Arbitrary Multiplicity Research Articles

All-multiplicity amplitudes in impulsive PP-waves from the worldline formalism

We use the worldline formalism to derive Bern-Kosower type Master Formulae for the tree-level scattering of a charged particle and an arbitrary number of photons on impulsive PP-waves, where the coupling of the PP-wave to matter is treated fully non-perturbatively. We show that, in a certain kinematic regime characterised by a semi-classical positive energy condition, both off-shell currents and scattering amplitudes exhibit two novel factorisation structures. First, they may be written as currents in vacuum but with a single additional photon, averaged over the momentum of that photon. This converts the all-orders interaction with the PP-wave into a single effective interaction. Second, the currents and amplitudes may be written as a weighted average of the corresponding quantities in an impulsive plane wave background, with the average taken over all possible field strengths of the plane wave. This generalises a known single-photon result to arbitrary multiplicity.

Representation computation for the hypergeometric function of a Hermitian matrix argument

We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.

Multi-geometric discrete spectral problem with several pairs of zeros for Sasa–Satsuma equation

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena.

Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of (C0,α0,r0)-almost area minimizing currents of arbitrary dimension m and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current T is an open dense set in the boundary.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

AbstractWe consider integral area-minimizing 2-dimensional currents $T$ T in $U\subset \mathbf {R}^{2+n}$ U ⊂ R 2 + n with $\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 , where $Q\in \mathbf {N} \setminus \{0\}$ Q ∈ N ∖ { 0 } and $\Gamma $ Γ is sufficiently smooth. We prove that, if $q\in \Gamma $ q ∈ Γ is a point where the density of $T$ T is strictly below $\frac{Q+1}{2}$ Q + 1 2 , then the current is regular at $q$ q . The regularity is understood in the following sense: there is a neighborhood of $q$ q in which $T$ T consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma $ Γ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$ Q = 1 . As a corollary, if $\Omega \subset \mathbf {R}^{2+n}$ Ω ⊂ R 2 + n is a bounded uniformly convex set and $\Gamma \subset \partial \Omega $ Γ ⊂ ∂ Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current $T$ T with $\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 is regular in a neighborhood of $\Gamma $ Γ .

Hankel transform, [formula omitted]-Bessel functions and zeta distributions in the Dunkl setting

We study analytic properties of a Hankel transform for the type A Dunkl setting with arbitrary multiplicity parameter k≥0 which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel functions and K-Bessel functions, generalizing those of a symmetric cone which occur when the multiplicity parameter k is chosen to be twice the Peirce dimension constant of the cone. In particular, we show that our K-Bessel function is a solution of a type B system of Dunkl operators which is a generalization of the Bessel system on a symmetric cone. Furthermore, we define zeta distributions and prove a functional equation relating them with their type B Dunkl transform. In the case of symmetric cones the aforementioned functional equation reduces to the known functional equation between zeta distributions and their Fourier transform. For the proof of this functional equation, the K-Bessel function and its analytic properties turn out to be crucial. Finally, we examine which of the zeta distributions are given by a positive measure.

Topology optimization of stability‐constrained structures with simple/multiple eigenvalues

AbstractThis work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.

Development of the mathematical apparatus of Boolean derivatives to improve the reliability and validity of information processing

The article substantiates the concept of the Boolean derivative of the generalized logic model (GLM) with respect to the fault parameter and the basis of its use for solving technical diagnostics problems. Since such a GLM describes the occurrence of faults of arbitrary multiplicity, the diagnostic tests built using this model allow quick identification of the technical condition of the logic circuit, and therefore reduce the recovery time and improve the reliability and validity of information processing

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments.

Асимптотика собственных значений оператора Лапласа с двумя разбегающимися возмущениями на плоскости (случай произвольной кратности)

The Laplacian with a pair of distant perturbations is studied in two-dimensional space. Perturbations are understood as real finite continuous potentials. The discrete spectrum of the perturbed Laplacian is studied when the distance between the potentials increases. The presence of its eigenvalues and eigen-functions that correspond to them is considered for various cases of multiplicities of the limiting eigenvalue. The first case of the considered multiplicity is the double limiting eigenvalue. By this we mean the simple and isolated Laplacian eigenvalue with the first potential, as well as the simple and isolated Laplacian eigenvalue with the second potential. The second case under consideration is the case of arbitrary multiplicity of the limiting eigenvalue. By this we mean the Laplacian eigenvalue with the first potential of arbitrary multiplicity and the Laplacian eigenvalue with the second potential also of arbitrary multiplicity. In both cases under consideration (of multiplicity two and arbitrary), the first terms of formal asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed Laplacian are constructed. A complex exponential-power structure of the constructed asymptotic is demonstrated. Also, in both cases under consideration, symmetry with respect to zero of the first corrections of the asymptotics of the eigenvalues of the perturbed Laplacian is shown

Sharp estimates for W-invariant Dunkl and heat kernels in the An case

In this article, we prove exact estimates for the W-invariant Dunkl kernel and heat kernel, for the root system of type A with arbitrary positive multiplicities. We apply the estimates of the W-invariant Dunkl heat kernel to compute sharp estimates for the Newton kernel and for the s-stable semigroups generated by a fractional power of the W-invariant Dunkl Laplacian.

The environment as a frontier of efficiency in tissue P systems with communication rules

Originally, in P systems the environment plays a passive role; that is, it can only receive objects, without having the ability to send objects to the system. Later, tissue P systems were introduced, where the cells are located in the environment in the sense that they can communicate between each other but also with the environment. In fact, a special alphabet was introduced as a way to symbolize the chemical elements available in it and that can interact with the cells. In the framework of membrane computing, all the objects of this alphabet are present in the environment with an arbitrary multiplicity at the beginning of the computation; that is, there are enough objects of this type in the environment to fire the rules that can be fired by these objects. From the computational complexity point of view, it seems to be a very strong ingredient, since it adds a virtually infinite number of objects to the system in the whole computation. In this paper, we demonstrate that the behaviour of this special alphabet can be simulated by a generation stage ruled by evolutional communication rules and/or division/separation rules, such that the ability of these systems to efficiently solve presumably hard problems is not changed if the environment does not play an active role.

Computational tools for the simulation and analysis of spin-polarized EPR spectra

The EPR spectra of paramagnetic species induced by photoexcitation typically exhibit enhanced absorptive and emissive features resulting from sublevel populations that differ from thermal equilibrium. The populations and the resulting spin polarization of the spectra are dictated by the selectivity of the photophysical process generating the observed state. Simulation of the spin-polarized EPR spectra is crucial in the characterization of both the dynamics of formation of the photoexcited state as well as its electronic and structural properties. EasySpin, the simulation toolbox for EPR spectroscopy, now includes extended support for the simulation of the EPR spectra of spin-polarized states of arbitrary spin multiplicity and formed by a variety of different mechanisms, including photoexcited triplet states populated by intersystem crossing, charge recombination or spin polarization transfer, spin-correlated radical pairs created by photoinduced electron transfer, triplet pairs formed by singlet fission and multiplet states arising from photoexcitation in systems containing chromophores and stable radicals. In this paper, we highlight EasySpin’s capabilities for the simulation of spin-polarized EPR spectra on the basis of illustrative examples from the literature in a variety of fields ranging across chemistry, biology, material science and quantum information science.

A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity

In this short note, we consider the elliptic problem λϕ+Δϕ=η|ϕ|σϕ,ϕ|∂Ω=0,λ,η∈C,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lambda \\phi + \\Delta \\phi = \\eta |\\phi |^\\sigma \\phi ,\\quad \\phi \\big |_{\\partial \\Omega }=0,\\quad \\lambda , \\eta \\in {{\\mathbb {C}}}, \\end{aligned}$$\\end{document}on a smooth domain Omega subset {{mathbb {R}}}^N, Ngeqslant 1. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature of such bifurcations in some specific cases. We conclude with the stability analysis of these branches under the complex Ginzburg-Landau flow.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Kinetic energy density for open-shell systems: analysis and development of a novel technique

The quest for an approximate yet accurate kinetic energy density (KED) functional is central to the development of orbital-free density functional theory. While a recipe for closed-shell systems has been proposed earlier, we have shown that it cannot be naïvely extended to open-shell atoms. In this present work, we investigated the efficacy of an ad-hoc recipe to compute the kinetic energy densities for open-shell atoms by extending the methodology used for closed-shell systems. We have also analysed the spin-dependent features of Pauli potentials derived from two previously devised enhancement factors. Further, we have proposed an alternate but exact methodology to systematically compute the KED for atoms of arbitrary spin multiplicity.

A Stochastic Transport Model for the Cumulative Number of Fissions and Deposited Fission Energy

The stochastic theory of neutron transport is extended to describe the cumulative distribution of fission numbers and deposited fission energy in a subvolume of a multiplying assembly. Solutions for the probability distributions are obtained using analytical approximations and Monte Carlo simulation in lumped geometry and in symmetric homogeneous and heterogeneous spheres. The results show the development of a power-law tail in the steady-state fission number and deposited energy distributions when the medium is critical, independent of the fission neutron multiplicity distribution and domain heterogeneity. In contrast, the asymptotic decay is faster than exponential in subcritical media due to rapid chain extinction and in supercritical media due to the increasing probability of chain divergence. A formal asymptotic analysis of the problem in lumped geometry with an arbitrary fission neutron multiplicity confirms the existence of power-law tails at critical.

Toric eigenvalue methods for solving sparse polynomial systems

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety X X . Our starting point is a homogeneous ideal I I in the Cox ring of X X , which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of I I . We study these properties and provide bounds on the size of the matrices in our approach when I I is a complete intersection.

Асимптотически робастный инвариантный алгоритм демодуляции DPSK-сигналов при воздействии внешних помех с априорно неопределенными параметрами

The study aims at synthesizing an asymptotically robust invariant demodulation algorithm for DPSK signals with arbitrary modulation multiplicity used in aircraft communication systems and observed against the background of noise with an unknown probability distribution and a noise complex with a priori uncertain parameters. The analysis of the efficiency of the proposed algorithm is carried out by a computer simulation modeling. The algorithm implies automatic detection and rejection of narrowband pulse noise. This type of interference is typical for radio-navigation systems. It has been shown that the narrowband pulse noise should be eliminated if the signal-to-interference ratio becomes less than -10 dB. The use of the asymptotic robustness principle ensures the stability of the algorithm characteristics to changes in the character of the noise probability distribution density. It was found that the joint application of narrowband pulse interference elimination and the asymptotically robust approach provides reliable demodulation performance both when receiving against the background Gaussian noise and under the action of pulse interference and noise with "heavy tails". In addition, it has been shown that the use of signals with differential phase modulation, direct sequence spread spectrum, and the sidelobe levels of less than 10% effectively suppresses interference caused by multipath signal propagation.