on the modular curves y e 7 let e denote an elliptic curve over q and y e 7 the modular curve classifying the elliptic curves e over q such that the representations of gal q q in the 7 torsion points of e and of e are symplectically isomorphic in case e is given by a weierstrass equation such that the c 4 invariant is a square we exhibit here nontrivial points of y e 7 q from this we deduce an infinite family of curves e for which y e 7 q has at least four nontrivial points

perfect powers expressible as sums of two cubes abstract let n 3 this paper is concerned with the equation a 3 b 3 c n which we attack using a combination of the modular approach via frey curves and galois representations with obstructions to the solutions that are of brauer manin type we shall show that there are no solutions in coprime non zero integers a b c for a set of prime exponents n having dirichlet density 28 219 44 928 0 628 and for a set of exponents n having natural density 1

on the uniqueness of solutions to linear complementarity problems this paper characterizes the classu of all realn n matricesm for which the linear complementarity problem q m has a unique solution for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all it is shown that restricting the uniqueness property to the interior ofk m is necessary because whenu the problem q m has infinitely many solutions ifq belongs to the boundary of intk m it is shown thatm must have nonnegative principal minors whenu andk m is convex finally it is shown that whenm has nonnegative principal minors only one of which is 0 andk m r n thenu andk m is a closed half space

the hessian of a genus one curve we continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve x n for n 2 3 4 5 our construction makes use of a covariant we call the hessian generalising the classical hessian that exists in degrees 2 and 3 in particular we give explicit formulae and algorithms for computing the hessian in degrees 4 and 5 this leads to a practical algorithm for computing equations for visible elements of order n in the tate shafarevich group of an elliptic curve taking jacobians we also recover the formulae of rubin and silverberg for families of n congruent elliptic curves

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

criteria for sufficient matrices abstract column sufficient row sufficient and sufficient matrices have recently arisen in connection with the linear complementarity problem we review and supplement the basic theory of these matrix classes propose new criteria for identifying them and compare these criteria with the existing ones our main mathematical tool is principal pivoting

fully copositive matrices the class of fully copositive c 0 f matrices introduced in g s r murthy t parthasarathy siam journal on matrix analysis and applications 16 4 1995 1268 1286 is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices it is shown that fully copositive matrices within the class ofq 0 matrices arep 0 matrices as a corollary of this main result we establish that a bisymmetricq 0 matrix is positive semidefinite if and only if it is fully copositive another important result of the paper is a constructive characterization ofq 0 matrices within the class ofc 0 f while establishing this characterization it will be shown that graves s principal pivoting method of solving linear complementarity problems lcps with positive semidefinite matrices is also applicable toc 0 f q 0 class as a byproduct of this characterization we observe that ac 0 f matrix is inq 0 if and only if it is completelyq 0 also from aganagic and cottle s m aganagic r w cottle mathematical programming 37 1987 223 231 result it is observed that lcps arising fromc 0 f q 0 class can be processed by lemke s algorithm 1998 the mathematical programming society inc published by elsevier science b v

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

almost copositive matrices abstract a real n n matrix is termed almost copositive if it is not copositive but all its principal submatrices of order n 1 are a real copositive n n matrix is called almost copositive plus almost strictly copositive if it is not copositive plus strictly copositive but all its principal submatrices of order n 1 are such matrices are of crucial importance in deriving criteria for copositivity the properties of these matrix classes are investigated the emphasis being on the less known class of almost copositive plus matrices the basic mathematical tools used are principal pivoting and quadratic programming

torsion points on curves and p adic abelian integrals theorem a let f c j be an albanese morphism defined over a number field k of a s mooth curve of genus g into its jacobian suppose j has potential complex multiplication let s denote the set of primes p of k satisfying i p does not divide 2 or 3 ii k q is unramified at p iii c has good ordinary reduction over k then the set t of torsion points of a defined over an algebraic closure of k which lie on the image of c is defined over an extension of k unramified above s moreover to pg

a partial characterization of a class of matrices defined by solutions to the linear complementarity problem in this paper we analyze the class of matrices q0 characterized as all n n matrices m for which the linear complementarity problem w mz q w z 0 w z 0 has a solution whenever q is feasible the method is to analyze the faces of the linear cone cone i m where i is the n n identity essentially a labelling requirement for the faces is shown necessary for m to be in q0 with proper restrictions and nondegenerate assumptions sufficiency is also shown these results arc related to the signs of the minors of m and some conjectures in this area are also discussed

an example of a nonregular semimonotone q matrix in their paper aganagic and cottle gave necessary and sufficient conditions for ap 0 matrix to be aq matrix in his paper pang showed that the same characterization holds for anl matrix in this paper we show that a similar characterization for ane 0 matrix semimonotone orl 1 matrix is not possible this is done by providing a counterexample to the inclusione 0 q r 0 thus answering in the negative a question first posed by pang

the basic theorem of complementarity revisited the basic theorm of linear complementarity was stated in a 1971 paper 6 by b c eaves who credited c e lemke for giving a constructive proof based on his almost complementary pivot algorithm this theorem asserts that associated with an arbitrary linear complementarity problem a certain augmented problem always possesses a solution many well known existence results pertaining to the linear complementarity problem are consequences of this fundamental theorem in this paper we explore some further implications of the basic theorem of complementarity and derive new existence results for the linear complementarity problem based on these results conditions for the existence of a solution to a linear complementarity problem with a fully semimonotone matrix are examined the class of the linear complementarity problems with ag matrix is also investigated

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

modular diagonal quotient surfaces the main objective of this paper is to analyze the geometry of the modular diagonal quotient surface zn e e x n x n which classifies pairs of elliptic curves e1 and e2 together with an isomorphism of determinant e between their associated modular representations mod n in particular we calculate some of the numerical invariants of its minimal desingularization zn e such as its betti and chern numbers and determine its place in the enriques kodaira classification table

on hidden z matrices and the linear complementarity problem abstract in this paper we explore various matrix theoretic aspects of the hidden z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results in fact we revisit various results obtained for hidden z class by mangasarian cottle and pang in context of solving linear complementarity problems as linear programs we identify hidden z matrices of special category and discuss the number of solutions of the associated linear complementarity problems we also present game theoretic interpretation of various results related to hidden z class and obtain proofs following the game theoretic approach of raghavan for a subclass of z matrices

on singular n0 matrices and the class q abstract the class of n and n 0 matrices arises in the theory of global univalence of functions multivariate analysis and in complementarity theory in this paper we study singular n 0 matrices with q property

invariant theory for the elliptic normal quintic i twists of x 5 a genus one curve of degree 5 is defined by the 4 times 4 pfaffians of a 5 times 5 alternating matrix of linear forms on mathbb p 4 we describe a general method for investigating the invariant theory of such models we use it to explain how we found our algorithm for computing the invariants and to extend our method for computing equations for visible elements of order 5 in the tate shafarevich group of an elliptic curve as a special case of the latter we find a formula for the family of elliptic curves 5 congruent to a given elliptic curve in the case the 5 congruence does not respect the weil pairing we also give an algorithm for doubling elements in the 5 selmer group of an elliptic curve and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree

on pseudomatroid property of matrices abstract a necessary and sufficient condition for a matrix to have pseudomatroid property is presented pseudosymmetric matrices which possess pseudomatroid property are introduced as a generalization of quasisymmetric matrices considered by a bouchet a duchamp linear algebra and appl 146 1991 67 78 principal unimodularity and perfect balancedness of these matrices are investigated a result on nonbinary pseudomatroids is presented in the end

the invariants of a genus one curve it was first pointed out by weil that we can use classical invariant theory to compute the jacobian of a genus one curve the invariants required for curves of degree n 2 3 4 were already known to the nineteenth century invariant theorists we have succeeded in extending these methods to curves of degree n 5 where although the invariants are too large to write down as explicit polynomials we have found a practical algorithm for evaluating them

more on positive subdefinite matrices and the linear complementarity problem abstract in this article we consider positive subdefinite matrices psbd recently studied by j p crouzeix et al siam j matrix anal appl 22 2000 66 and show that linear complementarity problems with psbd matrices of rank 2 are processable by lemke s algorithm and that a psbd matrix of rank 2 belongs to the class of sufficient matrices introduced by r w cottle et al linear algebra appl 114 115 1989 231 we also show that if a matrix a is a sum of a merely positive subdefinite copositive plus matrix and a copositive matrix and a feasibility condition is satisfied then lemke s algorithm solves lcp q a this supplements the results of jones and evers

integral solutions of linear complementarity problems we characterize the class of integral square matrices m having the property that for every integral vector q the linear complementarity problem with data m q has only integral basic solutions these matrices called principally unimodular matrices are those for which every principal nonsingular submatrix is unimodular as a consequence we show that if m is rank symmetric and principally unimodular and q is integral then the problem has an integral solution if it has a solution principal unimodularity can be regarded as an extension of total unimodularity and our results can be regarded as extensions of well known results on integral solutions to linear programs we summarize what is known about principally unimodular symmetric and skew symmetric matrices

explicit n descent on elliptic curves i algebra this is the first in a series of papers in which we study the n selmer group of an elliptic curve with the aim of representing its elements as genus one normal curves of degree n the methods we describe are practical in the case n 3 for elliptic curves over the rationals and have been implemented in magma

two characterizations of sufficient matrices abstract two characterizations are given for the class of sufficient matrices defined by cottle pang and venkateswaran the first is a direct translation of the definition into linear programming terms the second can be thought of as a generalization of a theorem of t d parsons on p matrices

on solution stability of the linear complementarity problem in the paper 5 c d ha introduced the notion of stability of a linear complementarity problem at a solution point and established several sufficient conditions for stability to hold in the present paper we derive some new stability results which significantly improve ha s results included in these new results are i a complete characterization of the stability condition under a nonsingularity assumption and ii further characterizations of solution stability in the case of a linear complementarity problem with a fully semimonotone matrix

visualising sha 2 in abelian surfaces given an elliptic curve e 1 over a number field k and an element s in its 2 selmer group we give two different ways to construct infinitely many abelian surfaces a such that the homogeneous space representing s occurs as a fibre of a over another elliptic curve e 2 we show that by comparing the 2 selmer groups of e 1 e 2 and a we can obtain information about iii e 1 k 2 and we give examples where we use this to obtain a sharp bound on the mordell weil rank of an elliptic curve as a tool we give a precise description of the m selmer group of an abelian surface a that is m isogenous to a product of elliptic curves e 1 x e 2 one of the constructions can be applied iteratively to obtain information about iii e 1 k 2 n we give an example where we use this iterated application to exhibit an element of order 4 in iii e 1 q

empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves this paper provides empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves the second of these conjectures relates six quantities associated to a jacobian over the rational numbers one of these six quantities is the size of the shafarevich tate group unable to compute that we computed the five other quantities and solved for the last one in all 32 cases the result is very close to an integer that is a power of 2 in addition this power of 2 agrees with the size of the 2 torsion of the shafarevich tate group which we could compute

computing twists of hyperelliptic curves abstract we give an efficient algorithm to compute equations of twists of hyperelliptic curves c of arbitrary genus over any perfect field k of characteristic different from 2 starting with a cocycle in h 1 gal k k aut c we also discuss some interesting examples

visible evidence for the birch and swinnerton dyer conjecture for modular abelian varieties of analytic rank zero this paper provides evidence for the birch and swinnerton dyer conjecture for analytic rank 0 abelian varieties a f that are optimal quotients of j 0 n attached to newforms we prove theorems about the ratio l a f 1 af develop tools for computing with af and gather data about certain arithmetic invariants of the nearly 20 000 abelian varieties a f of level 2333 over half of these a f have analytic rank 0 and for these we compute upper and lower bounds on the conjectural order of iii a f we find that there are at least 168 such af for which the birch and swinnerton dyer conjecture implies that iii a f is divisible by an odd prime and we prove for 37 of these that the odd part of the conjectural order of iii a f really divides iii a f by constructing nontrivial elements of iii a f using visibility theory we also give other evidence for the conjecture the appendix by cremona and mazur fills in some gaps in the theoretical discussion in their paper on visibility of shafarevich tate groups of elliptic curves

how to do a p descent on an elliptic curve in this paper we describe an algorithm that reduces the computation of the full p selmer group of an elliptic curve e over a number field to standard number field computations such as determining the p torsion of the s class group and a basis of the s units modulo pth powers for a suitable set s of primes in particular we give a result reducing this set s of bad primes to a very small set which in many cases only contains the primes above p as of today this provides a feasible algorithm for performing a full 3 descent on an elliptic curve over q but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory when the galois module structure of e p is favorable simplifications are possible and p descents for larger p are accessible even today to demonstrate how the method works several worked examples are included

on a subclass of p0 abstract in 1966 fiedler and ptak wrote the first systematic investigation of the matrix class p 0 consisting of all real square matrices with nonnegative principal minors in this note we focus on a particular subclass of p 0 that has arisen within the field of linear complementarity theory namely the class su of sufficient matrices our principal result is that su contains another class p we conjecture that the latter two classes are in fact the same

total dual integrality and integral solutions of the linear complementarity problem abstract this paper deals with the problem of finding an integer solution to a linear complementarity problem lcp chandrasekaran et al 1 introduced the class i of integral matrices for which the corresponding lcp has an integer solution for every integral vector q for which it has a solution and proved that for some well known matrix classes principal unimodularity forms a necessary and sufficient condition for inclusion in the class i in this paper we identify some more well known matrix classes for which principal unimodularity forms a necessary and sufficient condition for inclusion in the class i the concept of total dual integrality is utilized to obtain a necessary and sufficient condition for existence of an integer solution to lcp with a hidden k matrix we interconnect the concept of hilbert basis with principal unimodularity of a matrix and the corresponding complementary cones a necessary and sufficient condition is given for the existence of an integer solution of a linear fractional programming problem by using its lcp formulation

on the classes of fully copositive and fully semimonotone matrices abstract in this paper we consider the class c 0 f of fully copositive and the class e 0 f of fully semimonotone matrices we show that c 0 f matrices with positive diagonal entries are column sufficient we settle a conjecture made by murthy and parthasarathy to the effect that a c 0 f q 0 matrix is positive semidefinite by providing a counterexample we finally consider e 0 f matrices introduced by cottle and stone math program 27 1983 191 213 and partially address stone s conjecture to the effect that e 0 f q 0 p 0 by showing that e 0 f d c matrices are p 0 where d c is the doverspike class of matrices

two cover descent on hyperelliptic curves we describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve with the property that any rational point will lift to one of the covers in particular if the algorithm returns an empty set then the hyperelliptic curve has no rational points this provides a relatively efficiently tested criterion for solvability of hyperelliptic curves we also discuss applications of this algorithm to curves of genus 1 and to curves with rational points

structure properties of w matrices abstract we continue an investigation of the class w of real square matrices a matrix belongs to w if and only if certain pairs of its complementary cones intersect in the zero vector only we show that a w matrix with no zero diagonal entries is a p 0 matrix each principal minor is nonnegative and a w matrix with no nonzero diagonal entries is a nonnegative matrix which either is monomial or has a zero column lastly we show how these two results impose certain necessary conditions on the structure of a w matrix

explicit moduli spaces for congruences of elliptic curves we determine explicit birational models over mathbb q for the modular surfaces parametrising pairs of n congruent elliptic curves in all cases where this surface is an elliptic surface in each case we also determine the rank of the mordell weil lattice and the geometric picard number

on the reduction theory of binary forms cremona developed a reduction theory for binary forms of degree 3 and 4 with integer coefficients the motivation in the case of quartics being to improve 2 descent algorithms for elliptic curves over q in this paper we extend some of these results to forms of higher degree one application of this is to the study of hyperelliptic curves

the linear complementarity problem and a subclass of fully semimonotone matrices abstract in this paper we investigate a subclass w of the n n real matrices a matrix m belongs to w if and only if certain pairs of its complementary cones intersect only in the zero vector we show that w u where u is the class of matrices m for which lcp q m the linear complementarity problem has a unique solution whenever q belongs in the interior of the union of its complementary cones membership in w can be established by solving 2n 1 systems of linear equations finally several sufficient conditions are given on a matrix in w so that it possesses nonnegative principal minors

splitting families of sets in zfc abstract miller s 1937 splitting theorem was proved for every finite n 0 for all uniform families of sets in which is infinite a simple method for proving miller type splitting theorems is presented here and an extension of miller s theorem is proved in zfc for every cardinal for all uniform families in which the main ingredient in the method is an asymptotic infinitary lowenheim skolem theorem for anti monotone set functions as corollaries the use of additional axioms is eliminated from splitting theorems due to erd s and hajnal 1 komjath 7 hajnal juhasz and shelah 4 upper bounds are set on conflict free coloring numbers of families of sets and a general comparison theorem for uniform families of sets is proved which generalizes komjath s comparison theorem for 0 uniform families 8

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

the generalized continuum hypothesis can fail everywhere in 1874 cantor cl showed that every set has cardinality strictly smaller than the cardinality of its power set cantor asked c2 whether for infinite sets x there is a set y of cardinality strictly between cardinality x and cardinality 2x the special case of x z the integers was hilbert s first problem in his famous list hi the continuum hypothesis in this paper we show that it is consistent with zermelo frankel set theory with the full axiom of choice modulo large cardinals that for every set x there is a set y such that the cardinality of y lies strictly between the cardinality of x and the cardinality of the power set of x it was previously shown in gcdel g that the generalized continuum hypothesis g c h was consistent i e for every infinite cardinal x the cardinal successor to x was 2x in 1963 cohen co showed that it was consistent that the continuum hypothesis failed easton e showed that subject to relatively mild restrictions kdnig s theorem essentially arbitrary behavior of the power set operation could occur at regular cardinals singular cardinals presented a significantly more difficult matter the first work on them was done by prikry and silver p and sil who showed that the g c h can fail at singular strong limit cardinals magidor ml showed that it was consistent that it fail at the first singular strong limit and even that one could have the first failure of the g c h at a singular strong limit after magidor s work it was generally believed that arbitrary behavior was possible silver however showed si that if the g c h holds below a singular cardinal k of uncountable cofinality then it holds at k galvin and hajnal g h showed that under more general conditions the behavior of the power set below a singular cardinal k of uncountable cofinality strongly affects the power set at k these results are all local results the consistent global behavior of the power set operation was not settled we prove the following theorem

almost p 0 matrices and the class q this paper demonstrates that within the class of thosen n real matrices each of which has a negative determinant nonnegative proper principal minors and inverse with at least one positive entry the class ofq matrices coincides with the class of regular matrices each of these classes of matrices plays an important role in the theory of the linear complementarity problem lastly analogous results are obtained for nonsingular matrices which possess only nonpositive principal minors

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

some lcps solvable in strongly polynomial time with lemke s algorithm we identify a class of linear complementarity problems lcps that are solvable in strongly polynomial time by lemke s algorithm scheme 1 or by the parametric principal pivoting method pppm this algorithmic feature for the class of problems under consideration here is attributable to the proper selection of the covering vector in scheme 1 or the parametric direction vector in the pppm which leads to solutions of limited and monotonically increasing support size such solutions are sparse these and other lcps may very well have multiple solutions many of which are unattainable by either algorithm and thus are said to be elusive the initial conditions imposed on the new matrix class identified in sect 2 are subsequently relaxed in later sections

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

notes on sufficient matrices abstract we give some constructive characterizations for sufficient matrices and present some matrix classes whose elements are identified as column sufficient matrices in particular we give the structure of column row sufficient matrices which leads to efficient testing methods for column row sufficiency

linear complementarity problems with an invariant number of solutions this paper studies the class ins of all realn n matricesm for which the linear complementarity problem q m has exactlyk solutions k depending only onm for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all this generalizes the results in cottle and stone 1983 which deal with the subclassu in ins wherek equals one

on some classes of linear complementarity problems with matrices of order n and rank n 1 in this paper we generalize a fundamental result observed by b c eaves on the nature of solutions constructed by lemke s algorithm applied to solve the linear complementarity problem q m initiating it with the positive vector d rn we use this generalization and identify two subclasses of the class of n n matrices m of rank n 1 for which the set of q rn such that the lcp q m has a solution is convex we also present algorithms to compute a solution to q m if one exists for these classes of matrices