cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

kumjian pask algebras of locally convex higher rank graphs the kumjian pask algebra of a higher rank graph generalises the leavitt path algebra of a directed graph we extend the definition of kumjian pask algebra to row finite higher rank graphs with sources which satisfy a local convexity condition after proving versions of the graded uniqueness theorem and the cuntz krieger uniqueness theorem we study the kumjian pask algebra of rank 2 bratteli diagrams by studying certain finite subgraphs which are locally convex we show that the desourcification procedure of farthing and webster yields a row finite higher rank graph without sources such that the kumjian pask algebras of and are morita equivalent we then use the morita equivalence to study the ideal structure of the kumjian pask algebra of by pulling the appropriate results across the equivalence

kumjian pask algebras of finitely aligned higher rank graphs abstract we extend the definition of kumjian pask algebras to include algebras associated to finitely aligned higher rank graphs we show that these kumjian pask algebras are universally defined and have a graded uniqueness theorem we also prove the cuntz krieger uniqueness theorem to do this we use a groupoid approach as a consequence of the graded uniqueness theorem we show that every kumjian pask algebra is isomorphic to the steinberg algebra associated to its boundary path groupoid we then use steinberg algebra results to prove the cuntz krieger uniqueness theorem and also to characterize simplicity and basic simplicity

orthogonality relations and generating functions for the generalized bessel polynomials abstract in this paper the author begins by presenting a systematic historical account of the orthogonality of the generalized bessel polynomials that arise naturally in a number of seemingly diverse contexts the related expansion problem and various recent results involving generating functions for these polynomials are discussed next some relevant generalizations of many of these generating functions are also considered

realizations of su 1 1 and uq su 1 1 and generating functions for orthogonal polynomials positive discrete series representations of the lie algebra su 1 1 and the quantum algebra uq su 1 1 are considered the diagonalization of a self adjoint operator the hamiltonian in these representations and in tensor products of such representations is determined and the generalized eigenvectors are constructed in terms of orthogonal polynomials using simple realizations of su 1 1 uq su 1 1 and their representations these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials the relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials or to poisson kernels in particular a group theoretical derivation of the poisson kernel for meixner pollaczek and al salam chihara polynomials is obtained

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

fuzzy measures as extensions of stochastic measures abstract links between fuzzy measures cf hohle z wahrsch verw gebiete 36 1976 179 188 and stochastic measures cf morando lecture notes in mathematics no 88 pp 190 216 springer verlag berlin new york 1969 are specified in particular a class of stochastic measures from which fuzzy measures can be derived quite naturally is exhibited

the remarkable properties of the associated romanovski functions for and with n m q 0 a non negative integer we introduce a differential equation for the associated romanovski functions r q n m x so that its form remains invariant under the transformation n 2q n 1 introducing the rodrigues formula for both regions and we obtain simultaneously and separately realization of the laddering equations with respect to the labels n and m by means of two pairs of ladder operators also the square integrability and orthogonality of the functions are considered in cases 0 and 0 with respect to the weight functions and 1 x2 1 q respectively exponential generating functions for the infinite sequences with the same n in both regions and as well as for the infinite ones with given even and odd values of n m q are separately calculated as an application to supersymmetric quantum mechanics we show that there exist four different methods for generating the spectrum of the noncentral hyperbolic scarf potential

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

the interplay between steinberg algebras and skew rings abstract we study the interplay between steinberg algebras and skew rings for a partial action of a group in a hausdorff locally compact totally disconnected topological space we realize the associated partial skew group ring as a steinberg algebra over the transformation groupoid attached to the partial action we then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings over commutative algebras in terms of continuous orbit equivalence of the associated partial actions finally we show that any steinberg algebra associated to a hausdorff ample groupoid can be seen as a partial skew inverse semigroup ring

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

fractional skew monoid rings abstract given an action of a monoid t on a ring a by ring endomorphisms and an ore subset s of t a general construction of a fractional skew monoid ring s op a t is given extending the usual constructions of skew group rings and of skew semigroup rings in case s is a subsemigroup of a group g such that g s 1s we obtain a g graded ring s op a s with the property that for each s s the s component contains a left invertible element and the s 1 component contains a right invertible element in the most basic case where g z and s t z the construction is fully determined by a single ring endomorphism of a if is an isomorphism onto a proper corner pap we obtain an analogue of the usual skew laurent polynomial ring denoted by a t t examples of this construction are given and it is proven that several classes of known algebras including the leavitt algebras of type 1 n can be presented in the form a t t finally mild and reasonably natural conditions are obtained under which s op a s is a purely infinite simple ring

graph inverse semigroups their characterization and completion abstract graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of c algebras in this paper we characterize such semigroups and show how they may be completed under suitable conditions to form what we call the cuntz krieger semigroup of the graph this semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph and the semigroup analogue of the leavitt path algebra of the graph

an operator approach to the construction of generating functions for products of associated laguerre polynomials we introduce an iteratively defined operator sequence allowing the construction of new useful mathematical identities involving the product of associated laguerre polynomials possible physical applications as well as some methodological aspects of our approach are pointed out

strongly graded groupoids and strongly graded steinberg algebras abstract we study strongly graded groupoids which are topological groupoids g equipped with a continuous surjective functor g to a discrete group such that 1 1 1 for all we introduce the category of graded g sheaves and prove an analogue of dade s theorem g is strongly graded if and only if every graded g sheaf is induced by a g e sheaf the steinberg algebra of a graded ample groupoid is graded and we prove that the algebra is strongly graded if and only if the groupoid is applying this result we obtain a complete graphical characterisation of strongly graded leavitt path and kumjian pask algebras

representing finitely generated refinement monoids as graph monoids abstract graph monoids arise naturally in the study of non stable k theory of graph c algebras and leavitt path algebras they play also an important role in the current approaches to the realization problem for von neumann regular rings in this paper we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid the characterization is expressed in terms of the behavior of the structural maps of the associated i system at the free primes of the monoid

branching point processes with independent transformations we investigate branching point processes within an arbitrary number of point types based on the generating functional technique every point can be characterized by a set of parameters such as time space energy etc the analyses have been done for a random number of identical stages with independent transformations the set of functional equations has been obtained for the determination of either the statistical characteristics of the resulting point process or the parameters of the separate stage process the proposed approach lets us describe a wide range of random point processes and find mechanisms of their formation we considered some specific models of electron multiplication in a microchannel plate a photomultiplier tube and an avalanche photodiode

congruences on graph inverse semigroups abstract graph inverse semigroups were defined by ash and hall in 1975 they found necessary and sufficient conditions for the semigroups to be congruence free in this paper we give a classification of congruences on a graph inverse semigroup in terms of the underlying graph as a consequence we show that the inverse graph semigroup of a finite graph is congruence noetherian

c algebras associated with hilbert c quad modules of c textile dynamical systems abstract a c textile dynamical system a consists of a unital c algebra a two families of endomorphisms and a a of a and certain commutation relations among them it yields a two dimensional subshift and a multistructure hilbert c bimodule which we call a hilbert c quad module we introduce a c algebra from the hilbert c quad module as a two dimensional analogue of pimsner s construction of c algebras from hilbert c bimodules we study the c algebras defined by the hilbert c quad modules and prove that they have universal properties subject to certain operator relations we also present examples arising from commuting matrices

on graded irreducible representations of leavitt path algebras using the e algebraic branching systems various graded irreducible representations of a leavitt path k algebra l of a directed graph e are constructed the concept of a laurent vertex is introduced and it is shown that the minimal graded left ideals of l are generated by the laurent vertices or the line points leading to a detailed description of the graded socle of l following this a complete characterization is obtained of the leavitt path algebras over which every graded irreducible representation is finitely presented a useful result is that the irreducible representation v p induced by infinite paths tail equivalent to an infinite path p we call this a chen simple module is graded if and only if p is an irrational path we also show that every one sided ideal of l is graded if and only if the graph e contains no cycles supplementing the theorem of one of the co authors that every leavitt path algebra l is graded von neumann regular we show that l is graded self injective if and only if l is a graded semi simple algebra made up of matrix rings of arbitrary size over the field k or the graded field k xn x n where n n

certain classes of finite series relationships and generating functions involving the generalized bessel polynomials recently by making use of the familiar group theoretic lie algebraic method a certain mixed trilateral finite series relationship was proven for the generalized bessel polynomials the main object of this paper is to derive several substantially more general families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for these and other related classes of hypergeometric polynomials a duly corrected and modified analogue of the aforementioned trilateral finite series relationship is shown to follow by suitably specializing one of the general results presented here several closely related presumably new finite series relationships and generating functions some of which also involve for example the stirling numbers of the second kind are onsidered rather briefly

fractional operators and special functions i bessel functions most of the special functions of mathematical physics are connected with the representation of lie groups the action of elements d of the associated lie algebras as linear differential operators gives relations among the functions in a class for example their differential recurrence relations in this paper we define fractional generalizations d of these operators in the context of lie theory determine their formal properties and illustrate their use in obtaining interesting relations among the functions we restrict our attention here to the euclidean group e 2 and the bessel functions we show that the two variable fractional operator relations lead directly to integral representations for the bessel functions reproduce known fractional integrals for those functions when reduced to one variable and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure we extend the analysis to the associated legendre functions in a follo

an operator approach to the construction of generating functions for the product of laguerre polynomials the thermal average bandshape function of a molecule we introduce two different forms of mathematical identities involving the product of laguerre polynomials these identities are a direct reflection of an operator identity it is shown that these relations are useful in calculating the thermal average bandshape function of a molecular system for a model displaced oscillator system we derive how the displacement between two adiabatic potential surfaces affects the bandshape function

leavitt path algebras with finitely presented irreducible representations abstract let e be an arbitrary graph k be any field and let l l k e be the corresponding leavitt path algebra necessary and sufficient conditions both graphical and algebraic are given under which all the irreducible representations of l are finitely presented in this case the graph e turns out to be row finite and the cycles in e form an artinian partial ordered set under a defined relation when the graph is e is finite the above graphical conditions were shown in 6 to be equivalent to l k e having finite gelfand kirillov dimension examples are constructed showing that this equivalence no longer holds for infinite graphs and a complete description is obtained of leavitt path algebras over arbitrary graphs having finite gelfand kirillov dimensions the building blocks for these algebras seem to be von neumann rings and the laurent polynomial ring k x x 1

isomorphisms between leavitt algebras and their matrix rings let k be any field let ln denote the leavitt algebra of type 1 n 1 hav ing coe cients in k and let md ln denote the ring of dd matrices over ln in our main result we show that md ln g ln if and only if d and n 1 are coprime we use this isomorphism to answer a question posed in 14 regarding isomorphisms between various c algebras furthermore our result demonstrates that data about the k0 structure is suf ficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple k algebras

fuzzy measures assuming their values in the set of fuzzy numbers abstract fuzzy valued fuzzy measures are defined in an axiomatic way extending a result of klement j math anal appl 75 1980 330 339 it is shown that they can be characterized by a suitable family of ordinary measures and markov kernels

the dynamics of leavitt path algebras abstract recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the morita theory and the grothendieck group in the theory of leavitt path algebras abrams et al 2011 4 in this paper we show that the notion of the conjugacy of shifts of finite type is closely related to the graded morita theory and consequently the graded grothendieck group this fits into the general framework we have in these two theories conjugacy yields the flow equivalence and the graded morita equivalence can be lifted to the morita equivalence starting from a finite directed graph the observation that the graded grothendieck group of the leavitt path algebra associated to e coincides with the krieger dimension group of the shift of finite type associated to e provides a link between the theory of leavitt path algebras and symbolic dynamics it has been conjectured that the ordered graded grothendieck group as z x x 1 module we call this the graded dimension group classifies the unital leavitt path algebras completely hazrat 2013 20 via the above correspondence utilising the results from symbolic dynamics we prove that for two purely infinite simple unital leavitt path algebras if their graded dimension groups are isomorphic then the algebras are isomorphic

exponential generating functions for the associated bessel functions similar to the associated legendre functions the differential equation for the associated bessel functions bl m x is introduced so that its form remains invariant under the transformation l l 1 a rodrigues formula for the associated bessel functions as squared integrable solutions in both regions l 0 and l 0 is presented the functions with the same m but with different positive and negative values of l are not independent of each other while the functions with the same l m l m but with different values of l and m are independent of each other so all the functions bl m x may be taken into account as the union of the increasing decreasing infinite sequences with respect to l it is shown that two new different types of exponential generating functions are attributed to the associated bessel functions corresponding to these rearranged sequences

non simple purely infinite rings in this paper we introduce the concept of purely infinite rings which in the simple case agrees with the already existing notion of pure infiniteness we establish various permanence proper ties of this notion with respect to passage to matrix rings corners and behaviour under extensions so being purely infinite is preserved under morita equivalence we show that a wealth of examples falls into this class including important analogues of constructions commonly found in operator al gebras in particular for any s unital k algebra having enough nonzero idempotents for example for a von neumann regular algebra its tensor product over k with many non simple leavitt path algebras is purely infinite introduction the notion of pure infiniteness has proved key in the the ory of operator algebras since its conception in the early eighties by j cuntz see 27 this was done for simple algebras and provided a huge list of exam ples one of the milestones of the theory became the classification of separable nuclear unital purely infinite simple algebras by means of k theoretic invariants due to kirchberg 41 and phillips 58

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

structure of leavitt path algebras of polynomial growth we determine the structure of leavitt path algebras of polynomial growth and discuss their automorphisms and involutions

an algebraic generalization of the notion of tribe tribes became a successful basis of measure theory on fuzzy sets during their investigation new requirements arose we discuss possible modifications of the original definition of a tribe we suggest a more general structure which admits to apply different triangular norms in different parts emphasis is put on point free notions

the theory of prime ideals of leavitt path algebras over arbitrary graphs given an arbitrary graph e and a field k the prime ideals as well as the primitive ideals of the leavitt path algebra lk e are completely characterized in terms of their generators the stratification of the prime spectrum of lk e is indicated with information on its individual stratum necessary and sufficient conditions are given on the graph e under which every prime ideal of the leavitt path algebra lk e is primitive leavitt path algebras with krull dimension zero are characterized and those with various prescribed krull dimension are constructed the minimal prime ideals of lk e are described in terms of the graphical properties of e and using this complete descriptions of the height one as well as the co height one prime ideals of lk e are given

fractional operators and special functions ii legendre functions most of the special functions of mathematical physics are connected with the representation of lie groups the action of elements d of the associated lie algebras as linear differential operators gives relations among the functions in a class for example their differential recurrence relations in this paper we apply the fractional generalizations d of these operators developed in an earlier paper in the context of lie theory to the group so 2 1 and its conformal extension the fractional relations give a variety of interesting relations for the associated legendre functions we show that the two variable fractional operator relations lead directly to integral relations among the legendre functions and to one and two variable integral representations for those functions some of the relations reduce to known fractional integrals for the legendre functions when reduced to one variable the results enlarge the understanding of many properties of the associated legendre functions on the basis of the un

the graded grothendieck group and the classification of leavitt path algebras this paper is an attempt to show that parallel to elliott s classification of af c algebras by means of k theory the graded k0 group classifies leavitt path algebras completely in this direction we prove this claim at two extremes namely for the class of acyclic graphs graphs with no cycles and multi headed comets or rose graphs graphs in which each head is connected to a cycle or to a collection of loops or a mixture of these graphs i e polycephaly graphs

some families of generating functions for the bessel polynomials abstract the main object of this paper is to show how readily some general results on bilinear bilateral or mixed multilateral generating functions for the bessel polynomials would provide unifications and generalizations of numerous generating functions which were proven recently by using group theoretic techniques

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices