Integral Representation Research Articles

Large deviations for slow–fast processes on connected complete Riemannian manifolds

We consider a class of slow–fast processes on a connected complete Riemannian manifold M. The limiting dynamics as the scale separation goes to ∞ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with Hamilton–Jacobi–Bellman (HJB) equation techniques. Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on M and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.

Integral Representations and Zeros of the Lommel Function and the Hypergeometric 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1F_2$$\\end{document} Function

We give different integral representations of the Lommel function sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} involving trigonometric and hypergeometric 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2F_1$$\\end{document} functions. By using classical results of Pólya, we give the distribution of the zeros of sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} for certain regions in the plane (μ,ν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mu ,\ u )$$\\end{document}. Further, thanks to a well known relation between the functions sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} and the hypergeometric 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ _1F_2$$\\end{document} function, we describe the distribution of the zeros of 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1F_2$$\\end{document} for specific values of its parameters.

Trapped acoustic waves and raindrops: High-order accurate integral equation method for localized excitation of a periodic staircase

We present a high-order boundary integral equation (BIE) method for the frequency-domain acoustic scattering of a point source by a singly-periodic, infinite, corrugated boundary. We apply it to the accurate numerical study of acoustic radiation in the neighborhood of a sound-hard two-dimensional staircase modeled after the El Castillo pyramid. Such staircases support trapped waves which travel along the surface and decay exponentially away from it. We use the array scanning method (Floquet–Bloch transform) to recover the scattered field as an integral over the family of quasiperiodic solutions parameterized by on-surface wavenumber. Each such BIE solution requires the quasiperiodic Green's function, which we evaluate using an efficient integral representation of lattice sum coefficients. We avoid the singularities and branch cuts present in the array scanning integral by complex contour deformation. For each frequency, this enables a solution accurate to around 10 digits in a few seconds. We propose a residue method to extract the limiting powers carried by trapped modes far from the source. Finally, by computing the trapped mode dispersion relation, we use a simple ray model to explain an acoustic chirp-like time-domain response that is referred to in the literature as the “raindrop effect.”

FeynCalc 10: Do multiloop integrals dream of computer codes?

In this work we report on a new version of FeynCalc, a Mathematica package widely used in the particle physics community for manipulating quantum field theoretical expressions and calculating Feynman diagrams. Highlights of the new version include greatly improved capabilities for doing multiloop calculations, including topology identification and minimization, optimized tensor reduction, rewriting of scalar products in terms of inverse denominators, detection of equivalent or scaleless loop integrals, derivation of Symanzik polynomials, Feynman parametric as well as graph representation for master integrals and initial support for handling differential equations and iterated integrals. In addition to that, the new release also features completely rewritten routines for color algebra simplifications, inclusion of symmetry relations between arguments of Passarino–Veltman functions, tools for determining matching coefficients and quantifying the agreement between numerical results, improved export to ▪ and first steps towards a better support of calculations involving light-cone vectors.

Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

We consider an arbitrary deformation of the Gaussian matrix model parametrized by Miwa variables za. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a polynomial made from the values of skew Schur functions at the Gaussian locus pk=δk,2. This linear operation includes multiplication with an exponential eza2/2 and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary K-polynomials, which appeared in superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in great detail the generating function of correlators ⟨(TrX)k⟩ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian β-ensemble. Published by the American Physical Society 2024

Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field.

Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial Model

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} that are constrained to a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-sphere, equivalently a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.

Form factors, spectral and Källén-Lehmann representation in nonlocal quantum gravity

We discuss the conical region of convergence of exponential and asymptotically polynomial form factors and their integral representations. Then, we calculate the spectral representation of the propagator of nonlocal theories with entire form factors, in particular, of the above type. The spectral density is positive-definite and exhibits the same spectrum as the local theory. We also find that the piece of the propagator corresponding to the time-ordered two-point correlation function admits a generalization of the Källén-Lehmann representation with a standard momentum dependence and a spectral density differing from the local one only in the presence of interactions. These results are in agreement with what already known about the free theory after a field redefinition and about perturbative unitarity of the interacting theory. The spectral and Källén-Lehmann representations have the same standard local limit, which is recovered smoothly when sending the fundamental length scale ℓ* in the form factor to zero.

Measure theoretic aspects of the finite Hilbert transform

AbstractThe finite Hilbert transform , when acting in the classical Zygmund space (over ), was intensively studied in [8]. In this note, an integral representation of is established via the ‐valued measure for each Borel set . This integral representation, together with various non‐trivial properties of , allows the use of measure theoretic methods (not available in [8]) to establish new properties of . For instance, as an operator between Banach function spaces is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for plays a crucial role.

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions.

The continuous Choquet integral representation theorems in an abstract setting

The purpose of the paper is to formulate Choquet integral representation theorems for a monotone functional on a collection of functions in such a way that the representing measures are simultaneously inner and outer continuous on appropriate collections of sets such as open, closed, compact, and measurable. This type of theorem is referred to as the continuous Choquet integral representation theorem and will be discussed in a setting general enough for practical use. The benefits of our results are as follows:(i)the representing measures are simultaneously inner and outer continuous,(ii)the collections of sets for which the representing measures are inner and outer continuous are larger than those in previous studies,(iii)the regularity of the representing measures is also considered, and(iv)it is possible to handle not only σ-continuous but also τ-continuous functionals.

Quantum Rényi and f-Divergences from Integral Representations

Quantum Rényi and f-Divergences from Integral Representations

Expansion by regions meets angular integrals

We study the small-mass asymptotic behavior of so-called angular integrals, appearing in phase-space calculations in perturbative quantum field theory. For this purpose we utilize the strategy of expansion by regions, which is a universal method both for multiloop Feynman integrals and various parametric integrals. To apply the technique to angular integrals, we convert them into suitable parametric integral representations, which are accessible to existing automation tools. We use the code asy.m to reveal regions contributing to the asymptotic expansion of angular integrals. To evaluate the contributions of these regions in an epsilon expansion we apply the method of Mellin-Barnes representation. Our approach is checked against existing results on angular integrals revealing a connection between contributing regions and angular integrals constructed from an algebraic decomposition. We explicitly calculate the previously unknown asymptotics for angular integrals with three and four denominators and formulate a conjecture for the leading asymptotics and the pole part for a general number of denominators and masses.

Construction of multi‐bubble blow‐up solutions to the L2$L^2$‐critical half‐wave equation

AbstractThis paper concerns the bubbling phenomena for the ‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders and for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure.

Generalized form of 2D-Laguerre polynomials

The goal of this research is to present and investigate a novel class of 2D-Laguerre polynomials, which contains some well-known 2D-polynomials as special cases.We discuss its numerous features, including integral representations, operational identities, the monomiality principle, generating functions, and linkages to Appll functions of two variables and Lauricella functions of r-variables. Finally, we raise relevant issues for further research.

Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples

The fact that the famous basic reproduction number R0, i.e., the largest eigenvalue of the next generation matrix FV−1, sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, −V, is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F. Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R0 with respect to explicit “age kernels” a(t), which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.

Schrödinger equation with finitely many $$\delta $$-interactions: closed form, integral and series representations for solutions

Schrödinger equation with finitely many $$\delta $$-interactions: closed form, integral and series representations for solutions

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

Homogenization for convection-enhanced thermal transport in sea ice

Sea ice regulates heat exchange between the ocean and atmosphere in Earth’s polar regions. The thermal conductivity of sea ice governs this exchange, and is a key parameter in climate modelling. However, it is challenging to measure and predict due to its sensitive dependence on temperature, salinity and brine microstructure. Moreover, as temperature increases, sea ice becomes permeable, and fluid can flow through the porous microstructure. While models for thermal diffusion through sea ice have been obtained, advective contributions to transport have not been considered theoretically. Here, we homogenize a multiscale advection–diffusion equation that models thermal transport through porous sea ice when fluid flow is present. We consider two-dimensional models of convective flow and use an integral representation to derive bounds on the thermal conductivity as a function of the Péclet number. These bounds guarantee enhancement in the thermal conductivity due to the added flow. Further, we relate the Péclet number to temperature, making these bounds useful for global climate models. Our analytic approach offers a mathematical theory which can not only improve predictions of atmosphere–ice–ocean heat exchanges in climate models, but can provide a theoretical framework for a range of problems involving advection–diffusion processes in various fields of application.

Green’s functions in the presence of a bubble wall

Field theoretical tools are developed so that one can analyze quantum phenomena such as transition radiation that must have occurred during the Higgs condensate bubble expansion through plasma in the early universe. Integral representations of Bosonic and Fermionic propagators are presented for the case that particle masses are varied continuously during the passage through the bubble wall interface between symmetry-restored and symmetry-broken regions. The construction of propagators is based on the so-called eigenfunction expansion method associated with self-adjoint differential operators, developed by Weyl, Stone, Titchmarsh, Kodaira and several others. A novel method of field quantization in the presence of the bubble wall is proposed by using the spectral functions introduced in constructing the two-point Green’s functions.