At present, the construction of optimal quadrature formulas for the approximate computation of highly oscillatory integrals is gaining increasing importance worldwide. The development of optimal quadrature formulas for differentiable functions in different spaces to approximate highly oscillatory integrals is one of the most significant issues in computational mathematics. The approximation of highly oscillatory integrals using optimal numerical techniques constitutes an important area of research in computational mathematics. The significance of this problem is associated with its broad range of practical applications, including digital signal processing, image analysis, radar systems, medical technologies, and computer tomography. Since many tasks in these fields require the efficient evaluation of oscillatory integrals, the development of accurate computational approaches remains highly relevant. In this study, attention is focused on the construction of optimal quadrature formulas aimed at improving the numerical approximation of highly oscillatory integrals and enhancing the effectiveness of their computation. The formulation of optimal quadrature formulas with a derivative and an exponential weight function in the L_2^(2) space is examined in this study. The optimal quadrature formula yields an expression for the square of the norm of the error functional. A discrete system of the Wiener–Hopf type is created to assess the optimal coefficients. In particular, explicit representations for the coefficients are produced under the criteria ω h ∈ℤ and ω 0 , and the square of the error functional norm is then evaluated.
We prove uniform global regularity for the 2D micropolar Rayleigh–Bénard system for all thermal diffusivities δ≥ 0 , and establish the first explicit O(δ ^1/2) convergence rate in L^2 to the zero-diffusivity limit as δ→ 0 .
This paper investigates a mixed-type partial differential equation involving both classical and fractional time derivatives, where the spatial operator A(D) is assumed to be a homogeneous, symmetric, elliptic differential operator with constant coefficients. The equation transitions in time: for t<0 , it reduces to a classical evolution equation, while for t>0 , it becomes a subdiffusion equation with a Caputo fractional derivative of order 0<ρ <1 . The domain is taken to be the N -dimensional torus 𝕋^N , with periodicity conditions imposed on each spatial variable. The so-called Dezin problem is formulated under a gluing condition at t=0 and a nonlocal condition linking the values of the solution at t=-α and t=0 . The existence and uniqueness of a classical solution are discussed within an appropriate function space. Furthermore, an inverse problem is considered where the source term is separable as F(x,t) = f(x)g(t) , with g(t) being known, and the goal is to determine both the solution u(x,t) and the unknown spatial component f(x) from additional data at a fixed time t_0 . This study contributes to the theoretical understanding of time-switching diffusion processes and their inverse identification in periodic settings.
For the first time, this paper introduces a new superpolyconvolution operator for six Hartley integral transforms and studies its fundamental algebraic properties. We apply this superpolyconvolution operator to solve some classes of integral equations with complex kernel, integro-differential equations of Barbashin type, and obtain their solutions in closed form.
Ultrasound images are often degraded by speckle noise, which reduces their quality and makes clinical interpretation more difficult. Traditional methods based on total variation regularization can reduce noise but tend to create artifacts, especially in smooth regions. This study introduces a new image restoration model using fractional-order total variation, denoted TV^α , to improve denoising performance while preserving important image features. The proposed model combines a data fidelity term that reflects the multiplicative nature of speckle noise with a regularization term based on TV^α functional. To solve the resulting optimization problem, we implement the iterative primal–dual method. The model parameters are automatically tuned using Bayesian optimization, which allows an efficient and adaptive exploration of the parameter space and improving restoration performance without manual intervention. We also provide mathematical arguments to support the convergence of the procedure. In the numerical experiments, we compare the proposed model with the existing models. The proposed approach produces highly competitive results for preserving important structural details and effectively smooth images.
We study degenerate fixed-point configurations of a discrete Lotka–Volterra operator on the four-dimensional simplex S^4 . The analysis is based on the principal Pfaffian minors of the associated skew-symmetric interaction matrix, whose signs determine the orientation of the fixed-point card. We show that the vanishing of Pfaffian invariants produces non-oriented edges and generates continua of fixed points connecting cyclic face equilibria. The Jacobian spectrum along these continua is derived, and the corresponding local non-hyperbolic dynamics is classified. It is shown that each continuum has neutral directions associated with the fixed-point family itself, while a complex conjugate pair of eigenvalues has modulus greater than one. Hence, the degeneracy-generated continua possess unstable transversal directions and are not locally asymptotically attracting. The remaining degenerate configurations are classified according to the zero-patterns of the Pfaffian invariants. In the completely degenerate case, all cyclic face equilibria become mutually connected by continua of fixed points. Under a suitable Lyapunov condition, we also prove boundary localization of trajectories starting from the interior of the simplex.
In this paper, by decomposing the confluent hypergeometric function E_7 into eight parts, we demonstrate how some useful and generalized relations between the hypergeometric functions of Srivastava F^( 3) and E_7 can be obtained. It is shown that other main results are specified in order to derive certain relations between the functions F_1 , Ξ _1 , _4F_3 , _2F_3 , _1F_2 , _1F_1 and F_2:1;1^2:2;2 . Some other interesting functional relations involving the exponential function, hyperbolic functions, and modified Bessel functions are also considered.
In this paper, we derive the asymptotic properties of the generalized least squares estimator (GLSE) of autoregressive models endowed with fractional Gaussian noise (the so-called fractional autoregressive models). We establish the consistency and the asymptotic normality of the GLSE. Some simulation studies and a financial application are presented to corroborate our theoretical work.
In this paper, we investigate the dynamical behavior of a quasi-Volterra cubic stochastic operator defined on the two-dimensional simplex. First, an invariant set of the operator is explicitly characterized. Next, it is proved that the operator possesses a unique fixed point, which is shown to be non-hyperbolic. To analyze the global dynamics, a suitable Lyapunov function is constructed, by means of which it is established that the ω -limit set of every trajectory consists of a single point. As a consequence, it is shown that every trajectory originating in the simplex converges to the unique fixed point, thereby establishing the regularity of the operator.
In this manuscript, we examine the Elzaki transform method using multi-delay differential equations with power functions and history functions. Multi-delay differential equations are widely used in engineering, biology, epidemiology, and signal processing. The Laplace transform and Fourier transform are traditional analytical methods for solving multi-delay differential equations. This study utilised the Elzaki transform method to solve multi-delay differential equations, which converts these equations to algebraic equations. We solve second-order multi-delay differential equations using the Elzaki transform method, deriving exact analytical solutions by using power functions and incorporating them into the history functions. The Elzaki transform method is an alternative method to the Laplace transform method to analyse the properties and theoretical implementation for finding the solution and impact of stability conditions.
In this paper, the notions of p-convexity and q-concavity in lattice-normed spaces (E,‖·‖ _E) over the ring of measurable functions are introduced, and properties related to these notions are studied. It is proved that the p-convexification (E^p,‖·‖ _E^p) of a symmetric Banach–Kantorovich space (E,‖·‖ _E) is also a symmetric Banach–Kantorovich space. It is established that the Fatou property and order continuity are preserved when passing from (E,‖·‖ _E) to (E^p,‖·‖ _E^p) .
Geodesic mappings have important applications in Riemannian geometry, geodesic theory and cartography, modelling, physics, and mechanics. This paper examines the question of whether the sign of Gaussian curvature is preserved under geodesic mappings. We prove that nontrivial geodesic mappings preserve the sign of Gaussian curvature for surfaces of revolution with constant curvature.Bibliography: 15 titles.
In this study, we investigate two-step difference schemes of the first and second order of accuracy designed for the numerical approximation of a nonlocal boundary value problem governed by an elliptic equation with a strongly positive operator in the Banach space and the Samarskii–Ionkin condition. Stability and almost coercive stability estimates for the solution of these proposed difference schemes are established. Abstract results are applied to construct stable difference schemes for three boundary value problems for elliptic partial differential equations with the Samarskii–Ionkin condition.
In this paper we study the behavior of Π -completeness for Tychonoff maps under the functor of idempotent probability measures with finite support. We prove that a Tychonoff map is Π -complete if and only if the induced map between the corresponding spaces of idempotent probability measures is Π -complete. As a consequence, the functor of idempotent probability measures with finite support both preserves and reflects Π -completeness for maps. This provides a convenient criterion for verifying Π -completeness via induced mappings and supports the transfer of completeness-type properties to functorially constructed spaces. The obtained result yields a lifting of the functor to the category whose objects are Π -complete spaces and whose morphisms are Π -complete maps
In this paper, a Stefan-type problem with two free boundaries for a nonlinear heat equation in the one-dimensional case is considered. The study of nonlinear problems with free boundaries is carried out using a method based on constructing a priori estimates. In this regard, some initial a priori estimates are first established for solving the problem under consideration. Then, the problem is reduced to a problem with a fixed boundary through a change of variables. The resulting problem has time- and position-dependent coefficients with nonlinear terms. Next, a priori estimates of the Schauder type are constructed for solving the equation with nonlinear terms and a fixed boundary. Based on the estimates obtained, the solvability of the original problem is established.
We investigate an inverse problem of integral geometry in the four-dimensional space ℝ^4 . The problem consists in recovering a compactly supported function in a bounded domain from its line integrals measured along line segments contained in two-dimensional planes passing through a fixed point. In each such plane, however, the set of admissible directions is restricted to a prescribed angular sector. This restriction leads to an incomplete-data and limited-angle problem, which belongs to the class of strongly ill-posed inverse problems. Under natural geometric coverage assumptions, we prove a uniqueness theorem and derive a logarithmic stability estimate for the reconstruction. The main idea is to reduce the problem locally to two-dimensional limited-angle integral geometry on each plane and then combine these estimates by averaging over the Grassmannian G(2,4) , the manifold of two-dimensional linear subspaces of ℝ^4 . This approach reveals the essential role of the four-dimensional geometry and provides a framework for extending planar stability methods to higher-dimensional incomplete-data problems.
The geometric problem of recovering a convex surface from a given function is equivalent to solving a certain Monge–Ampere equation. In this case, the extrinsic curvature is defined as a function of Borel sets. I. Ya. Bakelman constructed this theory and proved the existence and uniqueness of the solution of the Monge–Ampere equation of elliptic type in a simply connected convex domain. A. Artykbaev generalized this solution for a non-simply connected domain applying of the geometry of Galilean space. This paper is devoted to the analytical solution of the Monge–Ampere equation in a non-simply connected domain. The extrinsic curvature of the surface is determined in a non-simply connected domain which is bounded by concentric circles. By applying the transformation which is the motion of Galilean space and the transition to the polar coordinate system, the equation is modified, in which it is possible to separate the variables of the solution, the equation is sought for the sum of three functions. As a result, an analytical form of the solution in a non-simply connected domain bounded by concentric circles is obtained.
This paper investigates the problem of local bifurcations in the vicinity of spatially homogeneous equilibrium states of reaction-diffusion systems in a bounded domain with homogeneous Neumann boundary conditions. The main results focus on studying Turing bifurcation and Andronov-Hopf bifurcation under conditions of multiple degeneracy in the linearized system. In the considered case the codimension of the bifurcation does not match the multiplicity of eigenvalues of the corresponding linear operators, which significantly complicates the analysis. The paper provides a detailed examination of cases leading to multiple bifurcations, establishes conditions for multiple degeneracy, and develops approaches for studying stability and bifurcations near equilibrium states under these conditions. The key result consists of the investigation and characterization of the solution manifold structure arising from bifurcations in reaction-diffusion systems. Potential directions for extending these results of the study of multiple bifurcations are also discussed.
This paper explores weak sharp solutions for a controlled variational inequality governed by a convex fractional curvilinear integral functional with path-independent properties. The fractional framework involves derivatives or integrals of non-integer order, which introduce non-local behavior by incorporating the influence of past states—commonly interpreted as memory-dependent effects in dynamic systems. This modeling approach captures phenomena with long-range temporal or spatial interactions more accurately than classical methods. Although the integral is taken along a curve, path-independence ensures that the functional depends only on the curve’s endpoints, while still being sensitive to geometric data such as orientation and boundary conditions. This structure provides a flexible and rigorous framework for representing system dynamics. The paper analyzes the properties of weak sharp solutions using a combination of variational techniques and a dual disparity functional. Under suitable conditions, it establishes an equivalence between the sufficiency of the minimum principle and the weak sharpness of the solution set for the controlled fractional variational inequality.
In this paper, we introduce and rigorously define the continuous Mehler–Fock–Clifford wavelet transform, exploring its essential properties and analytical structure. We establish and prove the associated Calderón reproducing formula, confirming its validity within the framework of the proposed transform. Furthermore, we represent the Mehler–Fock–Clifford convolution operator as a time-invariant filter and discuss several of its potential applications.
