Volume 66, Issue 4

It is shown that every effect algebra with a full set of states can be represented as a so-called numerical algebra introduced in the paper. For numerical algebras there are introduced tense operators which indicate dynamical changes of quantum events depending on variability of states. These operators enable to recognize an effect algebra with a full set of states as a temporal logic where events are quantified by these tense operators. The problem of representation of tense operators on a given numerical algebra is solved.

Several characterizations of orthomodular lattices based on properties of an antitone involution or on sectional antitone involutions are given. Another approach is based on properties of the implication operation which can be derived in every orthomodular lattice but can be also added to basic operations of a bounded lattice. Finally, we introduce the so-called weakly orthomodular lattice as a generalization of orthomodular lattices.

In this paper we show that the free BCI-algebra with one generator has an infinite number of branches and that every branch (as a poset) has the least and the greatest element and is of infinite length and width.

Narayana’s cows problem is a problem similar to the Fibonacci’s rabbit problem. We define the numbers which are the solutions of this problem as Narayana’s cows numbers. Narayana’s cows sequence satisfies the third order recurrence relation N r = N r −1 + N r −3 with initial condition N 0 = N 1 = N 2 = 1. In this paper, we define a new recurrence which is called generalized order- k Narayana’s cows sequence and by using this generalization and some matrix properties, we give some identities related to the Narayana’s cows numbers.

The main results of this paper give several criteria for certain infinite series of rational numbers to be irrational, transcendental or Liouville. Criteria are based on specific reccurence relations. Several examples are included.

In this paper we prove the following result. Let R be a n !-torsion free semiprime ring and let f : R → R be an additive mapping satisfying the relation f ( x ) x n + x n f ( x ) = 0 for all x ∈ R . In this case f = 0.

In this paper, we prove a general result that can be a sufficient condition for f to be in the some subclass of strongly starlike functions. Also, the related sandwich-type results are given.

In this paper we introduce a new class of analytic functions involving the integral operator I n defined by Sălăgean. For this class we provide inclusion relation, extreme points and coefficient estimates.

The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a value or small function. The results of this paper are generalizations of some problems studied in [BOUSSAF, K.—ESCASSUT, A.—OJEDA, J.: Complex meromorphic functions f′P′(f) , g′P′(g) sharing a small function , Indag. Math. (N.S.) 24 (2013), 15–41] and in [DYAVANAL, R. S.: Uniqueness and value-sharing of differential polynomials of meromorphic functions , J. Math. Anal. Appl. 374 (2011), 335–345].

The paper concerns inclusions of Bavrin families of complex valued functions which are hololomorphic in bounded complete n -circular domains G⊂ Cn$\mathcal{G} \subset \mathbb{C}^{n}$ and fulfil some geometric conditions. Such families XG$\mathcal{X}_{G}$ were applied later to research some families of locally biholomorphic mappings in Cn$\mathbb{C}^{n}$. (see for instance, J. A. Pfaltzgraff and T. J Suffridge, P. Liczberski, I. Graham and G. Kohr, H. Hamada, T. Honda, G. Kohr). Some results for functions of such families XG$\mathcal{X}_{G}$ were obtained by [BAVRIN, I. I.: A Class of Regular Bounded Functions in the Case of Several Complex Variables and Extreme Problems in That Class , Moskov Obl. Ped. Inst., Moscow, 1976 (Russian)], [DŁUGOSZ, R.: Embedding theorems for holomorphic functions of several complex variables , J. Appl. Anal. 19 (2013), 153--165], [DŁ UGOSZ, R.—LEŚ, E.: Embedding theorems and extremal problems for holomorphic functions on circular domains of Cn$\mathbb{C}^{n}$, Complex Var. Elliptic Equ. 59 (2014), 883--899], [LEŚ-BOMBA, E.—LICZBERSKI, P.: New properties of some families of holomorphic functions of several complex variables , Demonstratio Math. 42 (2009), 491--503], et al..

In this paper, sufficient conditions are established for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form 1a(t)((y(t)+p(t)y(t−τ))″)α″=q(t)f(y(t−σ1))+r(t)g(y(t+σ2))$$\left(\frac{1}{a(t)}\bigl((y(t)+p(t)y(t-\tau))''\bigr)^\alpha\right)''=q(t)f(y(t-\sigma_1))+r(t) g(y(t+\sigma_2))$$ under the assumption ∫0∞(a(t))1αdt<∞$$\int \limits_{0}^{\infty}(a(t))^{\frac{1}{\alpha}}\,{\rm d} t \lt \infty$$ for various ranges of p ( t ), where α is the ratio of odd positive integers.

In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear third order neutral differential equations with positive and negative coefficients of the form (E)(a(t)(b(t)(y(t)+p(t)y(σ(t)))′)′)′+q(t)G(y(α(t)))−h(t)H(y(β(t)))=0$$\begin{equation*}(a(t)(b(t)(y(t)+p(t)y(\sigma(t)))^{\prime})^{\prime})^{\prime} +q(t)G(y(\alpha(t)))-h(t)H(y(\beta(t)))=0 \tag{E}\end{equation*}$$ for 0 ⩽ p ( t ) ⩽ p 1 <1 and −1 < p 2 ⩽ p ( t ) ⩽ 0. The results in this paper generalize the results of [LI, T.—ZHANG, C.—XING, G.: Oscillation of third-order neutral delay differential equations , Abstr. Appl. Anal. 2012 (2012), Article ID 569201] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systems u¨t−Ltut+∇Wt,ut=0$$ \ddot u\left( t \right) - L\left( t \right)u\left( t \right) + \nabla W\left( {t,u\left( t \right)} \right) = 0$$(HS) where t ∈ ℝ, L ∈ C (ℝ, ℝ n 2 ) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C 1 (ℝ × ℝ n , ℝ) and ∇ W ( t,u ) is the gradient of W at u . The novelty of this paper is that, assuming that L meets some coercive condition and the potential W is of the form W ( t , u ) = W 1 ( t , u ) + W 2( t , u ), for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.

In this paper, we deal with the existence of solutions for the fractional impulsive functional differential equations with nonlocal conditions. Then we build a new comparison theorem and obtain the existence of extremal solutions and quasi-solutions by use of the monotone iterative technique and the method of lower and upper solutions.

A new type weighted reverse Poincaré inequality is established for a difference of two continuous weak parabolic subsolutions of a linear second order uniformly parabolic partial differential equation with constant coefficients in the cylindrical domain. This inequality asserts that if two continuous weak parabolic subsolutions are close in the uniform norm, then their gradients are close in the weighted L 2 norm.

In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.

In this paper certain integral operators of Szász-Mirakjan-Durrmeyer type are presented. Some approximation results and a Voronovskaya type theorem are obtained for these operators.

We study the relative position of two measure-theoretical conditional expectations treated as orthogonal projections on the L 2 -space. We show that any pair of orthogonal projections P and Q on a separable Hilbert space with dim( P ∧ Q ) = ∞ is unitarily equivalent to the conditional expectations under two sub-σ-fields of some probability space.

In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space ℓϱ2(Z)ℓϱ2(Z)(Z:={0,±1,±2,…}) $\ell_{\varrho}^{2}(\mathbb{Z}) (\mathbb{Z} :=\{0,\pm 1,\pm 2,\dots\})$ . Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at ± ∞. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called “dissipative at −∞” and “dissipative at ∞”. In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators.

Let G be a finite group. In this paper we give a new necessary condition for two real G -modules to be Smith equivalent if G has a normal Sylow 2-subgroup. We show that the condition is also sufficient under certain hypotheses. By results on the Smith equivalence obtained in this paper, the primary Smith sets are not subgroups of the real representation rings for various Oliver groups with normal Sylow 2-subgroups.

The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are subject to arbitrary transformations of variables in the widest possible sense. In this preparatory Part 1, the involutivity and the related standard bases are investigated as a technical tool within the framework of commutative algebra. The particular case of ordinary differential equations is briefly mentioned in order to demonstrate the strength of this approach in the study of the structure, symmetries and constrained variational integrals under the simplifying condition of one independent variable. In full generality, these topics will be investigated in subsequent Parts of this article.

In this paper we extend our recent results concerning the validity of the law of inertia for the factorization of cubic polynomials over the Galois field Fp$\Bbb F_p$, p being a prime. As the main result, the following theorem will be proved: Let D∈Z$D\in \Bbb Z$ and let C D be the set of all cubic polynomials x3+ax2+bx+c∈Z[x]$x^3+ax^2+bx+c\in\Bbb Z[x]$ with a discriminant equal to D . If D is square-free and 3∤h(−3D)$3\nmid h(-3D)$ where h(−3D)$h(-3D)$ is the class number of Q(−3D)$\Bbb Q(\sqrt {-3D})$, then all cubic polynomials in C D have the same type of factorization over any Galois field Fp$\Bbb F_p$ where p is a prime, p > 3.