Volume 72, Issue 6

We introduce the notion of ∧-projection in order to extend to d 0 -algebras the concept of Kalmbach measurable elements with respect to an outer measure μ . We prove, in case μ is faithful, that Kalmbach measurable ∧-projections are quasi-central, thus generalizing a result known for orthomodular lattices, and recently extended to D-lattices.

In this paper, we introduce the notion of monadic L-algebras and we study properties of these new structures. We define and investigate the monadic ideals of a monadic L-algebra, and we characterize the monadic ideal generated by a subset of a monadic L-algebra. We define the existential and universal quantifiers on semiregular L-algebras with negation, and we investigate certain properties of these structures. In this sense, we prove that there exists a one to one correspondence between existential and universal quantifiers. We show that any pair of strict quantifiers on a semiregular L-algebra with negation is a monadic operator.

For a zero-dimensional Hausdorff space X , denote, as usual, by C ( X , ℤ) the ring of continuous integer-valued functions on X . If f ∈ C ( X , ℤ), denote by Z( f ) the set of all points of X that are mapped to 0 by f . The set CK(X,Z)={f∈ C(X,Z)∣ clX(X∖ Z(f)) is compact} $$\begin{array}{} \displaystyle C_K(X,\mathbb Z)=\{f\in C(X,\mathbb Z)\mid \text{cl}_X(X\smallsetminus \mathsf Z(f))\text{ is compact}\} \end{array}$$ is the integer-valued analogue of the ideal of functions with compact support in C ( X ). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X , we also explore some properties of ideals of C ( X , ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y , subsets of β Y with ideals of C ( Y ).

P. Hájek introduced an S5-like modal fuzzy logic S5(BL) and showed that is equivalent to the monadic basic predicate logic mBL ∀ . Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic S5’(BL) , which is a simplified set of axioms of S5(BL) . In this paper, we review the algebraic semantics of S5’(BL) and obtain some new results regarding to monadic BL-algebras. First we recall that S5’(BL) is completeness with respect to the variety 𝕄𝔹𝕃 of monadic BL-algebras and obtain a necessary and sufficient condition for the logic S5’(BL) to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety 𝕄𝔹𝕃. These results also constitute a crucial first step for providing an equivalent algebraic foundation for mBL ∀ .

In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A 052529 that has strong relationships with triangular numbers.

In the present work, a pair of general inverse matrix series relations is established, and thereby a general class of matrix polynomials is introduced. This class generalizes the extended Jacobi polynomials and their particular cases such as the polynomials of Brafman, Jacobi, Chebyshev, and Legendre. It is further shown that this pair also gives rise to the matrix forms of the Wilson polynomials and the Racah polynomials. For these polynomials, the generating matrix function relations as well as the matrix summation formulas are deduced from their respective inverse pairs. Certain inverse pairs belonging to the Gould classes and the Legendre-Chebyshev classes due to John Riordan [ An Introduction to Combinatorial Identities , Wiley, 1968] are also extended to matrix forms.

In this paper, using the quantum derivatives and quantum integrals, we prove some new quantum Ostrowski’s type inequalities for pre-invex functions. Furthermore, in the special cases of newly developed inequalities, we obtain different new and existing Ostrowski’s type inequalities.

In this paper, first we define interval-valued left-sided and right-sided fractional integrals of a function with respect to the another function. Then, we handle Hermite-Hadamard type inequalities via these definitions.

The aim of the present paper is to study lattice outer measures on an unsharp quantum logic, viz. a difference poset P associated with a monotone function μ defined on a sub-difference poset L of P and an approximating family in L . Having proved a number of fundamental properties, relationships among these outer measures and eventually among corresponding families of measurable elements of P are investigated.

This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. We are interested in finding some structures in the solutions of these generalized Lamé-Navier systems. Using MATLAB we also implement algorithms to compute with such partial differential operators as well as to verify some theoretical results obtained in the paper.

In the present paper, we study the existence as well as the non-existence of some positive solutions for the equation −Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r under Robin boundary condition in a regular open bounded domain Ω of ℝ N , N ≥ 2. Δ p ( x ) is the p ( x )-Laplacian operator where p ∈ C 1 ( Ω ) and p > 1. Our proofs are based on the sub solution-super solution method and also on variational arguments.

Recently, the authors interested some new problems on multiplier spaces of Lorentz’ almost convergence and f λ -convergence as a generalization of almost convergence. f λ -convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X * . In the present paper, we deal with f λ -convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.

It was claimed in [Sadeqi, I.—Salehi Paydar, M.: A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results , J. Optim. Theory Appl. 165(2) (2015), 344–358] that the gradient of a convex Gâteaux differentiable function is Fanhemicontinuous. The aim of the present paper is to correct this implication by exemplifying for ∇‖⋅‖ $\nabla \|\cdot \|$in a Hilbert space. This goal was achieved by considering its domain to be scalarly-positive (see Definition 3). For a perturbation of a Fan-hemicontinuous operator with ∇‖⋅‖, $\nabla \|\cdot \|,$weak compactness for the set of solutions of the variational inequality problem is obtained.

In this study, the solvability of a problem of the heat conduction theory with two nonlocal boundary conditions is investigated. Systems of eigenfunctions of the corresponding operator with two nonlocal boundary conditions are taken into consideration. A theorem on the solvability of the problem of the theory of heat conduction with two nonlocal boundary conditions is given

Let ( M n , F ( t ), m ), t ∈ [0, T ], be a compact Finsler manifold with F ( t ) evolving by the Finsler-geometric flow ∂ g(x,t)∂ t=2h(x,t), $\begin{array}{} \displaystyle \frac{\partial g(x,t)}{\partial t}=2h(x,t), \end{array}$ where g ( t ) is the symmetric metric tensor associated with F , and h ( t ) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential ∂ tu(x,t)=Δ mu(x,t)− R(x,t)u(x,t)− au(x,t)log⁡ u(x,t),(x,t)∈ M× [0,T], $$\begin{array}{} \displaystyle \partial_{t}u(x,t)=\Delta_{m}u(x,t)-\mathcal{R}(x,t)u(x,t) -au(x,t)\log u(x,t),\quad(x,t)\in M\times [0,T], \end{array}$$ along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.

A space X is said to have the set-star-K-Menger property if for each nonempty subset A of X and for each sequence (𝓤 n : n ∈ ℕ) of collections of open sets in X such that for each n ∈ ℕ, A ⊆ ⋃ 𝓤 n , there is a sequence ( K n : n ∈ ℕ) of compact subsets of X such that A ⊆ ⋃ n∈ N $\begin{array}{} \bigcup\limits_{n \in \mathbb{N}} \end{array} $ St( K n , 𝓤 n ). In this paper, we prove that: There exists a T 1 set-star-Menger space which is not set-star-K-Menger and there exists a Tychonoff set-star-K-Menger space that is not set-star-Menger. Assuming 𝔡 = 𝔠, there exists a Tychonoff set-star-K-Menger space having a regular-closed G δ -subspace which is not set-star-K-Menger. If the Alexandroff duplicate of a space X is set-star-K-Menger, then X is set-star-K-Menger. The product of set-star-K-Menger space and a compact space is rectangular set-star-K-Menger space. The above-mentioned results answer to Problem 2.5 and Problem 3.6, and give a partial answer to Problem 3.11 in [SINGH, S.: On set-star-K-Menger spaces , Publ. Math. Debrecen 100 (2022), 87–100]. Further, we continue to study the topological properties of set-star-K-Menger spaces.

When data generating mechanism generates two correlated data sets both defined on the unit interval, a bivariate probabilistic distribution defined on the unit square is needed for modelling the data. For this purpose, we give a Marshall-Olkin type bivariate extension of an omega distribution in this paper. This is in fact a bivariate unit-exponentiated-half-logistic distribution. We study its mathematical properties in detail. The distribution contains neither an exponential term nor any special function which complicates the computations. Maximum likelihood estimation method and its large sample inference are considered for model parameters. Alternatively, we also propose an expectation- maximization algorithm to compute the estimates. To see the performances of the proposed estimators and validate the theoretical results obtained for estimation, we present the results of a simulation study. Data fitting demonstrations show its applicability in modelling random proportions.

A novel iterative numerical method is constructed for solving second kind Volterra fractional integral equations. The method uses at each iterative step a Bernstein spline interpolation procedure combined with the corresponding quadrature formula. In this way, based on the nice approximation and shape preserving properties of the Bernstein polynomials, we propose an alternative to the classical product integration technique that uses trapezoidal, Simpson, Gauss type and other well-known quadrature formulas. The convergence of the method is proved with the error estimate expressed in terms of the Lipschitz constants and the accuracy is illustrated on some numerical experiments.

This research advances the 4-player gambler’s ruin problem for the case of arbitrary initial stakes. The aim of the research is attained by offering simple expressions using the difference equation approach and thus providing closed form solution to the problem. Moreover, the existing technique of Chang [ A game with four players , Statist. Probab. Lett. 23(2) (1995), 111–115] dealing with equal initial stakes is demonstrated as a sub-case of the newly devised scheme. The legitimacy of the proposed formulation is further verified by considering various parametric settings.

Given a partial action, an enveloping action in the context of convergence spaces is studied. Whenever the enveloping action space is not Hausdorff ( T 3 ), a related enveloping action on a Hausdorff ( T 3 ) space is developed. Invariance of quotient (proper, open) maps to the related space is also discussed.