Volume 69, Issue 1

We introduce the concept of a translation ideal in pseudo-BCH algebras and investigate its properties. We prove that for any pseudo-BCH algebra 𝔛 there is a one-to-one order-preserving correspondence between the closed translation ideals of 𝔛 and the relative congruences on 𝔛.

In this paper, we investigate the variety RDP of regular double p -algebras and its subvarieties RDP n , n ≥ 1, of range n . First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP 1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP 1 , L V ( RDP 1 ), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP 1 and conclude that L V ( RDP 1 ) is countably infinite. An equational basis for each proper subvariety of RDP 1 is given. To study the subvarieties RDP n with n ≥ 2, Priestley duality as it applies to regular double p -algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2 ℵ 0 . We also use Priestley duality to prove that RDP and each of its subvarieties RDP n are generated by their finite members.

The aim of this paper is to study the class of distributive nearlattices with a necessity modal operator. We develop a full duality to the category of distributive nearlattices whose morphisms are applications that preserving the infimum when exists and, as special case, we obtain a representation and duality for distributive nearlattices with a necessity modal operator. We study certain particular subclasses and give some applications.

We present a characterization of states in generalized probabilistic models by appealing to a non-commutative version of geometric probability theory based on algebraic geometry techniques. Our theoretical framework allows for incorporation of invariant states in a natural way.

A power-associative n -ary groupoid is an n -ary groupoid G such that for every element a ∈ G , the n -ary subgroupoid of G generated by a is an n -ary subsemigroup of G . The class 𝓟 a of power-associative n -ary groupoids is a variety. A description of free objects in this variety and their characterization by means of injective n -ary groupoids in 𝓟 a are obtained.

We study the structure of the Galois group of the maximal unramified 2-extension of some family of number fields of large degree. Especially, we show that for each positive integer n , there exist infinitely many number fields with large degree, for which the defined Galois group is quaternion of order 2 n .

An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

We prove a periodic theorem of meromorphic functions of hyper-order ρ 2 ( f ) < 1. As an application, we obtain the corresponding uniqueness theorem on periodic meromorphic functions. In addition, we show the accuracy of the results by giving some examples.

The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

In this paper, we show that in dimension n ≥ 3, the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.

In this paper, we introduce the concept of entropy kernel operator for compact dynamical systems of finite Kolmogorov entropy. It is a compact positive operator on a Hilbert space. Then we state the Kolmogorov entropy in terms of the eigenvalues of the entropy kernel operator.

In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations xn+1=axnxn− 1bxn− 1+cxn− 2,n=0,1,… $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a , b , c are positive real numbers and the initial conditions x −2 , x −1 , x 0 are real numbers.

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ -additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

We show that the following properties are preserved under inverse limits: countable fan-tightness, q + , discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces.

This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

The notion of porouscontinuity was introduced by Borsík and Holos. We find classes of functions whose compositions with porouscontinuous functions are still porouscontinuous. Next, we study compositions of porouscontinuous functions with homeomorphisms. We investigate connections between 𝓘𝓒(𝓐) ∩ 𝓗𝓞𝓜 and 𝓘𝓒(𝓑) ∩ 𝓗𝓞𝓜 for different classes 𝓐,𝓑 ∈ 𝓟𝓒, where 𝓟𝓒 is the set of all classes of porouscontinuous functions and 𝓗𝓞𝓜 is the set of all homeomorphisms f : ℝ → ℝ.

Let F be a number field, O F the integral closure of ℤ in F , and P ( T ) ∈ O F [ T ] a monic separable polynomial such that P (0) ≠ 0 and P (1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of O F such that the reduction modulo 𝓟 of P ( T ) has a root in the residue field O F /𝓟, but the reduction modulo 𝓟 of P ( T k ) has no root in O F /𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.

In this paper, the complete convergence for the weighted sums of arrays of rowwise extended negatively dependent (END, for short) random variables is established under some mild conditions. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for arrays of rowwise END random variables is also obtained. The result obtained in the paper generalizes and improves some corresponding ones for independent random variables and some dependent random variables in some extent. By using the complete convergence that we established, we further study the complete consistency for the weighted estimator in a nonparametric regression model based on END errors.

In this paper, we establish fixed point theorems for one and two selfmaps in b -metric-like spaces, using ( s , q )-contractive and F -( ψ , φ , s , q )-contractive conditions, defined by means of altering distances and 𝓒- class functions. Our theorems unify, extend and generalize corresponding results in the literature.