not efficient. In the final part of this paper we give some applications of this result.     

On a class of lacunary series in BMOA

In this paper we consider a special class of lacunary series and we characterize their membership in the BMOA of the unit ball of Cn in terms of their Taylor coefficients. In addition, we show that a similar result holds for the class VMOA.     

Radar placement along banks of river

In this paper, we consider the Radar Placement and Power Assignment problem (RPPA) along a river. In this problem, a set of crucial points in the river are required to be monitored by a set of radars which are placed along the two banks. The goal is to choose the locations for the radars and assign powers to them such that all the crucial points are monitored and the total power is minimized. If each crucial point is required to be monitored by at least k radars, the problem is a k-Coverage RPPA problem (k-CRPPA). Under the assumption that the river is sufficiently smooth, one may focus on the RPPA problem along a strip (RPPAS). In this paper, we present an O(n ⁹) dynamic programming algorithm for the RPPAS, where n is the number of crucial points to be monitored. In the special case where radars are placed only along the upper bank, we present an O(kn ⁵) dynamic programming algorithm for the k-CRPPAS. For the special case that the power is linearly dependent on the radius, we present an O(n log n)-time \({2\sqrt 2}\)-approximation algorithm for the RPPAS.     

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.     

A new subclass of meromorphic functions with positive coefficients

In the present paper, we consider the new subclass of Meromorphic functions with positive coefficients and obtain a necessary and sufficient condition for a function f to be in this class. We obtain distortion properties, radius of convexity, starlikeness, convex linear combinations for the function f in this class.     

The convergence behavior of q-continued fractions on the unit circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, Y _G , on the unit circle such that if y ? Y _G then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.     

A note on uniform strong convergence of bivariate density estimates

In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.     

On the existence of a cycle in an asymmetric model of a molecular repressilator

In this paper, a nonlinear six-dimensional dynamic system, which is a model of functioning of a simple molecular repressilator, is considered. Sufficient conditions for the existence of a cycle C in the phase portrait of this system are found. An invariant neighborhood of C, which retracts to C, is constructed.     

A Characterization of Binary Bent Functions

A recent paper by Carlet introduces a general class of binary bent functions on (GF(2))ⁿ(neven) whose elements are expressed by means of characteristic functions (indicators) of (n/2)-dimensional vector-subspaces of (GF(2))ⁿ. An extended version of this class is introduced in the same paper; it is conjectured that this version is equal to the whole class of bent functions. In the present paper, we prove that this conjecture is true.     

Large strings of consecutive smooth integers

In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erdös that measures the minimal gap size f(k) in the sequence of integers at least one of whose prime factors exceeds k. This allows us to compute values of f(k) for larger k and obtain new values of f(k).     

Monotone optimal control for a class of Markov decision processes

This paper provides a unified framework to study monotone optimal control for a class of Markov decision processes through D-multimodularity. We demonstrate that each system in this class can be classified as either a substitution-type or a complement-type system according to the possible transition set, which can be used as a classification mechanism that integrates a variety of models in the literature. We develop a generic proof of the structural properties of both types of system. In particular, we show that D-multimodularity is a generally sufficient condition for monotone optimal control of different types of system in this class. With this unified theory, there is no need to pursue each problem ad hoc and the structural properties of this class of MDPs follow with ease.     

The Factorization of Queueing Equations and Their Interpretation

There are many queueing systems, including the M ^x /M ^y /c queue, the GI ^x /M/c queue and the M/D/c queue, in which the distribution of the queue length at certain epochs is determined by a Markov chain with the following structure. Except for a number of boundary states, all columns of the transition matrix are identical except for a shift which assures that there is always the same element occupying the main diagonal. This paper describes how one can find the equilibrium distribution for such Markov chains. Typically, this problem is solved by factorizing of a certain expression characterizing the repeated columns. In this paper, we show that this factorization has a probabilistic significance and we use this result to develop new approaches for finding the equilibrium distribution in question.     

On a novel eccentricity-based invariant of a graph

In this paper,for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs,a novel eccentricity-based invariant,named as non-self-centrality number(NSC number for short),of a graph G is defined as follows:N(G)=∑v_i,v_j∈V(G)|e_i-e_j| where the summation goes over all the unordered pairs of vertices in G and e_i is the eccentricity of vertex v_i in G,whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G.In this paper,we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained.Finally we propose some attractive research topics for this new invariant of graphs.     

An interpretation of the tame local Langlands correspondence for <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">PGSp</Emphasis>(4) from the perspective of real groups

Let F be a p-adic field. In this paper, we continue the work of the first author and give a new realization of the tame local Langlands correspondence for PGSp(4, F) that is analogous to the construction of the local Langlands correspondence for real groups.     

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α β]: When the phase f(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2: This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on R: In the present paper, however, these functions are only assumed to be continuously differentiable on [α β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.     

Orthogonal sets: The axiom of choice and proof of a fixed point theorem

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L ^p space.     

Arithmetical rank of binomial ideals

In this paper, we investigate the arithmetical rank of a binomial ideal J. We provide lower bounds for the binomial arithmetical rank and the J-complete arithmetical rank of J. Special attention is paid to the case where J is the binomial edge ideal of a graph. We compute the arithmetical rank of such an ideal in various cases.     

Analysis of a class of nonlinear and non-separable multiscale representations

In this paper, we introduce a particular class of nonlinear and non-separable multiscale representations which embeds most of these representations. After motivating the introduction of such a class on one-dimensional examples, we investigate the multi-dimensional and non-separable case where the scaling factor is given by a non-diagonal dilation matrix M. We also propose new convergence and stability results in L ^p and Besov spaces for that class of nonlinear and non-separable multiscale representations. We end the paper with an application of the proposed study to the convergence and the stability of some nonlinear multiscale representations.     

Pentavalent symmetric graphs of order 16p

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order 16p is given for each prime p. It follows from this result that a connected pentavalent symmetric graph of order 16p exists if and only if p = 2 or 31, and that up to isomorphism, there are three such graphs.     

Weighted-mean regions of a probability distribution

In this paper we investigate a new class of central regions for probability distributions on R^d, called weighted-mean regions. Their restrictions to an empirical distribution are the weighted-mean trimmed regions investigated by Dyckerhoff and Mosler (2011) for d-variate data. Furthermore a new class of stochastic orderings of variability, the weighted-mean orderings, is introduced.