a new condition for blow up solutions to discrete semilinear heat equations on networks abstract the purpose of this paper is to introduce a new condition c 0 u f s d s u f u u 2 u 0 for some 0 with 0 2 0 2 where 0 is the first eigenvalue of discrete laplacian with which we obtain blow up solutions to discrete semilinear heat equations u t x t u x t f u x t x t s 0 u x t 0 x t s 0 u x 0 u 0 0 n o n t r i v i a l x s on a discrete network s in fact it will be seen that the condition c improves the conditions known so far

ideal structure of uniform roe algebras of coarse spaces abstract let c u x e be the uniform roe algebra of a coarse space x e with uniformly locally finite coarse structure by a controlled truncation technique we show that the controlled propagation operators in an ideal i of c u x e are exactly the controlled truncations of elements in i it follows that the lattice of the ideals of the uniform roe algebra c u x e in which controlled propagation operators are dense the lattice of the invariant open subsets in the unit space of the groupoid g x introduced by skandalis tu and yu the lattice of the ideals of the coarse structure e and the lattice of the ideals of the coarse space x are mutually isomorphic these lattices also give rise to a type of classification for the ideals of c u x e

solving boundary value problems on networks using equilibrium measures abstract the purpose of this paper is to construct solutions of self adjoint boundary value problems on finite networks to this end we obtain explicit expressions of the green functions for all different boundary value problems the method consists of reducing each boundary value problem either to a dirichlet problem or to a poisson equation on a new network closely related with the former boundary value problem in this process we also get an explicit expression of the poisson kernel for the dirichlet problem in all cases we express the green function in terms of equilibrium measures solely which can be obtained as the unique solution of linear programming problems in particular we get analytic expressions of the green function for the following problems the poisson equation on a distance regular graph the dirichlet problem on an infinite distance regular graph and the neumann problem on a ball of an homogeneous tree

a complete characterization of the discrete p laplacian parabolic equations with q nonlocal reaction with respect to the blow up property abstract in this paper we consider the following discrete p laplacian nonlinear parabolic equations with q nonlocal reaction u t x t p u x t y s u y t q 1 u y t x t s 0 u x t 0 x t s 0 u x 0 u 0 0 x s here s is a network with boundary s the goal of this paper is to characterize completely the parameters p 1 and q 0 to see when the solution blows up vanishes or exists globally indeed the blow up rates for the blow up solutions are derived

folner sequences and finite operators abstract this article analyzes folner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by williams in the 70s we prove that each essentially hyponormal operator has a proper folner sequence i e an increasing folner sequence of projections strongly converging to 1 in particular any quasinormal any subnormal any hyponormal and any essentially normal operator has a proper folner sequence moreover we show that an operator is finite if and only if it has a proper folner sequence or if it has a non trivial finite dimensional reducing subspace we also analyze the structure of operators which have no folner sequence and give examples of them for this analysis we introduce the notion of strongly non folner operators which are far from finite block reducible operators in some uniform sense and show that this class coincides with the class of non finite operators

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

a generalization of pillen s theorem for principal series modules ii abstract let g be a connected semisimple and simply connected algebraic group and g q the corresponding finite chevalley group over the finite field of order q p r in a recent paper the author determined a direct sum decomposition of the k g q submodule generated by a highest weight vector of a certain weyl module when q is not too small which is a generalization of pillen s result in 1997 in this article we claim that the result does not need the assumption on q

extinction and positivity of solutions of the p laplacian evolution equation on networks abstract in this paper we consider a discrete version of the following p laplacian evolution equation u t p u with p 1 on a network using the discrete p laplacian operator in graphs which means a nonlinear diffusion phenomenon on networks we first introduce the p laplacian evolution equation on networks in fact spatial derivative in p laplacian evolution equation comparing the continuous case is replaced with the discrete p laplacian operator thus the resulting system is semi discrete discrete in space and continuous in time the main concern is to investigate the large time behaviors of nontrivial solutions of this equation whose initial data are nonnegative and the boundary data vanish in order to do so we use the analytic approaches such as vector calculus on networks maximum principle and comparison principle instead of numerical ones after deriving the basic properties of this equation we finally prove that the solution becomes extinct for 1 p 2 and remains strictly positive for p 2

omega harmonic functions and inverse conductivity problems on networks in this paper we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network in terms of an input output map to do t

maximal and reduced roe algebras of coarsely embeddable spaces in 7 gong wang and yu introduced a maximal or universal version of the roe c algebra associated to a metric space we study the relationship between this maximal roe algebra and the usual version in both the uniform and non uniform cases the main result is that if a uniformly discrete bounded geometry metric space x coarsely embeds into a hilbert space then the canonical map between the maximal and usual uniform roe algebras induces an isomorphism on k theory we also give a simple proof that if x has property a then the maximal and usual uniform roe algebras are the same these two results are natural coarse geometric analogues of certain well known implications of a t menability and amenability for group c algebras the techniques used are e theoretic building on work of higson kasparov and trout 11 12 and yu 28

box spaces group extensions and coarse embeddings into hilbert space abstract we investigate how coarse embeddability of box spaces into hilbert space behaves under group extensions in particular we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into hilbert space this provides a new class of examples of metric spaces with bounded geometry which coarsely embed into hilbert space but do not have property a generalising the example of arzhantseva guentner and spakula

determining the resistors in a network in this paper it is shown that the resistors in a rectangular network can be determined by measurements at the boundary of the voltages generated by imposed currents an algorithm for using the boundary measurements to compute the resistances is also given

the amenability of affine algebras abstract we introduce the notion of amenability for affine algebras we characterize amenability by folner sequences paradoxicality and the existence of finitely invariant dimension measures then we extend the results of rowen on ranks from affine algebras of subexponential growth to amenable affine algebras

amenability and uniform roe algebras abstract amenability for groups can be extended to metric spaces algebras over commutative fields and c algebras by adapting the notion of folner nets in the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform roe algebra c u x over a metric space x d with bounded geometry in particular we show that the following conditions are equivalent 1 x d is amenable 2 the translation algebra generating c u x is algebraically amenable 3 c u x has a tracial state 4 c u x is not properly infinite 5 1 0 0 0 in the k 0 group k 0 c u x 6 c u x does not contain the leavitt algebra as a unital subalgebra 7 c u x is a folner c algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm we also show that every possible tracial state of the uniform roe algebra c u x is amenable

decomposition numbers for generic iwahori hecke algebras of noncrystallographic type we compute the e modular decomposition matrices for the generic iwahori hecke algebras of typei2 m form n m 2 h3 andh4 for alle n leading to nontrivial decomposition maps the results are obtained by a combined use of different ideas from computational representation theory and by application of the computer algebra systemsgap chevie vectorenumeratorandmeataxe

on the critical set for fujita type blow up of solutions to the discrete laplacian parabolic equations with nonlinear source on networks abstract in this paper we are interested in long time behaviors of solutions to the discrete laplacian parabolic equations u t u f u with nonnegative and non trivial initial data in particular we assume that the function f is convex only on a short interval and f s f f s for 0 1 s 0 and we present a critical set depending on the function in the sense that if f is in the critical set then solutions blow up in finite time for any initial data and if not then solutions are global or blow up according to the size of initial data

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

extinction and positivity of the solutions of the heat equations with absorption on networks in this paper we propose a discrete version of the following semilinear heat equation with absorption ut u uq with q 1 which is said to be the heat equation with absorption on a network using the discrete laplacian operator on a weighted graph we define the heat equations with absorption on networks and give their physical interpretations the main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish it is proved that the asymptotic behaviors of the solutions u x t as t tends to strongly depend on the sign of q 1

the dirichlet to neumann map for a resistor network relations that characterize the dirichlet to neumann map for a square network of resistors are given using these relations a parametrization of the set of dirichlet to neumann maps is given

extinction and asymptotic behavior of solutions for the ω heat equation on graphs with source and interior absorption abstract in this paper we consider the extinction and asymptotic behavior of the solutions for the following heat equation on graphs with source and interior absorption u t x t u x t u q x t u p x t where is called the discrete weighted laplacian operator and p q 0 we first prove the local existence of the solutions and show that all solutions exist globally when q max p 1 then we obtain the following extinction and asymptotic properties of the solutions when q 1 and the initial datum are small enough the solution vanishes in infinite time when q 1 p 1 and is appropriately small the solution becomes extinct in finite time when q 1 p 1 and is appropriately small the solution vanishes in infinite time when q 1 and is appropriately large the solution is non extinction when q 1 q p is the critical extinction exponent moreover for the case that p q 1 and is appropriately large we show that the non extinction solution is asymptotically stable and lim t u x t u x where u x is the unique positive equilibrium solution finally we demonstrate our results through some numerical examples

on the solvability of the discrete conductivity and schroźdinger inverse problems we study the uniqueness question for two inverse problems on graphs both problems consist in finding possibly complex edge or nodal based quantities from boundary measurements of solutions to the dirichlet problem associated with a weighted graph laplacian plus a diagonal perturbation the weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete schro dinger potential we use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity or schro dinger potential then the linearized problem is solvable for almost all conductivities or schro dinger potentials in a suitable set we show that the conductivities or schro dinger potentials in a certain set are determined uniquely by boundary data except on a zero measure set this criterion for solvability is used in a statistical study of graphs where the conductivity or schro dinger inverse problem is solvable