the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

closed trajectories for quadratic differentials with an application to billiards on considere les orbites periodiques pour le systeme dynamique d une table polygonale avec des angles multiples rationnels de le billard rationnel

partiallyu 1 compactified scalar massless field on the compact riemann surface and the bosonic string amplitudes the theory of the partiallyu 1 compactified scalar massless field on the compact riemann surface with nambu goto action is defined the partition function is determined completely by a choice of the finite dimensional approximations the correlation functions are the only correctly defined objects of the theory the averages of the correlation function asymptotic values provide the amplitudes for the compact riemann surfaces of any genus the usual bosonic string amplitudes are the special cases of the above amplitudes

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

fiber spaces over teichmuller spaces for fuchsian groups with elliptic elements we determine the biholomorphic fiber preserving isomorphisms of fiber spaces over teichmuller spaces for fuchsian groups with elliptic elements we show that except in some special cases a biholomorphic fiber preserving isomorphism between two bers fiber spaces is always an allowable mapping we find that the situation is different for teichmuller curves showing that in general there are some other biholomorphic fiber preserving isomorphisms between teichmuller curves besides the allowable mappings

on the nielsen type and the classification for the mapping class group let s be a compact surface of genus g with n boundary components with 2g 2 n 0 and let m s be its mapping class group also known as the teichmiiller modular group using measurd foliations thurston 15 has shown that m s contains three different types of elements bers has shown this decomposition can be obtained using teichmiiller distance 3 subdividing one of the classes further he obtains four classes nielsen has a series of papers 7 8 parts i iii in which he develops an elaborate theory of fixed points of mapping classes if t is any homeomorphism of s and h a lift of t or of a power of 1 to the upper half plane nielsen s theory involves the assignment of a pair of integers to h called the nielsen type of h the purpose of this paper is to show how the thurston classes and the bers classes can be defined in terms of the nielsen types of the lifts of

electromagnetic propagators in hyperbolic robertson walker cosmologies green functions retarded advanced feynman and dyson propagators are calculated for the electromagnetic field in robertson walker cosmologies with hyperbolic 3 manifolds as spacelike slices the starting point is the proca equation i e the maxwell field with a finite photon mass for infrared regularization in a static cosmology with simply connected hyperbolic 3 sections the time and space components of the resolvent kernel are scalar and vectorial point pair invariants respectively and this symmetry allows for an explicit evaluation in the spectral representation it is found that the quantum propagators have a logarithmic infrared singularity which drops out in the zero curvature limit retarded and advanced green functions remain well defined in the limit of zero photon mass and they admit a simple generalization by conformal scaling to expanding 3 spaces in cosmologies with multiply connected hyperbolic 3 manifolds as spacelike sections the four enumerated propagators are constructed by means of poincare series the spectral decomposition of the green functions is given in terms of eisenstein series for a certain class of open hyperbolic 3 spaces including those with schottky covering groups corresponding to solid handle bodies as spacelike slices

the deligne mumford compactification of the real multiplication locus and teichmuller curves in genus 3 in the moduli space mathcal m g of genus g riemann surfaces consider the locus mathcal r mathcal m mathcal o of riemann surfaces whose jacobians have real multiplication by the order mathcal o in a totally real number field f of degree g if g 3 we compute the closure of mathcal r mathcal m mathcal o in the deligne mumford compactification of mathcal m g and the closure of the locus of eigenforms over mathcal r mathcal m mathcal o in the deligne mumford compactification of the moduli space of holomorphic 1 forms for higher genera we give strong necessary conditions for a stable curve to be in the boundary of mathcal r mathcal m mathcal o boundary strata of mathcal r mathcal m mathcal o are parameterized by configurations of elements of the field f satisfying a strong geometry of numbers type restriction

newton functions generating symmetric fields and irreducibility of schur polynomials abstract we prove theorem 1 1 that if e 0 e r 0 are coprime integers then the newton functions x 1 e i x r e i i 0 r generate over q the field of symmetric rational functions in x 1 x r this generalizes a previous result of us for r 2 this extension requires new methods including i a study of irreducibility and galois theoretic properties of schur polynomials theorem 3 1 and ii the study of the dimension of the varieties obtained by intersecting fermat hypersurfaces theorem 4 1 we shall also observe how these results have implications to the study of zeros of linear recurrences over function fields in particular we give theorem 4 2 a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1

the minimum rank problem over the finite field of order 2 minimum rank 3 our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2 we also list all 62 such minimal forbidden subgraphs we conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

on fiedler s characterization of tridiagonal matrices over arbitrary fields abstract fiedler proved in linear algebra appl 2 1969 191 197 that the set of real n by n symmetric matrices a such that rank a d n 1 for every real diagonal matrix d is the set of matrices ptpt where p is a permutation matrix and t an irreducible tridiagonal matrix we show that this result remains valid for arbitrary fields with some exceptions for 5 by 5 matrices over z 3

on the minimum rank of a graph over finite fields abstract in this paper we deal with two aspects of the minimum rank of a simple undirected graph g on n vertices over a finite field f q with q elements which is denoted by mr f q g in the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on n vertices over f 2 is 1 e n n were lim n e n 0 in the second part of this paper we assume that g contains a clique k k on k vertices we show that if q is not a prime then mr f q g n k 1 for 4 k n 1 and n 5 it is known that mr f q g 3 for k n 2 n 4 and q 4 we show that for k n 2 and each n 10 there exists a graph g such that mr f 3 g 3 for k n 3 n 5 and q 4 we show that mr f q g 4

on the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature abstract we consider geodesic motion on three dimensional riemannian manifolds of constant negative curvature topologically equivalent to s 0 1 s a compact surface of genus two to those trajectories which are bounded and recurrent in both directions of the time evolution t t a fractal limit set is associated whose hausdorff dimension is intimately connected with the quantum mechanical energy ground state determined by the schrodinger operator on the manifold we give a rather detailed and pictorial description of the hyperbolic spaces we have in mind discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and hausdorff dimension and give finally some examples of ground state calculations in terms of hausdorff dimensions of limit sets of classical trajectories

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

on the irreducibility of irreducible characters of simple lie algebras we establish an irreducibility property for the characters of finite di mensional irreducible representations of simple lie algebras or simple algebraic groups over the complex numbers i e that the characters of irreducible represen tations are irreducible after dividing out by generalized weyl denominator type factors for sl r the irreducibility result is the following let a1 a2 ar 1 0 be the highest weight of an irreducible rational representation vof sl r as sume that the integers a1 r 1 a2 r 2 ar 1 1 are relatively prime then the character of vis strongly irreducible in the following sense for any natural number d the function g d g sl r c is irreducible in the ring of regular functions of sl r c