Band 31, Heft 2

A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian Δ. In this paper, we classify two types of affine homothetical surfaces of finite type in isotropic 3-space 𝕀 3 {\mathbb{I}^{3}} under the condition Δ ⁢ r i = λ i ⁢ r i {\Delta r_{i}=\lambda_{i}r_{i}} , where ( r i ) {(r_{i})} is the i -component function of the position vector r ( i = 1 , 2 , 3 {i=1,2,3} ), λ i ∈ ℝ {\lambda_{i}\in\mathbb{R}} and Δ denotes the Laplace operator.

In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator Δ ı ȷ \Delta^{\jmath}_{\imath} and the de la Vallée–Poussin mean, denoted as [ ( V , λ ) w , Δ ı ȷ ] m [(\mathscr{V},\lambda)_{w},\Delta^{\jmath}_{\imath}]_{m} for m = 0 m=0 , 1, and ∞. Further, we explore some algebraic and topological properties of these spaces, including their nature as linear, normed, Banach, and BK spaces. Additionally, we examine properties such as solidity, symmetry, and monotonicity. Finally, we define and establish some inclusion relations among generalized difference weak statistical convergence, generalized difference weak 𝜆-statistical convergence, and generalized difference weak [ V , λ ] [\mathscr{V},\lambda] -convergence.

In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent.

We investigate a stochastic coupled model of the Cahn–Hilliard equations and the stochastic magnetohydrodynamic equations in a bounded domain of ℝ 2 {\mathbb{R}^{2}} . The model describes the flow of the mixture of two incompressible and immiscible fluids under the influence of an electromagnetic field with stochastic perturbations. We prove the existence, uniqueness and regularity of a probabilistic strong solution. The proof of the existence is based on the Galerkin approximation, the stopping time technique and some weak convergence principles in functional analysis.

In this paper, we first consider the 𝕊 ⁢ ℙ ⁢ 𝔽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M . Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 𝕊 ⁢ ℙ ⁢ 𝔽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold.

This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

This study investigates existence result of a nontrivial weak solutions for a system of nonlinear elliptic equations involving Hardy Potentials. The problem arises in the mathematical modeling of complex physical phenomena.

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

In theory of generalized measure, Egorov-like theorems have been investigated. In this article, we introduce generalized concepts of almost everywhere convergence and almost uniform convergence. Next Egorov-like theorems with respect to the generalized convergences are provided. Finally, versions of Egorov condition for the more general context of convergence are also established.

This paper presents an existence result for a novel nonlinear Caputo q -fractional p -Laplacian epidemiological model on time scales. The proof of the main result leverages Arzelà–Ascoli and Schauder’s fixed point theorems. Additionally, a set of open questions is addressed to further enrich this field of valuable scientific research

In this paper, we prove a Heisenberg uncertainty inequality and a local uncertainty inequality for the Dunkl–Weinstein–Stockwell transform. We give also a Shapiro-type uncertainty inequality for this transform.

The goal of this study is to derive the generalized Levinson-type inequalities in the form of Taylor representation for higher order convex functions. This study uses Taylor’s formula and several novel types of 3-convex Green functions, to establish the novel identities associated with Bullen-type inequalities for higher order convex functions. Moreover, various Levinson-type inequalities are proved for positive real weights using Green functions and Taylor’s formula.

In this paper, an attempt has been made to establish the sufficient conditions for the existence and multiplicity of positive solutions by using the fixed point index theory and the Avery–Peterson fixed point theorem respectively for the following class of nonlinear singular fractional differential equation D 0 + β ⁢ ( ϕ p ⁢ ( D 0 + α ⁢ u ⁢ ( t ) ) ) + f ⁢ ( t , u ⁢ ( t ) , … , u ( n - 2 ) ⁢ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D_{0^{+}}^{\beta}(\phi_{p}(D_{0^{+}}^{\alpha}u(t)))+f(t,u(t),\dots,u^{(n-2)}(t% ))=0,\quad t\in(0,1), with the boundary conditions u ( k ) ⁢ ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , D 0 + α ⁢ u ⁢ ( 0 ) = 0 , D 0 + α - 1 ⁢ u ⁢ ( 1 ) = ∫ 0 1 D 0 + α - 1 ⁢ u ⁢ ( t ) ⁢ 𝑑 A ⁢ ( t ) , u^{(k)}(0)=0,\quad 0\leq k\leq n-2,\qquad D_{0^{+}}^{\alpha}u(0)=0,\qquad D_{0% ^{+}}^{\alpha-1}u(1)=\int_{0}^{1}D_{0^{+}}^{\alpha-1}u(t)\,dA(t), where 0 < β ≤ 1 {0<\beta\leq 1} , n - 1 < α ≤ n {n-1<\alpha\leq n} , n ≥ 3 {n\geq 3} , A : [ 0 , 1 ] → ℝ {A:[0,1]\to\mathbb{R}} is a function of bounded variation, ϕ p ⁢ ( s ) = | s | p - 2 ⁢ s {\phi_{p}(s)=|s|^{p-2}s} for p > 1 {p>1} and ϕ q ⁢ ( s ) {\phi_{q}(s)} is the inverse of ϕ p ⁢ ( s ) {\phi_{p}(s)} , where p , q {p,q} satisfies the relation 1 p + 1 q = 1 {\frac{1}{p}+\frac{1}{q}=1} , f may have singularity at t = 0 {t=0} .

Let u ′ + A ⁢ u = h ⁢ ( u , t ) + f ⁢ ( x , t ) {u^{\prime}+Au=h(u,t)+f(x,t)} , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) {u(x,0)=u_{0}(x)} , where u ∈ H {u\in H} , u ′ := u t := d ⁢ u d ⁢ t {u^{\prime}:=u_{t}:=\frac{du}{dt}} , H is a Hilbert space, ∥ h ⁢ ( u , t ) ∥ ≤ a ⁢ ∥ u ∥ p ⁢ ( 1 + t ) - b \|h(u,t)\|\leq a\|u\|^{p}(1+t)^{-b} , ( A ⁢ u , u ) ≥ γ ⁢ ( t ) ⁢ ( u , u ) {(Au,u)\geq\gamma(t)(u,u)} , γ ⁢ ( t ) = c ⁢ ( 1 + t ) - c 1 {\gamma(t)=c(1+t)^{-c_{1}}} . Here p , a , b , c , c 1 {p,a,b,c,c_{1}} are positive constants. Sufficient conditions are given for the above problem to have a solution bounded or going to zero as t → ∞ {t\to\infty} .

When there are no constraints upon the solutions of the equation 𝑨 ⁢ 𝝃 = 𝒚 {{\boldsymbol{A}}{\boldsymbol{\xi}}={\boldsymbol{y}}} , where 𝑨 {{\boldsymbol{A}}} is a K × N - {K\times N-} matrix, 𝝃 ∈ ℝ N {{\boldsymbol{\xi}}\in{\mathbb{R}}^{N}} and 𝒚 ∈ ℝ K {{\boldsymbol{y}}\in{\mathbb{R}}^{K}} a given vector, the description of the set of solutions as 𝒚 {{\boldsymbol{y}}} varies in ℝ K {{\mathbb{R}}^{K}} is well known. But this is not so when the solutions are required to satisfy 𝝃 ∈ 𝒦 ⁢ ∏ i ≤ j ≤ N [ a j , b j ] {{\boldsymbol{\xi}}\in{\mathcal{K}}\prod_{i\leq j\leq N}[a_{j},b_{j}]} , for finite a j ≤ b j : 1 ≤ j ≤ N {a_{j}\leq b_{j}:1\leq j\leq N} . To solve this problem we bring in a strictly convex, Fermi-Dirac entropy function Ψ ⁢ ( 𝝃 ) {\Psi({\boldsymbol{\xi}})} , and find the solution as a ⁢ r ⁢ g ⁢ m ⁢ i ⁢ n ⁢ { Ψ ⁢ ( 𝝃 ) : 𝝃 ∈ 𝒦 , 𝑨 ⁢ 𝝃 = y } {argmin\{\Psi({\boldsymbol{\xi}}):{\boldsymbol{\xi}}\in{\mathcal{K}},\,{% \boldsymbol{A}}{\boldsymbol{\xi}}=y\}} . If λ denotes the Lagrange multipliers of the optimization problem, we study the properties of the parametric surface 𝝀 → 𝝃 ⁢ ( 𝝀 ) {{\boldsymbol{\lambda}}\to{\boldsymbol{\xi}}({\boldsymbol{\lambda}})} in the geometry on 𝒦 {{\mathcal{K}}} defined by the Hessian metric derived from Ψ ⁢ ( 𝝃 ) {\Psi({\boldsymbol{\xi}})} . In particular, we prove that the surface 𝝀 → 𝝃 ⁢ ( 𝝀 ) {{\boldsymbol{\lambda}}\to{\boldsymbol{\xi}}({\boldsymbol{\lambda}})} is contained in ker ( 𝑨 ) ⊥ {\ker({\boldsymbol{A}})^{\perp}} in the Hessian metric derived from Ψ.

The Whittaker Gabor transform (WGT) is a novel addition to the class of Gabor transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time. Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the aim of this article is to explore two more aspects of the time-frequency analysis associated with the WGT including the spectral analysis associated with the concentration operators and the scalogram.