free idempotent generated semigroups and endomorphism monoids of free g acts abstract the study of the free idempotent generated semigroup ig e over a biordered set e began with the seminal work of nambooripad in the 1970s and has seen a recent revival with a number of new approaches both geometric and combinatorial here we study ig e in the case e is the biordered set of a wreath product g t n where g is a group and t n is the full transformation monoid on n elements this wreath product is isomorphic to the endomorphism monoid of the free g act end f n g on n generators and this provides us with a convenient approach we say that the rank of an element of end f n g is the minimal number of free generators in its image let e e 2 end f n g for rather straightforward reasons it is known that if rank e n 1 respectively n then the maximal subgroup of ig e containing e is free respectively trivial we show that if rank e r where 1 r n 2 then the maximal subgroup of ig e containing e is isomorphic to that in end f n g and hence to g s r where s r is the symmetric group on r elements we have previously shown this result in the case r 1 however for higher rank a more sophisticated approach is needed our current proof subsumes the case r 1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some ig e on the other hand varying r again and taking g to be trivial we obtain an alternative proof of the recent result of gray and ruskuc for the biordered set of idempotents of t n

crossed products by endomorphisms of c_0 x algebras abstract in the first part of the paper we develop a theory of crossed products of a c algebra a by an arbitrary not necessarily extendible endomorphism a a we consider relative crossed products c a j where j is an ideal in a and describe up to morita rieffel equivalence all gauge invariant ideals in c a j and give six term exact sequences determining their k theory we also obtain certain criteria implying that all ideals in c a j are gauge invariant and that c a j is purely infinite in the second part we consider a situation where a is a c 0 x algebra and is such that f a f a a a f c 0 x where is an endomorphism of c 0 x pictorially speaking is a mixture of a topological dynamical system x dual to c 0 x and a continuous field of homomorphisms x between the fibers a x x x of the corresponding c bundle for systems described above we establish efficient conditions for the uniqueness property gauge invariance of all ideals and pure infiniteness of c a j we apply these results to the case when x prim a is a hausdorff space in particular if the associated c bundle is trivial we obtain formulas for k groups of all ideals in c a j in this way we constitute a large class of crossed products whose ideal structure and k theory is completely described in terms of x x x x y where y is a closed subset of x

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

maximal subgroups of free idempotent generated semigroups over the full linear monoid we show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear semigroup full of n times n matrices over a division ring q has maximal subgroup isomorphic to the general linear group gl r q where n and r are positive integers with r n 3

fast evaluation of gaunt coefficients recursive approach solution of electromagnetic multisphere scattering problems involves the determination of a large number of gaunt coefficients that are connected with definite integrals of products of three associated legendre functions this paper presents general recursion formulae cycling a single index for the gaunt coefficient with the use of the recursive scheme gaunt coefficients of either low or high degree and order can be evaluated accurately and expeditiously

hecke algebras and semisimplicity of monoid algebras abstract we begin by constructing hecke algebras for arbitrary finite regular monoids m we then show that the semisimplicity of the complex monoid algebra c m is equivalent to the semisimplicity of the associated hecke algebras and a condition on induced group characters we apply these results to finite lie type monoids m we show that the monoid algebra fm over a field f is semisimple if and only if the characteristic of f does not divide the order of the unit group g this is accomplished by developing formulas for the unities of c j j a j class of m the unity is explicitly given when g is a simply connected chevalley group and j is associated with a borel subgroup of g

subgroups of free idempotent generated semigroups need not be free abstract we study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods these subgroups are realized as the fundamental groups of a number of 2 complexes naturally associated to the biorder structure of the set of idempotents we use this to construct the first example of a free idempotent generated semigroup containing a non free subgroup

representation theory of finite semigroups semigroup radicals and formal language theory in this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field generalizing results of rhodes for the field of complex numbers applications are given to obtain many new results as well as easier proofs of several results in the literature involving triangularizability of finite semigroups which semigroups have split basic semigroup algebras two sided semidirect product decompositions of finite monoids unambiguous products of rational languages products of rational languages with counter and cerny s conjecture for an important class of automata

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

group c algebras as decreasing intersection of nuclear c algebras we prove that for every exact discrete group gamma there is an intermediate rm c ast algebra between the reduced group rm c ast algebra and the intersection of the group von neumann algebra and the uniform roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the cuntz algebra cal o 2 in particular when gamma has the ap approximation property the reduced group rm c ast algebra is realized in this way we also study extensions of the reduced free group rm c ast algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear rm c ast algebra is realized in this way

maximal subgroups of free idempotent generated semigroups over the full transformation monoid let tn be the full transformation semigroup of all mappings from the set 1 n to itself under composition let e e tn denote the set of idempotents of tn and let e e be an arbitrary idempotent satisfying im e r n 2 we prove that the maximal subgroup of the free idempotent generated semigroup over e containing e is isomorphic to the symmetric group sr 2000 mathematics subject classification 20m05 05e15 20f05

an extremely efficient approach for accurate and rapid evaluation of three centre two electron coulomb and hybrid integrals over b functions analytic expressions of three centre two electron coulomb and hybrid integrals over b functions are obtained using the fourier transform method thoroughly explored by steinborn s group these analytic expressions involve semi infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical bessel functions in the integrands we have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the h approach which greatly simplifies the application of the nonlinear transformation this work presents a rapid and accurate evaluation of these integrals obtained by using a new approach which we called s this new method is based on the h and methods and some practical properties of spherical bessel reduced bessel and sine functions the s method has greatly simplified the calculations avoiding the long and difficult implementation of the successive zeros of the spherical bessel function and a method for solving linear systems which are required by h and

complex representations of matrix semigroups let m be a finite monoid of lie type these are the finite analogues of linear algebraic monoids with group of units g the multiplicative semigroup 4 f where f is a finite field is a particular example using harishchandra s theory of cuspidal representations of finite groups of lie type we show that every complex representation of m is completely reducible using this we characterize the representations of g extending to irreducible representations of m as being those induced from the irreducible representations of certain parabolic subgroups of g we go on to show that if f is any field and s any multiplicative subsemigroup of 4 f then the semigroup algebra of s over any field of characteristic zero has nilpotent jacobson radical if s 4 f then this algebra is jacobson semisimple finally we show that the semigroup algebra of 4 f over a field of characteristic zero is regular if and only if ch f p 0 and f is algebraic over its prime field

amenable minimal cantor systems of free groups arising from diagonal actions we study amenable minimal cantor systems of free groups arising from the diagonal actions of the boundary actions and certain cantor systems it is shown that every virtually free group admits continuously many amenable minimal cantor systems whose crossed products are mutually non isomorphic kirchberg algebras in the uct class with explicitly determined k theory the technique developed in our study also enables us to compute the k theory of certain amenable minimal cantor systems we apply it to the diagonal actions of the boundary actions and the products of the odometer transformations and determine their k theory then we classify them in terms of the topological full groups continuous orbit equivalence strong orbit equivalence and the crossed products

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

efficient evaluation of vector translation coefficients in multiparticle light scattering theories vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle scattering problems these theorems include a large number of vector addition coefficients there exist three basic types of analytical expressions for vector translation coefficients stein s quart appl math 19 15 1961 cruzan s quart appl math 20 33 1962 and xu s j comput phys 127 285 1996 stein s formulation relates vector translation coefficients with scalar translation coefficients cruzan s formulas use the wigner 3jm symbol xu s expressions are based on the gaunt coefficient since the scalar translation coefficient can also be expressed in terms of the gaunt coefficient the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the wigner 3jm symbol or the gaunt coefficient we present highly efficient recursive approaches to accurately evaluating wigner 3jm symbols and gaunt coefficients armed with these recursive approaches we discuss several schemes for the calculation of the vector translation coefficients using the three general types of formulation respectively our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of stein s type is less accurate in some particular cases these extensive test calculations also show that the scheme using the formulation based on the gaunt coefficient is the most efficient in practical computations

biordered sets come from semigroups information about a semigroup can often be gleaned from its partial algebra of idempotents for example the idempotents of an inverse semigroup form a semilattice all isomorphisms between principal ideals of a semilattice e form the munn inverse semigroup t which contains the fundamental image of every inverse semigroup whose idempotents form the semilattice e ll 123 thus semilattices give rise to all fundamental inverse semigroups successful efforts have been made to generalize the munn construction to the wider class of regular semigroups 1 6 9 13 141 nambooripad achieved this using the concept of a regular biordered set the biordered set of a semigroup s means simply the partial algebra consisting of the set e e s of idempotents of s with multiplication restricted to

representations of monoids arising from finite groups of lie type a class of finite monoids m constructed from a group g of lie type is considered we describe the irreducible complex representations and prove the complete reducibility of the representations of m the sandwich matrix of m is decomposed into a product of matrices corresponding to maximal parabolic subgroups of g

ring c algebras we associate reduced and full c algebras to arbitrary rings and study the inner structure of these ring c algebras as a result we obtain conditions for them to be purely infinite and simple we also discuss several examples originally our motivation comes from algebraic number theory

the canonical compactification of a finite group of lie type let g be a finite group of lie type we construct a finite monoid m having g as the group of units m has properties analogous to the canonical compactification of a reductive group the complex representation theory of m yields harish chandra s philosophy of cuspidal representations of g the main purpose of this paper is to determine the irreducible modular representations of m we then show that all the irreducible modular representations of g come via the 1942 work of clifford from the one dimensional representations of the maximal subgroups of m this yields a semigroup approach to the modular representation theory of g via the full rank factorizations of the sandwich matrices of m we then determine the irreducible modular representations of any finite monoid of lie type

presentations for subgroups of monoids abstract the main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation it is a modification of the well known reidemeister schreier theorem for groups some consequences of this result are explored it is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented an inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid an example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

a new algorithm for accurate and fast numerical evaluation of hybrid and three centre two electron coulomb integrals over slater type functions recently we developed a new algorithm for a fast and accurate numerical evaluation of three centre nuclear attraction integrals over slater type functions the results obtained were very satisfactory now it is shown that this new algorithm can also be applied to hybrid and three centre two electron coulomb integrals over slater type functions these integrals which are numerous are very difficult to evaluate to a high accuracy because of the presence of spherical bessel functions and hypergeometric series in the integrands we have proved that the integrands that occur in the analytic expressions of the integrals under consideration satisfy all the conditions to apply the approach the hypergeometric functions which occur in the semi infinite integrals can be expressed as a finite expansion and the semi infinite integrals involving the spherical bessel functions can be transformed into semi infinite integrals involving the simple sine function the numerical results obtained with linear and non linear systems illustrate clearly a further improvement of accuracy and a substantial reduction in calculation times comparisons with existing codes stop developed by bouferguene et al 1996 int j quantum chem 57 801 and adggstngint developed by rico et al 1997 comp phys commun 105 216 are listed

classification of semisimple rank one monoids consider the classification problem for irreducible normal algebraic monoids with unit group g we obtain complete results for the groups s12 k x k g12 k and pgi2 k x k if g is one of these groups let 4 g denote the set of isomorphy types of normal algebraic monoids with zero element and unit group g our main result establishes a canonical one to one correspondence g q where q is the set of positive rational numbers the classification is achieved in two steps first we construct a class of monoids from linear representations of g that done we show that any other e must already be one of those constructed to do this we devise an extension principle analogous to the big cell construction of algebraic group theory this yields a birational comparison morphism qp er e for some r e q which is ultimately an isomor phism because the monoid er e g is regular

conjugacy classes in algebraic monoids let m be a connected linear algebraic monoid with zero and a reductive group of units g the following theorem is established theorem there exist affne subsets ml mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms 0t mi gi such that 1 every element of m is conjugate to an element of some ms and 2 two elements a b in mi are conjugate in m if and only if there exists x e gi such that x0 a x si b as a consequence it is shown that m is a union of its inverse submonoids introduction the objects of study in this paper are connected linear algebraic monoids m with zero this means by definition that the underlying set of m is an irreducible affine variety and that the product map is a morphism i e a polynomial map we will further assume that the group of units g of m is reductive this means 1 3 that the unipotent radical of g is trivial then by 6 10 m is unit regular i e m e m g where e e m e e mle2 e in this paper we study the conjugacy classes of m an initial study was made by the author 8 where the general problem was reduced to nilpotent elements the approach here is quite different yielding a more complete answer to be precise we show that there exist affine subsets m1 mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms si mi gi i 1 k such that 1 every element of m is conjugate to an element of some mi and 2 if a b e mi then a is conjugate to b in m if and only if there exists x e gi such that xsi a x si b as an application of this result we show that m is a union of its inverse submonoids an inverse semigroup is a semigroup s with the property that for each a e s there exists a unique a e s such that aaa a and aaa a see 2 finally in 3 we use our main results to briefly analyze the conjugacy classes of nilpotent elements 1 preliminaries throughout this paper z will denote the set of all positive integers and k an algebraically closed field let g be a connected linear algebraic group defined over k the radical r g is the maximal closed connected normal solvable subgroup of g and the unipotent radical ru g is the group of unipotent elements of r g we will assume throughout that g is a reductive group i e ru g 1 then r g c c g the center of g moreover g r g go where go g g is a semisimple group i e r go 1 also 3 theorem 27 5 go is a product of the simple closed normal subgroups of g in particular we have the following received by the editors november 3 1986 1980 mathematics subject classification 1985 revision primary 20g99 20m10 c 1987 american mathematical society 0002 9947 87 1 00 25 per page

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

k theoretic duality for hyperbolic dynamical systems the k theoretic analog of spanier whitehead duality for noncommutative c algebras is shown to hold for the ruelle algebras associated to irreducible smale spaces this had previously been proved only for shifts of finite type implications of this result as well as relations to the baum connes conjecture and other topics are also considered hn i x h i s n 1 x given by slant product this was generalized by spanier and whitehead to allow s n 1 x to be replaced by a space of the homotopy type of a finite complex dn x for which the analog of the above relations hold it is in this form that these notions extend naturally to the noncommutative setting in noncommutative topology the roles of homology and cohomology are played by k theory and k homology these have been combined into a bivariant theory by kasparov for which one has k theory k a kk c a and k homology k a kk a c kasparov s theory comes equipped with a product that is the analog of the slant product used above of course noncommutative c algebras play the role of algebras of continuous functions on ordinary spaces with this in hand the generalization of spanier whitehead duality to this setting is easily accomplished let a be a c algebra we will consider indices in k theory to be modulo 2 a c algebra dn a is a spanier whitehead n dual of a or simply a dual when the context

a new approach to recent constructions of c algebras from modular index theory abstract we present a new approach to c algebras recently constructed in the context of modular index theory by carey phillips putnam and rennie it turns out that their constructions can be identified with full corners of ideals in semigroup c algebras this new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

on regularity and the word problem for free idempotent generated semigroups the category of all idempotent generated semigroups with a prescribed structure e of their idempotents e called the biordered set has an initial object called the free idempotent generated semigroup over e defined by a presentation over alphabet e and denoted by ig e recently much effort has been put into investigating the structure of semigroups of the form ig e especially regarding their maximal subgroups in this paper we take these investigations in a new direction by considering the word problem for ig e we prove two principal results one positive and one negative we show that for a finite biordered set e it is decidable whether a given word w e represents a regular element if in addition one assumes that all maximal subgroups of ig e have decidable word problems then the word problem in ig e restricted to regular words is decidable on the other hand we exhibit a biorder e arising from a finite idempotent semigroup s such that the word problem for ig e is undecidable even though all the maximal subgroups have decidable word problems this is achieved by relating the word problem of ig e to the subgroup membership problem in finitely presented groups

simple purely infinite c algebras and n filling actions let n be a positive integer we introduce a concept which we call the n filling property for an action of a group on a separable unital c algebra a if a c is a commutative unital c algebra and the action is induced by a group of homeomorphisms of then the n filling property reduces to a weak version of hyperbolicity the n filling property is used to prove that certain crossed product c algebras are purely infinite and simple a variety of group actions on boundaries of symmetric spaces and buildings have the n filling property an explicit example is the action of sln z on the projective n space

k theory of certain purely infinite crossed products it was shown by rordam and the second named author that a countable group g admits an action on a compact space such that the crossed product is a kirchberg algebra if and only if g is exact and non amenable this construction allows a certain amount of choice we show that different choices can lead to different algebras at least with the free group

minimal ambient nuclear c algebras abstract we provide examples of ambient nuclear c algebras of non nuclear c algebras with no proper intermediate c algebras in particular this gives the first examples of minimal ambient nuclear c algebras of non nuclear c algebras for this purpose we study generic cantor systems of infinite free product groups

split subvarieties of group embeddings let g be a connected reductive group and x an equivariant com pactifiction of g in x we study generalised and opposite generalised schu bert varieties their intersections called generalised richardson varieties and projected generalised richardson varieties any complete g embedding has a canonical frobenius splitting and we prove that the compatibly split subvari eties are the generalised projected richardson varieties extending a result of knutson lam and speyer to the situation

multicentre two electron coulomb and exchange integrals over slater functions evaluated using a generalized algorithm based on nonlinear transformations when calculating molecular electronic energies the contributions involving the coulomb operator for bielectronic terms are required rapidly and to high chemically significant accuracy the atomic orbital basis functions chosen in the present work are slater type functions stfs these functions can be expressed as finite linear combinations of b functions which are suitable to apply the fourier transform method the difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of two or three dimensional integral representations in this work we present a generalized algorithm for a precise and fast numerical evaluation of molecular integrals over stfs numerical results obtained with c2h2 c2h4 and ch4 molecules show the efficiency of the approach presented in this work comparisons with the existing codes are also listed

a group theoretical interpretation of the word problem for free idempotent generated semigroups abstract the set of idempotents of any semigroup carries the structure of a biordered set which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question this leads to the construction of a free idempotent generated semigroup ig e the free est semigroup with a given biordered set e of idempotents we show that when e is finite the word problem for ig e is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of ig e as an application we obtain decidability of the word problem for an important class of examples also we prove that for finite e ig e is always a weakly abundant semigroup satisfying the congruence condition

calculation of the addition coefficients in electromagnetic multisphere scattering theory one of the most intractable problems in electromagnetic multisphere scattering theory is the formulation and evaluation of vector addition coefficients introduced by the addition theorems for vector spherical harmonics this paper presents an efficient approach for the calculation of both scalar and vector translational addition coefficients which is based on fast evaluation of the gaunt coefficients the paper also rederives the analytical expressions for the vector translational addition coefficients and discusses the strengths and limitations of other formulations and numerical techniques found in the literature numerical results from the formulation derived in this paper agree with those of a previously published recursion scheme that completely avoids the use of the gaunt coefficients but the method of direct calculation proposed here reduces the computing time by a factor of 4 6

fast evaluation of the gaunt coefficients addition theorems for vector spherical harmonics require the determination of the gaunt coefficients that appear in a linearization expansion of the product of two associated legendre functions this paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate lower triangular linear system and derives all relevant recurrence relations needed in the calculation this algorithm is also applicable to the calculation of the clebsch gordan coefficients that are closely related to the gaunt coefficients and are frequently encountered in the quantum theory of angular momentum the new method described in this paper reduces the computing time to 1 compared to the existing formulation that is widely used this new method can be applied to the calculation of both low and high degree coefficients while the existing formulation works well only for low degrees

the properties of sine spherical bessel and reduced bessel functions for improving convergence of semi infinite very oscillatory integrals the evaluation of three centre nuclear attraction integrals over b functions this paper focuses on the use of useful properties of sine spherical bessel and reduced bessel functions to simplify the application of the nonlinear d and transformations for accelerating the convergence of semi infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy three centre nuclear attraction integrals which are one of the most difficult type involved in density functional theory methods when using a basis set of b functions are evaluated using the new approach the numerical results show the efficiency of the new method compared with other alternatives

the fourier transform method and the sd approach for the analytical and numerical treatment of multicenter overlap like quantum similarity integrals molecular overlap like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by dirac delta function when the electronic densities are expanded over atomic orbitals using the usual lcao mo approach linear combination of atomic orbitals overlap like quantum similarity integrals could be expressed in terms of four center overlap integrals it is shown that by introducing the fourier transform of delta dirac function in the integrals and using the fourier transform approach combined with the so called b functions one can obtain analytic expressions of the integrals under consideration these analytic expressions involve highly oscillatory semi infinite spherical bessel functions which are the principal source of severe numerical and computational difficulties in this work we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap like quantum similarity integrals over slater type functions this algorithm is based on the s d approach due to safouhi recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm the numerical result section shows the efficiency of our algorithm compared with the alternatives using the one center two range expansion method which led to very complicated analytic expressions the epsilon algorithm and the nonlinear d transformation

a class of c algebras generalizing both graph algebras and homeomorphism c algebras iv pure infiniteness this is the final one in the series of papers where we introduce and study the c algebras associated with topological graphs in this paper we get a sufficient condition on topological graphs so that the associated c algebras are simple and purely infinite using this result we give one method to construct all kirchberg algebras as c algebras associated with topological graphs

boundary quotients and ideals of toeplitz c algebras of artin groups abstract we study the quotients of the toeplitz c algebra of a quasi lattice ordered group g p which we view as crossed products by a partial actions of g on closed invariant subsets of a totally disconnected compact hausdorff space the nica spectrum of g p our original motivation and our main examples are drawn from right angled artin groups but many of our results are valid for more general quasi lattice ordered groups we show that the nica spectrum has a unique minimal closed invariant subset which we call the boundary spectrum and we define the boundary quotient to be the crossed product of the corresponding restricted partial action the main technical tools used are the results of exel laca and quigg on simplicity and ideal structure of partial crossed products which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets when there exists a generalised length function or controlled map defined on g and taking values in an amenable group we prove that the partial action is amenable on arbitrary closed invariant subsets the topological freeness of the boundary action depends on topological freeness of the restriction to a certain lattice subgroup of g the core of g p which often turns out to be trivial our main results are obtained for right angled artin groups with trivial centre that is those with no cyclic direct factor they include a presentation of the boundary quotient in terms of generators and relations that generalises cuntz s presentation of o n a proof that the boundary quotient is purely infinite and simple and a parametrisation of the ideals of the toeplitz c algebra in terms of subsets of the standard generators of the artin group

three centre two electron coulomb and hybrid integrals evaluated using nonlinear d and bard transformations of the two electron integrals occuring in the molecular context the three centre coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work accelerates the convergence of these integrals by first manipulating the indices of the hypergeometric function and exploiting relationships to express this function as a finite expansion and exploiting the properties of bessel functions which satisfy second order linear differential equations this is a suitable form of the integrand to apply the nonlinear d due to d levin and a sidi and due to a sidi transformations the extensive numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

analog of the bruhat decomposition for algebraic monoids ii the length function and the trichotomy 0 introduction the purpose of this paper is to continue the study of b b orbits on a reductive monoidm this was first taken up systematically by the author in 9 where several basic properties analogous to the case of groups were established the b b orbits were identified with a finite monoid r defined in terms of the normalizer of a maximal torus and a direct analysis of these orbits led to an extension of tits axiom br brb b rb to the case of reductive monoids however there is more to this axiom when dealing with monoids obviously

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

the structure ofc algebras associated with hyperbolic dynamical systems we consider the stable unstablec algebras and the ruelle algebras associated to a mixing smale space in the case of a shift of finite type these are the af algebras studied by w krieger and the stabilized cuntz krieger algebras in the general case we show that the stable and unstable algebras are simple and amenable we also show that the ruelle algebras are simple amenable and purely infinite

efficient and rapid numerical evaluation of the two electron four center coulomb integrals using nonlinear transformations and useful properties of sine and bessel functions two electron four center coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations millions of such integrals are required for molecules of interest therefore rapidity is the primordial criterion when the precision has been reached this work presents an extremely efficient approach for improving convergence of semi infinite very oscillatory integrals based on the nonlinear d transformation and some useful properties of spherical bessel reduced bessel and sine functions the new method is now shown to be applicable to evaluating the two electron four center coulomb integrals over b functions the section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest

families of type iii kms states on a class of c algebras containing on and qn abstract we construct a family of purely infinite c algebras q for 0 1 that are classified by their k groups there is an action of the circle t with a unique kms state on each q for 1 n q 1 n o n with its usual t action and kms state for p q rational in lowest terms q o n n q p 1 with uhf fixed point algebra of type p q for any n 1 q o n for infinitely many with distinct kms states and uhf fixed point algebras for any 0 1 q o for irrational the fixed point algebras are not af and the q are usually not cuntz algebras for transcendental k 1 q k 0 q z so that q is cuntz q n cuntz 2008 16 if and 1 are both algebraic integers the only o n which appear are those for which n 3 mod 4 for each the representation of q defined by the kms state generates a type iii factor these algebras fit into the framework of modular index theory twisted cyclic theory of carey et al 2010 8 carey et al 2009 12 carey et al in press 5

regular article efficient and rapid evaluation of three center two electron coulomb and hybrid integrals using nonlinear transformations among the two electron integrals occurring in the molecular context the three center coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work pursues the acceleration of convergence for three center two electron coulomb and hybrid integrals we have proven that the hypergeometric function can be expressed as a finite expansion and that the integrand involving this series and a product of bessel functions satisfies a linear differential equation with coefficients having a power series expansion in the reciprocal of the variable suitable for application of the nonlinear d and d transformations these transformations depend strongly on the order of the differential equation that the integrand of interest satisfies this work concentrates on reduction of this order to two exploiting properties of spherical and reduced bessel functions leading to greatly simplified calculations to evaluate the integrals precisely by reducing the order of linear systems to be solved it also avoids the long and difficult implementations of successive derivatives of the integrands the numerical results section illustrates clearly the reduction of the calculation time we obtained for a high predetermined accuracy

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

reductive monoids are von neumann regular let k be an algebraically closed field an a e algebraic monoid e is an affme algebraic variety defined over k together with an associative morphism m e x e e and a two sided unit 1 e e for m in categorical terminology an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von neumann regular monoids the main result of this paper provides a fundamental link between these two objects