arXiv subjects

Boris Shapiro

Publications and source records attributed to Boris Shapiro.

136 records · Page 8Linked to original sources

arXiv1999

The problem of phase breaking in the electronic conduction in mesoscopic systems: a linear-response theory approach

We study the problem of electronic conduction in mesoscopic systems when the electrons are allowed to interact not only with static impurities, but also with a scatterer (a phase breaker(PB)) that possesses internal degrees of freedom. We first analyze the role of the PB in reducing the coherent interference effects in a one-electron quantum-mechanical system. In the many-electron system we can make a number of quite general statements within the framework of linear-response theory and the random-phase approximation. We cannot calculate the conductivity tensor in full generality: we thus resort to a model, in which that tensor can be expressed entirely in a single-electron picture. The resulting zero-temperature conductance can be written in terms of the total transmission coefficient at the Fermi energy, containing an additional trace over the states of the PB.

cond-mat.mes-hall↗

arXiv1999

Simply-laced Coxeter groups and groups generated by symplectic transvections

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, the number of Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F_2 is the two-element field.

math.AG↗

arXiv1999

Algebras of curvature forms on homogeneous manifolds

Let C(X) be the algebra generated by the curvature 2-forms of the standard hermitian line bundles over the complex homogeneous manifold X=G/B. We calculate the Hilbert polynomial of C(X) and give its presentation as a quotient of a polynomial ring. In particular, we show the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. As a tool we study a more general algebra associated with a point on a Grassmannian and calculate its Hilbert polynomial as well as its presentation in terms of generators and relations.

math.AG↗

arXiv1998

Comment on ``Critical Behavior in Disordered Quantum Systems Modified by Broken Time--Reversal Symmetry''

In a recent Letter [Phys. Rev. Lett. 80, 1003 (1998)] Hussein and Pato employed the maximum entropy principle (MEP) in order to derive interpolating ensembles between any pair of universality classes in random matrix theory. They apply their formalism also to the transition from random matrix to Poisson statistics of spectra that is observed for the case of the Anderson-type metal-insulator transition. We point out the problems with the latter procedure.

cond-mat.dis-nn↗

arXiv1997

Tree-like curves and their inflection points

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points on such curves unremovable by diffeomorphisms of the plane in terms of their combinatorics.

dg-ga↗

arXiv1997

Perturbation Theory for the Rosenzweig-Porter Matrix Model

We study an ensemble of random matrices (the Rosenzweig-Porter model) which, in contrast to the standard Gaussian ensemble, is not invariant under changes of basis. We show that a rather complete understanding of its level correlations can be obtained within the standard framework of diagrammatic perturbation theory. The structure of the perturbation expansion allows for an interpretation of the level structure on simple physical grounds, an aspect that is missing in the exact analysis (T. Guhr, Phys. Rev. Lett. 76, 2258 (1996), T. Guhr and A. Müller-Groeling, cond-mat/9702113).

cond-mat.mes-hall↗

arXiv1997

Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.

dg-ga↗

arXiv1997

Real Space Renormalization of the Chalker-Coddington Model

We study a number of hierarchical network models related to the Chalker-Coddington model of quantum percolation. Our aim is to describe the physics of the quantum Hall transition. The hierarchical network models are constructed by combining series and parallel composition of quantum resistors. The localization-delocalization transition occurring in these models is treated by real space renormalization techniques. Essentially, the localization-delocalization transition is due to a competition between two one-dimensional localization mechanisms.

cond-mat.mes-hall↗

arXiv1996

Combinatorics and topology of stratifications of the space of monic polynomials with real coefficients

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity $k$. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.

math.CO↗

arXiv1994

A generalized ensemble of random matrices

A random matrix ensemble incorporating both GUE and Poisson level statistics while respecting $U(N)$ invariance is proposed and shown to be equivalent to a system of noninteracting, confined, one dimensional fermions at finite temperature.

cond-mat↗