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Effect algebras were introduced in order to describe the structure of effects, i.e. events in quantum mechanics. They are partial algebras describing the logic behind the corresponding events. It is natural to ask how to introduce the logical connective implication in effect algebras. For lattice-ordered effect algebras this task was already solved by several authors, including the present ones. We concentrate on effect algebras that need not be lattice-ordered since these can better describe the events occurring in the quantum physical system. Although an effect algebra is only partial, we try to find a logical connective implication which is everywhere defined. But such a connective can be “unsharp” or “inexact” because its outputs for given pairs of entries need not be elements of the underlying effect algebra, but may be subsets of (mutually incomparable) maximal elements. We introduce such an implication together with its adjoint functor representing conjunction. Then we consider so-called tense operators on effect algebras. Of course, also these operators turn out to be “unsharp” in the aforementioned sense, but they are in a certain relation with the operators implication and conjunction. Finally, for given tense operators and given time set T , we describe two methods how to construct a time preference relation R on T such that the given tense operators are either comparable with or equivalent to those induced by the time frame ( T , R ).

Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled (1 × n ) and (2 × n ) square grids by covering them with squares and dominoes. We describe these numbers not only recursively, but also as rational polynomial linear combinations of Fibonacci numbers.

We introduce the notion of CFL ew -algebras, which are bounded commutative integral residuated lattices equipped with an additional operation that mimics the composition of functions. We obtain and study some of the basic properties of CFL ew -algebras. In particular, we characterize semi-divisible CFL ew -algebras and prove that Boolean algebras are the only CFL ew -algebras having sub-CFL ew -algebras of order 4. Finally, we study the effect of the composition on the filters/congruences of CFL ew -algebras and also investigate the usual constructions of subalgebras, quotient algebras, as well as related classes of filters.

Let λ f ( n ), σ ( n ) and ϕ ( n ) denote n th Fourier coefficient of the primitive holomorphic cusp form f , the sum-of-divisors function, and the Euler totient function, respectively. In this paper, we first refine related previous results on the power sum ∑ n≤ xλ fj(n)σ b(n)ϕ c(n) $\begin{array}{} \displaystyle \sum\limits_{n\leq x} \lambda_{f}^{j}(n)\sigma^{b}(n)\phi^{c}(n) \end{array}$ and then investigate the average behavior of the power sum ∑ n=a2+b2+c2+d2≤ x,(a,b,c,d)∈ Z4λ fj(n)σ b(n)ϕ c(n). $\begin{array}{} \displaystyle \sum\limits_{\substack{n=a^2+b^2+c^2+d^2\leq x, \\ (a,b,c,d)\in {\mathbb{Z}}^4}} \lambda_{f}^{j}(n)\sigma^{b}(n)\phi^{c}(n). \end{array}$

The goal of this paper is to calculate explicitly the field index of any quintic number field K generated by a complex root α of a monic irreducible trinomial F ( x ) = x 5 + ax + b ∈ ℤ[ x ]. In such a way we provide a complete answer to the Problem 22 of Narkiewicz [36] for this class of number fields. Namely, for every prime integer p , we evaluate the highest power of p dividing i ( K ). In particular, we give sufficient conditions on a and b , which guarantee the non-monogenity of K .

The article deals with the study of the existence of a fundamental system of solutions to the differential equations: ( D 2 − 2 Dα + α 2 + β 2 ) y = 0, ( D 2 − 2 Dα + α 2 − β 2 ) y = 0, which are created using higher derivatives of the appropriate functions which are solutions to these equations. Special attention is paid to the solution of the generalized Cauchy problem with initial conditions.

In the paper, in view of two monotonicity rules for the ratios of two functions and of two Maclaurin power series expansions, the authors establish several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals.

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper.

The present paper concerns the periodic problem u″=p(t)u− q(t,u)u+f(t);u(0)=u(ω ),u′(0)=u′(ω ), $$\begin{array}{} \displaystyle u''=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\, u'(0)=u'(\omega), \end{array}$$ where p , f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general results obtained are applied to the non-autonomous Duffing type equation with a super-linear power non-linearity.

The aim of the study is to establish sufficient conditions to ensure the existence and uniqueness of solutions for nonlinear Riemann-Liouville fractional dynamic equations under impulse effects. The current state of the literature reveals a visible gap in the investigation of the existence-uniqueness aspects of such equations, and this research makes a significant contribution to filling this gap. To highlight the practical implications of our results, we present an illustrative example that exemplifies the applicability of the established conditions.

This paper studies a new type of conformable non-autonomous non-instantaneous impulsive differential equations. We present the solution by a new kinds of conformable Cauchy matrix. Also, we present its some properties. Next, we respectively discuss about the existence and uniqueness of 𝓒-periodic solutions of linear homogeneous and nonhomogeneous problems. Further, we study the nonlinear problem via fixed point theorem. Examples are also given to verify theory results.

In this note, we study three differential problems with a dynamic, which are be represented by a self referred equation and a boundary condition, which are expressed as an integral constraint. We prove that under certain assumptions, there exists at least one solution of for all of these problems by using Schauder’s fixed point theorem. In the end, we propose briefly some open problems.

In the present paper, we discuss the rate of the approximation by the matrix transform of partial sums of double Vilenkin-Fourier series in L p ( Gm2 $\begin{array}{} \displaystyle G_{m}^{2} \end{array}$) space (1 ≤ p < ∞) and in C ( Gm2 $\begin{array}{} \displaystyle G_{m}^{2} \end{array}$). We give an application for Lipschitz functions.

In this paper, we give the complete description of maps on self-adjoint bounded operators on Hilbert space which preserve a triadic relation involving the difference of operators and either commutativity or quasi-commutativity in both directions. We show that those maps are implemented by unitary or antiunitary equivalence and possible additive perturbation by a scalar operator.

The aim of this paper is to construct chains of length (2 κ ) + in the Rudin-Frolík order of β κ for every infinite κ .

Let CL( X ) be the collection of all non-empty closed subsets of X , and Δ any subfamily of CL( X ). By τΔ=τV−∨τΔ+ $\tau_\Delta = \tau_V^- \lor \tau_\Delta^+$ , we denote the hit-and-miss topology on CL( X ). In 1995, Arhangel’skii focused his attention on the following general problem was formulated: given any subspace Y of X , study the subspace Y + of CL( X ) in the Vietoris topology and the other natural topologies, where Y + is the subspace of CL( X ) consisting of all non-empty closed subsets of X which are contained in Y . In this paper, we investigate when Y + is first (second) countable in (CL( X ), τ ), for τ∈{τΔ,τV−,τΔ+} $ \tau\in \{\tau_\Delta,\tau_V^- ,\tau_\Delta^+\} $ . All content of this work extend some results of the theory of hyperspaces, when Y = X or when Δ coincides with CL( X ) or when is equal to the collection of all non-empty compact subsets of X .

Estimating the proportion of individuals having a disease in a given population is a common problem in medicine. This is simply done by drawing a random sample from the target population, and computing the proportion of positive results based on a suitable diagnostic test. In some situations, the number of quantified units is limited because measuring the variable of interest is difficult or expensive. In this setting, one can utilize alternative designs that allow achieving the desired precision level with a smaller sample size. Multistage pair ranked set sampling (MSPRSS) is such a design that can be used instead of simple random sampling. It is a rank-based sampling method that incorporates auxiliary information in order to collect an informative sample. This article deals with the proportion estimation in MSPRSS. Some results about the proposed estimator are proved. A simulation experiment and a real data set in the context of breast cancer are used to demonstrate the finite sample properties of the new estimator.

In this research paper, we introduce a new three-parameter model called Marshall-Olkin Extended Gamma Lindley, which provides greater flexibility for modeling lifetime data. We investigate some of its structural properties, including moments and moment generating function. Furthermore, we prove that this model is identifiable. We corroborate that the Marshall-Olkin Extended Gamma Lindley distribution is characterized by its truncated moments of order statistics. Relying on this characterization and the least squares, we set forward a new method in order to estimate the unknown parameters of the proposed model. A comparative study is conducted between this approach and different classical methods of estimation. A real data set is applied to show the flexibility of the suggested model against other models.