cellular networks as models for planck scale physics starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics we base our own approach on what we call cellular networks consisting of cells nodes interacting with each other via bonds figuring as elementary interactions according to a certain local law geometrically our dynamical networks are living on graphs hence a substantial amount of the investigation is devoted to the development of various versions of discrete functional analysis and geometry on such almost random webs another important topic we address is a suitable concept of intrinsic fractal dimension on erratic structures of this kind in the course of the investigation we make comments concerning both different and related approaches to quantum gravity as say the spin network framework it may perhaps be said that certain parts of our programme seem to be a realization of ideas sketched by smolin some time ago

a geometric renormalization group in discrete quantum space time we model quantum space time on the planck scale as dynamical networks of elementary relations or time dependent random graphs the time dependence being an effect of the underlying dynamical network laws we formulate a kind of geometric renormalization group on these random networks leading to a hierarchy of increasingly coarse grained networks of overlapping lumps we provide arguments that this process may generate a fixed limit phase representing our continuous space time on a mesoscopic or macroscopic scale provided that the underlying discrete geometry is critical in a specific sense geometric long range order our point of view is corroborated by a series of analytic and numerical results which allow us to keep track of the geometric changes taking place on the various scales of the resolution of space time of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality

martin points on open manifolds of non positive curvature the martin boundary of a cartan hadamard manifold describes a fine geometric structure at infinity which is a sub space of positive harmonic functions we describe conditions which ensure that some points of the sphere at infinity belong to the martin boundary as well in the case of the universal cover of a compact manifold with ballmann rank one we show that martin points are generic and of full harmonic measure the result of this paper provides a partial answer to an open problem of s t yau

on the measure of maximal entropy for finite horizon sinai billiard maps the sinai billiard map t on the two torus i e the periodic lorentz gas is a discontinuous map assuming finite horizon we propose a definition h for the topological entropy of t we prove that h is not smaller than the value given by the variational principle and that it is compatible with the definitions of bowen using spanning or separating sets if h ge log 2 our actual condition is weaker then we get more first using a transfer operator acting on a space of anisotropic distributions we construct an invariant probability measure mu of maximal entropy for t i e h mu t h we show that mu has full support and is bernoulli and we prove that mu is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to h second h is compatible with the bowen pesin pitskel topological entropy of the restriction of t to a non compact domain of continuity last applying results of lima and matheus the map t has at least c e pnh periodic points of period pn for all n and some p ge 1

dirac operators and the calculation of the connes metric on arbitrary infinite graphs as an outgrowth of our investigation of non regular spaces within the context of quantum gravity and non commutative geometry we develop a graph hilbert space framework on arbitrary infinite graphs and use it to study spectral properties of graph laplacians and graph dirac operators we define a spectral triplet sharing most of the properties of what connes calls a spectral triple with the help of this scheme we derive an explicit expression for the connes distance function on general directed or undirected graphs we derive a series of a priori estimates and calculate it for a variety of examples of graphs as a possibly interesting side we show that the natural setting for approaching such problems may be the framework of non linear programming or optimization we compare our results arrived at within our particular framework with those of other authors and show that the seeming differences depend on the use of different graph geometries and or dirac operators

on the parameterized differential inverse galois problem over k t x abstract in this article we consider the inverse galois problem for parameterized differential equations over k t x with k any field of characteristic zero and use the method of patching over fields due to harbater and hartmann we show that every connected semisimple k t split linear algebraic group is a parameterized differential galois group over k t x

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

dimension theory of graphs and networks starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics a core concept is the notion of dimension in the following we develop such a notion for irregular structures such as large graphs and networks and derive a number of its properties among other things we show its stability under a wide class of perturbations which is important if one has dimensional phase transitions in mind furthermore we systematically construct graphs with almost arbitrary fractal dimension which may be of some use in the context of dimensional renormalization or statistical mechanics on irregular sets

the differential galois theory of strongly normal extensions differential galois theory the theory of strongly normal extensions has unfortunately languished this may be due to its reliance on kolchin s elegant but not widely adopted axiomatization of the theory of algebraic groups this paper attempts to revive the theory using a differential scheme in place of those axioms we also avoid using a universal differential field instead relying on a certain tensor product we identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product thus identifying the galois group with the closed points of an affine differential scheme moreover the tensor product has a natural coring structure which translates into the galois group operation composition of automorphisms this affine differential scheme splits i e is obtained by base extension from a not differential not necessarily affine group scheme as a consequence the galois group is canonically isomorphic to the closed or rational points of a group scheme defined over constants we obtain the fundamental theorem of differential galois theory giving a bijective correspondence between subgroup schemes and intermediate differential fields on the way to this result we study certain aspects of differential algebraic geometry e g closed immersions products local ringed space of constants and split differential schemes

specification and towers in shift spaces we show that a shift space on a finite alphabet with a non uniform specification property can be modeled by a strongly positive recurrent countable state markov shift to which every equilibrium state lifts in addition to uniqueness of the equilibrium state this gives strong statistical properties including the bernoulli property exponential decay of correlations central limit theorem and analyticity of pressure which are new even for uniform specification we give applications to shifts of quasi finite type synchronised and coded shifts and factors of beta shifts and s gap shifts

new classes of parameterized differential galois groups this paper is on the inverse parameterized differential galois problem we show that surprisingly many groups do not occur as parameterized differential galois groups over k x even when k is algebraically closed we then combine the method of patching over fields with a suitable version of galois descent to prove that certain groups do occur as parameterized differential galois groups over k t x this class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

generalized differential galois theory a galois theory of differential fields with parameters is developed in a manner that generalizes kolchin s theory it is shown that all connected differential algebraic groups are galois groups of some appropriate differential field extension

calculating differential galois groups of parametrized differential equations with applications to hypertranscendence the main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations via the parameterized and differential galois theories to achieve this we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential galois groups for differential equations whose coefficients are rational functions berman and singer presented an algorithm calculating the differential galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators we use their algorithm as a part of our algorithm as a result we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter

the uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds in this paper we prove a conjecture of a katok stating that the geodesic flow on a compact rank 1 manifold admits a uniquely determined invariant measure of maximal entropy this generalizes previous work of r bowen and g margulis as an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy

linear algebraic groups as parameterized picard vessiot galois groups we show that a linear algebraic group is the galois group of a parameterized picard vessiot extension of k x x 1 for certain differential fields k if and only if its identity component has no one dimensional quotient as a linear algebraic group

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

a galois theoretic proof of the differential transcendence of the incomplete gamma function abstract we give simple necessary and sufficient conditions for the t transcendence of the solutions to a parameterized second order linear differential equation of the form 2 y x 2 p y x 0 where p f x is a rational function in x with coefficients in a t field f our criteria imply in particular the t transcendence of the incomplete gamma function t x generalizing a result of johnson reinhart and rubel this result is also an important part of an efficient algorithm to compute the parameterized picard vessiot group of an arbitrary parameterized second order linear differential equation over f x

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

genericity of contracting elements in groups we establish that for statistically convex cocompact actions contracting elements are exponentially generic in counting measure we obtain as corollaries results on the exponential genericity for the set of hyperbolic elements in relatively hyperbolic groups the set of rank 1 elements in cat 0 groups and the set of pseudo anosov elements in mapping class groups for a proper action with purely exponential growth we show that the set of contracting elements is generic in particular for mapping class groups the set of pseudo anosov elements is generic in a sufficiently large subgroup provided that the subgroup has purely exponential growth by roblin s work we obtain that the set of hyperbolic elements is generic in any discrete group action on cat 1 space with finite bms measure we present applications to the number of conjugacy classes of non rank 1 elements in cat 0 groups with rank 1 elements