scaling limit of the loop erased random walk green s function we consider loop erased random walk lerw running between two boundary points of a square grid approximation of a planar simply connected domain the lerw green s function is the probability that the lerw passes through a given edge in the domain we prove that this probability multiplied by the inverse mesh size to the power 3 4 converges in the lattice size scaling limit to a constant times an explicit conformally covariant quantity which coincides with the hbox sle 2 green s function the proof does not use sle techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a spinor observable for random walk started from one of the vertices of the marked edge

circular planar graphs and resistor networks abstract we consider circular planar graphs and circular planar resistor networks associated with each circular planar graph there is a set p q of pairs of sequences of boundary nodes which are connected through a graph is called critical if removing any edge breaks at least one of the connections p q in we prove that two critical circular planar graphs are y equivalent if and only if they have the same connections if a conductivity is assigned to each edge in there is a linear from boundary voltages to boundary currents called the network response this linear map is represented by a matrix we show that if is any circular planar resistor network whose underlying graph is critical then the values of all the conductors in may be calculated from finally we give an algebraic description of the set of network response matrices that can occur for circular planar resistor networks

inverse problem in cylindrical electrical networks in this paper we study the inverse dirichlet to neumann problem for certain cylindrical electrical networks we define and study a birational transformation acting on cylindrical electrical networks called the electrical r matrix we use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder this conjecture extends work of curtis ingerman and morrow linear algebra appl 283 1998 pp 115 150 and of de verdiere gitler and vertigan comment math helv 71 1996 pp 144 167 for circular planar electrical networks we show that our conjectural solution holds for certain purely cylindrical networks here we apply the grove combinatorics introduced by kenyon and wilson trans amer math soc 363 2011 pp 1325 1364

superprojective banach spaces we study superprojective banach spaces we show that they cannot contain copies of l1 which restricts the search for non reflexive examples of these spaces we also show that the class of superprojective spaces is stable under finite products certain unconditional sums certain tensor products and other operations providing new examples

the perturbation classes problem for subprojective and superprojective banach spaces abstract we show that the perturbation class for the upper semi fredholm operators between two banach spaces x and y coincides with the strictly singular operators when x is subprojective and that the perturbation class for the lower semi fredholm operators coincides with the strictly cosingular operators when y is superprojective similar results were previously obtained under stronger conditions for x and y

soft sets and soft groups molodtsov introduced the concept of soft set theory which can be used as a generic mathematical tool for dealing with uncertainty in this paper we introduce the basic properties of soft sets and compare soft sets to the related concepts of fuzzy sets and rough sets we then give a definition of soft groups and derive their basic properties using molodtsov s definition of the soft sets

diffuse scattering on graphs abstract we formulate and analyze difference equations on graphs analogous to time independent diffusion equations arising in the study of diffuse scattering in continuous media moreover we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous background problem using born series providing necessary conditions for convergence and demonstrating the process through numerous examples in addition we outline a method for finding green s functions for cayley graphs for both abelian and non abelian groups finally we conclude with a discussion of the effects of sparsity on our method and results outlining the simplifications that can be made provided that the scatterers are weak and well separated

topological structure of fuzzy soft sets in this paper the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set interior fuzzy soft set fuzzy soft basis fuzzy soft subspace topology are studied

potential kernel for two dimensional random walk it is proved that the potential kernel of a recurrent aperiodic random walk on the integer lattice z 2 admits an asymptotic expansion of the form 2 q 1 ln q x 2 x 1 const x u 1 x x 2 u 2 x where q and q are respectively the determinant and the quadratic form of the covariance matrix of the increment x of the random walk x x x and the u k are smooth functions of 1 provided that all the moments of x are finite explicit forms of u 1 and u 2 are given in terms of the moments of x

the perturbation classes problem in fredholm theory abstract we give a negative answer to the perturbation classes problem the perturbation class of the upper semi fredholm operators contains properly the strictly singular operators and the perturbation class of the lower semi fredholm operators contains properly the strictly cosingular operators

subprojective banach spaces abstract a banach space x is called subprojective if any of its infinite dimensional subspaces contains a further infinite dimensional subspace complemented in x this paper is devoted to systematic study of subprojectivity we examine the stability of subprojectivity of banach spaces under various operations such as direct or twisted sums tensor products and forming spaces of operators along the way we obtain new classes of subprojective spaces

soft semirings molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we initiate the study of soft semirings by using the soft set theory the notions of soft semirings soft subsemirings soft ideals idealistic soft semirings and soft semiring homomorphisms are introduced and several related properties are investigated

on some new operations in soft set theory molodtsov introduced the theory of soft sets which can be seen as a new mathematical approach to vagueness in this paper we first point out that several assertions proposition 2 3 iv vi proposition 2 4 and proposition 2 6 iii iv in a previous paper by maji et al p k maji r biswas a r roy soft set theory comput math appl 45 2003 555 562 are not true in general by counterexamples furthermore based on the analysis of several operations on soft sets introduced in the same paper we give some new notions such as the restricted intersection the restricted union the restricted difference and the extended intersection of two soft sets moreover we improve the notion of complement of a soft set and prove that certain de morgan s laws hold in soft set theory with respect to these new definitions

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

lifting non topological divisors of zero modulo the compact operators this paper provides the first examples of non semifredholm operators s on a banach space such that the left or right multiplication operators r sr or r rs define linear embeddings of the corresponding calkin algebra into itself for instance if s is a bounded linear operator on c 0 1 with closed range such that ker s l1 then there is a constant c 0 with dist sr k c 0 1 c dist r k c 0 1 for all bounded operators r l c 0 1 here k c 0 1 stands for the compact operators on c 0 1 moreover if s l1 l1 has closed range and l1 im s contains no copies of l1 then there is a constant c 0 such that dist rs w l1 c dist r w l1 for all r l l1 here w l1 denotes the weakly compact operators on l1