on the order of approximation in approximative triadic decompositions of tensors abstract based on alder s 1984 result on the equivalence of algebraic and topological border rank of tensors over algebraically closed ground fields we give an upper bound for the order of approximation in approximative triadic decompositions we also treat the analogous question for real closed fields in particular we carry over alder s result to the case of real closed fields

local properties of self similar solutions to smoluchowski s coagulation equation with sum kernels the regularity of the scaling profiles to smoluchowski s coagulation equation is studied when the coagulation kernel k is given by k x y x y with 0 1 more precisely is c smooth on 0 and decays exponentially fast for large x furthermore the singular behaviour of x as x 0 is identified thus giving a rigorous proof of physical conjectures

realizing stable categories as derived categories abstract in this paper we discuss a relationship between representation theory of graded self injective algebras and that of algebras of finite global dimension for a positively graded self injective algebra a such that a 0 has finite global dimension we construct two types of triangle equivalences first we show that there exists a triangle equivalence between the stable category of z graded a modules and the derived category of a certain algebra of finite global dimension secondly we show that if a has gorenstein parameter l then there exists a triangle equivalence between the stable category of z l z graded a modules and a derived orbit category of which is a triangulated hull of the orbit category of the derived category

the asymptotic spectrum of tensors a un ensemble x d informations bilineaires entre des espaces vectoriels de dimension finie on associe un espace de hausdorff compact x avec une application de x a l anneau des fonctions reelles continues sur x appele le spectre asymptotique de x qui contient toute l information exponentiellement asymptotique sur la complexite relative de calcul de f x

asymptotic entanglement transformation between w and ghz states we investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory results well known in that field imply that ghz states can be transformed into w states at rate 1 for any number of parties as a generalization we find that the asymptotic conversion rate from ghz states to dicke states is bounded as the number of subsystems increases and the number of excitations is fixed by generalizing constructions of coppersmith and winograd and by using monotones introduced by strassen we also compute the conversion rate from w to ghz states

the happel functor and homologically well graded iwanaga gorenstein algebras abstract happel constructed a fully faithful functor h d b mod mod z t for a finite dimensional algebra he also showed that this functor h gives an equivalence precisely when gldim thus if h gives an equivalence then it provides a canonical tilting object h of mod z t in this paper we generalize the happel functor h in the case where t is replaced with a finitely graded ig algebra a we study when this functor is fully faithful or is an equivalence for this purpose we introduce the notion of homologically well graded hwg ig algebra which can be characterized as an algebra posses a homological symmetry which a posteriori guarantee that the algebra is ig we prove that hwg ig algebras is precisely the class of finitely graded ig algebras that the happel functor is fully faithful we also identify the class that the happel functor gives an equivalence as a consequence of our result we see that if h gives an equivalence then it provides a canonical tilting object h t of cm z a for some special classes of finitely graded ig algebras our tilting objects h t coincide with tilting object constructed in previous works

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

a counterexample in coagulation theory two models of interacting polymers both within smoluchovski s theory are considered the first model has purely diagonal reaction rates and shows gelation the second model has the same diagonal reaction rates but also some nondiagonal ones the second model does not show gelation hence contrary to intuition gelation can be destroyed by an increase in the reaction rates the example also disproves a conjecture stating that the diagonal reaction rates are the determining factor for gelation

tilting and cluster tilting for quotient singularities we shall show that the stable categories of graded cohen macaulay modules over quotient singularities have tilting objects in particular these categories are triangle equivalent to derived categories of finite dimensional algebras our method is based on higher dimensional auslander reiten theory which gives cluster tilting objects in the stable categories of ungraded cohen macaulay modules

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

the asymptotic spectrum of locc transformations we study the exact non deterministic conversion of multipartite pure quantum states into one another via local operations and classical communication locc and asymptotic entanglement transformation under such channels in particular we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability known as the converse error exponent we give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of locc conversion a full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum we present a characterization of spectral points and use it to describe the spectrum in the bipartite case this leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states given a converse error exponent this extends the result on entanglement concentration in 1 where the target state is fixed as the bell state in the limit of vanishing converse error exponent the rate formula provides an upper bound on the exact asymptotic extraction rate between two states when the probability of success goes to 1 in the bipartite case we prove that this bound holds with equality

a note on the gap between rank and border rank we study the tensor rank of the tensor corresponding to the algebra of n variate complex polynomials modulo the dth power of each variable as a result we find a sequence of tensors with a large gap between rank and border rank and thus a counterexample to a conjecture of rhodes at the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised w state tensor in addition we exactly determine the tensor rank of the tensor cube of the three party w state tensor thus answering a question of chen et al

exact solutions for the coagulation fragmentation equation the authors present new exact equilibrium and time dependent solutions to the two main versions of the coagulation fragmentation equation equilibria are first found for the pure coagulation and pure fragmentation equations equilibrium solutions for the full coagulation fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form time dependent solutions are also derived for the pure fragmentation equation

uniqueness of self similar solutions to smoluchowski s coagulation equations for kernels that are close to constant we consider self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 1 we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small which is the first such uniqueness result for kernels that are not solvable our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures

a correction orthogonal representations and connectivity of graphs abstract this note corrects an error in the proof of the main result of the authors paper orthogonal representations and connectivity of graphs which appeared in linear algebra and its applications 114 115 1989 439 454

lower bounds in minimum rank problems abstract the minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph the minimum semidefinite rank of a graph is the minimum rank among hermitian positive semidefinite matrices with the given graph we explore connections between os sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large

grothendieck constant is norm of strassen matrix multiplication tensor we show that two important quantities from two disparate areas of complexity theory strassen s exponent of matrix multiplication omega and grothendieck s constant k g are different measures of size for the same underlying object the matrix multiplication tensor i e the 3 tensor or bilinear operator mu l m n mathbb f l times m times mathbb f m times n rightarrow mathbb f l times n a b mapsto ab defined by matrix matrix product over mathbb f mathbb r or mathbb c it is well known that strassen s exponent of matrix multiplication is the greatest lower bound on the log of the tensor rank of mu l m n we will show that grothendieck s constant is the least upper bound on a tensor norm of mu l m n taken over all l m n in mathbb n aside from relating the two celebrated quantities this insight allows us to rewrite grothendieck s inequality as a norm inequality begin aligned mu l m n 1 2 infty max x y m ne 0 frac mathrm tr xmy x 1 2 y 2 infty m infty 1 leqslant k g end aligned we prove that grothendieck s inequality is unique in the sense that if we generalize the 1 2 infty norm to arbitrary p q r in 1 infty begin aligned mu l m n p q r max x y m ne 0 frac mathrm tr xmy x p q y q r m r p end aligned then p q r 1 2 infty is up to cyclic permutations the only choice for which mu l m n p q r is uniformly bounded by a constant independent of l m n

graded self injective algebras are trivial extensions abstract for a positively graded artin algebra a n 0 a n we introduce its beilinson algebra b a we prove that if a is well graded self injective then the category of graded a modules is equivalent to the category of graded modules over the trivial extension algebra t b a consequently there is a full exact embedding from the bounded derived category of b a into the stable category of graded modules over a it is an equivalence if and only if the 0 th component algebra a 0 has finite global dimension

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

maximum generic nullity of a graph abstract for a graph g of order n the maximum nullity of g is defined to be the largest possible nullity over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively a new parameter maximum generic nullity is introduced maximum generic nullity provides insight into the structure of the null space of a matrix realizing maximum nullity of a graph it is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity vertex connectivity edge connectivity and minimum degree

entanglement distillation from greenberger horne zeilinger shares we study the problem of converting a product of greenberger horne zeilinger ghz states shared by subsets of several parties in an arbitrary way into ghz states shared by every party such a state can be described by a hypergraph on the parties as vertices and with each hyperedge corresponding to a ghz state shared among the parties incident with it our result is that if slocc transformations are allowed then the best asymptotic rate is the minimum of bipartite log ranks of the initial state which in turn equals the minimum cut of the hypergraph this generalizes a result by strassen on the asymptotic subrank of the matrix multiplication tensor

on p vs np and geometric complexity theory dedicated to sri ramakrishna this article gives an overview of the geometric complexity theory gct approach towards the p vs np and related problems focusing on its main complexity theoretic results these are 1 two concrete lower bounds which are currently the best known lower bounds in the context of the p vs nc and permanent vs determinant problems 2 the flip theorem which formalizes the self referential paradox in the p vs np problem and 3 the decomposition theorem which decomposes the arithmetic p vs np and permanent vs determinant problems into subproblems without self referential difficulty consisting of positivity hypotheses in algebraic geometry and representation theory and easier hardness hypotheses

error exponents for entanglement concentration consider entanglement concentration schemes that convert n identical copies of a pure state into a maximally entangled state of a desired size with success probability being close to one in the asymptotic limit we give the distillable entanglement the number of bell pairs distilled per copy as a function of an error exponent which represents the rate of decrease in failure probability as n tends to infinity the formula fills the gap between the least upper bound of distillable entanglement in probabilistic concentration which is the well known entropy of entanglement and the maximum attained in deterministic concentration the method of types in information theory enables the detailed analysis of the distillable entanglement in terms of the error rate in addition to the probabilistic argument we consider another type of entanglement concentration scheme where the initial state is deterministically transformed into a possibly mixed final state whose fidelity to a maximally entangled state of a desired size converges to one in the asymptotic limit we show that the same formula as in the probabilistic argument is valid for the argument on fidelity by replacing the success probability with the fidelity furthermore we also discuss entanglement yield when optimal success probability or optimal fidelity converges to zero in the asymptotic limit strong converse and give the explicit formulae for those cases

entangled pure state transformations via local operations assisted by finitely many rounds of classical communication we consider generic pure n qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems we characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication locc n for n qubits with n 3 we show that this set of states is of measure zero which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of locc n moreover we identify a class of states for which any locc n protocol can be realized via a concatenation of deterministic steps we show however that in general there exist state transformations which require a probabilistic step within the protocol which highlights the difference between bipartite and multipartite locc

stable categories of graded maximal cohen macaulay modules over noncommutative quotient singularities abstract tilting objects play a key role in the study of triangulated categories a famous result due to iyama and takahashi asserts that the stable categories of graded maximal cohen macaulay modules over quotient singularities have tilting objects this paper proves a noncommutative generalization of iyama and takahashi s theorem using noncommutative algebraic geometry namely if s is a noetherian as regular koszul algebra and g is a finite group acting on s such that s g is a gorenstein isolated singularity then the stable category cm z s g of graded maximal cohen macaulay modules has a tilting object in particular the category cm z s g is triangle equivalent to the derived category of a finite dimensional algebra

improved bound for complexity of matrix multiplication in 1968 strassen 16 described a method of multiplying two 2 2 matrices using only seven multiplications by applying it recursively it was shown that two n n matrices could be multiplied using o n operations in all where log2 7 2 81 this discovery led to a period of activity devoted to the improvement of this bound one can define an exponent of complexity which is in effect the smallest number such that two n n matrices can be multiplied using o n operations using an method of the type described by strassen a precise definition is given in 2 then strassen s result says that log2 7 so it is natural to wish to determine the true value of in the decade or so after 1978 several successive reductions in the upper bound for were achieved 2 3 6 8 10 12 14 17 leading to the bound 2 375477 obtained by coppersmith and winograd 7 it is an open question whether 2 we present a small improvement of the coppersmith winograd bound namely 2 3736897 which was first obtained in stothers 15 the proof uses an extension of the ideas of 7 and relies heavily on a combinatorial construction from 7 which depends on probabilistic arguments and a theorem of salem and spencer on sets of integers containing no three terms in arithmetical progression we have tried to organize the proof so that it is reasonably concise but also self contained to make it more accessible in the belief that at least parts of the argument may be of wider interest outside the algebraic complexity community to this end we derive the results we need from algebraic complexity theory in 2 which also contains general background material in 3 we present the essence of the combinatorial construction as a lemma which we subsequently apply several times we first apply it in 4 to recover the coppersmith winograd bound although our presentation is rather different the proof is essentially the same as that of 7 except for the proof of lemma 4 1 for which we give a somewhat simpler argument using 4 as preparation and motivation we obtain our new bound in 5 the argument is similar but more complicated and technical

entanglement classification with algebraic geometry we approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry its natural language we show that the class of symmetric separable states has the structure of a veronese variety and that its k secant varieties are slocc invariants thus slocc classes gather naturally into families this classification presents useful properties such as a linear growth of the number of families with the number of particles and nesting i e upward consistency of the classification we attach physical meaning to this classification through the required interaction length of parent hamiltonians we show that the states w n and ghz n are in the same secant family and that effectively the former can be obtained in a limit from the latter this limit is understood in terms of tangents leading to a refinement of the previous families we compute explicitly the classification of symmetric states with qubits in terms of both secant families and its refinement using tangents this paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory

graph expansion and communication costs of fast matrix multiplication the communication cost of algorithms also known as i o complexity is shown to be closely related to the expansion properties of the corresponding computation graphs we demonstrate this on strassen s and other fast matrix multiplication algorithms and obtain the first lower bounds on their communication costs in the sequential case where the processor has a fast memory of size m too small to store three n by n matrices the lower bound on the number of words moved between fast and slow memory is for a large class of matrix multiplication algorithms n m 0 m where 0 is the exponent in the arithmetic count e g 0 lg 7 for strassen and 0 3 for conventional matrix multiplication with p parallel processors each with fast memory of size m the lower bound is asymptotically lower by a factor of p these bounds are attainable both for sequential and for parallel algorithms and hence optimal

tripartite to bipartite entanglement transformation by stochastic local operations and classical communication and the structure of matrix spaces we study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication slocc it is known that the tripartite to bipartite slocc convertibility is characterized by the maximal schmidt rank of the given tripartite state i e the largest schmidt rank over those bipartite states lying in the support of the reduced density operator in this paper we further study this problem and exhibit novel results in both multi copy and asymptotic settings utilizing powerful results from the structure of matrix spaces in the multi copy regime we observe that the maximal schmidt rank is strictly super multiplicative i e the maximal schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal schmidt ranks we then provide a full characterization of those tripartite states whose maximal schmidt rank is strictly super multiplicative when taking tensor product with itself notice that such tripartite states admit strict advantages in tripartite to bipartite slocc transformation when multiple copies are provided in the asymptotic setting we focus on determining the tripartite to bipartite slocc entanglement transformation rate computing this rate turns out to be equivalent to computing the asymptotic maximal schmidt rank of the tripartite state defined as the regularization of its maximal schmidt rank despite the difficulty caused by the super multiplicative property we provide explicit formulas for evaluating the asymptotic maximal schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces including the study of matrix semi invariants these formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under slocc in the asymptotic setting applying the recent progress on the non commutative rank problem we can verify this condition in deterministic polynomial time

approach to self similarity in smoluchowski s coagulation equations we consider the approach to self similarity or dynamical scaling in smoluchowski s equations of coagulation for the solvable kernels k x y 2 x y and xy in addition to the known self similar solutions with exponential tails there are one parameter families of solutions with algebraic decay whose form is related to heavy tailed distributions well known in probability theory for k 2 the size distribution is mittag leffler and for k x y and k xyit is a power law rescaling of a maximally skewed stable levy distribution we characterize completely the domains of attraction of all self similar solutions under weak convergence of measures our results are analogous to the classical characterization of stable distributions in probability theory the proofs are simple relying on the laplace transform and a fundamental rigidity lemma for scaling limits c 2004 wiley periodicals inc

self similar solutions with fat tails for smoluchowski s coagulation equation with locally bounded kernels the existence of self similar solutions with fat tails for smoluchowski s coagulation equation has so far only been established for the solvable kernels and the diagonal one in this paper we prove the existence of such self similar solutions for continuous kernels k that are homogeneous of degree gamma in infty 1 and satisfy k x y c x y more precisely for any rho in gamma 1 we establish the existence of a continuous weak nonnegative self similar profile with decay x 1 rho as x

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

a uniqueness result for self similar profiles to smoluchowski s coagulation equation revisited in this note we indicate how to correct the proof of a uniqueness result in 6 for self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 frac 1 2 under the additional assumption in comparison to 6 that k has an analytic extension to mathbb c setminus infty 0 and that the precise asymptotic behaviour of k at the origin is prescribed we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small the complete details of the proof are available in 4 in addition we give here the proof of a uniqueness result for a related but simpler problem that appears in the description of self similar solutions for x rightarrow infty

vector representations of graphs and distinguishing quantum product states with one way locc abstract distinguishing sets of quantum states shared by two parties using only local operations and classical communication measurements is a fundamental topic in quantum communication and quantum information theory we introduce a graph theoretic approach based on the theory of vector representations of graphs to the core problem of distinguishing product states with one way locc we establish a number of results that show how distinguishing such states can be framed in terms of properties of the underlying graphs associated with a set of vector product states we also present a number of illustrative examples

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

existence of self similar solutions to smoluchowski s coagulation equation the existence of self similar solutions to smoluchowski s coagulation equation has been conjectured for several years by physicists and numerical simulations have confirmed the validity of this conjecture still there was no existence result up to now except for the constant and additive kernels for which explicit formulae are available in this paper the existence of self similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels

three connected graphs whose maximum nullity is at most three for a graph g v e with vertex set v 1 2 n let s g be the set of all n n real valued symmetric matrices a which represent g the maximum nullity of a graph g denoted by m g is the largest possible nullity of any matrix a s g fiedler showed that a graph g has m g 1 if and only if g is a path johnson et al gave a characterization of all graphs g with m g 2 independently hogben and van der holst gave a characterization of all 2 connected graphs with m g 2 in this paper we show that k connected graphs g have m g k that k connected partial k graphs g have m g k and that for 3 connected graphs g m g 3 if and only if g is a partial 3 path

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

second order asymptotics of conversions of distributions and entangled states based on rayleigh normal probability distributions we discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space to derive the second order conversion rate we introduce new probability distributions named rayleigh normal distributions the family of rayleigh normal distributions includes a rayleigh distribution and coincides with the standard normal distribution in the limit case using this family of probability distributions we represent the asymptotic second order rates for the distribution conversion as an application we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications locc are allowed this problem contains entanglement concentration entanglement dilution and a kind of cloning problem with locc restriction as special cases

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

local unitary equivalence of multipartite pure states necessary and sufficient conditions for the equivalence of arbitrary n qubit pure quantum states under local unitary lu operations are derived first an easily computable standard form for multipartite states is introduced two generic states are shown to be lu equivalent iff their standard forms coincide the lu equivalence problem for nongeneric states is solved by presenting a systematic method to determine the lu operators if they exist which interconvert the two states

size distribution in the polymerisation model afrbg the polymer size distribution in flory s polymerisation model afrbg is in equilibrium and non equilibrium of the general form ck ank xi k both below and above the gelation transition here explicit expressions are derived for a alpha and xi alpha as a function of the extent of reaction alpha the combinatorial factors nk are calculated from the recursion relation 2 k 1 nk sigma i j kkijninj with kij si f sj g si g sj f and sk f f 1 k 1 using a generating function technique the authors express nk in terms of laguerre polynomials g to infinity and jacobi polynomials g finite whose large k behaviour is of the form nk approximately bk tau xi c k with tau 3 2 no gelation occurs if f or g equals 1 and with tau 5 2 gelation occurs if f 1 and g 1

distillation of greenberger horne zeilinger states by combinatorial methods we prove a lower bound on the rate of greenberger horne zeilinger states distillable from pure multipartite states by local operations and classical communication locc our proof is based on a modification of a combinatorial argument used in the fast matrix multiplication algorithm of coppersmith and winograd previous use of methods from algebraic complexity in quantum information theory concerned transformations with stochastic locc slocc resulting in an asymptotically vanishing success probability in contrast our new protocol works with an asymptotically vanishing error

entanglement of pure states for a single copy an optimal local conversion strategy between any two pure states of a bipartite system is presented it is optimal in that the probability of success is the largest achievable if the parties which share the system and which can communicate classically are only allowed to act locally on it the study of optimal local conversions sheds some light on the entanglement of a single copy of a pure state we propose a quantification of such an entanglement by means of a finite minimal set of new measures from which the optimal probability of conversion follows

the asymptotic entanglement cost of preparing a quantum state we give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state is equal to limn ef n n where ef is the entanglement of formation

general solutions and scaling violation for fragmentation with mass loss explicit solutions of a linear rate equation lead to improved understanding of fragmenlatian with discrete and continuous mass loss discrete mass loss provides a general expression for the overall mass 10 rate which accounts for mass loss in the shattering regime where runaway fragmentation for small panicles producer a phase of zero mass panicles an explicit solution far the recession regime where small panicles lose large fractionsoftheir mass to surface recession shows that the total number ofpaniclesincreases with time until the typical panicles become small enough to lose all of their mass and disappear rather than to break a general series solution is presented for fragmentation with continuous and discrete mass loss a proof that continuous mass loss precludes dynamic scaling is presented

orthogonal representations and connectivity of graphs abstract it is proved that a graph on n nodes is k connected if and only if its nodes can be represented by real vectors in dimension n k such that a nonadjacent nodes are represented by orthogonal vectors and b any n k of them are linearly independent we show that the closure of the set of all representations with properties a and b is irreducible as an algebraic variety and study the question of irreducibility of the variety of all representations with property a

bilinear mincing rank note abstract we define here the bilinear mincing rank of a bilinear form over a field of the characteristic zero and we demonstrate that this invariant represents a new in fact the first known general and nontrivial lower bound to the bilinear circuit size complexity of a bilinear form

post gelation solutions to smoluchowski s coagulation equation it is proven how the post gelation behaviour originally suggested by flory 1993 can be obtained as a result of a limiting process passing from a finite to an infinite system in a previous paper by the authors it was shown how the post gelation behaviour first suggested by stockmayer 1943 can be obtained by passing to the limit of an infinite system in a different way it is thus demonstrated that different post gelation solutions of smoluchowski s coagulation equation can be obtained by different limiting processes

tensor rank is not multiplicative under the tensor product the tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors let s be a k tensor and let t be an l tensor the tensor product of s and t is a k l tensor tensor rank is sub multiplicative under the tensor product we revisit the connection between restrictions and degenerations a result of our study is that tensor rank is not in general multiplicative under the tensor product this answers a question of draisma and saptharishi specifically if a tensor t has border rank strictly smaller than its rank then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power the tensor kronecker product from algebraic complexity theory is related to our tensor product but different namely it multiplies two k tensors to get a k tensor nonmultiplicativity of the tensor kronecker product has been known since the work of strassen it remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product interestingly lower bounds on border rank obtained from generalized flattenings including young flattenings multiply under the tensor product

an explicit solution to a discrete fragmentation model the discrete binary fragmentation equation is solved explicitly far a model where tnenetrateoibreakupoiaparlicie oishek o equals k lj k l andthedaughter ri e distribution b lx equals 2 k 1 this system is closely related to a model of polymer degradation considered by simha in which ax i and b is as above in the continuum limit both of these models go over to a continuous fragmentation model in which all particles break with an equal rate a x i and the daughter size distribution is uniform b yl x z x which is at the borderline of the shattering transition

graphs associated with vector spaces of even dimension a link with differential geometry abstract we define a new association between graphs and orthonormal bases of even dimensional euclidean vector spaces endowed with an special isomorphism motivated by the recently introduced theory of submanifolds associated with graphs we provide several interesting examples and we analyze the shape of such graphs by proving some general results

tverberg s theorem is 50 years old a survey this survey presents an overview of the advances around tverberg s theorem focusing on the last two decades we discuss the topological linear algebraic and combinatorial aspects of tverberg s theorem and its applications the survey contains several open problems and conjectures 2018 american mathematical society

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

everything you always wanted to know about locc but were afraid to ask in this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication locc while locc emerges as the natural class of operations in many important quantum information tasks its mathematical structure is complex and difficult to characterize here we provide a precise description of locc and related operational classes in terms of quantum instruments our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds while the set of locc is not topologically closed we show that finite round locc constitutes a compact subset of quantum operations additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of locc implementable maps finally we demonstrate a two qubit map whose action can be approached arbitrarily close using locc but nevertheless cannot be implemented perfectly

classical quantum and nonsignalling resources in bipartite games we study bipartite games that arise in the context of nonlocality with the help of graph theory our main results are alternate proofs that deciding whether a no communication classical winning strategy exists for certain games called forbidden edge and covering games is np complete while the problem of deciding if these games admit a nonsignalling winning strategy is in p we discuss relations between quantum winning strategies and orthogonality graphs we also show that every pseudotelepathy game yields both a proof of the bell kochen specker theorem and an instance of a two prover interactive proof system that is classically sound but that becomes unsound when provers use shared entanglement

expected values of parameters associated with the minimum rank of a graph abstract we investigate the expected value of various graph parameters associated with the minimum rank of a graph including minimum rank maximum nullity and related colin de verdiere type parameters let g v p denote the usual erd s renyi random graph on v vertices with edge probability p we obtain bounds for the expected value of the random variables mr g v p m g v p g v p and g v p which yield bounds on the average values of these parameters over all labeled graphs of order v

n representation finite algebras and n apr tilting we introduce the notion of n representation finiteness generalizing representationfinite hereditary algebras we establish the procedure of n apr tilting and show that it preserves n representation finiteness we give some combinatorial description of this procedure and use this to completely describe a class of n representation finite algebras called type a

the minimum rank problem for circulants abstract the minimum rank problem is to determine for a graph g the smallest rank of a hermitian or real symmetric matrix whose off diagonal zero nonzero pattern is that of the adjacency matrix of g here g is taken to be a circulant graph and only circulant matrices are considered the resulting graph parameter is termed the minimum circulant rank of the graph this value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set and in this case is shown to coincide with the usual minimum rank under the additional restriction to positive semidefinite matrices the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph this value is then determined when the number of vertices is prime the analogous parameter over r is also investigated

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway