We introduce the short-time fractional Fourier transform (STFRFT) in the direction of u. Then, following the duality approach, we develop a distributional framework for the STFRFT in the direction of u on the space of Schwartz tempered distributions S? (Rn). We provide several Abelian-and Tauberian-type results that characterize the quasiasymptotic behavior of tempered distributions at the origin in terms of the asymptotic behavior of their STFRFT in the direction of u.

We study the properties of the class of charming spaces. It is proved that if X is the preimage of a metrizable locally Lindel?f p-space(respectively, locally s-space) under a perfect mapping, then every remainder bX \ X of X in any compactification bX is 1-strong charming(respectively, charming). Some corollaries related to this statement are presented. It is shown that if X is a metrizable space, and X is a locally Lindel?f p-space(respectively, locally s-space), then for any compactification bX of X, the remainder bX \ X of X is 1-strong charming (respectively, charming). It is also proved that if X is a nowhere locally compact metrizable space, then X is a locally s-space (respectively, locally Lindel?f p-space) if and only if for any (or some) compactification bX of X, the remainder bX \ X of X is charming (respectively, 1-strong charming). Some related propositions are proved within this section. In addition, some properties of s-space are investigated.

As a probabilistic extension of the theorem obtained by Pant et al. [R.P. Pant, V. Rakocevic, D. Gopal, A. Pant, M. Ram, A General Fixed Point Theorem, Filomat 35(12) (2021), 4061-4072] in this paper, we prove existence and uniqueness of a fixed point for a wide class of self-mapping defined on complete Menger PM-space. Our theorem generalizes well-known fixed point theorems proved for Menger PM spaces. Also, this theorem characterizes probabilistic metric completeness. Some examples and comments are provided based on the obtained results.

In this article, we present the concept of one sided x-projection, explore and investigate left and right x-projection elements, and introduce new characterizations of one sided x-idempotent elements in a +-ring. We also propose numerous equivalent conditions for left x-projections within a given ring. Next, several equivalent conditions about Hermitian, one sided x-projection, one sided x-idempotent, and strongly EP elements in a +-ring are given.

In comparison to PDF-based measures which provided by extropy, the CDF-based measures of cumulative residual and past extropies are generally more stable. In this paper, we extend the concept of cumulative extropy which includes both cumulative inaccuracy and dynamic versions, such as cumulative residual inaccuracy and cumulative past inaccuracy. By using these measures, we characterize specific lifetime distributions and explore some generalized results. Additionally, we find that these inaccuracy measures can uniquely identify the underlying distributions, and we also characterize a few specific lifetime distributions. Two non-parametric estimators for this measure are provided and their performances are compared via some simulation studies. Finally, we illustrate our method in a real data set.

We consider the singularity formation of strong solutions to the three-dimensional incompress-ible fractional Navier-Stokes equations in the whole space. By making use of the Bony decomposition technique, we prove that a unique local strong solution does not blow-up at time T if deformation tensor belongs to nonhomogeneous Besov spaces. As a bi-product, the result improves some well-known results on regularity for the particular case of classical Navier-Stokes equations.

We investigate sums of exceptional units in a quaternion ring H(R) over a finite commutative ring R. We prove that in order to find the number of representations of an element in H(R) as a sum of k exceptional units for some integer k ? 2, we can limit ourselves to studying the quaternion rings over local rings. For a local ring R of even order, we find the number of representations of an element of H(R) as a sum of k exceptional units for any integer k ? 2. For a local ring R of odd order, we find either the number or the bounds for the number of representations of an element of H(R) as a sum of 2 exceptional units.

In this paper, the notions of upper order-preserving and lower order-preserving set-valued mappings are given. By using these notions, some fixed point theorems in the setting of partially ordered sets equipped with the hull-kernel topology for set-valued mappings are established. As an application of these theorems, an existence theorem for a solution of vector equilibrium problem is provided. The main results of this article can be viewed as the set-valued version of the main theorems given in [8, 12, 17, 18] with mild assumptions.

In this paper, we consider the three-space properties in paratopological gyrogroups. The fol-lowing are the established conclusions: (1) metrizability of compact (resp., sequentially compact, countably compact) subsets is a three-space property in the class of k-gentle paratopological gyrogroups; (2) let G be a strongly paratopological gyrocommutative gyrogroup and let H be a second-countable invariant topo-logical subgyrogroup of G. If the paratopological gyrogroup G/H has a countable network, then so does G; (3) let H be a compact strongly L-subgyrogroup of a paratopological gyrogroup G. If H and G/H have countable tightness, then G has countable tightness.

In this paper, we consider the generalized Delannoy paths with steps N_i = (0, j) , j \ge 1 and H_i = (1, j) , j \ge 0 , where all steps are weighted by u_i for N_i and v_j ( v_0 = 1 ) for H_j . By the Riordan array theory, we provide a counting formula for the number of all generalized Delannoy paths dominated by a cyclically shifting piecewise linear boundary of varying slopes. Our main result can be viewed as a unified generalization of the well-known enumerative formulas for the generalized Dyck and Schr?der paths from (0,0) to (kn,n) staying above the line x = ky . We also study the number of generalized Delannoy path boundary pairs (P, a_{n,k}) with m -flaws, where P is a generalized Delannoy path (with steps N_i and H_0 ) from (0,0) to (n,k) , a_{n,k} is a k -part composition of n , and a flaw is a horizontal step (1,0) of P below the boundary \partial a_{n,k} .