Arbitrary Order Research Articles

Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions.

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo ([Formula: see text]) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers ([Formula: see text]) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

Algorithm for solving a pump-probe model for an arbitrary number of energy levels.

We describe a generalized algorithm for evaluating the steady-state solution of the density matrix equation of motion, for the pump-probe scheme, when two fields oscillating at different frequencies couple the same set of atomic transitions involving an arbitrary number of energy levels, to an arbitrary order of the harmonics of the pump-probe frequency difference. We developed a numerical approach and a symbolic approach for this algorithm. We have verified that both approaches yield the same result for all cases studied, but require different computation time. The results are further validated by comparing them with the analytical solution of a two-level system to first order. We have also used both models to produce results up to the third order in the pump-probe frequency difference, for two-level systems, and up to first order for three- and four-level systems. In addition, we have used this model to determine accurately, the gain profile for a self-pumped Raman laser, for a system involving 16 Zeeman sublevels in the D1 manifold of ^{87}Rb atoms. We have also used this model to determine the behavior of a single-pumped superluminal laser. In many situations involving the applications of multiple laser fields to atoms with many energy levels, one often makes the approximation that each field couples only one transition, because of the difficulty encountered in accounting for the effect of another field coupling the same transition but with a large detuning. The use of the algorithm presented here would eliminate the need for making such approximations, thus improving the accuracy of numerical calculations for such schemes.

Brauer and Geršgorin Locations of Zeros of Perturbed Chebyshev Polynomials of the Second Kind by Dilation

We consider some perturbations of Chebyshev polynomials of the second kind obtained by modifying by dilation one of its recurrence coefficients at an arbitrary order. By applying Brauer and Geršgorin theorems to Jacobi matrices associated with such perturbed sequences we obtain some locations of their zeros.

Influential simplices mining via simplicial convolutional networks

The identification of influential simplices is crucial for understanding higher-order network dynamics. Yet, despite relatively mature research on influential nodes (0-simplices) mining, characterizing simplices’ influence and identifying influential simplices remain challenging due to notable discrepancies in vital nodes and vital simplices mining. In this paper, we propose a higher-order graph learning model, named influential simplices mining neural networks (ISMnet), to identify vital simplices in simplicial complexes. ISMnet leverages novel higher-order representations: hierarchical bipartite graphs and higher-order hierarchical (HoH) Laplacians, where target simplices are grouped into a hub set and can interact with other simplices. It also employs learnable graph convolution operators in each HoH Laplacian domain to capture interactions among simplices and can identify influential simplices of arbitrary order by changing the hub set. Notably, ISMnet addresses the limitations inherent in traditional graph neural networks that struggle with higher-order tasks, while seamlessly retaining the capability to exploit network topology and node features concurrently. Numerical results on empirical and synthetic datasets demonstrate that ISMnet significantly outperforms existing methods by at least 12% and 4%, respectively, in ranking 2-simplices. In general, this novel framework promises to serve as a potent tool in higher-order network analysis.

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

Method for Generating Closely Packed Orbital Shells and Orbital Capacity Implications

Shell-wise orbital coordination in low Earth orbit can improve space safety, simplify space traffic coordination and management, and optimize orbital capacity. This work describes two methods to generate 2D lattice flower constellations (2D-LFCs) that are defined with respect to either an arbitrary zonal, or zonal, sectoral, and tesseral Earth geopotential. By generating orbital shells that are periodic with respect to the Earth geopotential, it is possible to safely stack shells with vertical separation distances smaller than the osculating variation in semimajor axis and radial distance of each shell, or a corresponding Keplerian 2D-LFC propagated under an aspherical geopotential. These methods exploit previous work on constellations based on time distributions and designs of closed 2D-LFCs under arbitrary Earth geopotentials using repeating ground track orbits. Factors that influence the widths and shapes of these shells are identified. Additionally, simplified formulas for estimating shell geometry and thickness are presented. Based on these results, it is shown that geopotential-aware periodic orbital shells significantly increase the number of admissible orbital shells while preserving shell separation. Finally, this work shows that sequencing shells to group similar or ascending inclinations can further improve capacity versus arbitrary inclination ordering, particularly for smaller shell separation distances.

The Analitical Solution of Linear and Non-Linear Differential- Algebraic Equations (DAEs) with Laplace-Padé Series Method

In this paper, we apply Laplace-Padé Series method to solve linear and non-linear differentialalgebraic equations (DAEs). Firstly, The basic properties of the Laplace-Padé Series method are given. Secondly, we calculate the arbitrary order power series of differential-algebraic equations (DAEs), then convert it to the series form Laplace-Padé. Then, the three differential-algebraic equations (DAEs) are solved by Laplace-Padé Series method. It was seen that the method gave effective and fast results. Therefore, the method can be easily applied to linear and non-linear differential-algebraic equations (DAEs) problems in different fields.

The eXtended virtual element method for elliptic problems with weakly singular solutions

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method’s accuracy on various mesh families, demonstrating optimal convergence rates in the L2- and H1-norms on fractured or L-shaped domains.

Optimal multiple functional regression analysis using perturbation theory

ABSTRACT A regression analysis is presented based on several functions. A normalized data are used which enables the usage of magnitude of orders in perturbation theory. The criteria to eliminate unnecessary base functions are derived with the aid of order of magnitudes in perturbations. The analysis generalizes the previous work on polynomial regression of arbitrary orders. Properties of the regression coefficients are outlined via theorems. Numerical tests are conducted to outline the usage of the analysis to eliminate unnecessary functions and obtain a reasonable approximation of the data in a continuous form.

Sumset Estimates in Convex Geometry

Abstract Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

Geometric properties of generalized Bessel function of arbitrary order and degree

Geometric properties of generalized Bessel function of arbitrary order and degree

Quantum Nonlinear Spectroscopy via Correlations of Weak Faraday‐Rotation Measurements

AbstractThe correlations of fluctuations are key to studying fundamental quantum physics and quantum many‐body dynamics. They are also useful information for understanding and combating decoherence in quantum technology. Nonlinear spectroscopy and noise spectroscopy are powerful tools to characterize fluctuations, but they can access only very few among the many types of higher‐order correlations. A systematic quantum sensing approach, called quantum nonlinear spectroscopy (QNS), is recently proposed for extracting arbitrary types and orders of time‐ordered correlations, using sequential weak measurement via a spin quantum sensor. However, the requirement of a central spin as the quantum sensor limits the versatility of the QNS since usually a central spin interacts only with a small number of particles in proximity and the measurement of single spins needs stringent conditions. Here, the aim is to employ the polarization (a pseudo‐spin) of a coherent light beam as a quantum sensor for QNS. After interacting with a target system (such as a transparent magnetic material), the small Faraday rotation of the linearly polarized light can be measured, which constitutes a weak measurement of the magnetization in the target system. Using a Mach–Zehnder interferometer with a designed phase shift, one can post‐select the effects of the light–material interaction to be either a quantum evolution or a quantum measurement of the material magnetization. This way, the correlated difference photon counts of a certain number of measurement shots, each with a designated interference phase, can be made proportional to a certain type and order of correlations of the magnetic fluctuations in the material. The analysis of the signal‐to‐noise ratios shows that the second‐order correlations are detectable in general under realistic conditions and higher‐order correlations are significant when the correlation lengths of the fluctuations are comparable to the laser spot size (such as in systems near the critical points). Since the photon sensor can interact simultaneously with many particles and interferometry is a standard technique, this protocol of QNS is advantageous for studying quantum many‐body systems.

Non-numerical weakly relational domains

The weakly relational domain of Octagons offers a decent compromise between precision and efficiency for numerical properties. Here, we are concerned with the construction of non-numerical relational domains. We provide a general construction of weakly relational domains, which we exemplify with an extension of constant propagation by disjunctions. Since for the resulting domain of 2-disjunctive formulas satisfiability is NP-complete, we provide a general construction for a further, more abstract, weakly relational domain where the abstract operations of restriction and least upper bound can be efficiently implemented. In the second step, we consider a relational domain that tracks conjunctions of inequalities between variables, and between variables and constants for arbitrary partial orders of values. Examples are sub(multi)sets, as well as prefix, substring or scattered substring orderings on strings. When the partial order is a lattice, we provide precise polynomial algorithms for satisfiability, restriction, and the best abstraction of disjunction. Complementary to the constructions for lattices, we find that, in general, satisfiability of conjunctions is NP-complete. We therefore again provide polynomial abstract versions of restriction, conjunction, and join. By using our generic constructions, these domains are extended to weakly relational domains that additionally track disjunctions. For all our domains, we indicate how abstract transformers for assignments and guards can be constructed.

On the Cauchy problem for p-evolution equations with variable coefficients: A necessary condition for Gevrey well-posedness

In this paper we consider a class of p-evolution equations of arbitrary order with variable coefficients depending on time and space variables (t,x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Joint equalization and decoding with per-survivor processing based on super-trellis in time-varying underwater acoustic channels.

This Letter proposes a low-complexity joint equalization and decoding reception scheme based on super-trellis per-survivor processing, making it possible to apply maximum likelihood sequence estimation in high-order underwater acoustic communications under fast time-varying channels. The technique combines trellis-coded modulation states and intersymbol interference states and uses per-survivor processing to track channel parameters. Furthermore, a general trellis configuration for arbitrary order quadrature amplitude modulation signal is provided when truncate the channel is used to describe the intersymbol interference state to 1. Sea trials results show that the performance of proposed method can be more than 1.4 dB superiority than conventional schemes.

On Damping a Control System of Arbitrary Order with Global Aftereffect on a Tree

On Damping a Control System of Arbitrary Order with Global Aftereffect on a Tree

Extensions and investigations of space‐time generalized Riemann problems numerical schemes for linear systems of conservation laws with source terms

AbstractThe space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied.

Sobolev spaces on arbitrary domains and semigroups generated by the fractional Laplacian

We describe a procedure to introduce on an arbitrary domain of Rd Sobolev spaces of arbitrary order s∈R and with an intrinsic characterization via fractional powers of the Dirichlet Laplacian. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup generated by the fractional Dirichlet Laplacian.