Integral Formula Research Articles

Some integral inequalities on weighted Riemannian manifolds with boundary

In this paper, we continue to study some applications with respect to a Reilly type integral formula associated with the ϕ \phi -Laplacian. Some inequalities of Brascamp-Lieb type and Colesanti type are provided.

Ob approksimatsii poverkhnostnykh proizvodnykh funktsiy s primeneniem integral'nykh operatorov

Integral formulas are presented for approximating the surface gradient (of a scalar function given on a surface) and divergence (of a tangent vector field given on a surface) that are analogs of the well-known formulas for the derivatives of a function on a plane. Estimates of the error in the approximation of these functions are obtained. The question of subsequent approximation of the integrals that give expression for the surface gradient and divergence by quadrature sums over the values of the function under study at the nodes selected on the cells of the unstructured grid approximating the surface is also considered.

High-order methods for the simulation of unsteady counterflow flames subject to stochastic forcing of large amplitude

Unsteady counterflows are employed to understand and model the effect of turbulence on flames. We present a novel numerical approach for the simulation of one-dimensional unsteady counterflow flames with fourth order spatial discretization and up to fourth order time discretization. The approach couples a three-stage Lobatto IIIa formula for boundary value problems and variable-order, variable time step size Backward Differentiation Formulas for time integration. The framework is explained in detail, its computational performance is analysed, and its use is demonstrated for the case of stochastic forcing of premixed counterflow flames, whereby the imposed rate of strain is a multi-scale lognormal discrete random process with exponential autocorrelation. High-order spatial and temporal discretization make the approach well-suited for the accurate and computationally efficient simulation of the effect of turbulence on flames, which are characterised by large amplitude stochastic fluctuations of the local rate of strain.

Weighted Brunn-Minkowski Theory I: On weighted surface area measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in Rn – in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

Adapting a new formula to generalize multidimensional transforms

In this article, we present a new general formula of a multidimensional general integral transform; this new formula covers mostly all types of multi‐integral transforms of the Laplace class. Moreover, single, double, and triple integral transforms of the Laplace kind can be derived from the proposed formula. So, the obtained results in this study can be utilized to reduce computations when using these transforms of same category in other studies. Fundamental properties of the new transform, including linearity, uniqueness, and the inverse, are introduced. In addition, we establish the multi‐convolution theorem and some important results related to partial derivatives. To show the applicability of new multidimensional transform, we employ the new proposed transform to investigate the solutions of some partial differential equations and partial integro‐differential equations.

Classification of Solutions to Conformally Invariant Systems with Mixed Order and Exponentially Increasing or Nonlocal Nonlinearity

In this paper, without any assumption on and under the extremely mild assumption at for some arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in where , and that satisfies the finite total curvature condition . In order to show the integral representation formula and the crucial asymptotic property for , we derive and use an inequality, which is itself of independent interest. When , the system is closely related to single conformally invariant equations and on , which have been quite extensively studied (cf. [H. Brezis and F. Merle, Comm. Partial Differential Equations, 16 (1991), pp. 1223–1253; D. Cao, Comm. Partial Differential Equations, 17 (1992), pp. 407–435; S.-Y. A. Chang and P. C. Yang, Math. Res. Lett., 4 (1997), pp. 91–102; W. Chen and C. Li, Duke Math. J., 63 (1991), pp. 615–622; W. Chen, C. Li, and Y. Li, Adv. Math., 308 (2017), pp. 404–437; W. Chen, Y. Li, and R. Zhang, J. Funct. Anal., 272 (2017), pp. 4131–4157], etc.). We also derive classification results for nonnegative solutions to a conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in . Extensions to mixed order conformally invariant systems in with general dimensions are also included.

A Boundary Integral Equation Method for Computing Numerical Conformal Mappings onto the Disk with Rectilinear Slit and Spiral Slits Regions

This article proposes a boundary integral equation method for computing numerical conformal mappings of bounded multiply connected region Ω onto the disk with rectilinear slit and spiral slits regions, Ω1 and Ω2 Initially, the process involves calculating the boundary value of the canonical region. Cauchy’s integral formula can then be used to compute the mapping of the interior values. The effectiveness of the proposed method is demonstrated using several numerical examples.

Experimental and probabilistic study of fatigue crack growth behavior of steel Q345qC under different stress ratios

The stress ratio effects and the scatter of the fatigue test results are the key issues in the design of steel fatigue performance. This paper reports an experimental and probabilistic investigation of the fatigue crack growth (FCG) behavior of steel Q345qC. Firstly, the microstructural characteristics and mechanical properties of steel Q345qC were examined via an optical microscope and tensile tests. Next, the FCG behavior of steel Q345qC was investigated under different stress ratios. The fatigue crack growth rate (FCGR) equations incorporating survival probabilities were proposed based on Paris and Walker laws. Finally, the theoretical FCG life under different survival probabilities in the crack stable growth region is calculated based on the integral formulas of Paris and Walker law and compared with the tested FCG life. The experimental results demonstrate that the larger the stress ratio, the shorter the length of stress intensity factor (SIF) amplitude experienced in the stable growth region and the smaller the SIF at the end of this region. Both probabilistic Paris and Walker equations can be used for FCGR calculations when the survival probability is set as 50%. The FCG life is calculated using the integral formula based on the Walker law with a high safety tolerance and a wide application interval of stress ratio when the survival probability is set as 90%.

On the Splitting Tensor of the Weak f-Contact Structure

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT.

On Mneimneh's binomial sum involving harmonic numbers

Quite recently, Mneimneh introduced the remarkable result whereby∑k=0nHk(nk)pk(1−p)n−k=∑i=1n1−(1−p)ii for 0≤p≤1 as the main result of a 2023 Discrete Mathematics article, where Hk=1+12+⋯+1k denotes the kth harmonic number. Mneimneh's formula, as above, generalizes a result due to Paule and Schneider and to Spivey, and naturally arises as an expected value of a sum involving permutations chosen uniformly at random. In contrast to Mneimneh's probabilistic methods employed to prove the above formula, we generalize the above result so as to not require the given bounds on p, according to our two much simplified proofs of this generalized version of Mneimneh's formula introduced in this article. Our new and simplified proofs demonstrate how our previously introduced methods concerning harmonic sums may be applied systematically to prove and generalize identities as above. Our first proof is based on a way of extending Zeilberger's algorithm so as to be applicable to non-hypergeometric summands; our second proof is based on a beta-type integral formula. We also consider how Mneimneh's identity may be proved and generalized using the Mathematica package Sigma.

Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems

Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems

Some finite integrals involving Mittag-Leffler confluent hypergeometric function

Abstract In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented.

The path integral formula for the stochastic evolutionary game dynamics

Although the long-term behavior of stochastic evolutionary game dynamics in finite populations has been fully investigated, its evolutionary characteristics in a limited period of time is still unclear. In order to answer this question, we introduce the formulation of the path integral approach for evolutionary game theory. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability is expected to be a new mathematical tool to explore the stochastic game evolutionary dynamics. As an example, we derive the transition probability for stochastic evolutionary game dynamics by the path integral in a limited period of time with the updating rule of the Wright-Fisher process.

A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain.

A note on the initial motion of a hanging string with an initial velocity at its free end

The initial movement of a hanging string is revisited through an integral transformation, the analysis of the integral solution allows to obtain some exact solutions for the string profile and to understand the physical meaning of the solutions. Finally, approximate solutions to the integral formula are found and compared with the numerical solution of the problem.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

A novel method for calculating the mutual inductance between two perpendicular coils in wireless power transfer

SummaryThe perpendicular coil is increasingly applied in some wireless power transfer (WPT) systems where space position is limited. The existing method of calculating the mutual inductance between perpendicular coils is mainly applied to rectangular and circular coils. However, the practical rectangular coil often has rounded corners, so it unavoidably introduces systematic errors. Concerning this issue, this paper proposed an analytical method to calculate the mutual inductance between mutually perpendicular round‐corner rectangular (RCR) coils. The proposed method is based on a second‐order vector potential (SOVP) model where the mutual inductance between the coils is described as a double integral formula. In this formula, a novel shape and position function is introduced. This shape and position function describes not only RCR coils but also rectangular (including square), circular, and runway ones. A series of experiments were used to validate the proposed method. According to the consistency validation, the proposed method can be more generalized for calculating the mutual inductance of rectangular and circular coils. Through experimental comparison, the proposed method computes the mutual inductance between the perpendicular coil in excellent agreement with the measured ones. Therefore, the proposed method unifies the mutual inductance calculation of mutually perpendicular circular, rectangular (square), runway, and RCR coils and can provide an important basis for designing WPT systems using the above coils.

Complex Analysis and Residue Theorem

The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a functions Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

An advanced discrete‐time RNN for handling discrete time‐varying matrix inversion: Form model design to disturbance‐suppression analysis

AbstractTime‐varying matrix inversion is an important field of matrix research, and lots of research achievements have been obtained. In the process of solving time‐varying matrix inversion, disturbances inevitably exist, thus, a model that can suppress disturbance while solving the problem is required. In this paper, an advanced continuous‐time recurrent neural network (RNN) model based on a double integral RNN design formula is proposed for solving continuous time‐varying matrix inversion, which has incomparable disturbance‐suppression property. For digital hardware applications, the corresponding advanced discrete‐time RNN model is proposed based on the discretisation formulas. As a result of theoretical analysis, it is demonstrated that the advanced continuous‐time RNN model and the corresponding advanced discrete‐time RNN model have global and exponential convergence performance, and they are excellent for suppressing different disturbances. Finally, inspiring experiments, including two numerical experiments and a practical experiment, are presented to demonstrate the effectiveness and superiority of the advanced discrete‐time RNN model for solving discrete time‐varying matrix inversion with disturbance‐suppression.

Projections of orbital measures and quantum marginal problems

This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn’s problem, the randomized Schur’s problem, and the orbital corners process. In this general setting, we prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to multiplicity problems in representation theory as well as to known results in the symplectic geometry literature. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities.