Bohmian measures and their classical limit

We consider a class of phase space measures, which naturally arise in the Bohmian interpretation of quantum mechanics. We study the classical limit of these so-called Bohmian measures, in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration. The obtained results are consequently compared to those derived via semi-classical Wigner measures. To this end, we shall also give a connection to the theory of Young measures and prove several new results on Wigner measures themselves. Our analysis gives new insight on oscillation and concentration effects in the semi-classical regime.     

Unitary spherical spectrum for p-adic classical groups

Let G be a split reductive p-adic group. Then the determination of the unitary representations with nontrivial Iwahori fixed vectors can be reduced to the determination of the unitary dual of the corresponding Iwahori-Hecke algebra. In this paper we study the unitary dual of the Iwahori-Hecke algebras corresponding to the classical groups. We determine all the unitary spherical representations.     

The Gaussian limit for high-dimensional spherical means

We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.     

Harmonic analysis on a class of spherical homogeneous spaces

The spectrum of representations of a semisimple algebraic group in spaces of sections of homogeneous linear bundles on a certain class of spherical homogeneous spaces is studied; the algebra of invariant functions on the cotangent bundles of spaces from this class and invariant differential operators are described.     

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:

$\dot{x}=ax?y+P_n(x,y),\dot{y}=x+ay+Q_n(x,y),$

where $a\in \mathbb{R}$ and $P_n,Q_n$ are homogeneous polynomials of degree $n\ge 2$. Denote $\psi (\theta )=\cos ?(\theta )?Q_n(\cos ?(\theta ),\sin ?(\theta ))?\sin ?(\theta )?P_n(\cos ?(\theta ),\sin ?(\theta ))$. We prove that the system has at most 1 limit cycle surrounding the origin provided $(n?1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.     

Functional equations of spherical functions on p-adic homogeneous spaces

LetG be a connected reductive linear algebraic group andX aG-homogeneous affine algebraic variety both defined over a p-adic field k, where we assume a minimalk-parabolic subgroup ofG acts with open orbit. We are interested in spherical functions onX =X(k). In the present papaer, we give a unified method to obtain functional equations of spherical functions on X under the condition (AF) in the introduction, and explain functional equations are reduced to those ofp-adic local zeta functions of small prehomogeneous vector spaces of limited type.     

Harmonic analysis on spherical homogeneous spaces with solvable stabilizer

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of representations of G on spaces of regular sections of homogeneous line bundles over G/H.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

On embeddings of certain spherical homogeneous spaces in prime characteristic

Let $ \mathcal{G} $ be a reductive group over an algebraically closed field of characteristic p?>?0. We study embeddings of homogeneous $ \mathcal{G} $ -spaces that are induced from the G?×?G-space G, G a suitable reductive group, along a parabolic subgroup of $ \mathcal{G} $ . We give explicit formulas for the canonical divisors and for the divisors of B-semi-invariant functions. Furthermore, we show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen?CMacaulay. Examples are the G?×?G-orbits in normal reductive monoids with unit group G. Further examples are the open $ \mathcal{G} $ -orbits of the well known determinantal varieties and the varieties of (circular) complexes. Finally, we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.     

Decay of matrix coefficients on reductive homogeneous spaces of spherical type

Let \(Z\) be a homogeneous space \(Z=G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H\) . The investigation concerns the quantitative decay of matrix coefficients on \(Z\) under the assumption that \(Z\) is of spherical type, that is, minimal parabolic subgroups have open orbits on \(Z\) .     

The classical limit of a quantum kinetic equation for quarks

The classical limit of the exact quantum kinetic equation for quarks with spin is obtained on the basis of a spinor decomposition. A calculation scheme for the Lenard-Balescu-type collision term is presented. The quantum correction to the classical matter equation is calculated for Abelian plasma.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 159–175, July, 1996.     

On almost-sure versions of classical limit theorems for dynamical systems

The purpose of this article is to support the idea that “whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average”. We follow three different approaches: martingale methods, spectral methods and induction arguments. Our results apply, among others, to Axiom A maps or flows, to systems inducing a Gibbs–Markov map, and to the stadium billiard.