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

i v fuzzy relations and sanchez s approach for medical diagnosis abstract this paper introduces the concept of i v fuzzy relations i v fuzzy tolerance relations composition of i v fuzzy relations etc several useful propositions are proved finally sanchez s approach for medical diagnosis is considered and discussed using the idea of i v fuzzy relations

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

on maximum typical and generic ranks we show that for several notions of rank including tensor rank waring rank and generalized rank with respect to a projective variety the maximum value of rank is at most twice the generic rank we show that over the real numbers the maximum value of the real rank is at most twice the smallest typical rank which is equal to the complex generic rank

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

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

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

fast matrix multiplication without apa algorithms abstract the method of trilinear aggregating with implicit canceling for the design of fast matrix multiplication mm algorithms is revised and is formally presented with the use of generating tables and of linear transformations of the problem of mm it is shown how to derive the exponent of mm below 2 67 even without the use of approximation algorithms

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

maximal rank of an element of a tensor product abstract upper bounds are given for the maximal rank of an element of the tensor product of three vector spaces

a stability result for sectorial operators in branch spaces in this paper the author considers approximations of holomorphic semigroups on general banach spaces based on a acceptable rational functions a general stability theorem is proved it covers rational functions r z with r infty 1 and can be applied even to variable step cases in particular the question of the text l p stability p ne 2 of crank nicolson schemes for parabolic initial value problems is settled

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

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

applications of interval valued fuzzy n ary polygroups with respect to t norms t conorms using the triangular norm t conorm s in the context of n ary polygroups we introduce the concept of an interval valued anti fuzzy n ary subpolygroup with respect to t s respectively a necessary and sufficient condition for an interval valued fuzzy subset in order to be an interval valued anti fuzzy n ary subpolygroup is established and some important results are presented this study is useful for the most computational part of fuzzy control which is the defuzzification

the homomorphic image of a fuzzy subgroup is always a fuzzy subgroup let l be a completely distributive complete lattice with universal bound 0 and 1 and x any group we will prove that the image of a fuzzy group x l under a homomorphism f x y is always a fuzzy group without any restriction on and f

a hybrid finite difference scheme for pricing asian options in this paper we apply a hybrid finite difference scheme to evaluate the prices of asian call options with fixed strike price we use the crank nicolson method to discretize the time variable and a hybrid finite difference scheme to discretize the spatial variable the hybrid difference scheme uses the central difference approximation whenever the mesh points are sufficiently away from the left hand side of the domain to ensure the stability of the scheme otherwise a midpoint upwind difference scheme is used the matrix associated with the discrete operator is an m matrix which ensures that the spatial discretization scheme is maximum norm stable it is proved that the scheme is second order convergent with respect to both time and spatial variables numerical experiments support these theoretical results

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

rosenfeld s fuzzy subgroups with interval valued membership functions abstract in this paper we define interval valued fuzzy subgroups i e i v fuzzy subgroups of rosenfeld s nature a necessary and sufficient condition for an i v fuzzy subset to be an i v fuzzy subgroup is stated intersection of two i v fuzzy subgroups is also an i v fuzzy subgroup the images and inverse images of i v fuzzy subgroups are defined and how the images or inverse images of i v fuzzy subgroups become i v fuzzy subgroups is studied

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

a new view of fuzzy hypernear rings in this survey article we first introduce the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set this concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we consider the interval valued q fuzzy sub hypernear rings hyperideals of a hypernear ring and hence a generalization of a fuzzy sub near ring ideal is given some related properties of fuzzy hypernear rings are described finally we consider the concept of implication based interval valued fuzzy sub hypernear rings hyperideals in a hypernear ring in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

some types of generalized fuzzy filters of bl algebras the aim of this paper is to introduce the notions of interval valued q fuzzy boolean implicative positive implicative and fantastic filters in bl algebras and to investigate some related properties some characterizations of these generalized fuzzy filters are derived finally the relations among these generalized fuzzy filters are discussed it is proved that an interval valued q fuzzy filter in bl algebras is an interval valued q fuzzy boolean implicative filter if and only if it is both an interval valued q fuzzy positive implicative filter and an interval valued q fuzzy fantastic filter

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

grassmann secants identifiability and linear systems of tensors abstract for any irreducible non degenerate variety x p r we give a criterion for the k s identifiability of x if k s 1 r then the k s identifiability holds for x if and only if the s identifiability holds for the segre product seg p k x moreover if the s th secant variety of x is not defective and it does not fill the ambient space then we can produce a family of pairs k s for which the k s identifiability holds for x

optimization techniques for small matrix multiplication the complexity of matrix multiplication has attracted a lot of attention in the last forty years in this paper instead of considering asymptotic aspects of this problem we are interested in reducing the cost of multiplication for matrices of small size say up to 30 following the previous work of probert fischer smith and mezzarobba in a similar vein we base our approach on the previous algorithms for small matrices due to strassen winograd pan laderman and others and show how to exploit these standard algorithms in an improved way we illustrate the use of our results by generating multiplication codes over various rings such as integers polynomials differential operators and linear recurrence operators

on the asymptotic complexity of rectangular matrix multiplication abstract the numher of essential multiplications required to multiply matrices of size n x n and n x n is studied as a function f s 1 bounds to f x sharper than trivial ones are presented and the asymptotic behawour of f s 1 is studied an analogous investigation is performed for the problem of multiplying matrices of size n x n and n x n 1 introduction and preliminaries the complexity of square n x n matrix multiplication has been investigated by several authors l lo a lower bound to the number of operations is n and the best known upper bound is o n2 16h 8 some results on rectangular matrix multiplication have been obtained by brockett and dobkin a who have shown the upper bound n o n to the non scalar complexity of n x n by n x log n matrix multiplication recently coppersmith s found the upper bound o n log n for the n x n by n x n matrix multipli cation prnblem with n s 2 log 2 5 log 5 0 1722

a network that learns strassen multiplication we study neural networks whose only non linear components are multipliers to test a new training rule in a context where the precise representation of data is paramount these networks are challenged to discover the rules of matrix multiplication given many examples by limiting the number of multipliers the network is forced to discover the strassen multiplication rules this is the mathematical equivalent of finding low rank decompositions of the n n matrix multiplication tensor mn we train these networks with the conservative learning rule which makes minimal changes to the weights so as to give the correct output for each input at the time the input output pair is received conservative learning needs a few thousand examples to find the rank 7 decomposition of m2 and 105 for the rank 23 decomposition of m3 the lowest known high precision is critical especially for m3 to discriminate between true decompositions and border approximations

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

on commutativity and approximation abstract the action of commutativity and approximation is analyzed for some problems in computational complexity lower bound criteria to the approximate complexity are given in terms of border rank and commulative border rank of a given tensor upper bounds for the approximate complexity of the matrix vector product are given in particular 1 2 m n 1 multiplications are necessary and sufficient to approximate n m matrix vector product 6 multiplications are sufficient 5 are needed to approximate a 2 2 matrix product by using commutativity an application to polynomial evaluation shows that 1 2 n 2 multiplications are sufficient to approximate any n degree polynomial at a point for what concerns matrix multiplication complexity a number is introduced such that is the exponent of matrix multiplication complexity this number measures the degree of complexity of the best commutative approximate algorithm for matrix multiplication the bound 2 3211 and conditions under which are shown

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

matrix multiplication a little faster strassen s algorithm 1969 was the first sub cubic matrix multiplication algorithm winograd 1971 improved the leading coefficient of its complexity from 6 to 7 there have been many subsequent asymptotic improvements unfortunately most of these have the disadvantage of very large often gigantic hidden constants consequently strassen winograd s o nlog27 algorithm often outperforms other fast matrix multiplication algorithms for all feasible matrix dimensions the leading coefficient of strassen winograd s algorithm has been generally believed to be optimal for matrix multiplication algorithms with a 2 2 base case due to the lower bounds by probert 1976 and bshouty 1995 surprisingly we obtain a faster matrix multiplication algorithm with the same base case size and asymptotic complexity as strassen winograd s algorithm but with the leading coefficient reduced from 6 to 5 to this end we extend bodrato s 2010 method for matrix squaring and transform matrices to an alternative basis we also prove a generalization of probert s and bshouty s lower bounds that holds under change of basis showing that for matrix multiplication algorithms with a 2 2 base case the leading coefficient of our algorithm cannot be further reduced and is therefore optimal we apply our method to other fast matrix multiplication algorithms improving their arithmetic and communication costs by significant constant factors

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

finite difference scheme with a moving mesh for pricing asian options in this paper we propose a stable numerical method for pricing asian call options which is based on a central difference scheme with a moving mesh in the spatial discretization and the rannacher time stepping scheme in the time discretization at each time mesh point we make a piecewise uniform mesh to discretise the space interval which ensures that the matrix associated with the discrete operator is an m matrix hence the spatial discretization scheme is maximum norm stable for arbitrary volatility and arbitrary interest rate we show that the scheme is second order convergent with respect to both time and spatial variables numerical results support the theoretical results

a fourth order numerical method based on b spline functions for pricing asian options abstract this paper is concerned with the construction of a higher order numerical method for pricing asian options with fixed strike price we first transform the two dimensional partial differential equation governing the value of an asian option into a one dimensional partial differential equation in this method the temporal variable is discretized by means of crank nicolson scheme and the spatial variable is discretized by means of quartic b spline collocation approach the discretization is based on uniform mesh we prove that the method is fourth order convergent with respect to space variable and second order convergent with respect to time variable in the maximum norm stability of the method is analyzed numerical experiment is carried out to demonstrate the applicability and efficiency of the method it is shown that the rates of convergence predicted theoretically are same as that obtained numerically and the method is unconditionally stable the approximate solution corresponding to the present method has been compared with the analytical result moreover our results have been compared with those obtained by other methods the option and delta values for various values of volatilities and interest rates are computed

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

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

analytical pricing of geometric asian power options on an underlying driven by a mixed fractional brownian motion abstract in this paper we study the pricing problem of the continuously monitored fixed and floating strike geometric asian power options in a mixed fractional brownian motion environment first we derive both closed form solutions and mixed fractional partial differential equations for fixed and floating strike geometric asian power options based on delta hedging strategy and partial differential equation method second we present the lower and upper bounds of the prices of fixed and floating strike geometric asian power options under the assumption that both risk free interest rate and volatility are interval numbers finally numerical studies are performed to illustrate the performance of our proposed pricing model

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

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

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

generalized fuzzy hyperideals of hyperrings the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set is considered in fact this is a generalized concept of the quasi coincidence of a fuzzy point with a fuzzy set by using this new idea the notion of interval valued a b fuzzy hyperideals in a hyperring is introduced and consequently a generalization of a fuzzy hyperideal is defined in this paper we study the related properties of the fuzzy hyperideals and in particular the interval valued q fuzzy hyperideals in a hyperring will be investigated moreover we also consider the concept of implication based interval valued fuzzy hyperideals in a hyperring some recent results obtained by b davvaz and j m zhan on hyperstructures are extended and consolidated

the techniques of trilinear aggregating and the recent progress in the asymptotic acceleration of matrix operations abstract the recent progress in the asymptotic acceleration of matrix multiplication and of related matrix operations is surveyed the techniques of trilinear aggregating and their applications to the above problems and to the disproval of the direct sum conjecture over some rings of constants are presented in some detail

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

multiplicative complexity of direct sums of quadratic systems abstract we consider the quadratic complexity of certain sets of quadratic forms we study classes of direct sums of quadratic forms for these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct sum algorithm

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

some kinds of c ccq interval valued fuzzy ideals of bci algebras in this paper we first introduce the notions of positive implicative implicative and commutative interval valued fuzzy ideals of bci algebras which are generalizations of positive implicative implicative and commutative fuzzy ideals respectively and investigate some of their related properties the concept of quasi coincidence of an interval valued fuzzy point within an interval valued fuzzy set is introduced in fact the concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we further introduce the notions of positive implicative implicative and commutative q interval valued fuzzy ideals of bci algebras and investigate some of their related properties some characterization theorems of these generalized interval valued fuzzy ideals are derived the relationship among these generalized interval valued fuzzy ideals of bci algebras is also considered

how can we speed up matrix multiplication due to the new algebraic methods of algorithm design recently it became possible to perform multiplication and inversion of n times n matrices using o n 2 496 rather than o n 3 arithmetical operations consequently algorithms for several other computational problems of linear algebra and combinatorics have been accelerated the major ideas and techniques that have led to that progress are surveyed

fast parallel algorithms for polynomial division over an arbitrary field of constants abstract it is shown that the division of an mth degree polynomial by an nth degree polynomial can be performed over an arbitrary field of constants f involving o log m n 1 loglog m n 1 parallel steps and a polynomial in m number of processors if the field f has more than 2 2 m n 1 min n m n distinct elements then o log m n 1 parallel steps and the polynomial number of processors suffice the values of the constants for these asymptotic bounds are also presented the number of parallel steps can be reduced to o log m n 1 over an arbitrary field using polynomial circuits for the operations in the extended field of constants these methods can be extended to the design of parallel algorithms of the same asymptotic complexity for the inversion of circulant matrices even where the field of constants does not support fft the latter result does not follow from the recent methods by the authors and eberly which otherwise lead to similar results by different means

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

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

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

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

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

trilinear aggregating with implicit canceling for a new acceleration of matrix multiplication abstract a new improvement of author s techniques of trilinear aggregating uniting and canceling is presented and applied to accelerate multiplication of matrices of moderate sizes this results in the exact computing algorithms for n n matrix multiplication in only n 2 1 75 n 2 n 2 4 n 3 3 essential multiplication steps for arbitrary even n also the new techniques allow us to simplify the design of the fastest known apa algorithms for matrix multiplication

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

the s h interval valued fuzzy subgroup abstract in this paper the idempotent interval conorm s h induced by a t conorm on the space of the interval valued fuzzy sets on groups is introduced the s h interval valued fuzzy subgroup is defined in the meantime some of its basic properties and structural characteristics are discussed and studied furthermore the theorems of the homomorphic image and the inverse image are given

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

the bit operation complexity of matrix multiplication and of all pair shortest path problem abstract an n n matrix product can be evaluated with precision e 0 in o n s log m e log log m e log log log m e bit operations 0 arbitrary m the maximum absolute value of the entries of given matrices o n s the arithmetic complexity of n n matrix multiplication s h can be solved in o n s h bit operations

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

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

nonuniform acc circuit lower bounds the class acc consists of circuit families with constant depth over unbounded fan in and or not and modm gates where m 1 is an arbitrary constant we prove the following nexp the class of languages accepted in nondeterministic exponential time does not have nonuniform acc circuits of polynomial size the size lower bound can be slightly strengthened to quasipolynomials and other less natural functions enp the class of languages recognized in 2o n time with an np oracle doesn t have nonuniform acc circuits of 2no 1 size the lower bound gives an exponential size depth tradeoff for every d m there is a 0 such that enp doesn t have depth d acc circuits of size 2n with modm gates previously it was not known whether expnp had depth 3 polynomial size circuits made out of only mod6 gates the high level strategy is to design faster algorithms for the circuit satisfiability problem over acc circuits then prove that such algorithms entail these lower bounds the algorithms combine known properties of acc with fast rectangular matrix multiplication and dynamic programming while the second step requires a strengthening of the author s prior work

rapid multiplication of rectangular matrices the number of essential multiplications required to multiply matrices of size n times n and n times n 0 172 is bounded by cn 2 log 2 n

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

a reliable numerical method to price arithmetic asian options abstract we design and analyze two numerical methods for pricing asian options the first one is an explicit finite difference method and therefore as usual only conditionally stable the second method is an implicit finite difference method and unconditionally stable to explore the basic ideas of analysis we discuss the explicit method in detail and then highlight the crucial steps in the analysis of the implicit method numerical results obtained by these two methods are compared with those obtained by an improved monte carlo method the proposed implicit method is very robust as is evident from the comparative numerical results we also provide additional numerical results that confirm theoretical investigations done by many other researchers

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

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