Search Results (42)

The main aim of this study is to create appropriate criteria for the existence of a unique $\mu$-pseudo almost periodic solution to a particular type of stochastic differential equation, utilizing Bochner’s double sequences criterion, improved Gronwall’s lemma, Hölder’s inequality, and measure theory techniques. By applying the inequalities analysis condition and the fixed point theorem for contraction mapping, we can establish the existence of a single $\mu$-pseudo almost periodic solution in distribution to the given stochastic equation. Finally, we use an example to demonstrate our stochastic processes. Full article

In this short note, we focus on complete translating solitons with a bounded $L_f^n$-norm of the second fundamental form and obtain two results. First, based on a Sobolev-type inequality and a Simons-type inequality, we establish a rigidity theorem of complete translating solitons. Second, based on the same Sobolev-type inequality and a Bochner-type inequality, a vanishing theorem regarding $L_f^p$ weighted harmonic 1-forms is proved. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference $\alpha$, an integer with the Voronoĭ function weight $V_k$. In the case of $V_1\left(x\right)=e^{-x}$, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The $\alpha =0$ case is the divisor function, while the $\alpha =1$ case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan. Full article

Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive relation of an electromagnetic Zener-type fractional 2D and 3D anisotropic model so that a weak form of the thermodynamic (entropy) inequality is satisfied. Moreover, for such models, we analyze the corresponding thermodynamic constraints for field reconstruction and regularity in the 2D anisotropic case. This is carried out by the use of the matrix version of the Bochner theorem in the most general form, including generalized functions as elements of a matrix, which appear in that theorem. The given numerical results confirm the calculus presented in the paper. Full article

This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner–Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs. Full article

In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids $\rho ({\partial }_{t}u+u{\partial }_{x}u)=\mu {\partial }_x\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right),$ augmented with the initial condition $u(0,x)=u_0$, $0<x<L$, and the boundary condition $u(t,0)=u(t,L)=0$, where $\rho$ is the density, $\mu$ the viscosity, u the velocity of the fluid, $1<p<2$, $L>0$, and $T>0$. We show that this initial boundary problem has an unique solution in the Buchner space $L^2\left(0,T;{W_0}^{1,p}(0,1)\right)$ for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions. Full article

This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong convergence of the sequence generated by the iterative scheme to a common fixed point of two operators. The method is illustrated with some examples of application. The procedure is used to approach a common solution of two Fredholm integral equations of the second kind. In the second part of the article, the existence of a fractal function coming from two different Read–Bajraktarević operators is proved. Afterwards, a study of the approximation of fixed points of a fractal convolution of operators is performed, in the framework of Lebesgue or Bochner spaces. Full article

We study the characteristics of affine vector fields on pseudo-Riemannian manifolds and provide tensorial formulas that characterize these vector fields. Using our approach, we present a straightforward proof that any affine vector field on a compact Riemannian manifold is a Killing vector field. Moreover, we establish several necessary and sufficient conditions for an affine vector field on a Riemannian manifold to be classified as a Killing or parallel vector field. Full article

In this paper, we offer Ostrowski-type inequalities that extend the findings that have been proven for functions of one variable with values in Banach spaces, conducted in a remarkable study by Dragomir, to functions of two variables containing values in the product Banach spaces. Our findings are also an extension of several previous findings that have been established for functions of two variable functions. Prior studies on Ostrowski-type inequalities incriminated functions that have values in Banach spaces or Hilbert spaces. This study is unique and significant in the field of mathematical inequalities, and specifically in the study of Ostrowski-type inequalities, because they have been established for functions having values in a product of two Banach spaces. Full article

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals. Full article

We consider an operator ${\{S\left(t\right)\}}_{t\ge 0}$ on a Banach space X with generator A, characterized by being an $\alpha$-times-integrated C-regularized semigroup. The adjoint family $S^{*}\left(t\right):X^{*}\to X^{*}$ is introduced for analysis. ${\{S^{*}\left(t\right)\}}_{t\ge 0}$ maintains the characteristics of an $\alpha$-times-integrated C-regularized semigroup, though with strong continuity and Bochner integrals being substituted by weak* continuity and weak* integrals, respectively. Our investigation focuses on the closed subspace $X^{\odot }$, where ${\{S^{*}\left(t\right)\}}_{t\ge 0}$ exhibits strong continuity. Additionally, a comparison between the adjoint $A^{*}$ of A and the generator of the adjoint family is conducted. Full article

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions. Full article

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms ${\mathbb{D}}^{2n+1}\left(ϵ\right)$ and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants ($\delta$-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern. Full article

This review focuses on the principles, applications, and performance of mpMRI for bladder imaging. Quantitative imaging biomarkers (QIBs) derived from mpMRI are increasingly used in oncological applications, including tumor staging, prognosis, and assessment of treatment response. To standardize mpMRI acquisition and interpretation, an expert panel developed the Vesical Imaging–Reporting and Data System (VI-RADS). Many studies confirm the standardization and high degree of inter-reader agreement to discriminate muscle invasiveness in bladder cancer, supporting VI-RADS implementation in routine clinical practice. The standard MRI sequences for VI-RADS scoring are anatomical imaging, including T₂w images, and physiological imaging with diffusion-weighted MRI (DW-MRI) and dynamic contrast-enhanced MRI (DCE-MRI). Physiological QIBs derived from analysis of DW- and DCE-MRI data and radiomic image features extracted from mpMRI images play an important role in bladder cancer. The current development of AI tools for analyzing mpMRI data and their potential impact on bladder imaging are surveyed. AI architectures are often implemented based on convolutional neural networks (CNNs), focusing on narrow/specific tasks. The application of AI can substantially impact bladder imaging clinical workflows; for example, manual tumor segmentation, which demands high time commitment and has inter-reader variability, can be replaced by an autosegmentation tool. The use of mpMRI and AI is projected to drive the field toward the personalized management of bladder cancer patients. Full article