a sufficient criterion for integrability of stochastic many body dynamics and quantum spin chains we propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded volume interaction and the quantum mechanics of the associated quantum spin chains respectively the time dependent algebra which is obtained from the action of the markov generator of the exclusion process or quantum hamiltonian of the spin chain respectively is given in terms of a set of quadratic relations by analysing the permutation consistency of the induced cubic relations we obtain sufficient conditions on the hopping rates i e the quantum mechanical interaction constants which allow us to identify integrable models from the dynamical algebra we construct the quadratic algebra of zamolodchikov type the associativity of which is a yang baxter equation we also obtain directly from the dynamical matrix product ansatz the bethe ansatz equations for the spectra of these models

exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

a new algebraic geometry algorithm for integer programming we propose a new algorithm for solving integer programming ip problems that is based on ideas from algebraic geometry the method provides a natural generalization of the farkas lemma for ip leads to a way of performing sensitivity analysis offers a systematic enumeration of all feasible solutions and gives structural information of the feasible set of a given ip we provide several examples that offer insights on the algorithm and its properties

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

on computing hilbert bases via the elliot macmahon algorithm the ways of using the elliot macmahon algorithm to compute the hilbert base of a system of linear diophantine equations known so far are either not efficient or can fail to terminate we present a version of an algorithm exploiting this range of ideas which however is reasonably efficient as well as finite

neighborhood systems for production sets with indivisibilities a production set with indivisibilities is described by an activity analysis matrix with activity levels which can assume arbitrary integral values a neighborhood system is an association with each integral vector of activity levels of a finite set of neighboring vectors the neighborhood relation is assumed to be symmetric and translation invariant each such neighborhood system can be used to define a local maximum for the associated integer programs obtained by selecting a single commodity whose level is to be maximized subject to specified factor endowments of the remaining commodities it is shown that each technology matrix subject to mild regularity assumptions has a unique minimal neighborhood system for which a local maximum is global the complexity of such minimal neighborhood systems is examined for several examples

stationary distributions of multi type totally asymmetric exclusion processes we consider totally asymmetric simple exclusion processes with n types of particle and holes n taseps on z and on the cycle z n angel recently gave an elegant construction of the stationary measures for the 2 tasep based on a pair of independent product measures we show that angel s construction can be interpreted in terms of the operation of a discrete time m m 1 queueing server the two product measures correspond to the arrival and service processes of the queue we extend this construction to represent the stationary measures of an n tasep in terms of a system of queues in tandem the proof of stationarity involves a system of n 1 taseps whose evolutions are coupled but whose distributions at any fixed time are independent using the queueing representation we give quantitative results for stationary probabilities of states of the n tasep on z n and simple proofs of various independence and regeneration properties for systems on z

small cancellation theory and automatic groups part ii the main results obtained in part i can be rephrased in the language of root systems and euclidean planar tessellations as saying that finite a1 x a1 and a 2 piecewise euclidean abbreviated pe complexes of non positive curvature have automatic fundamental groups a 2 dimensional complex is a a 1 a1 respectively a2 pe complex if every 2 cell can be identified with a unit square resp equilateral triangle in the euclidean plane in such a way that the induced metrics agree on intersections such a complex has non positive curvature if the link of each vertex contains no circuits without backtracking of length less than 2 n here we prove using similar techniques that finite 2 complexes modelled on the other two root systems b 2 and g2 corresponding to tessellations by 45 right angle triangles and by 30 60 right angle triangles also have automatic fundamental groups theorems 2 3 and 3 2 to be precise

the complexity of generic primal algorithms for solving general integer programs primal methods constitute a common approach to solving combinatorial optimization problems starting from a given feasible solution they successively produce new feasible solutions with increasingly better objective function value until an optimal solution is reached from an abstract point of view an augmentation problem is solved in each iteration that is given a feasible point these methods find an augmenting vector if one exists usually augmenting vectors with certain properties are sought to guarantee the polynomial running time of the overall algorithm in this paper we show that one can solve every integer programming problem in polynomial time provided one can efficiently solve the directed augmentation problem the directed augmentation problem arises from the ordinary augmentation problem by splitting each direction into its positive and its negative part and by considering linear objectives on these parts our main result states that in order to get a polynomial time algorithm for optimization it is sufficient to efficiently find for any linear objective function in the positive and negative part an arbitrary augmenting vector this result also provides a general framework for the design of polynomial time algorithms for specific combinatorial optimization problems we demonstrate its applicability by considering the min cost flow problem by giving a novel algorithm for linear programming over unimodular spaces and by providing a different proof that for 0 1 integer programming an efficient algorithm solving the ordinary augmentation problem suffices for efficient optimization our main result also implies that directed augmentation is at least as hard as optimization in other words for an np hard problem already the task of finding a directed augmenting vector in polynomial time is hopeless unless p np we illustrate this kind of consequence with the help of the knapsack problem

the bethe ansatz as a matrix product ansatz the bethe ansatz in its several formulations is a common tool for the exact solution of one dimensional quantum hamiltonians this ansatz asserts that several eigenfunctions of the hamiltonians are given in terms of a sum of permutations of plane waves we present results that induce us to expect that alternatively the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz in this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices this ansatz allows an unified formulation of several exact integrable hamiltonians we show how to formulate this ansatz for a large family of quantum chains such as the anisotropic heisenberg model fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc

bounded rank subgroups of coxeter groups artin groups and one relator groups with torsion we obtain a number of results regarding the freeness of subgroups of coxeter groups artin groups and one relator groups with torsion in the case of coxeter groups we also obtain results on quasiconvexity and subgroup separability

small cancellation groups and translation numbers in this paper we prove that c 4 t 4 p c 3 t 6 p and c 6 p small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any g and any n there is an algorithm deciding whether or not the equation xn g has a solution there is also an algorithm for calculating for each g the maximum n such that g is an n th power of some element we also note that these groups cannot contain isomorphic copies of the group of p adic fractions and so in particular of the group of rational numbers besides we show that for c 4 t 4 and c 3 t 6 groups all translation numbers are rational and have bounded denominators

undecidability of the word problem for one relator inverse monoids via right angled artin subgroups of one relator groups we prove the following results 1 there is a one relator inverse monoid mathrm inv langle a w 1 rangle with undecidable word problem and 2 there are one relator groups with undecidable submonoid membership problem the second of these results is proved by showing that for any finite forest the associated right angled artin group embeds into a one relator group combining this with a result of lohrey and steinberg j algebra 320 2 728 755 2008 we use this to prove that there is a one relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable to prove 1 a new construction is introduced which uses the one relator group and submonoid in which membership is undecidable from 2 to construct a one relator inverse monoid mathrm inv langle a w 1 rangle with undecidable word problem furthermore this method allows the construction of an e unitary one relator inverse monoid of this form with undecidable word problem the results in this paper answer a problem originally posed by margolis et al in semigroups and their applications reidel dordrecht pp 99 110 1987

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

multiline queues with spectral parameters using the description of multiline queues as functions on words we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues we show that the spectral weight of a word is invariant under a natural action of the symmetric group giving a proof of the commutativity conjecture of arita ayyer mallick and prolhac we give a determinant formula for the spectral weight of a word which gives a proof of a conjecture of the first author and linusson

variation of cost functions in integer programming we study the problem of minimizingc x subject toa x b x 0 andx integral for a fixed matrixa two cost functionsc andc are considered equivalent if they give the same optimal solutions for eachb we construct a polytopest a whose normal cones are the equivalence classes explicit inequality presentations of these cones are given by the reduced grobner bases associated witha the union of the reduced grobner bases asc varies called the universal grobner basis consists precisely of the edge directions ofst a we present geometric algorithms for computingst a the graver basis and the universal grobner basis

twisted quantum affine superalgebra invariant r matrices and a new integrable electronic model we describe the twisted affine superalgebra sl 2 2 2 and its quantized version u q sl 2 2 2 we investigate the tensor product representation of the four dimensional grade star representation for the fixed point sub superalgebra u q osp 2 2 we work out the tensor product decomposition explicitly and find that the decomposition is not completely reducible associated with this four dimensional grade star representation we derive two u q osp 2 2 invariant r matrices one of them corresponds to u q sl 2 2 2 and the other to u q osp 2 2 1 using the r matrix for u q sl 2 2 2 we construct a new u q osp 2 2 invariant strongly correlated electronic model which is integrable in one dimension interestingly this model reduces in the q 1 limit to the one proposed by essler et al which has a larger sl 2 2 symmetry

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

on the finite presentation of subdirect products and the nature of residually free groups we establish it virtual surjection to pairs vsp as a general criterion for the finite presentability of subdirect products of groups if gamma 1 ldots gamma n are finitely presented and s we use the vsp criterion to characterize the finitely presented residually free groups we prove that the class of such groups is recursively enumerable we describe an algorithm that given a finite presentation of a residually free group constructs a canonical embedding into a direct product of finitely many limit groups we solve the multiple conjugacy problem and membership problem for finitely presentable subgroups of residually free groups we also prove that there is an algorithm that given a finite generating set for such a subgroup will construct a finite presentation new families of subdirect products of free groups are constructed including the first examples of finitely presented subgroups that are neither rm fp infty nor of stallings bieri type

exact solution of the asymmetric exclusion model with particles of arbitrary size a generalization of the simple exclusion asymmetric model is introduced in this model an arbitrary mixture of molecules with distinct sizes s 0 1 2 dots in units of lattice space diffuses asymmetrically on the lattice a related surface growth model is also presented variations of the distribution of the molecules sizes may change the excluded volume almost continuously we solve the model exactly through the bethe ansatz and the dynamical critical exponent z is calculated from the finite size corrections of the mass gap of the related quantum chain our results show that for an arbitrary distribution of molecules the dynamical critical behavior is on the kardar parizi zhang universality

multispecies tasep and the tetrahedron equation we introduce a family of layer to layer transfer matrices in a three dimensional 3d lattice model which can be viewed as partition functions of the q oscillator valued six vertex model on an m n square lattice by invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions at q 0 and m n they ultimately yield a new proof of the steady state formula for the n species totally asymmetric simple exclusion process obtained recently by the authors revealing the 3d integrability in the matrix product construction

complex hyperbolic hyperplane complements we study spaces obtained from a complete finite volume complex hyperbolic n manifold m by removing a compact totally geodesic complex n 1 submanifold s the main result is that the fundamental group of m setminus s is relatively hyperbolic relative to fundamental groups of the ends of m setminus s and m setminus s admits a complete finite volume a regular riemannian metric of negative sectional curvature it follows that for n 1 the fundamental group of m setminus s satisfies mostow type rigidity has solvable word and conjugacy problems has finite asymptotic dimension and rapid decay property satisfies borel and baum connes conjectures is co hopf and residually hyperbolic has no nontrivial subgroups with property t and has finite outer automorphism group furthermore if m is compact then the fundamental group of m setminus s is biautomatic and satisfies strong tits alternative

genericity the arzhantseva ol shanskii method and the isomorphism problem for one relator groups we show that the isomorphism problem is solvable in at most exponential time for a class of one relator groups which is exponentially generic in the sense of ol shanskii this is obtained by applying the arzhantseva ol shanskii graph minimization method to prove the general result that for fixed integers m 2 and n 1 there is an exponentially generic class of non free m generator n relator groups with the property that there is only one nielsen equivalence class of m tuples which generate a non free subgroup in particular every m generated subgroup in such a generic group g is either free or is equal to g itself and such groups are thus co hopfian these results are obtained by elementary methods without using the deep results of sela about co hopficity and the isomorphism problem for torsion free hyperbolic groups

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

permutohedra and minimal matrices the notions of minimality uniqueness and additivity originated in discrete tomography they have applications to kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems here we introduce the notion of real minimality and give geometric characterizations of all these notions for a matrix a by considering the intersection of the permutohedron determined by a with the transportation polytope in which a lies we also study the computational complexity of deciding if the properties of being additive real minimal unique and minimal hold for a given matrix and show how to efficiently construct some matrix with any of these properties

test sets for integer programs in this paper i discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrix a if the matrix satisfies some mild conditions and is generic the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row of a as the objective function and using the remaining rows to impose constraints on the integer variables 1997 the mathematical programming society inc published by elsevier science b v

neighbors of the origin for four by three matrices scarf has defined a neighborhood system for families of integer programs where the right hand side is allowed to vary this system depends on a matrix a of constraint and objective function coefficients of the integer programs this paper characterizes the set of neighbors of the origin when a is four by three showing that it may be described as the set of integer vectors in a union of two dimensional polyhedra where the number of polyhedra is quadratic in the bit size of a

matrix representation of the stationary measure for the multispecies tasep in this work we construct the stationary measure of the n species totally asymmetric simple exclusion process in a matrix product formulation we make the connection between the matrix product formulation and the queueing theory picture of ferrari and martin in particular in the standard representation the matrices act on the space of queue lengths for n 2 the matrices in fact become tensor products of elements of quadratic algebras this enables us to give a purely algebraic proof of the stationary measure which we present for n 3

automatic structures rational growth and geometrically finite hyperbolic groups we show that the set s mathfrak a g of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupg is dense in the product of the sets s mathfrak a p over all maximal parabolic subgroupsp the set bs mathfrak a g of equivalence classes of biautomatic structures ong is isomorphic to the product of the sets bs mathfrak a p over the cusps conjugacy classes of maximal parabolic subgroups ofg each maximal parabolicp is a virtually abelian group so s mathfrak a p and bs mathfrak a p were computed in ns1

exact solution to integrable open multi species ssep and macroscopic fluctuation theory we introduce a multi species generalization of the symmetric simple exclusion process with open boundaries this model possesses the property of being integrable and appears as physically relevant because the boundary conditions can be interpreted as the interaction with particles reservoirs with fixed densities of each species the system is driven out of equilibrium by these reservoirs the steady state is analytically computed in a matrix product form this algebraic structure allows us to obtain exact expressions for the mean particle currents and for the one and two point correlation functions an additivity principle is also derived from the matrix ansatz and permits the computation of the large deviation functional of the density profile we also propose a description of the model in the context of the macroscopic fluctuation theory and we check the consistency with the exact computations from the finite size lattice

stallings graphs for quasi convex subgroups abstract we show that one can define and effectively compute stallings graphs for quasi convex subgroups of automatic groups e g hyperbolic groups or right angled artin groups these stallings graphs are finite labeled graphs which are canonically associated with the corresponding subgroups we show that this notion of stallings graphs allows a unified approach to many algorithmic problems some which had already been solved like the generalized membership problem or the computation of a quasi convexity constant kapovich 1996 and others such as the computation of intersections the conjugacy or the almost malnormality problems our results extend earlier algorithmic results for the more restricted class of virtually free groups we also extend our construction to relatively quasi convex subgroups of relatively hyperbolic groups under certain additional conditions

completeness of the bethe ansatz solution of the open xxz chain with nondiagonal boundary terms a bethe ansatz solution of the open spin xxz quantum spin chain with nondiagonal boundary terms has recently been proposed using a numerical procedure developed by mccoy et al we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation moreover our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit

rational subgroups of biautomatic groups centralizers of finite subsets in biautomatic groups are them selves biautomatic every polycyclic subgroup of a biautomatic group is abelian by finite

decomposition of test sets in stochastic integer programming we study graver test sets for linear two stage stochastic integer programs and show that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program we present a finite algorithm to compute the building blocks directly without prior knowledge of test set vectors once computed building blocks can be employed to solve the stochastic program by a simple augmentation scheme again without explicit knowledge of test set vectors finally we report preliminary computational experience

structure and finiteness properties of subdirect products of groups we investigate the structure of subdirect products of groups particularly their finiteness properties we pay special attention to the subdirect products of free groups surface groups and hnn extensions we prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands this leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups we obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup a useful structure theory for subgroups of finite direct products of groups has yet to be developed to begin to study such subgroups it is natural to assume one knows about the subgroups of the direct factors and to concentrate on subdirect products recall that g is termed a subdirect product of the groups a1 an if g a1 an is a subgroup that projects surjectively to each factor work by various authors has exposed the surprisingly rich structure to be found amongst the subdirect products of superficially tame groups for example in contrast to the fact that subdirect products of abelian or nilpotent groups are again in the specified class nonabelian free groups harbour a great diversity of subdirect products including some with unsolvable decision problems 21 this diversity has long been known but it is only as a result of more recent work by baumslag roseblade 5 and bridson howie miller short 9 that it has been understood as a phenomenon that is intimately tied to the failure of various homological finiteness conditions from this background we pick out the three strains of thought to be developed in this article the subtlety of subdirect products in general 1991 mathematics subject classification 20f05 20e07 20f67 20j05 20f10

a geometric buchberger algorithm for integer programming let ip a c denote the family of integer programs of the form min cx ax b x nn obtained by varying the right hand side vector b but keeping a and c fixed a test set for ipa c is a set of vectors in zn such that for each nonoptimal solution to a program in this family there is at least one element g in this set such that g has an improved cost value as compared to we describe a unique minimal test set for this family called the reduced grobner basis of ip a c an algorithm for its construction is presented which we call a geometric buchberger algorithm for integer programming and we show how an integer program may be solved using this test set the reduced grobner basis is then compared with some other known test sets from the literature we also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix a zn 2 n of full row rank

subgroup properties of fully residually free groups we prove that fully residually free groups have the howson property that is the intersection of any two finitely generated subgroups in such a group is again finitely generated we also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup

matrix product formula for u _ q a_ 2 1 zero range process the zero range processes introduced recently by mangazeev maruyama and the authors are integrable discrete and continuous time markov processes associated with the stochastic r matrix derived from the well known quantum r matrix by constructing a representation of the relevant zamolodchikov faddeev algebra we present for n 2 a matrix product formula for the steady state probabilities in terms of q boson operators

garside groups are strongly translation discrete abstract in this paper we show that all garside groups are strongly translation discrete that is the translation numbers of non torsion elements are strictly positive and for any real number r there are only finitely many conjugacy classes of elements whose translation numbers are less than or equal to r it is a consequence of the inequality inf s g inf s g n n inf s g 1 for a positive integer n and an element g of a garside group g where inf s denotes the maximal infimum for the conjugacy class we prove the inequality by studying the semidirect product g n z g n of the infinite cyclic group z and the cartesian product g n of a garside group g which turns out to be a garside group we also show that the root problem in a garside group g can be reduced to a conjugacy problem in g n hence the root problem is solvable for garside groups

average case complexity and decision problems in group theory abstract we investigate the average case complexity of decision problems for finitely generated groups in particular the word and membership problems using our recent results on generic case complexity we show that if a finitely generated group g has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non elementary word hyperbolic quotient group then the average case complexity of the word problem of g is linear time uniformly with respect to the collection of all length invariant measures on g this results applies to many of the groups usually studied in geometric group theory for example all braid groups b n all groups of hyperbolic knots many coxeter groups and all artin groups of extra large type

matrix product solution to a 2 species tasep with open integrable boundaries we present an explicit representation for the matrix product ansatz for sometwo species tasep with open boundary conditions the construction relies on theintegrability of the models a property that constrains the possible rates atthe boundaries the realisation is built on a tensor product of copies of thedehp algebras using this explicit construction we are able to calculate thepartition function of the models the densities and currents in the stationarystate are also computed it leads to the phase diagram of the models dependingon the values of the boundary rates we obtain for each species shock waves maximal current or low high densities phases

small cancellations over relatively hyperbolic groups and embedding theorems we generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings this generalization is then used to prove various embedding theorems for countable groups for instance we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes in particular this gives the affirmative answer to the well known question of the existence of a finitely generated group g other than 2 such that all nontrivial elements of g are conjugate

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

exact solutions of exactly integrable quantum chains by a matrix product ansatz most of the exact solutions of quantum one dimensional hamiltonians are obtained thanks to the success of the bethe ansatz on its several formulations according to this ansatz the amplitudes of the eigenfunctions of the hamiltonian are given by a sum of permutations of appropriate plane waves in this paper alternatively we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product the eigenvalue equation for the hamiltonian defines the algebraic properties of the matrices defining the amplitudes the consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model the matrix product ansatz we propose allows an unified and simple formulation of several exact integrable hamiltonians in order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the xxz chain spin 1 fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc formulation of the matrix product ansatz for quantum chains with open ends is also possible as an illustration we present the exact solution of an extended xxz chain with z magnetic fields at the surface and arbitrary hard core exclusion among the spins

the matrix product solution of the multispecies partially asymmetric exclusion process we find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles the solution is in the form of a matrix product representation where the matrices for a system of n species are defined recursively in terms of the matrices for a system of n 1 species a complete proof is given based on the quadratic relations verified by these matrices this matrix product construction is interpreted in terms of the action of a transfer matrix

equations in groups that are virtually direct products abstract in this note we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

two stage integer programs with stochastic right hand sides a superadditive dual approach we consider two stage pure integer programs with discretely distributed stochastic right hand sides we present an equivalent superadditive dual formulation that uses the value functions in both stages we give two algorithms for finding the value functions to solve the reformulation after obtaining the value functions we develop a global branch and bound approach and a level set approach to find an optimal tender we show that our method can solve randomly generated instances whose extensive forms are several orders of magnitude larger than the extensive forms of those instances found in the literature

integrable boundary conditions for multi species asep the first result of the present paper is to provide classes of explicitsolutions for integrable boundary matrices for the multi species asep with anarbitrary number of species all the solutions we have obtained can be seen as representations of a newalgebra that contains the boundary hecke algebra the boundary hecke algebra isnot sufficient to build these solutions this is the second result of ourpaper

the generalised word problem in hyperbolic and relatively hyperbolic groups abstract we prove that for a finitely generated group hyperbolic relative to virtually abelian subgroups the generalised word problem for a parabolic subgroup is the language of a real time turing machine then for a hyperbolic group we show that the generalised word problem for a quasiconvex subgroup is a real time language under either of two additional hypotheses on the subgroup by extending the muller schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context free conversely we prove that a hyperbolic group must be virtually free if it has a torsion free quasiconvex subgroup of infinite index with context free generalised word problem

tetrahedron equation and generalized quantum groups we construct families of solutions of the yang baxter equation from n products of three dimensional r and l operators satisfying the tetrahedron equation they are identified with the quantum r matrices for the hopf algebras known as generalized quantum groups depending on the number of r s and l s involved in the product the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of whereas a boundary vector construction interpolates the q oscillator representation of and the spin representation of the intermediate cases are associated with an affinization of quantum superalgebras

solving stochastic programs with integer recourse by enumeration a framework using grobner basis in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

zamolodchikov s tetrahedron equation and hidden structure of quantum groups the tetrahedron equation is a three dimensional generalization of the yang baxter equation its solutions define integrable three dimensional lattice models of statistical mechanics and quantum field theory their integrability is not related to the size of the lattice therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices obviously any such three dimensional model can be viewed as a two dimensional integrable model on a square lattice where the additional third dimension is treated as an internal degree of freedom therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this hidden third dimension in this paper we construct a new solution of the tetrahedron equation which provides in this way the two dimensional solvable models related to finite dimensional highest weight representations for all quantum affine algebra where the rank n coincides with the size of the hidden dimension these models are related to an anisotropic deformation of the sl n invariant heisenberg magnets they were extensively studied for a long time but the hidden 3d structure was hitherto unknown our results lead to a remarkable exact rank size duality relation for the nested bethe ansatz solution for these models note also that the above solution of the tetrahedron equation arises in the quantization of the resonant three wave scattering model which is a well known integrable classical system in 2 1 dimensions

a primal all integer algorithm based on irreducible solutions this paper introduces an exact primal augmentation algorithm for solving general linear integer programs the algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities we prove that various versions of our algorithm are finite it is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively an implementation of the presented algorithms is given computational results for a number of hard 0 1 integer programs from the miplib demonstrate the practical power of the method

conjugacy growth of finitely generated groups abstract the conjugacy growth function of a finitely generated group measures the number of conjugacy classes in balls with respect to a word metric we study the following natural question which functions can occur as the conjugacy growth function of finitely generated groups our main result answers the question completely namely we prove that a function f n n can be realized up to a natural equivalence as the conjugacy growth function of a finitely generated group if and only if f is non decreasing and bounded from above by a n for some a 1 we also construct a finitely generated group g and a subgroup h g of index 2 such that h has only 2 conjugacy classes while the conjugacy growth of g is exponential in particular conjugacy growth is not a quasi isometry invariant

short rational generating functions for lattice point problems abstract we prove that for any xed d the generating function of the projectionof the set of integer points in a rational d dimensional polytope can be computed inpolynomial time as a corollary we deduce that various interesting sets of latticepoints notably integer semigroups and minimal hilbert bases of rational cones have short rational generating functions provided certain parameters the dimensionand the number of generators are xed it follows then that many computationalproblems for such sets for example nding the number of positive integers notrepresentable as a non negative integer combination of given coprime positive integersa 1 a d admit polynomial time algorithms we also discuss a related problem ofcomputing the hilbert series of a ring generated by monomials 1 introduction and main resultsour main motivation is the following question which goes back to frobenius andsylvester 1 1 the frobenius problem let a 1 a d be positive coprime integers andlets n

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

an algebraic geometry algorithm for scheduling in presence of setups and correlated demands we study here a problem of schedulingn job types onm parallel machines when setups are required and the demands for the products are correlated random variables we model this problem as a chance constrained integer program

generic case complexity decision problems in group theory and random walks abstract we give a precise definition of generic case complexity and show that for a very large class of finitely generated groups the classical decision problems of group theory the word conjugacy and membership problems all have linear time generic case complexity we prove such theorems by using the theory of random walks on regular graphs

algorithms detecting stability and morseness for finitely generated groups abstract the notions of stable and morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word hyperbolic group for a word hyperbolic group g kapovich 31 provided a partial algorithm which on input a finite set s of g halts if s generates a quasiconvex subgroup of g and runs forever otherwise in this paper we give various detection and decidability algorithms for stability and morseness of a finitely generated subgroup of mapping class groups right angled artin groups toral relatively hyperbolic groups and finitely generated groups discriminated by a locally quasiconvex torsion free hyperbolic group for example ordinary limit groups

braid groups of imprimitive complex reflection groups abstract we obtain new presentations for the imprimitive complex reflection groups of type d e e r and their braid groups b d e e r for d r 2 diagrams for these presentations are proposed the presentations have much in common with coxeter presentations of real reflection groups they are positive and homogeneous and give rise to quasi garside structures diagram automorphisms correspond to group automorphisms the new presentation shows how the braid group b d e e r is a semidirect product of the braid group of affine type a r 1 and an infinite cyclic group elements of b d e e r are visualised as geometric braids on r 1 strings whose first string is pure and whose winding number is a multiple of e we classify periodic elements and show that the roots are unique up to conjugacy and that the braid group b d e e r is strongly translation discrete

shock profiles for the asymmetric simple exclusion process in one dimension the asymmetric simple exclusion process asep on a one dimensional lattice is a system of particles which jump at rates p and 1 p here p 1 2 to adjacent empty sites on their right and left respectively the system is described on suitable macroscopic spatial and temporal scales by the inviscid burgers equation the latter has shock solutions with a discontinuous jump from left density rho to right density rho rho sqrt rho 1 rho rho 1 rho for a special value of the asymmetry given by p 1 p rho 1 rho rho 1 rho the measure is bernoulli with density rho on the left and rho on the right in the weakly asymmetric limit 2p 1 to0 the microscopic width of the shock diverges as 2p 1 1 the stationary measure is then essentially a superposition of bernoulli measures corresponding to a convolution of a density profile described by the viscous burgers equation with a well defined distribution for the location of the second class particle