scholarly journals Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

2018 ◽  
Vol 70 (2) ◽  
pp. 565-602 ◽  
Author(s):  
Rita Jiménez Rolland ◽  
Jennifer C H Wilson
Keyword(s):  
Asymptotic Stability ◽  
Stability Result ◽  
Constant Term ◽  
Convergence Result ◽  
Type B ◽  
Convergence Conditions ◽  
Convergence Results ◽  
Polynomials Over Finite Fields ◽  
Proof Of Convergence ◽  
Complex Hyperplane

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras H*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FIW-algebras finitely generated inFIW-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

scholarly journals A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021 ◽  
Author(s):  
Stefano Almi ◽  
Marco Morandotti ◽  
Francesco Solombrino
Keyword(s):  
Mean Field ◽  
Evolutionary Games ◽  
Type Systems ◽  
Convergence Result ◽  
Performance Criterion ◽  
Mean Field Limit ◽  
Spatially Inhomogeneous ◽  
Convergence Results ◽  
Special Cases ◽  
Lagrangian Scheme

AbstractA multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

scholarly journals Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 708-720 ◽  
Author(s):  
Michael Brannan
Keyword(s):  
Strong Convergence ◽  
Quantum Group ◽  
Quantum Groups ◽  
Convergence Theorem ◽  
Convergence Result ◽  
Operator Norm ◽  
Matrix Coefficient ◽  
Strong Convergence Theorem ◽  
Convergence Results ◽  
Noncommutative Polynomials

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

Stability of planar rarefaction wave to the 3D bipolar Vlasov–Poisson–Boltzmann system

2019 ◽  
Vol 30 (01) ◽  
pp. 23-104 ◽  
Author(s):  
Shu Wang ◽  
Teng Wang
Keyword(s):  
Wave Propagation ◽  
Boltzmann Equation ◽  
Asymptotic Stability ◽  
Rarefaction Wave ◽  
Stability Result ◽  
Three Dimensions ◽  
Wave Patterns ◽  
Shock Profiles ◽  
Poisson Boltzmann ◽  
Math Phys

We investigate the time-asymptotic stability of planar rarefaction wave for the 3D bipolar Vlasov–Poisson Boltzmann (VPB) system, based on the micro–macro decompositions introduced in [T. P. Liu and S. H. Yu, Boltzmann equation: Micro–macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004) 133–179; Energy method for the Boltzmann equation, Physica D 188 (2004) 178–192] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of basic wave patterns for bipolar VPB system in three dimensions.

scholarly journals Strong convergence of multivariate maxima

2020 ◽  
Vol 57 (1) ◽  
pp. 314-331
Author(s):  
Michael Falk ◽  
Simone A. Padoan ◽  
Stefano Rizzelli
Keyword(s):  
Strong Convergence ◽  
Higher Dimensions ◽  
Convergence Result ◽  
Common Distribution ◽  
Convergence Results ◽  
Negative Exponential Distribution ◽  
Common Distribution Function ◽  
Variation Distance ◽  
Negative Exponential ◽  
Variational Distance

AbstractIt is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

Asymptotic behaviour of some reaction-diffusion systems modelling complex combustion on bounded domains

1993 ◽  
Vol 123 (6) ◽  
pp. 1151-1163
Author(s):  
Joel D. Avrin
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Convergence Result ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Systems ◽  
Convergence Results ◽  
Mass Fractions

SynopsisWe consider three models of multiple-step combustion processes on bounded spatial domains. Previously, steady-state convergence results have been established for these models with zero Neumann boundary conditions imposed on the temperature as well as the mass fractions. We retain here throughout the same boundary conditions on the mass fractions, but in our first set of results we establish steady-state convergence results with fixed Dirichlet boundary conditions on the temperature. Next, under certain physically reasonable assumptions, we develop, for two of the models, estimates on the decay rates of both mass fractions to zero, while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction. These results hold under either set of boundary conditions, but when the Dirichlet conditions are imposed on the temperature, we are able to obtain estimates on the rate of convergence of the temperature to its (generally nonconstant) steady-state. Finally, we improve the results of a previous paper by adding a temperature convergence result.

scholarly journals Asymptotic Stability of the Golden-Section Control Law for Multi-Input and Multi-Output Linear Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Duo-Qing Sun ◽  
Zhu-Mei Sun
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Golden Section ◽  
Stability Result ◽  
Least Square ◽  
Control Law ◽  
Phase System ◽  
Characteristic Model ◽  
Model Based ◽  
The Stability

This paper is concerned with the problem of the asymptotic stability of the characteristic model-based golden-section control law for multi-input and multi-output linear systems. First, by choosing a set of polynomial matrices of the objective function of the generalized least-square control, we prove that the control law of the generalized least square can become the characteristic model-based golden-section control law. Then, based on both the stability result of the generalized least-square control system and the stability theory of matrix polynomial, the asymptotic stability of the closed loop system for the characteristic model under the control of the golden-section control law is proved for minimum phase system.

DISCONTINUOUS FINITE ELEMENT APPROXIMATION OF QUASISTATIC CRACK GROWTH IN NONLINEAR ELASTICITY

2006 ◽  
Vol 16 (01) ◽  
pp. 77-118 ◽  
Author(s):  
ALESSANDRO GIACOMINI ◽  
MARCELLO PONSIGLIONE
Keyword(s):  
Finite Element Approximation ◽  
Convergence Result ◽  
Element Approximation ◽  
General Notion ◽  
Elastic Bodies ◽  
Time Space ◽  
Convergence Results ◽  
Discontinuous Finite Element ◽  
Brittle Fractures ◽  
Space Discretization

We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed by Dal Maso, Francfort and Toader,14 which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we also prove a convergence result for the deformations.

scholarly journals Asymptotics for multilinear averages of multiplicative functions

2016 ◽  
Vol 161 (1) ◽  
pp. 87-101 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Unit Disc ◽  
Convergence Result ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Multiplicative Functions ◽  
Structural Result ◽  
Convergence Results ◽  
The Mean

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

scholarly journals A ROBUST ITERATIVE APPROACH FOR SOLVING NONLINEAR VOLTERRA DELAY INTEGRO–DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 7 (2) ◽  
pp. 59
Author(s):  
Austine Efut Ofem ◽  
Unwana Effiong Udofia ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Differential Equations ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Stability Result ◽  
Iterative Approach ◽  
Weak And Strong Convergence ◽  
Mild Conditions ◽  
Convergence Results ◽  
The Stability

This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized \(\alpha\)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak \(w^2\)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized \(\alpha\)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.

scholarly journals Convergence Results on the Boundary Conditions for 2D Large-Scale Primitive Equations in Oceanic Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Yuanfei Li
Keyword(s):  
Boundary Conditions ◽  
Large Scale ◽  
Weather Prediction ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Result ◽  
Primitive Equations ◽  
Convergence Results ◽  
Initial Boundary Value

In this paper, the initial boundary value problem for the two-dimensional large-scale primitive equations of large-scale oceanic motion in geophysics is considered, which are fundamental models for weather prediction. By establishing rigorous a priori bounds with coefficients and deriving some useful inequalities, the convergence result for the boundary conditions is obtained.

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