fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

on the pointwise limits of bivariate lagrange projectors a linear algebra proof is given of the fact that the nullspace of a finite rank linear projector on polynomials in two complex variables is an ideal if and only if the projector is the bounded pointwise limit of lagrange projectors i e projectors whose nullspace is a radical ideal i e the set of all polynomials that vanish on a certain given finite set a characterization of such projectors is also given in the real case more generally a characterization is given of those finite rank linear projectors on polynomials in d complex variables with nullspace an ideal that are the bounded pointwise limit of lagrange projectors the characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences

commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

isomorphism classes of short gorenstein local rings via macaulay s inverse system in this paper we study the isomorphism classes of artinian gorenstein local rings with socle degree three by means of macaulay s inverse system we prove that their classification is equivalent to the projective classification of the hypersurfaces of p n of degree three this is an unexpected result because it reduces the study of this class of local rings to the homogeneous case the result has applications in problems concerning the punctual hilbert scheme hilbd p n and

eigenvalue based algorithms for testing positive realness of siso systems a computational test for scalar positive real and strictly positive real functions is proposed in this paper the input for this algorithm is state space representation not necessarily minimal it is derived for the discrete time problem based on numerical stable eigenvalue computations the test can be used in the continuous counterpart which is in fact an improvement of an algorithm posed by bai and freund differences between the discrete and continuous time problems are also pointed out

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

approximating commuting operators the problem of approximating m tuples of commuting n n complex matrices by commuting m tuples of generic matrices is studied we narrow the gap for commuting triples by showing that they can be perturbed if n 29

a general approach for optimizing regular criteria in the job shop scheduling problem even though a very large number of solution methods has been developed for the job shop scheduling problem a majority has been designed for the makespan criterion in this paper we propose a general approach for optimizing any regular criterion in the job shop scheduling problem the approach is a local search method that uses a disjunctive graph model and neighborhoods generated by swapping critical arcs the connectivity property of the neighborhood structure is proved and a novel efficient method for evaluating moves is presented besides its generality another prominent advantage of the proposed approach is its simple implementation that only requires to tune the range of one parameter extensive computational experiments carried out on various criteria makespan total weighted flow time total weighted tardiness weighted sum of tardy jobs maximum tardiness show the efficiency of the proposed approach best results were obtained for some problem instances taken from the literature

on approximately simultaneously diagonalizable matrices abstract a collection a1 a2 ak of n n matrices over the complex numbers c has the asd property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable such a collection must perforce be commuting we show by a direct matrix proof that the asd property holds for three commuting matrices when one of them is 2 regular dimension of eigenspaces is at most 2 corollaries include results of gerstenhaber and neubauer sethuraman on bounds for the dimension of the algebra generated by a1 a2 ak even when the asd property fails our techniques can produce a good bound on the dimension of this subalgebra for example we establish dim c a 1 a k 5 n 4 for commuting matrices a1 ak when one of them is 2 regular this bound is sharp one offshoot of our work is the introduction of a new canonical form the h form for matrices over an algebraically closed field the h form of a matrix is a sparse jordan like upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular the jordan form itself does not accommodate this

computing point enclosures given a set of n rectangles in the plane the point enclosure query is the question to determine for any point p which rectangles of the set it is contained in it is the dual of the well known range query in computational geometry it is shown that the point enclosure query in the plane can be answered in 0 log n k time where k is the number of rectangles reported the solution makes use of a new data structure called the s tree the data structure can be generalized to obtain an efficient algorithm for the point enclosure problem in d dimensional space d 2

witness sets of projections abstract elimination is a basic algebraic operation which geometrically corresponds to projections this article describes using the numerical algebraic geometric concept of witness sets to compute the projection of an algebraic set the ideas described in this article apply to computing the image of an algebraic set under any linear map

polynomials in a matrix and its commutant abstract given any norm u on the space of linear transformations matrices over a finite dimensional complex vector space and given commuting transformations t s and any e 0 there is a transformation t e such that u t e t p e such that p e t e s this result has implications for the theory of the numerical radius and other operator radii its proof involves the following fact of interest in itself if the jordan canonical form of s has k blocks then the transformations with k distinct eigenvalues are dense in the commutant of s

on the irreducibility of commuting varieties of nilpotent matrices abstract given an n n nilpotent matrix over an algebraically closed field k we prove some properties of the set of all the n n nilpotent matrices over k which commute with it then we give a proof of the irreducibility of the variety of all the pairs a b of n n nilpotent matrices over k such that a b 0 if either char k 0 or char k n 2 we get as a consequence a proof of the irreducibility of the local hilbert scheme of n points of a smooth algebraic surface over k if either char k 0 or char k n 2

the sliding method for the nonlocal monge ampere operator abstract we develop a sliding method for the nonlocal monge ampere operator we first establish a narrow region principle in bounded domains which plays an important role in the sliding method then we consider the equation with the nonlocal monge ampere operator and derive the monotonicity of solutions in both bounded domains and the whole space we use a new idea estimating the singular integrals defining the nonlocal monge ampere operator along a sequence of approximate maximum points

free extensions and jordan type abstract free extensions of graded artinian algebras were introduced by t harima and j watanabe and were shown to preserve the strong lefschetz property the jordan type of a multiplication map m by a nilpotent element of an artinian algebra is the partition determining the sizes of the blocks in a jordan matrix for m we show that a free extension c of the artinian algebra a with fiber b is a deformation of the usual tensor product this has consequences for the generic jordan types of a b and c we show that the jordan type of c is at least that of the usual tensor product in the dominance order theorem 2 5 in particular this gives a different proof of the t harima and j watanabe result concerning the strong lefschetz property of a free extension examples illustrate that a non strong lefschetz graded gorenstein algebra a with non unimodal hilbert function may nevertheless have a non homogeneous element with strong lefschetz jordan type and may have an a free extension that is strong lefschetz we apply these results to algebras of relative coinvariants of linear group actions on a polynomial ring

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

two generated commutative subalgebras of mn f abstract for two commuting matrices a b m n k k a field it has been known that dim k k a b n we show here that dim k cent m n k a b n and that equality for one implies equality for the other we also give both a module theoretic and a geometric characterization of those algebras for which equality is obtained furthermore we present a new proof of a result of gerstenhaber

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

a variety of commuting triples abstract the problem of approximating triples of commuting n n matrices by generic matrices is equivalent to the problem of whether the variety c 3 n of commuting triples is irreducible the answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well all the proofs are given over an arbitrary algebraically closed field of characteristic zero

jordan forms for mutually annihilating nilpotent pairs abstract in this paper we completely characterize all possible pairs of jordan canonical forms for mutually annihilating nilpotent pairs i e pairs a b of nilpotent matrices such that ab ba 0

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

monotonicity of positive solutions for fractional p systems in unbounded lipschitz domains abstract in this paper without imposing any decay assumptions on the solutions at infinity we first develop an unbounded narrow region principle for the fractional p laplacian systems an essential ingredient to carry on the direct method of moving planes then we combine this direct method with the sliding method to obtain the monotonicity of positive bounded solutions for cooperative systems involving the fractional p laplacian in unbounded lipschitz domains we believe that the new ideas introduced here will become useful tools in studying qualitative properties of solutions for other types of nonlinear nonlocal systems in various unbounded domains

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

vector bundles give equations of cactus varieties abstract we prove that vector bundles give equations of cactus varieties we derive from it that equations coming from vector bundles are not enough to define secant varieties of veronese varieties in general

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

on fractional elliptic equations in lipschitz sets and epigraphs regularity monotonicity and rigidity results we consider a nonlocal equation set in an unbounded domain with the epigraph property we prove symmetry monotonicity and rigidity results in particular we deal with halfspaces coercive epigraphs and epigraphs that are flat at infinity these results can be seen as the nonlocal counterpart of the celebrated article berestycki et al commun pure appl math 50 11 1089 1111 1997

a nonlocal anisotropic model for phase transitions where u is a scalar density function on a domain of r and takes values in 1 1 w is a positive double well potential which vanishes at 1 and j is a positive possibly anisotropic interaction potential which vanishes at infinity see paragraph 1 2 for precise definitions the scalar function u represents the macroscopic density profile of a system which has two equilibrium pure phases described by the profiles u 1 and u 1 the integral w u at the right side of 1 1 forces a minimizer of f to take values close to 1 and 1 phase separation while the double integral represents an interaction energy which penalizes the spatial inhomogeneity of the system surface tension in equilibrium statistical mechanics functionals of the form 1 1 arise as free energies of continuum limits of ising spin systems on lattices in this setting u plays the role of a macroscopic magnetization density and j is a ferromagnetic kac potential see for instance 2 and references therein we underline the analogy with the more familiar gradient theory for phase transition proposed in 9 where the free energy of the system is of the form

root vectors for geometrically simple two parameter eigenvalues a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied bases for lower order root subspaces at geometrically simple eigenvalues of fredholm type of arbitrary finite index are computed in terms of the underlying multiparameter system

on varieties of commuting triples abstract the problem of approximating triples of commuting n n matrices by triples of generic matrices is equivalent to that whether the variety c 3 n of triples of commuting matrices is irreducible it is known that the variety is irreducible for n 7 and reducible for n 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that c 3 8 is irreducible

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

novel transformation techniques using q heaps with applications to computational geometry using the notions of q heaps and fusion trees developed by fredman and willard we develop general transformation techniques to reduce a number of computational geometry problems to their special versions in partially ranked spaces in particular we develop a fast fractional cascading technique which uses linear space and enables sublogarithmic iterative search on catalog trees in the case when the degree of each node is bounded by o log epsilon n for some constant epsilon 0 where n is the total size of all the lists stored in the tree we apply the fast fractional cascading technique in combination with the other techniques to derive the first linear space sublogarithmic time algorithms for two fundamental geometric retrieval problems orthogonal segment intersection and rectangular point enclosure

non local gradient dependent operators abstract in this paper we study a general class of quasilinear non local equations depending on the gradient which arises from tug of war games we establish a c c 1 c 2 regularity theory for these equations the kind of regularity depending on the assumptions on the kernel and we construct different non local approximations of the p laplacian

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

nilpotent matrices having a given jordan type as maximum commuting nilpotent orbit abstract the jordan type of a nilpotent matrix is the partition giving the sizes of its jordan blocks we study pairs of partitions p q where q q p is the jordan type of a generic nilpotent matrix a commuting with a nilpotent matrix b of jordan type p t kosir and p oblak have shown that q has parts that differ pairwise by at least two such partitions which are also known as super distinct or rogers ramanujan are exactly those that are stable or self large in the sense that q q q in 2012 p oblak formulated a conjecture concerning the cardinality of q 1 q when q has two parts and proved some special cases r zhao refined this to posit that the partitions in q 1 q for q u u r with u r 1 could be arranged in an r 1 u r table t q where the entry in the k th row and l th column has k l parts we prove this table theorem and then generalize the statement to propose a box conjecture for the set of partitions q 1 q for an arbitrary partition q whose parts differ pairwise by at least two

efficient data structures for adaptive remeshing with the fem abstract the meshing and interpolation problems encountered during adaptive remeshing on unstructured finite element grids are analyzed both problems are solved independently in 2d and 3d with cost efficient data structures issued from recent advances in computational geometry implementation algorithms and timings are presented in 2d the performance of commonly used schemes is also reviewed

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

dynamization of decomposable searching problems yielding good worsts case bounds a technique is described for dynamizing data structures of common types of decomposable searching problems yielding good worst case bounds rather than merely good average bounds on all operations to be performed on the resulting structures

maximum principles for the fractional p laplacian and symmetry of solutions abstract in this paper we consider nonlinear equations involving the fractional p laplacian p s u x c n s p p v r n u x u y p 2 u x u y x y n s p d y f x u we prove a maximum principle for anti symmetric functions and obtain other key ingredients for carrying on the method of moving planes such as a variant of the hopf lemma a boundary estimate lemma which plays the role of the narrow region principle then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p laplacian in a unit ball and in the whole space we believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators

commuting pairs in the centralizers of 2 regular matrices inmn k kan algebraically closed field we call a matrixl regular if each eigenspace is at mostl dimensional we prove that the variety of commuting pairs in the centralizer of a 2 regular matrix is the direct product of various affine spaces and various determinantal varieties zl mobtained from matrices over truncated polynomial rings we prove that these varieties zl mare irreducible and apply this to the case of thek algebra generated by three commuting matrices we show that if one of the three matrices is 2 regular then the algebra has dimension at mostn we also show that such an algebra is always contained in a commutative subalgebra ofmn k of dimension exactlyn

on stability of invariant subspaces of commuting matrices we study the stability of joint invariant subspaces of a finite set of commuting matrices we generalize some of the results of gohberg lancaster and rodman for the single matrix case for sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case we show that each root subspace is a stable invariant subspace that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable and that even in the derogatory case the eigenspace is stable if it is one dimensional we prove that a pair of commuting matrices has only finitely many stable invariant subspaces at the end we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem

k3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds the main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the fano variety of lines of a cubic 4 fold f in general different from f z f a general k3 surface s of genus 8 determines uniquely a pair of cubic 4folds the apolar cubic f s and the dual pfaffian cubic f s or for simplicity f and f as beauville and donagi have shown the fano variety tf of lines on the cubic f is isomorphic to the hilbert scheme hilb2 s of length two subschemes of s the first main result of this paper is that hilb2 s parametrizes the variety vsp f 10 of presentations of the cubic form f with f z f as a sum of 10 cubes which yields an isomorphism between nff and vsp f 10 furthermore we show that vsp f 10 sets up a 6 10 correspondence between f and ff the main result follows by a deformation argument 1 pfaffian and apolar cubic 4 folds associated to k3 surfaces of genus 8 1 1 let v be a 6 dimensional vector space over c fix a basis e e5 for v then ei a ej for 0 i j aijei aej e a2v a skewsymmetric matrix m g aij with aji aj with pliicker coordinates xij the embedding of the grassmannian g g 2 v in p14 p a2v is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices 0 x01 x02 x03 x04 x05 x0i 0 x12 x13 x14 x15 m x02 x12 0 x23 x24 x25 x03 x13 x23 0 x34 x35 x04 x14 x24 x34 0 x45 x05 x15 x25 x35 x45 0 since the sum of two rank 2 matrices has rank at most 4 and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices the secant variety of g is the cubic hypersurface k defined by the 6 x 6 pfaffian m of the matrix m the dual variety of g in p14 p a2v is a cubic hypersurface k k cf 10 k is the secant variety of g g v 2 the grassmannian of rank 2 quotient spaces of v and of course g c g received by the editors july 5 1999 2000 mathematics subject classification primary 14j70 secondary 14m15 14n99 2000 american mathematical society

counting and reporting intersections of d ranges we present a solution to the problem of reporting and counting all pairs of intersecting or overlapping d ranges in a given set of n d ranges a d range is a rectilinearly oriented rectangular d dimensional box or brick the three dimensional case is presented in some detail and an 0 n log2 2 n k time and 0 n log2 2 n space algorithm is developed for the reporting problem where k is the number of reported intersecting pairs this gives an 0 n log2 2 n time and 0 n log 2 n space algorithm for the counting problem apart from its usefulness wherever a bounding boxes technique is used the development of the algorithm is of interest in its own right because of the data structures that are introduced and because of the technique used to reduce the time needed for preprocessing finally the algorithm is generalized to give an 0 2 d 1 n log 2 d n k time and an 0 2d 1n log 2 d n space algorithm for the d range intersection reporting problem for d 2 which again reduces to an 0 2d 1n log 2 d 1n time and 0 2d 1n log 2 d 2n space algorithm for the counting problem finally we sketch how the reporting problem can be carried out with the same space requirements as the counting problem the bounds presented are the best known to date

a groebner basis for the 2 2 determinantal ideal mod t2 abstract in an earlier paper t kosir b a sethuraman determinantal varieties over truncated polynomial rings j pure appl algebra 195 2005 75 95 we had begun a study of the components and dimensions of the spaces of k 1 th order jets of the classical determinantal varieties these are the varieties z r k m n obtained by considering generic m n m n matrices over rings of the form f t t k and for some fixed r setting the coefficients of powers of t of all r r minors to zero in this paper we consider the case where r k 2 and provide a groebner basis for the ideal i 2 2 m n which defines the tangent bundle to the classical 2 2 determinantal variety we use the results of these groebner basis calculations to describe the components of the varieties z r 4 m n where r is arbitrary the components of z r 2 m n and z r 3 m n were already described in the above cited paper

the sliding methods for the fractional p laplacian abstract in this paper we develop a sliding method for the fractional p laplacian we first obtain the key ingredient needed in the sliding method in a bounded domain the narrow region principle then using nonlinear equations involving the fractional p laplacian in both bounded domains and in the whole space we illustrate how this new sliding method can be employed to obtain monotonicity of solutions during these processes we introduce a new idea estimating the singular integrals defining the fractional p laplacian along a sequence of approximate maximum points we believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

centralizers in endomorphism rings we prove that the centralizer cen endr m of a nilpotent endomorphism of a finitely generated semisimple left r module mr over an arbitrary ring r is the homomorphic image of the opposite of a certain z r subalgebra of the full m m matrix algebra mm r z where m is the dimension of ker if r is a local ring then we give a complete characterization of cen and of the containment cen cen where is a not necessarily nilpotent element of endr m for a k linear map a of a finite dimensional vector space over a field k we determine the pi degree of cen a

vsps of cubic fourfolds and the gorenstein locus of the hilbert scheme of 14 points on a6 abstract to a cubic fourfold one may associate a geometric object a hyperkahler manifold via the theory of vsp and an algebraic object a finite gorenstein algebra via apolarity we prove that the associated algebra is smoothable if and only if the fourfold lies on the iliev ranestad divisor which parameterizes certain cubics whose vsp is isomorphic to the hilbert scheme of two points on a k3 surface this bridge allows us to give a detailed description of the algebraic side i e the gorenstein locus of 14 points on a 6 and also to identify the equation of the iliev ranestad divisor as the unique degree 10 invariant of sl 6 as our main technical tool we develop a relative version of apolarity building on ideas of elias rossi

the tropical commuting variety we study tropical commuting matrices from two viewpoints linear algebra and algebraic geometry in classical linear algebra there exist various criteria to test whether two square matrices commut

nonexistence of positive solutions for an indefinite fractional p laplacian equation abstract in this paper we consider the following fractional p laplacian equation with indefinite nonlinearities p s u x x 1 u q x x r n where 0 s 1 p q 1 2 we use the direct method of moving planes to prove the solution of above fractional p laplacian equation is monotone increasing along x 1 direction then obtain that the above equation possess no positive bounded solution on the whole space there is no decay condition at infinity on solution for the fractional p laplacian and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p laplacian here we also cannot find a suitable auxiliary function for this fractional p laplacian to circumvent these difficulties we use the following two estimate to study the property of solution i we estimate the singular integrals defining the fractional p laplacian along a sequence of approximate maxima ii we estimate the lower and upper bound of the fractional p laplacian equation

new classes of examples satisfying the three matrix analog of gerstenhaber s theorem abstract in 1961 gerstenhaber proved the following theorem if k is a field and x and y are commuting d d matrices with entries in k then the unital k algebra generated by these matrices has dimension at most d the analog of this statement for four or more commuting matrices is false the three matrix version remains open we use commutative algebraic techniques to prove that the three matrix analog of gerstenhaber s theorem is true for some new classes of examples in particular we translate this three commuting matrix statement into an equivalent statement about certain maps between modules and prove that this commutative algebraic reformulation is true in special cases we end with ideas for an inductive approach intended to handle the three matrix analog of gerstenhaber s theorem more generally

the structure of the inverse system of gorenstein k algebras abstract macaulay s inverse system gives an effective method to construct artinian gorenstein k algebras to date a general structure for gorenstein k algebras of any dimension and codimension is not understood in this paper we extend macaulay s correspondence characterizing the submodules of the divided power ring in one to one correspondence with gorenstein d dimensional k algebras we discuss effective methods for constructing gorenstein graded rings several examples illustrating our results are given

dynamic c oriented polygonal intersection searching a set of polygons is called c oriented if the edges of all polygons are oriented in a constant number of previously defined directions the intersection searching problem is studied for such objects namely given a set of c oriented polygons p and a c oriented query polygon q find all polygons in p that intersect q it is shown that this problem can be solved in o log 2 n t time with o n log n space and o n log 2 n preprocessing where n is the cardinality of p and t the number of answers to a query furthermore the solution is extended to the cases in which p is a semidynamic or dynamic set of polygons whereas planar intersection searching can be carried out more efficiently for orthogonal objects e g rectangles it is expensive for arbitrary polygons this suggests that the c oriented solution be used in appropriate areas of application for instance in vlsi design

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

on the separation property of orbits in representation spaces a subset x of a vector space v is said to have the separation property if it separates linear forms in the following sense given a pair of linearly independent linear forms on v there is a point x on x such that x 0 and x 0 a more geometric way to express this is the following every linear subspace h v of codimension 1 is linearly spanned by its intersection with x the separation property was first asked for conjugacy classes in simple lie algebras in connection with some classification problems we give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases we also introduce a strong and a weak separation property which come up very naturally in our setting it turns out that these separation properties have a number of very nice features for example we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit and we show that a generic orbit always has the separation property

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

indefinite fractional elliptic problem and liouville theorems abstract in this paper we consider an indefinite fractional elliptic problem a corresponding liouville type theorem is established furthermore we obtain a priori estimate for solutions in a bounded domain by blowing up and rescaling we also classify the solutions of some degenerate elliptic equation originated from the fractional laplacian

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

the number of monodromy representations of abelian varieties of low p rank abstract let a g be an abelian variety of dimension g and p rank 1 over an algebraically closed field of characteristic p 0 we compute the number of homomorphisms from 1 et a g a to g l n f q where q is any power of p we show that for fixed g and n the number of such representations is polynomial in q and give an explicit formula for this polynomial we show that the set of such homomorphisms forms a constructible set and use the geometry of this space to deduce information about the coefficients and degree of the polynomial in the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups as a corollary we deduce that when 0 hom 1 et a g a g l n f q g l n f q is a laurent polynomial in q

some thoughts on gerstenhaber s theorem abstract we weigh the current evidence for and against an extension of gerstenhaber s 1961 theorem to three commuting matrices over a field

one dimensional symmetry of bounded entire solutions of some elliptic equations this paper is about one dimensional symmetry properties for some entire and bounded solutions of u f u 0 in ir we consider solutions u such that 1 u 1 and u x1 xn 1 as xn uniformly with respect to x1 xn 1 under some conditions on f we prove that the solutions only depend on the variable xn we also discuss more general elliptic operators the qualitative properties then strongly depend on the coefficients of the operator these results extend to higher dimensions and to more general operators a result of ghoussoub and gui 21 proved for n 3 ams classification 35b05 35b40 35b50 35j60

liouville theorems involving the fractional laplacian on a half space abstract let r n be the upper half euclidean space and let be any real number between 0 and 2 consider the following dirichlet problem involving the fractional laplacian 1 2 u u p x r n u 0 x r n instead of using the conventional extension method of caffarelli and silvestre 8 we employ a new and direct approach by studying an equivalent integral equation 2 u x r n g x y u p y d y applying the method of moving planes in integral forms we prove the non existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions