Arbitrary Polynomial Research Articles

Insight on the Nonhomogeneous Pantograph Equation with an Arbitrary Polynomial of Degree n: Explicit Solution

This paper solves a nonhomogeneous version of the pantograph equation. The nonhomogeneous term is taken as a polynomial of degree n with arbitrary coefficients. The nonhomogeneous pantograph equation is successfully converted to the standard homogeneous version by means of a simple transformation. An explicit formula is derived for the coefficients of the assumed transformation. Accordingly, the solution of the nonhomogeneous version is obtained in different forms in terms of power series, in addition to exponential functions. The obtained solution in power-series form is investigated to produce exact solutions for several examples under specific relationships between the involved parameters. In addition, exact solutions in terms of trigonometric and hyperbolic functions are determined at a certain value of the proportional delay parameter. The obtained results may be reported for the first time for the present nonhomogeneous version of the pantograph equation and can be further applied to include other versions with different nonhomogeneous terms.

On von Neumann’s inequality on the polydisc

AbstractGiven a d-tuple T of commuting contractions on Hilbert space and a polynomial p in d-variables, we seek upper bounds for the norm of the operator p(T). Results of von Neumann and Andô show that if $$d=1$$ d = 1 or $$d=2$$ d = 2 , the upper bound $$\Vert p(T)\Vert \le \Vert p\Vert _\infty $$ ‖ p ( T ) ‖ ≤ ‖ p ‖ ∞ , holds, where the supremum norm is taken over the polydisc $$\mathbb {D}^d$$ D d . We show that for $$d=3$$ d = 3 , there exists a universal constant C such that $$\Vert p(T)\Vert \le C \Vert p\Vert _\infty $$ ‖ p ( T ) ‖ ≤ C ‖ p ‖ ∞ for every homogeneous polynomial p. We also show that for general d and arbitrary polynomials, the norm $$\Vert p(T)\Vert $$ ‖ p ( T ) ‖ is dominated by a certain Besov-type norm of p.

On some matrix counting problems

AbstractWe estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime . In particular, in the integer case, we improve a recent bound of V. Blomer and J. Li (2022).

Data-free non-intrusive model reduction for nonlinear finite element models via spectral submanifolds

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.

Arithmetic Ramsey theory over the primes

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$ , where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

A new property of arbitrary complex polynomials

A new property of an arbitrary complex polynomial P ( z ) related to its derivative is established. It turns out that for arbitrary three values a 1 , a 2 and a 3 , there are lower bounds for | P ( z ) | when z belongs to the set of a 1 , a 2 , a 3 points of P ( z ) .

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

Compact FE for unbounded attribute-weighted sums for logspace from SXDH

This paper presents the first functional encryption (FE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extsf{FE})$$\\end{document} scheme for the attribute-weighted sum functionality that supports the uniform model of computation. In such an FE scheme, encryption takes as input a pair of attributes (x, z) where x is public and z is private. A secret key corresponds to some weight function f, and decryption recovers the weighted sum f(x)z. In our scheme, both the public and private attributes can be of arbitrary polynomial lengths that are not fixed at system setup. The weight functions are modelled as Logspace Turing machines\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Logspace Turing machines}$$\\end{document}. Prior schemes could only support non-uniform Logspace. The proposed scheme is proven adaptively simulation secure under the well-studied symmetric external Diffie–Hellman assumption against an arbitrary polynomial number of secret key queries both before and after the challenge ciphertext. This is the best possible security notion that could be achieved for FE. On the technical side, our contributions lie in extending the techniques of Lin and Luo [EUROCRYPT 2020] devised for indistinguishability-based payload hiding attribute-based encryption for uniform Logspace access policies and the “three-slot reduction” technique for simulation-secure attribute-hiding FE for non-uniform Logspace devised by Datta and Pal [ASIACRYPT 2021] to the context of simulation-secure attribute-hiding FE for uniform Logspace.

A2GCN: Graph Convolutional Networks with Adaptive Frequency and Arbitrary Order

Graph Neural Networks (GNNs) gain remarkable success in various graph learning tasks under homophily graph assumption. This assumption is extremely fragile since real-world graphs with heterophily are ubiquitous. Under this circumstance, existing GNNs attempt to design or learn graph spectral filters for observed graphs. The representation ability of them, however, is limited due to: (1) Constant frequency response is incapable of simulating complicated filters that real-world applications require. (2) Fixed polynomial order fails to effectively uncover the node label patterns concealing different order neighborhoods. To this end, we propose a novel Graph Convolutional Networks with Adaptive Frequency and Arbitrary Order (A2GCN) to learn various graph spectral filters suitable for distinct networks. Specifically, a simple but elegant filter with adaptive frequency response is designed to span across multiple layers for capturing different frequency components hiding in varying orders, producing A2GCN filter bases. Afterward, the coefficients of the A2GCN basis for each node are learned to achieve A2GCN filters with arbitrary polynomial order. The resulting A2GCN filters possess flexible frequency response that can automatically adapt to the node label pattern, as such, it empowers A2GCN with stronger expressiveness and naturally alleviates the over-smoothing problem. Theoretical analysis is provided to show the superiority of the proposed A2GCN. Additionally, extensive experiments on both node-level and graph-level tasks validate that the proposed A2GCN accomplishes highly competitive performance and improves classification accuracy. Codes are available at https://github.com/AIG22/A2GCN.

Generalizing the Telescoping Procedures for Power Series to Reduce the Degree of Polynomials

By using the Telescoping Procedures for Power Series (TPPS), the degree of any given polynomial can be decreased by one degree on the interval [-1; 1], with only a slight accuracy loss, the reason for this is that, the (TPPS) depends on the Chebyshev polynomials, which have a minimum maximum-absolute value that is distributed uniformly over [-1; 1]. In this paper, we generalized the (TPPS) with the aim to reduce the degree of an arbitrary polynomial for any degree and in any interval [a; b]. The generalization was in the form of a theorem and the theorem was proven using the Induction. The new technique called Generalized Telescoping Procedures for Power Series (GTPPS). We also conducted a study of the error bound of the theorem in the intervals [-1; 1], and [a; b]. The authors believe that this paper paves the way for a comprehensive set of researches and applications in the fields of approximation theory, optimization theory, computer simulation and other computation techniques to issues in numerous scientific disciplines.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.

Dynamic characteristics of vertically irregular structures with random fields of different probability distributions based on stochastic homotopy method

Considering the uncertainty of elastic modulus in different materials as well as the strong nonlinearity between the structural frequency and elastic modulus, it is a great challenge for computing the dynamic characteristics of vertically irregular structures. In this research, a new stochastic homotopy approach is presented to compute the basic dynamic characteristics of the vertically irregular structures, where the elastic modulus of different materials are assumed as the random fields with different probability distributions. To obtain the dynamic characteristics of the vertically irregular structures, a stochastic eigenvalue equation is established firstly. Then the stochastic eigenvalue and eigenvector are represented by the homotopy series, and the stochastic residual error in relation to the stochastic eigenvalue equation is minimized to obtain the coefficients of the homotopy series. Afterwards, the basic dynamic characteristics, including stochastic natural frequencies and modal shapes, of the vertically irregular structures are obtained. In comparison to the sampling-based stochastic surrogate model methods like the Kriging and non-intrusive arbitrary polynomial chaos methods, the presented approach can efficiently yield more stable statistical moments of the natural frequencies. Meanwhile the proposed method can generate the statistical moments of the modal shapes, which is difficult to achieve with the stochastic surrogate model methods. Finally, the effectiveness of the suggested method is testified through two examples of a reinforced concrete-steel column and a realistic reinforced concrete-steel frame structure.

Inf-sup stabilized Scott–Vogelius pairs on general shape-regular simplicial grids for Navier–Stokes equations

This paper considers the discretization of the time-dependent Navier–Stokes equations with the family of inf-sup stabilized Scott–Vogelius pairs recently introduced in [John/Li/Merdon/Rui, Math. Models Methods Appl. Sci., 2024] for the Stokes problem. Therein, the velocity space is obtained by enriching the H1-conforming Lagrange element space with some H(div)-conforming Raviart–Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used.In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a Pk−P0-like system for arbitrary velocity polynomial degree k. Numerical studies verify the theoretical findings.

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

Polynomial chaos expansion of SAR and temperature increase variability in 3 T MRI due to stochastic input data

Objective. Numerical simulations are largely adopted to estimate dosimetric quantities, e.g. specific absorption rate (SAR) and temperature increase, in tissues to assess the patient exposure to the radiofrequency (RF) field generated during magnetic resonance imaging (MRI). Simulations rely on reference anatomical human models and tabulated data of electromagnetic and thermal properties of biological tissues. However, concerns may arise about the applicability of the computed results to any phenotype, introducing a significant degree of freedom in the simulation input data. In addition, simulation input data can be affected by uncertainty in relative positioning of the anatomical model with respect to the RF coil. The objective of this work is the to estimate the variability of SAR and temperature increase at 3 T head MRI due to different sources of variability in input data, with the final aim to associate a global uncertainty to the dosimetric outcomes. Approach. A stochastic approach based on arbitrary Polynomial Chaos Expansion is used to evaluate the effects of several input variability’s (anatomy, tissue properties, body position) on dosimetric outputs, referring to head imaging with a 3 T MRI scanner. Main results. It is found that head anatomy is the prevailing source of variability for the considered dosimetric quantities, rather than the variability due to tissue properties and head positioning. From knowledge of the variability of the dosimetric quantities, an uncertainty can be attributed to the results obtained using a generic anatomical head model when SAR and temperature increase values are compared with safety exposure limits. Significance. This work associates a global uncertainty to SAR and temperature increase predictions, to be considered when comparing the numerically evaluated dosimetric quantities with reference exposure limits. The adopted methodology can be extended to other exposure scenarios for MRI safety purposes.

Optimization study of lead-based reactor heat exchangers based on Dakota coupling CFD

The lead-bismuth fast reactor is one of the fourth-generation reactor types.The microchannel heat exchangers are used as intermediate heat exchangers, which significantly influence the efficiency of supercritical carbon dioxide (S–CO2) Brayton cycle power generation. It is necessary to quantify the uncertainties and optimize the design of heat transfer and pressure drop of the microchannel heat exchangers. Due to the high cost and time-consuming problem of numerical simulation, this study employs the Arbitrary polynomial chaos expansion method to generate surrogate models for simulation calculations. Specifically, the microchannel parameters of the heat exchanger, namely flow direction spacing, transverse spacing, staggered spacing, and inlet velocity, are studied for the simulation. The pressure drop and convective heat transfer coefficient of the heat exchanger are considered as the objective functions for both simulation and optimization models. Uncertainty quantification analysis, sensitivity analysis, and optimization design of those parameters are conducted. By coupling DAKOTA with CFD, an optimization objective function is established to obtain optimal design data within specified ranges.

Uncertainty quantification and global sensitivity analysis of bearing capacity of push-out specimen considering randomness in bond-slip behaviour

As an artificial composite material, the uncertainty in mechanical property of concrete significantly impacts the bond-slip behaviour of the shaped steel concrete structure. To investigate the effect of the uncertainty of concrete mechanical properties on the bearing capacity of push-out specimens, based on the data of a small number of push-out test with identical design parameters, the improved Bootstrap method is utilized to execute sample augmentation for the bond stress-slip curves, and the stochastic bond slip constitutive model is established after considering five parameters (K1, …, K5) in the deterministic bond-slip model as random variables. Simultaneously, considering the randomness in elastic modulus of concrete (Ec), uncertainty quantification (UQ) of the bearing capacities of the push-out specimens is conducted using the arbitrary polynomial chaos (aPC) method. Subsequently, the global sensitivity analysis of the ultimate and residual bearing capacities is accomplished through the aPC surrogate model. The results indicate that the probability distribution types of the parameters in the stochastic bond-slip constitutive model can be arbitrary when considering the randomness of concrete material, and the aPC method with at least a third-order expansion is recommended to execute UQ for the push-out test due to the strong nonlinearty between the bearing capacities and the random parameters. The ultimate bearing capacity is primarily affected by the parameter K2, followed by K4, K3, and K5. Whereas the residual bearing capacity is predominantly influenced by the parameter K4, followed by K5, the remaining parameters exhibit negligible influence. In contrast, K1 and Ec have the greatest influence on equivalent plastic strain of the concrete near the loading end of the push-out specimen.