syzygies of bounded rank symmetric tensors are generated in bounded degree we study the syzygies of secant ideals of veronese subrings of a fixed commutative graded algebra over a field of characteristic 0 one corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any veronese embedding of a projective scheme x can be bounded by a constant that only depends on i r and x and not on the choice of the veronese embedding

hadamard products of linear spaces abstract we describe properties of hadamard products of algebraic varieties we show any hadamard power of a line is a linear space and we construct star configurations from products of collinear points tropical geometry is used to find the degree of hadamard products of other linear spaces

condition numbers for the tensor rank decomposition abstract the tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low rank tensor these parameters are analyzed and interpreted in many applications as tensors are often perturbed by measurement errors in practice one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis the magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor this paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of terracini s matrix which represents the tangent space to the set of tensors of fixed rank a practical algorithm for computing this condition number is presented the latter s elementary properties such as scaling and orthogonal invariance are established rank 1 tensors are always well conditioned the class of weak 3 orthogonal tensors which includes orthogonally decomposable tensors contains both well conditioned and ill conditioned problems numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed

secants of minuscule and cominuscule minimal orbits we study the geometry of the secant and tangential variety of a cominuscule and minuscule variety e g a grassmannian or a spinor variety using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the zariski closure of the image of a map defined by generalized determinants in particular equations of the secant or tangential variety correspond to relations among generalized determinants we also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any grassmannian

relations between the minors of a generic matrix abstract it is well known that the plucker relations generate the ideal of relations of the maximal minors of a generic m n matrix in this paper we discuss the relations of t minors for t min m n we will exhibit minimal relations in degrees 2 non plucker in general and 3 and give some evidence for our conjecture that we have found the generating system of the ideal of relations the approach is through the representation theory of the general linear group

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

the stability of the kronecker product of schur functions in the late 1930 s murnaghan discovered the existence of a stabilization phenomenon for the kronecker product of schur functions for n large enough the values of the kronecker coefficients appearing in the product of two schur functions of degree n do not depend on the first part of the indexing partitions but only on the values of their remaining parts we compute the exact value of n when this stable expansion is reached we also compute two new bounds for the stabilization of a particular coefficient of such a product given partitions alpha and beta we give bounds for all the parts of any partition gamma such that the corresponding kronecker coefficient is nonzero finally we also show that the reduced kronecker coefficients are structure coefficients for the heisenberg product introduced by aguiar ferrer and moreira

even partitions in plethysms abstract we prove that for all natural numbers k n d with k d and every partition of size kn with at most k parts there exists an irreducible gl d c representation of highest weight 2 in the plethysm sym k sym 2 n c d this gives an affirmative answer to a conjecture by weintraub steven h weintraub some observations on plethysms j algebra 129 1 1990 103 114 our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory

typical tensorial rank abstract upper bounds on the typical rank r n m l of tensors maximal border rank rank of almost all tensors of a given shape n m l are presented these improve previous results by a tkinson and lloyd for cubic shape tensors the typical rank is determined exactly r n n n n3 3n 2 n 3

induction for secant varieties of segre varieties this paper studies the dimension of secant varieties to segre varieties the problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry an inductive procedure is built around the ideas of successive specializations of points and projections this reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases as applications of the technique we give a complete classification of defective p secant varieties to segre varieties for p 6 we generalize a theorem of catalisano geramita gimigliano on non defectivity of tensor powers of pn we determine the set of p for which unbalanced segre varieties have defective p secant varieties in addition we completely describe the dimensions of the secant varieties to the deficient segre varieties p1 x p1 x pn x pn and p2 x p3 x p3 in the final section we propose a series of conjectures about defective segre varieties

characterization of division algebras of minimal rank and the structure of their algorithm varieties the purpose of this paper is to show that every finite dimensional division algebra of minimal rank i e of minimal complexity with respect to the noncommutative model of computation is a finite simple field extension moreover we investigate the structure of the variety of optimal algorithms for the computation of the multiplication in such fields modulo the isotropy group of the problem

generalized foulkes modules and maximal and minimal constituents of plethysms of schur functions this paper proves a combinatorial rule giving all maximal and minimal partitions lambda such that the schur function s lambda appears in a plethysm of two arbitrary schur functions determining the decomposition of these plethysms has been identified by stanley as a key open problem in algebraic combinatorics as corollaries we prove three conjectures of agaoka on the partitions labeling the lexicographically greatest and least schur functions appearing in an arbitrary plethysm we also show that the multiplicity of the schur function labelled by the lexicographically least constituent may be arbitrarily large the proof is carried out in the symmetric group and gives an explicit non zero homomorphism corresponding to each maximal or minimal partition

secant varieties of toric varieties arising from simplicial complexes abstract motivated by the study of the secant variety of the segre veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes we prove that every such secant is toric which gives a way to use combinatorial tools to study singularities we focus on the segre veronese variety for which we completely classify their secants that give gorenstein or q gorenstein varieties we conclude providing the explicit description of the singular locus

nonvanishing of kronecker coefficients for rectangular shapes we prove that for any partition 1 d2 of size d there exists k 1 such that the tensor square of the irreducible representation of the symmetric group sk d with respect to the rectangular partition k k contains the irreducible representation corresponding to the stretched partition k 1 k d 2 we also prove a related approximate version of this statement in which the stretching factor k is eectively bounded in terms of d we further discuss the consequences for geometric complexity theory which provided the motivation for this work

nondeterminism and infinite computations in constraint programming we investigate the semantics of concurrent constraint programming and of various sublanguages with particular emphasis on nondeterminism and infinite behavior the aim is to find out what is the minimal structure which a domain must have in order to capture these two aspects we show that a notion of observables obtained by the upward closure of the results of computations is relatively easy to model even in presence of synchronization on the contrary modeling the exact set of results is problematic even for the simple sublanguage of constraint logic programming we show that most of the standard topological techniques fail in capturing this more precise notion of observables the analysis of these failed attempts leads us to consider a categorical approach

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

a quasi one dimensional model of thermoacoustics in the presence of mean flow abstract in thermoacoustic regenerators the interaction of thermo viscous boundary layers and axial temperature gradients causes a conversion from thermal energy to acoustic power or vice versa in this paper an improved analytical model for thermoacoustic boundary layer effects in the presence of mean flow is derived and analyzed previous formulations of the thermo acoustic effect take into account effects of mean flow on acoustic propagation only implicitly i e in as much as mean flow influences the mean temperature field the new model however includes additional terms in the perturbation equations which describe explicitly the interaction between steady mean flow and acoustics for a parallel plate pore the three dimensional thermoacoustic equations are derived and reduced to a transversally averaged system of differential equations by applying green s function technique and suitable assumptions the resulting one dimensional perturbation equations are then solved numerically for two sets of boundary conditions to obtain the linear scattering matrix coefficients the solutions generated for a wide range of frequencies can be applied in a low order network model context to study the stability of thermoacoustic devices the impact of mean flow on the thermoacoustic interaction is investigated and validated against full computational fluid dynamics simulations of laminar compressible flow for one specific configuration it is shown that at low frequencies womersley number 1 the new formulation predicts the acoustic behavior more accurately than the earlier formulations finally the ideas and benefit of further improved and more complex models for higher mach numbers are discussed

typical ranks for m n m 1 n tensors with m n abstract in various application fields tensor type data are used recently and then a typical rank is important there may be more than one typical ranks over the real number field it is well known that the set of 2 n n tensors has two typical ranks n n 1 for n 2 that the set of 3 4 8 tensors has two typical ranks 8 9 and that the set of 4 4 12 tensors has two typical ranks 12 13 in this paper we show that the set of m n m 1 n tensors with m n has two typical ranks m 1 n m 1 n 1 if m n where is the hurwitz radon function defined as n 2 b 8 c for nonnegative integers a b c such that n 2 a 1 2 b 4 c and 0 b 4

cocharacter closure and the rational hilbert mumford theorem for a field k let g be a reductive k group and v an affine k variety on which g acts using the notion of cocharacter closed g k orbits in v we prove a rational version of the celebrated hilbert mumford theorem from geometric invariant theory we initiate a study of applications stemming from this rationality tool a number of examples are discussed to illustrate the concept of cocharacter closure and to highlight how it differs from the usual zariski closure

rectangular kronecker coefficients and plethysms in geometric complexity theory abstract we prove that in the geometric complexity theory program the vanishing of rectangular kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial moreover we prove the positivity of rectangular kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition we also compare rectangular kronecker coefficients with their corresponding plethysm coefficients which leads to a new lower bound for rectangular kronecker coefficients moreover we prove that the saturation of the rectangular kronecker semigroup is trivial we show that the rectangular kronecker positivity stretching factor is 2 for a long first row and we completely classify the positivity of rectangular limit kronecker coefficients that were introduced by manivel in 2011

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

border rank of m n mn q tensors abstract the border rank of a nondegenerate m n mn q tensor over the complex field is mn q provided that q max m n

decidability of infinite state timed ccp processes and first order ltl the ntcc process calculus is a timed concurrent constraint programming ccp model equipped with a first order linear temporal logic ltl for expressing process specifications a typical behavioral observation in ccp is the strongest postcondition sp the ntcc sp denotes the set of all infinite output sequences that a given process can exhibit the verification problem is then whether the sequences in the sp of a given process satisfy a given ntcc ltl formula this paper presents new positive decidability results for timed ccp as well as for ltl in particular we shall prove that the following problems are decidable 1 the sp equivalence for the so called locally independent ntcc fragment unlike other fragments for which similar results have been published this fragment can specify infinite state systems 2 verification for locally independent processes and negation free first order formulae of the ntcc ltl 3 implication for such formulae 4 satisfiability for a first order fragment of manna and pnueli s ltl the purpose of the last result is to illustrate the applicability of ccp to well established formalisms for concurrency

kronecker coefficients for some near rectangular partitions abstract we give formulae for computing kronecker coefficients occurring in the expansion of s s where both and are nearly rectangular and have smallest parts equal to either 1 or 2 in particular we study s n n 1 1 s n n s n 1 n 1 1 s n n 1 s n 1 n 1 2 s n n s n 1 n 1 1 1 s n n and s n n 1 s n n 1 our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group mainly employing the pieri rule and a useful identity of littlewood as a consequence of these formulae we also derive an expression enumerating certain standard young tableaux of bounded height in terms of the motzkin and catalan numbers

on waringʼs problem for systems of skew symmetric forms abstract this paper is devoted to a problem of finding the smallest positive integer s m n k such that m 1 generic skew symmetric k 1 forms in n 1 variables as linear combinations of the same s m n k decomposable skew symmetric k 1 forms this problem is analogous to a well known problem called waring s problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry in this paper we will go through some basics of algebraic geometry describe how objects in algebraic geometry can be associated to systems of skew symmetric forms and discuss algebro geometric approaches to establish the existence of triples m n k where s m n k is more than expected

bounded rank tensors are defined in bounded degree matrices of rank at most k are defined by the vanishing of polynomials of degree k 1 in their entries namely their k 1 k 1 subdeterminants regardless of the size of the matrix we prove a qualitative analogue of this statement for tensors of arbitrary dimension where matrices correspond to two dimensional tensors more specifically we prove that for each k there exists an upper bound d d k such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d regardless of the dimension of the tensor and regardless of its size in each dimension our proof involves passing to an infinite dimensional limit of tensor powers of a vector space whose elements we dub infinite dimensional tensors and exploiting the symmetries of this limit in crucial ways

eigenvalue distributions of reduced density matrices given a random quantum state of multiple distinguishable or indistinguishable particles we provide an effective method rooted in symplectic geometry to compute the joint probability distribution of the eigenvalues of its one body reduced density matrices as a corollary by taking the distribution s support which is a convex moment polytope we recover a complete solution to the one body quantum marginal problem we obtain the probability distribution by reducing to the corresponding distribution of diagonal entries i e to the quantitative version of a classical marginal problem which is then determined algorithmically this reduction applies more generally to symplectic geometry relating invariant measures for the coadjoint action of a compact lie group to their projections onto a cartan subalgebra and can also be quantized to provide an efficient algorithm for computing bounded height kronecker and plethysm coefficients

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

the uniqueness of plethystic factorisation we prove that a plethysm product of two schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products

a probabilistic algorithm for the secant defect of grassmann varieties abstract in this paper we study the higher secant varieties of grassmann varieties in relation to the functional waring s problem for alternating tensors and to the alexander hirschowitz theorem we show how to identify defective higher secant varieties of grassmannians using a probabilistic method involving terracini s lemma and we describe an algorithm which can compute by numerical methods dim ssg k n for n 14 our main result is that except for grassmannians of lines if n 14 and k n 1 2 if n 14 we have studied the case k 5 there are only the four known defective cases s2g 2 6 s2g 3 7 s3g 3 7 and s3g 2 8

characteristic based non linear simulation of large scale standing wave thermoacoustic engine a few linear theories swift j acoust soc am 84 4 1145 1180 1988 swift j acoust soc am 92 3 1551 1563 1992 olson and swift j acoust soc am 95 3 1405 1412 1994 and numerical models based on low mach number analysis worlikar and knio j comput phys 127 2 424 451 1996 worlikar et al j comput phys 144 2 199 324 1996 hireche et al canadian acoust 36 3 164 165 2008 describe the flow dynamics of standing wave thermoacoustic engines but almost no simulation results are available that enable the prediction of the behavior of practical engines experiencing significant temperature gradient between the stack ends and thus producing large amplitude oscillations here a one dimensional non linear numerical simulation based on the method of characteristics to solve the unsteady compressible euler equations is reported formulation of the governing equations implementation of the numerical method and application of the appropriate boundary conditions are pres

on tensors of border rank l in cm n l notice this is the author s version of a work that was accepted for publication in linear algebra and its applications changes resulting from the publishing process such as peer review editing corrections structural formatting and other quality control mechanisms may not be reflected in this document changes may have been made to this work since it was submitted for publication a definitive version was subsequently published in linear algebra and its applications june 23 2011 doi 10 1016 j laa 2011 05 013 the original publication is available at www elsevier com

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

the complementation problem for bu chi automata with applications to temporal logic the problem of complementing buchi automata arises when developing procedures for temporal logics of programs unfortunately previously known constructions for complementing buchi automata involve a doubly exponential blow up in the size of the automaton we present a construction that involves only an exponential blow up we use this construction to prove a polynomial space upper bound for the propositional temporal logic of regular events and to prove a complexity hierarchy result for quantified propositional temporal logic

tensor decomposition for signal processing and machine learning tensors or multiway arrays are functions of three or more indices i j k ldots similar to matrices two way arrays which are functions of two indices r c for row column tensors have a rich history stretching over almost a century and touching upon numerous disciplines but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing statistics data mining and machine learning this overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors as such it focuses on fundamentals and motivation using various application examples aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and or developing tensor algorithms and software some background in applied optimization is useful but not strictly required the material covered includes tensor rank and rank decomposition basic tensor factorization models and their relationships and properties including fairly good coverage of identifiability broad coverage of algorithms ranging from alternating optimization to stochastic gradient statistical performance analysis and applications ranging from source separation to collaborative filtering mixture and topic modeling classification and multilinear subspace learning

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

binary determinantal complexity abstract we prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros ones and variables as entries a 7 by 7 matrix is required our proof is computer based and uses the enumeration of bipartite graphs furthermore we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros ones and variables we show that these are exactly the sequences in the complexity class of constant free polynomially sized weakly skew circuits

vanishing hessian wild forms and their border vsp wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank the discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero dimensional schemes and the span of their flat limit for concise forms of minimal border rank we show that the condition of vanishing hessian is equivalent to being wild this is proven by making a detour through structure tensors of smoothable and gorenstein algebras the equivalence fails in the non minimal border rank regime we exhibit an infinite series of minimal border rank wild forms of every degree d ge 3 as well as an infinite series of wild cubics inspired by recent work on border apolarity of buczynska and buczynski we study the border varieties of sums of powers underline mathrm vsp of these forms in the corresponding multigraded hilbert schemes

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

kronecker products characters partitions and the tensor square conjectures abstract we study the remarkable saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups s n contain all irreducibles as their constituents our main result is that for sufficiently large n they contain representations corresponding to young diagrams of hooks two row and diagrams obtained from hooks and two rows by adding a finite number of squares for that we develop a new sufficient condition for the positivity of kronecker coefficients in terms of characters and apply this tool using combinatorics of rim hook tableaux in combination with known results on unimodality of certain partition functions we also present connections and speculations on random characters of s n

flexible affine cones and flexible coverings we provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties we apply this criterion to prove flexibility of affine cones over secant varieties of segre veronese embeddings and over certain fano threefolds we further prove flexibility of total coordinate spaces of cox rings of del pezzo surfaces

ideals of bounded rank symmetric tensors are generated in bounded degree we study the syzygies of secant ideals of veronese subrings of a fixed commutative graded algebra over a field of characteristic 0 one corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any veronese embedding of a projective scheme x can be bounded by a constant that only depends on i r and x and not on the choice of the veronese embedding

generalized counting constraint satisfaction problems with determinantal circuits abstract generalized counting constraint satisfaction problems include holant problems with planarity restrictions polynomial time algorithms for such problems include matchgates and matchcircuits which are based on pfaffians in particular they use gates which are expressible in terms of a vector of sub pfaffians of a skew symmetric matrix we introduce a new type of circuit based instead on determinants with seemingly different expressive power in these determinantal circuits a gate is represented by the vector of all minors of an arbitrary matrix determinantal circuits permit a different class of gates applications of these circuits include proofs of theorems from algebraic graph theory including the chung langlands formula for the number of rooted spanning forests of a graph and computing tutte polynomials of certain matroids they also give a strategy for simulating quantum circuits with closed timelike curves monoidal category theory provides a useful language for discussing such counting problems turning combinatorial restrictions into categorical properties we introduce the counting problem in monoidal categories and count preserving functors as a way to study fp subclasses of problems in settings which are generally p hard using this machinery we show that surprisingly determinantal circuits can be simulated by pfaffian circuits at quadratic cost

the ideal of the trifocal variety techniques from representation theory symbolic com putational algebra and numerical algebraic geometry are used to nd the minimal generators of the ideal of the trifocal variety an eective test for determining whether a given tensor is a trifocal tensor is also given

on generic identifiability of symmetric tensors of subgeneric rank we prove that the general symmetric tensor in sdcn 1 of rank r is identifiable provided that r is smaller than the generic rank that is its waring decomposition as a sum of r powers of linear forms is unique only three exceptional cases arise all of which were known in the literature our original contribution regards the case of cubics d 3 while for d 4 we rely on known results on weak defectivity by ballico ciliberto chiantini and mella

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

cook s versus valiant s hypothesis we first investigate the dependency of our complexity concepts on the field k we prove that for algebraically closed fields k the truth of valiant s hypothesis vp vnp depends at most on the characteristic of k

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

geometric complexity theory i an approach to the p vs np and related problems we suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size specifically we introduce the notion of a partially stable point in a reductive group representation which generalizes the notion of stability in geometric invariant theory due to mumford geometric invariant theory springer verlag berlin 1965 then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points we also suggest an approach to tackle the latter class of problems via construction of explicit obstructions

fundamental invariants of orbit closures abstract for several objects of interest in geometric complexity theory namely for the determinant the permanent the product of variables the power sum the unit tensor and the matrix multiplication tensor we introduce and study a fundamental sl invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure for the power sums we can write down this fundamental invariant explicitly in most cases our constructions generalize the two aronhold invariants on ternary cubics for the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well known alon tarsi conjecture on latin squares we provide computer calculations in small dimensions for these cases as a main tool for our analysis we determine the stabilizers and we establish the polystability of all the mentioned forms and tensors including the generic ones

inequalities for the ranks of multipartite quantum states abstract we investigate relations between the ranks of marginals of multipartite quantum states we show that there exist inequalities constraining the possible distribution of ranks this is perhaps surprising since it was recently discovered that the renyi entropies for 0 1 1 satisfy only one trivial linear inequality non negativity and the distribution of entropies for 0 1 is completely unconstrained beyond non negativity our results resolve an important open question by showing that the case of 0 logarithm of the rank is restricted by nontrivial linear relations and thus the cases of von neumann entropy i e 1 and 0 renyi entropy are exceptionally interesting measures of entanglement in the multipartite setting we close the paper with an intriguing open problem which has a simple statement but is seemingly difficult to resolve

smoothed analysis for tensor methods in unsupervised learning smoothed analysis is a powerful paradigm in overcoming worst case intractability in high dimensional data analysis and unsupervised learning while polynomial time smoothed analysis guarantees have been obtained for worst case intractable problems like tensor decomposition and learning mixtures of gaussians such guarantees have been hard to obtain for several other important problems in data analysis a core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries that are given by low degree polynomials of a few base underlying random variables in this work we address this challenge by obtaining high confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind we then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in high dimensional data analysis

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

ranks of tensors and a generalization of secant varieties abstract we introduce subspace rank as a tool for studying ranks of tensors and x rank more generally we derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in c 2 c b c b we review the literature from a geometric perspective

hyperdeterminants of polynomials abstract the hyperdeterminant of a polynomial interpreted as a symmetric tensor factors into several irreducible factors with multiplicities using geometric techniques these factors are identified along with their degrees and their multiplicities the analogous decomposition for the discriminant of polynomial is found

an overview of mathematical issues arising in the geometric complexity theory approach to mathbf vp neq mathbf vnp we discuss the geometry of orbit closures and the asymptotic behavior of kronecker coefficients in the context of the geometric complexity theory program to prove a variant of valiant s algebraic analogue of the mathbf p neq mathbf np conjecture we also describe the precise separation of complexity classes that their program proposes to demonstrate

maximal bilinear complexity and codes abstract the connection between bilinear complexity and error correcting codes discovered by brockett and dobkin in 1973 yields lower bounds on the maximal ranks of tensors with a given shape the resulting bounds are linear and thus interesting only for unbalanced shapes like n n 2 and n n n 2 k with k n as an example for odd n the maximal rank of n n 2 tensors is larger over z 2 than over an algebraic closure of z 2

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

on the algorithmic complexity of associative algebras abstract the complexity l a of a finite dimensional associative algebra a is the number of non scalar multiplications divisions of an optimal algorithm to compute the product of two elements of the algebra we show l a 2ddim a t where t is the number of maximal two sided ideals of a

subtracting a best rank 1 approximation may increase tensor rank it has been shown that a best rank r approximation of an order k tensor may not exist when r 2 and k 3 this poses a serious problem to data analysts using tensor decompositions it has been observed numerically that generally this issue cannot be solved by consecutively computing and subtracting best rank 1 approximations the reason for this is that subtracting a best rank 1 approximation generally does not decrease tensor rank in this paper we provide a mathematical treatment of this property for real valued 2 2 2 tensors with symmetric tensors as a special case regardless of the symmetry we show that for generic 2 2 2 tensors which have rank 2 or 3 subtracting a best rank 1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank 2 and rank 3 sets hence for a typical tensor of rank 2 subtracting a best rank 1 approximation increases the tensor rank

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

the complexity of propositional linear temporal logics the complexity of satisfiability and determination of truth in a particular finite structure are considered for different propositional linear temporal logics it is shown that these problems are np complete for the logic with f and are pspace complete for the logics with f x with u with u s x operators and for the extended logic with regular operators given by wolper

rank and optimal computation of generic tensors abstract the typical rank maximal border rank of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated for the size n n 3 with n odd the complement of the set of tensors of maximal border rank is a hypersurface its equation is given

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

nonlinear two dimensional model for thermoacoustic engines a two dimensional model and efficient solution algorithm are developed for studying nonlinear effects in thermoacoustic engines there is no restriction on the length or location of the stack and the cross sectional area of the resonator may vary with position along its axis reduced model equations are obtained by ordering spatial derivatives in terms of rapid variations across the pores in the stack versus slow variations along the resonator axis high efficiency is achieved with the solution algorithm because the stability condition for numerical integration of the model equations is connected with resonator length rather than pore diameter computation time is reduced accordingly by several orders of magnitude without sacrificing spatial resolution the solution algorithm is described in detail and the results are verified by comparison with established linear theory two examples of nonlinear effects are investigated briefly the onset of instability through to saturation and steady state and nonlinear waveform distortion as a function of resonator shape

geometric aspects of iterated matrix multiplication abstract this paper studies geometric properties of the iterated matrix multiplication polynomial and the hypersurface that it defines we focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial the dual variety and the jacobian loci of the hypersurface that are computed with the aid of representation theory of quivers

the power of temporal proofs abstract some methods for reasoning about concurrent programs and hardware devices have been based on proof systems for temporal logic unfortunately all effective proof systems for temporal logic are incomplete for the standard semantics in the sense that some formulas hold in every intended model but cannot be proved we evaluate and compare the power of several proof systems for temporal logic specifically we relate temporal systems to classical systems with explicit time parameters a typical temporal system turns out to be incomplete in a strong sense we exhibit a short valid formula it fails to prove we suggest the addition of new rules to define auxiliary predicates with these rules we obtain nonstandard soundness and completeness results in particular one of the simple temporal systems we describe is as powerful as peano arithmetic

low mach number analysis of idealized thermoacoustic engines with numerical solution a model of an idealized thermoacoustic engine is formulated coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load correct coupling results in an asymptotically consistent global model in the small mach number approximation a well resolved numerical solution is obtained for two dimensional heat exchangers and stack the model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative mach number the model is well suited for simulation of the entire startup process whereby as a result of some excitation an initially specified temperature profile in the stack evolves toward a near steady profile eventually reaching stationary operation a validation analysis is presented together with results showing the early amplitude growth and approach of a stationary regime two types of initial excitation are used random noise and a small periodic wave the set of assumptions made leads to a heat exchanger section that acts as a source of volume but is transparent to pressure and to a local heat exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed