12,550 Results Found

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series appro...

In this paper, we introduce an integral transform with a kernel defined on the Wiener space. We first establish the existence of the integral transform and present several illustrative examples. As the main result, we derive an approximation theorem...

Here we study the quantitative multivariate approximation of perturbed hyperbolic tangent-activated singular integral operators to the unit operator. The engaged neural network activation function is both parametrized and deformed, and the related ke...

Let $\mathcal{K}\left(r\right)$ be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for $\mathcal{K}\left(r\right)$ . More precisely, we establish the inequality $\frac{2}{\pi }\mathcal{K}(r)>\frac{5{\left(r^{\prime }\right)}^2+126r^{}}{}$...

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both $s=t$ and $s=0$. We propose a Galerkin approximate scheme to solve the equation numerically, a...

We develop a PDE and boundary integral framework for quaternion-valued fields on product domains $\Omega \subset \mathbb{H}\times \mathbb{H}$ governed by the mixed left/right Cauchy–Fueter system We identify the natural compatibility condition and prove local solvabi...

In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation...

In this paper, a generalized sequence of Baskakov operators connected to the Riemann–Liouville fractional integral is introduced. These sequences of operators deal with smooth approximation behavior in a wider class, i.e., a class of measurable...

Let $\mathcal{K}\left(r\right)$ be the complete elliptic integral of the first kind. Then, the inequality $2\mathcal{K}\left(r\right)/\pi >{tanh}^{-1}\left(r\right)/{sin}^{-1}\left(r\right)$ holds for all $r\in (0,1)$. This conclusion does not match those in the existing literature.

Spatially dense operative deflection shapes and receptances, acquired in broad frequency bands, increase the detail in the spatial and frequency domains of the responses of parts in actual dynamic loading, manufacturing and mounting conditions. This...

In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems fo...

To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory integral which has one or more saddle points. Int...

The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear...

Approximate Noether symmetries of a Hamiltonian system with two-degrees of freedom have been determined by incorporating the resonances. Approximate first integrals corresponding to these symmetries have been obtained by utilizing the approximate. ve...

We study the universal approximation properties of transformers and neural integral operators for operators in Banach spaces. In particular, we show that the transformer architecture is a universal approximator of integral operators between Höld...

Decomposition integrals on finite spaces have a natural linear programming representation, but their evaluation may require exponentially many variables indexed by subsets of the underlying space. Restricting the admissible sets or collections produc...

In this article we present the $L_p$, $p\ge 1$, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation fu...

Background/Objectives: Identification of human remains is of utmost importance for criminal investigations and providing closure to the families. The reconstruction of a biological profile of the individual will narrow down the list of candidates for...

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an id...

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fun...

This paper introduces a Gegenbauer-based fractional approximation (GBFA) method for high-precision approximation of the left Riemann–Liouville fractional integral (RLFI). By using precomputable fractional-order shifted Gegenbauer integration ma...

In this article, we employ the uniform and $L_p$, $1\le p<\infty$ approximation properties of general smooth multivariate singular integral operators over ${\mathbb{R}}^N$, $N\ge 1$. It is a trigonometric relief approach with detailed applications to the correspondin...

Covering granular reduction is an important issue in multi-covering information systems. The main methods to solve this problem are set operators. How to solve this problem by quantitative analysis is an interesting topic. Furthermore, as a type of n...

In this paper, a neural adaptive fault-tolerant control scheme is proposed for the integrated attitude and position control of spacecraft proximity operations in the presence of unknown parameters, disturbances, and actuator faults. The proposed cont...

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion the...

This study introduces a multivariate extension to the class of stochastic volatility models, employing integrated nested Laplace approximations (INLA) for estimation. Bayesian methods for estimating stochastic volatility models through Markov Chain M...

Solving mixed-integer nonlinear programs (MINLPs) is hard from both a theoretical and practical perspective. Decomposing the nonlinear and the integer part is promising from a computational point of view. In general, however, no bounds on the objecti...

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS for...

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. W...

We investigate a nonlinear integral equation derived through moment approximation from the individual-based representation of spatial logistic dynamics. The equation describes how the densities of pairs of individuals represented by points in continu...

Hydrodynamic models of river networks are commonly used for flood disaster simulation, and the accuracy of model parameter settings directly affects the reliability of simulation results. Among these, Manning’s roughness coefficient is the core...

This research investigates spatio-temporal patterns of police calls-for-service in the Region of Waterloo, Canada, at a fine spatial and temporal resolution. Modeling was implemented via Bayesian Integrated Nested Laplace Approximation (INLA). Tempor...

In this paper, we consider a collocation method for solving a nonlinear Volterra integral equation involving the solution-dependent singular kernel ${(t-s)}^{-\alpha \left(u\right(s\left)\right)}F(s,u\left(s\right))$, where $F(s,u\left(s\right))=s^{\sigma }\varphi (s,u\left(s\right))$, $0<\sigma <1$, a...

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin...

An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with...

This paper presents a numerical iterative method for the approximate solutions of nonlinear Volterra integral equations of the second kind, with weakly singular kernels. We derive conditions so that a unique solution of such equations exists, as the...

Loaded Volterra integral equations represent a novel class of integral equations that have attracted considerable attention in recent years due to their numerous applications in various fields of science and engineering. This class of Volterra integr...

This paper aims to consider approximation-estimation tests for decision-making by machine-learning methods, and integral-estimation tests are defined, which is a generalization for the continuous case. Approximation-estimation tests are measurable sa...

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstr...

In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtain...

This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton’s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operati...

A summary of the relationship between the Langevin equation, Fokker-Planck-Kolmogorov forward equation (FPKfe) and the Feynman path integral descriptions of stochastic processes relevant for the solution of the continuous-discrete filtering problem i...

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants&m...

In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of $\mathcal{O}\left(n^2\right)$ interaction...

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuab...

The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has alrea...

This paper proposes a novel mixed numerical scheme for approximating linear Fredholm integral equation systems (LFISs) using a combination of Bernoulli polynomials (BPs) and enhanced block-pulse functions $\left(\mathcal{E}\mathcal{B}\mathcal{P}Fs\right)$. This suggested representation makes...

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast an...

Great attention has been paid to the propagation of electromagnetic (EM) waves across the sea surface due to its important applications. Most of the previous research, however, focuses on the half-space model illustrating the deep sea environment. In...

The transponder-style deception jamming implemented by Digital Radio Frequency Memory (DRFM) exhibits high similarity to real target radar echoes, while traditional detection methods suffer severe performance degradation under low signal-to-noise rat...