Volume 68, Issue 3

We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgroup of ℝ, while it is consistent that there there is no non-null meager subgroup of ℝ.

Let S be a set of states of a physical system. The probabilities p ( s ) of the occurrence of an event when the system is in different states s ∈ S define a function from S to [0, 1] called a numerical event or, more precisely, an S - probability . A set of S -probabilities comprising the constant functions 0 and 1 which is structured by means of the addition and order of real functions in such a way that an orthomodular partially ordered set arises is called an algebra of S - probabilities , a structure significant as a quantum-logic with a full set of states. The main goal of this paper is to describe algebraic properties of algebras of S -probabilities through operations with real functions. In particular, we describe lattice characteristics and characterize Boolean features. Moreover, representations by sets are considered and pertinent examples provided.

The objective of the present paper is to introduced and study the notion of μ -approximation by a subfamily K of a difference poset E . Various properties are proved and then applied to obtain some crucial results including a generalization of the Marczewski Theorem which states that countable compactness is sufficient for σ -additivity of a supermodular measure μ .

Recently Marques and Trojovsky [On some new identities for the Fibonomial coefficients, Math. Slovaca 64 (2014), 809–818] presented interesting two sum identities including the Fibonomial coefficients and Fibonacci numbers. These sums are unusual as they include a rare sign function and their upper bounds are odd. In this paper, we give generalizations of these sums including the Gaussian q -binomial coefficients. We also derive analogue q -binomial sums whose upper bounds are even. Finally we give q -binomial sums formulæ whose weighted functions are different from the earlier ones. To prove the claimed results, we analytically use q -calculus.

Given an additive function f and a multiplicative function g , let E ( f , g ; x ) = #{ n ≤ x : f ( n ) = g ( n )}. We study the size of E ( ω , g ; x ) and E (Ω, g ; x ), where ω ( n ) stands for the number of distinct prime factors of n and Ω( n ) stands for the number of prime factors of n counting multiplicity. In particular, we show that E ( ω , g ; x ) and E (Ω, g ; x ) are Oxloglog⁡ x$\begin{array}{} \displaystyle O\left(\frac{x}{\sqrt{\log\log x}}\right) \end{array}$ for any integer valued multiplicative function g . This improves an earlier result of De Koninck, Doyon and Letendre.

We define a truncated Euler polynomial E m , n ( x ) as a generalization of the classical Euler polynomial E n ( x ). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

The class of weakly U -abundant semigroups is an important source of non-regular semigroups, and it is well studied by semigroup theorists in recent years. An important subclass, called Ehresmann monoids, is deeply investigated by Branco, Gomes and Gould in 2014. In this paper, we are concerned with weakly U -abundant semigroups with strong Ehresmann transversals. Our aim is to give a structure theorem for such semigroups following the standard “Rees Theorem” type approach. As a direct application of the main result, we reobtain the structure theorem of abundant semigroups with quasi ideal adequate transversals by Chen in 2000.

In this paper, we explore PD-type ILC schemes of fractional version with pulse compensation for single-input-single-output fractional differential systems. More precisely, we design a new type of pulse-based ILC schemes involving fractional derivative and sign function for a class of fractional order linear systems with initial state shift. In order to tracking discrepancy incurred by the initial state shift effectively, a new function of pulse compensation is introduced. The effectiveness of the result is illustrated by numerical simulations.

We study a geometric kind of asymptotic behaviour of every C 1 solution of a class of nonautonomous systems of half-linear differential equations with continuous coefficients. We give necessary and sufficient conditions such that the image of every solution (solution curve) has finite length (rectifiable curve) and infinite length (nonrectifiable, possible fractal curve). A particular attention is paid to systems having attractive zero solution. The main results are proved by using a new result for the nonrectifiable plane curves.

We use the notion of weighted sharing of sets to improve a number of results in the uniqueness theory of meromorphic functions sharing three sets. We have also presented a new idea in the formation of one of the range sets in connection with the three shared set problems, which ultimately yields a better result than many existing result in some sense.

In this article, a class of analytic functions is investigated and their some properties are established. Several recurrence relations and various classes of bilinear and bilateral generating functions for these analytic functions are also derived. Examples of some members belonging to this family of analytic functions are given and differential equations satisfied by these functions are also obtained.

We deal with the interest rate model proposed by Schaefer and Schwartz, which models the long rate and the spread, defined as the difference between the short and the long rates. The approximate analytical formula for the bond prices suggested by the authors requires a computation of a certain constant, defined via a nonlinear equation and an integral of a solution to a system of ordinary differential equations. A quantity entering the nonlinear equation is expressed in a closed form, but it contains infinite sums and evaluations of special functions. In this paper we use perturbation methods to compute the constant of interest as an asymptotic serie with coefficients given in closed form and expressed using elementary functions. A quick computation of the bond prices, which our approach allows, is essential for example in calibration of the model by means of fitting the observed yields, where the theoretical bond prices need to be recalculated for every observed date and maturity, as well as every combination of parameters considered. The first step of our derivation is identification of a small parameter in the problem, since it is not immediately clear. We verify our choice by numerical experiments using the values of parameters from the literature.

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation xn+1=α xn− 1+β γ xnxn− 1,n∈ N0,$$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$ where ℕ 0 = ℕ ∪ {0}, α , β , γ ∈ ℝ + , and the initial conditions x –1 and x 0 are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by xn+1=α xn− 1+β γ ynxn− 1,yn+1=α yn− 1+β γ xnyn− 1,n∈ N0.$$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$

We discuss the existence and uniqueness of fixed points for a self-mapping defined on a C ∗ -algebra valued b -metric space endowed with a graph. Our results extend and supplement several recent results in the literature. Some examples are provided to illustrate our results. Finally, as an application of G -contraction mapping theorem, existence of unique solution for a type of operator equation is given.

In this paper a general fixed point theorem for mappings with a new type of common limit range property satisfying a mixed implicit relation is proved. In the last part of the paper, as application, some fixed point results for mappings satisfying contractive conditions of integral type and for φ -contractive mappings are obtained.

We characterize productively sequential spaces, that is, spaces whose product with every strongly sequential space is sequential, equivalently strongly sequential. It turns out that a regular topology is productively sequential if and only if it is sequential and bi-quasi- k .

We show that when k ≠ 2, 4, 8 the Euler class of any vector bundle over Σ k ℝℙ m is zero if the rank of the bundle is not m + k , provided that m ≠ 3 when k = 6. If k = 2, 4, 8 we show that the Euler class of any vector bundle over Σ k ℝℙ m is zero whenever the rank of the bundle is not kr + k , provided that m ≠ 6, 7 when k = 2, where r is the largest integer such that kr ≤ m .

This paper is concerned with a stochastic mutualism system with toxicant substances and saturation terms. We obtain the sufficient conditions for the existence of a unique stationary distribution to the equation and it has an ergodic property. It is interesting and surprising that toxicant substances have no effect on the stationary distribution of the stochastic model. Simulations are also carried out to confirm our analytical results.

In this paper, we study the small restricted injective dimension over local ring homomorphisms. Some of known results are generalized. For example, the Bass formula for the small restricted injective dimension of complexes is extended.