Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

Contributions to the theory of first‐exit times of some compound processes in queueing theory

We consider compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process. These processes are basic in queueing, dam and risk theory. For positive and for negative slope we derive the distribution of the first crossing time of a prespecified level. The related problem of busy periods of single‐server queueing systems is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Inverse theory of Schrödinger matrices

In this note we discuss the inverse spectral theory for Schrödinger matrices, in particular a conjecture of Gesztesy-Simon [1] on the number of distinct iso-spectral Schrödinger matrices. We consider 3 × 3 matrices and obtain counter examples to their conjecture.     

More about empirical evaluation of formal theory†

The purpose of this discussion is to transform the implicit equilibrium assumption endemic to network analysis into an explicit instrument for such analysis. I propose a formal model that brings together Coleman's restriction of Walras’ general equilibrium model and recent developments in describing the “social topology” of a multiple network system of actors such that a class of relational equilibria is defined. The specific equilibrium expected in a system is a function of the previously existing stratification of actors in the system. Corresponding to multiple observed networks, the model generates multiple equilibrium networks. The structural analysis of the observed networks can therefore be repeated on the equilibrium networks so as to assess the extent to which the analysis would differ if the observed relations were actually in an equilibrium state. Numerical illustration is provided by an analysis of alternative relational equilibria in the system of elite experts in methodological and mathematical sociology as such a system existed in 1975.     

Interpolation theory of anisotropic finite elements and applications

Interpolation theory is the foundation of finite element methods.In this paper,after reviewing some existed interpolation theorems of anisotropic finite element methods,we present a new way to analyse the interpolation error of anisotropic elements based on Newton's formula of polynomial interpolation as well as its applications.     

Representation theory of the vertex algebraW 1+∞

In our paper [KR] we began a systematic study of representations of the universal central extension of the Lie algebra of differential operators on the circle. This study was continued in the paper [FKRW] in the framework of vertex algebra theory. It was shown that the associated to simple vertex algebraW _1+∞,N with positive integral central chargeN is isomorphic to the classical vertex algebraW(gl _N), which led to a classification of modules overW _1+∞,N . In the present paper we study the remaing nontrivial case, that of a negative central charge-N. The basic tool is the decomposition ofN pairs of free charged bosons with respect togl _N and the commuting withgl _N Lie algebra of infinite matricesĝl. To Alexander Alexandrovich Kirillov on his 60-th birthday Supported in part by NSF grant DMS-9103792.     

Modularity of Calabi-Yau varieties and conformal field theory

We study the modularity problem of Calabi-Yau varieties from the conformal field theo- retic point of view.We express the modular forms associated to all 1-dimensional Calabi-Yau orbifolds in terms of products of Dedekind eta functions,which is hoped to shed light on the modularity questions for higher dimensional Calabi-Yau varieties.     

From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules

Density functional theory is used to describe the phase behaviors of rigid molecules. The construction of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C _2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.     

Representation theory of the Yokonuma–Hecke algebra

We develop an inductive approach to the representation theory of the Yokonuma–Hecke algebra _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$, based on the study of the spectrum of its Jucys–Murphy elements which are defined here. We give explicit formulas for the irreducible representations of _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$ in terms of standard d  -tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$ and calculate the Schur elements with respect to that form.     

Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.     

Goodisman's correction of the Thomas-Fermi-von Weizsäcker theory of atoms

Summary Following a suggestion of J. Goodisman, we substitute the therm by in the Thomas-Fermi-von Weizsäcker energy functional for atoms.f _z³ [0,1] is a function depending on the nuclear chargez.We establish conditions for the functionsf _z such that the ratio of this modified TFW-energyE _kz ^/TFWG (z) (kz is the total number of electrons) and the exact quantum mechanical energy converges to 1 asz. Moreover, we prove thatE _kz ^/TFWG (z)=E _kz ^/TFW (z)+Dz ²+o(z ²) (z) and determineD. Here,E _kz ^/TFW (z) is the unmodified TFW energy.

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Zusammenfassung Einem Vorschlag J. Goodismans folgend, ersetzen wir in dem Thomas-Fermi-von Weizsäcker Energiefunktional für Atome den Term durch .f _z³ [0,1] ist hierbei eine Funktion, die von der Kernladungszahlz abhängt.Wir geben dann Bedingungen für die Funktionenf _z an, unter denen der Quotient der so modifizierten TFW-energieE _kz ^/TFWG (z) (kz=Anzahl der Elektronen) und der exakten quantenmechanischen Energie fürz gegen 1 konvergiert. Darüber hinaus beweisen wir, daßE _kz ^/TFWG (z)=E _kz ^/TFW (z)+D·o(itz) ² (z) gilt und bestimmmenD. E _kz ^/TFW (z) ist hierbei die nicht-modifizierte TFW-Energie.

     

Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

“Geometry of Numbers” in a context of algebraic theory of numbers

As noted in circumstantial monograph “Geometry of Numbers” by P. M. Gruber and C. G. Lekkerkerker, lately there is no considerable contribution to classical theory of calculation and estimates of critical determinants. In this paper, in a certain sense, we fill a gap in a retardation occurred in developing of this theme. We propose a new theoretical-numerical approach with numerous consequences which, in fact, allows to revise all these subjects from another position.     

The deformation theory of representations of fundamental groups of compact Kähler manifolds

Let Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ?(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ?(Γ, G) it is shown that there exists a neighborhood of ρ in ?(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space \(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in \(H^2 (\Gamma ,g_{Ad\rho } )\) . We prove that ?(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ?(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.     

A prominence theory of context‐sensitive choice behavior

A simple theory of context‐sensitive choice behavior is proposed which focuses on the role of perceived prominence among alternatives. The fundamental hypothesis is that the similarity (or lack of similarity) among alternatives determines their relative prominence, and that, other things being equal, more prominent alternatives have a better chance of being chosen. This prominence theory of choice is formalized in terms of two testable axioms, which are shown to characterize a wide class of prominence choice models. In addition, a number of refinements and extensions of the basic theory are explored which incorporate a richer variety of perceived‐prominence behavior.     

Local computations in Dempster–Shafer theory of evidence

When applying any technique of multidimensional models to problems of practice, one always has to cope with two problems: the necessity to represent the models with a ”reasonable” number of parameters and to have sufficiently efficient computational procedures at one’s disposal. When considering graphical Markov models in probability theory, both of these conditions are fulfilled; various computational procedures for decomposable models are based on the ideas of local computations, whose theoretical foundations were laid by Lauritzen and Spiegelhalter.The presented contribution studies a possibility of transferring these ideas from probability theory into Dempster-Shafer theory of evidence. The paper recalls decomposable models, discusses connection of the model structure with the corresponding system of conditional independence relations, and shows that under special additional conditions, one can locally compute specific basic assignments which can be considered to be conditional.