scholarly journals On the First Order Cohomology of Infinite-Dimensional Unitary Groups

2018 ◽  
Vol 68 (5) ◽  
pp. 2149-2176
Author(s):  
Manuel Herbst ◽  
Karl-Hermann Neeb
Keyword(s):  
Unitary Groups ◽  
Infinite Dimensional ◽  
First Order

scholarly journals Particles, conformal invariance and criticality in pure and disordered systems

2021 ◽  
Vol 94 (3) ◽  
Author(s):  
Gesualdo Delfino
Keyword(s):  
Conformal Invariance ◽  
Disordered Systems ◽  
Short Range ◽  
Recent Progress ◽  
Quenched Disorder ◽  
Exact Methods ◽  
Special Position ◽  
Infinite Dimensional ◽  
First Order ◽  
The Continuum

AbstractThe two-dimensional case occupies a special position in the theory of critical phenomena due to the exact results provided by lattice solutions and, directly in the continuum, by the infinite-dimensional character of the conformal algebra. However, some sectors of the theory, and most notably criticality in systems with quenched disorder and short-range interactions, have appeared out of reach of exact methods and lacked the insight coming from analytical solutions. In this article, we review recent progress achieved implementing conformal invariance within the particle description of field theory. The formalism yields exact unitarity equations whose solutions classify critical points with a given symmetry. It provides new insight in the case of pure systems, as well as the first exact access to criticality in presence of short range quenched disorder. Analytical mechanisms emerge that in the random case allow the superuniversality of some critical exponents and make explicit the softening of first-order transitions by disorder.Graphic abstract

scholarly journals Integral operators, bispectrality and growth of Fourier algebras

2020 ◽  
Vol 2020 (766) ◽  
pp. 151-194 ◽  
Author(s):  
W. Riley Casper ◽  
Milen T. Yakimov
Keyword(s):  
Differential Operator ◽  
Integral Kernel ◽  
Exact Estimate ◽  
Airy Functions ◽  
Direct Computation ◽  
Infinite Dimensional ◽  
First Order ◽  
Rank 2 ◽  
Commuting Differential Operator ◽  
Algorithmic Procedure

AbstractIn the mid 1980s it was conjectured that every bispectral meromorphic function {\psi(x,y)} gives rise to an integral operator {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions {\psi(x,y)} where the commuting differential operator is of order {\leq 6}. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

Statistical mechanics of the melting transition in lattice models of polymers

1974 ◽  
Vol 337 (1611) ◽  
pp. 569-589 ◽  
Keyword(s):  
Infinite Order ◽  
Lattice Models ◽  
Melting Transition ◽  
Excluded Volume ◽  
Approximate Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
First Order ◽  
Exactly Solvable ◽  
Cross Links

Five two-dimensional lattice models, four with rotational isomeric and excluded volume interactions and one with cross links, are used to discuss the theory of the melting transition in polymers. The models have been chosen because they are isomorphic to exactly solvable six vertex and dimer models. The orders of the thermodynamic transitions are extremely varied from model to model, including first-order, 3/2 order and infinite order transitions. These models are used to test and reveal the shortcomings of the Flory–Huggins approximate theory, which is most aptly described as an infinite dimensional theory.

Invariant solutions of two models of evolution of turbulent bursts

1999 ◽  
Vol 10 (3) ◽  
pp. 237-249 ◽  
Author(s):  
VICTOR A. GALAKTIONOV
Keyword(s):  
Order Equation ◽  
Limit Set ◽  
Invariant Solutions ◽  
Infinite Dimensional ◽  
First Order ◽  
Finite Dimensional ◽  
Finite Time Extinction ◽  
Dimensional Dynamical System ◽  
Dimensional Equation ◽  
Time Extinction

We consider two problems related to the b–l and b–ε models of propagation of turbulent bursts. We show that these equations admit some particular exact solutions which reduce to a finite-dimensional dynamical system. This makes it possible to describe a singular effect of finite-time extinction, and in particular, nonsymmetric solutions which do not exhibit symmetrization in the asymptotic behaviour. We show that in the multi-dimensional equation related to the b–l model, the nonsymmetric extinction behaviour is governed by the first-order equation. For the b–ε model with α=β=1 and γ<1, using such particular solutions, we establish that the ω-limit set of all the rescaled extinction orbits is essentially infinite-dimensional.

scholarly journals Spectral theory of first-order systems: From crystals to Dirac operators

2021 ◽  
pp. 2150014
Author(s):  
Matania Ben-Artzi ◽  
Tomio Umeda
Keyword(s):  
Dirac Operator ◽  
Spectral Theory ◽  
Spectral Gap ◽  
Unitary Groups ◽  
Decay Estimates ◽  
Homogeneous Part ◽  
Homogeneous Case ◽  
First Order ◽  
Partial Differential System ◽  
Potential Perturbation

Let [Formula: see text] be a constant coefficient first-order partial differential system, where the matrices [Formula: see text] are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: • Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated. • The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

Chaos Behavior and Control of Stay-Cables

2012 ◽  
Vol 446-449 ◽  
pp. 1109-1114
Author(s):  
Jin Hai Li ◽  
Qing Li Yan ◽  
Jin Shuan Liu
Keyword(s):  
Dynamic Behavior ◽  
Chaotic Motion ◽  
Nonlinear Dynamical System ◽  
Melnikov Method ◽  
Periodic Excitation ◽  
Stay Cable ◽  
Nonlinear Dynamical ◽  
Infinite Dimensional ◽  
First Order ◽  
And Control

Stay-cable is infinite dimensional nonlinear dynamical system with a very complex vibration types and mechanism which are not described reasonably yet. In order to better control its dynamic behavior, it is necessary to study complex dynamic behavior carefully. Fistly, partial differential equation of the cable motion is established based on the parabolic initial configuration and is simplified into n Duffing-equations by using Galerkin method. Secondly, the chaos behaviors of the first order Duffing-equation under periodic excitation are studied by taking advantage of Melnikov method. At last , parameters may lead to chaotic motion of a true cable in laboratory are calculated and the methods of chaos control are discussed briefly. The study shows that: 1. First order vibration of cable under periodic excitation has much more complex behaviors than the freedom vibration; 2. The Melnikov method can be very effective and convenient for the analysis of chaotic motion of cable.

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