Some further studies on strong ℐλ-statistical convergence in probabilistic metric spaces

We prove some basic properties of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence of sequences in probabilistic metric spaces and introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical cluster point. We also introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences in probabilistic metric spaces. Further, we establish a connection between strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence and strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences.

Certain results on trans-paraSasakian 3-manifolds

Let M be a trans-paraSasakian 3-manifold. In this paper, the necessary and sufficient condition for the Reeb vector field of a trans-paraSasakian 3-manifold to be harmonic is obtained. Also, it is proved that the Ricci operator of M is invariant along the Reeb flow if and only if M is a paracosymplectic manifold, an α-paraSasakian manifold or a space of negative constant sectional curvature.

Gradient estimates for Monge–Ampère type equations on compact almost Hermitian manifolds with boundary

We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

Regularity criteria for 3D Navier–Stokes equations in terms of a mid frequency part of velocity in B˙-1 ∞,∞

We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular in ( 0 , T ] {(0,T]} if the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , the mid frequency part of Fourier modes k 2 ≤ | ξ | < k {\frac{k}{2}\leq|\xi|<k} , is small depending on the kinematic viscosity ν, initial value u 0 {u_{0}} and the maximum of an averaged energy dissipation rate A ≡ sup t ∈ ( 0 , T ) ⁡ ( ν ⁢ t - 1 ⁢ ∫ 0 t ∥ ∇ ⁡ u ∥ 2 ⁢ 𝑑 τ ) A\equiv\sup_{t\in(0,T)}\biggl{(}\nu t^{-1}\int_{0}^{t}\lVert\nabla u\rVert^{2}% \,d\tau\biggr{)} for some k ≥ k 0 ⁢ ( ν , u 0 , A ) {k\geq k_{0}(\nu,u_{0},A)} . In particular, when a sufficiently high frequency part of u 0 {u_{0}} decays fast at an exponential rate, then we obtain regularity conditions in terms of smallness of the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , which involve only the known data ν and u 0 {u_{0}} .

Slices of Hewitt–Stromberg measures and co-dimensions formula

This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto ( n - m ) {(n-m)} -dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and q lies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula.

On the domain of difference double sequence spaces of arbitrary orders

The prime objective of this paper is to define a new double difference operator with arbitrary order via which new classes of difference double sequences are introduced. Results on topological structures, dual spaces and four-dimensional matrix mappings related to the proposed difference double sequence spaces are discussed. As an application of this work, the proposed operator is being used to approximate partial derivatives of fractional orders. Some numerical examples are also given in support of the validity or the clear visualization of the results obtained.

1,2,31,2,3, some inductive real sequences and a beautiful algebraic pattern

By rewriting the relation 1 + 2 = 3 {1+2=3} as 1 2 + 2 2 = 3 2 {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with 1 2 + 2 2 = 3 2 {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} as initial one converging to the segment [ 0 , 1 ] {[0,1]} of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra.

Generating functions involving the incomplete H-functions

In this article, for the incomplete H-functions, we obtain a set of new generating functions. The bilateral along with linear generating relations are derived for the incomplete H-functions. Many of the generating functions readily accessible in the literature are often deemed as implementations of the main findings. All the derived findings are of a natural type and can produce a variety of new results in generating function theory.