bandwidth truncation for chebyshev polynomial and ultraspherical chebyshev galerkin discretizations of differential equations the petrov galerkin ultraspherical polynomial chebyshev polynomial discretization of the highest derivative of a differential equation is a diagonal matrix the same is true for fourier galerkin discretizations nevertheless the spectral discretizations of simple problems like u x x q x u f x are usually dense matrices the villain is the multiplication matrix the galerkin representation of a term like q x u x unfortunately this part of the galerkin matrix is dense however if the ode coefficient q x has a chebyshev or fourier series that converges much more rapidly than u x then it is possible to realize great cost savings at no loss of accuracy by truncating the full n n galerkin matrix to a banded matrix where the bandwidth m n one of our themes is that when the spectral series for q x and u x have similar rates of convergence as is almost universal when a nonlinear equation is linearized for a newton krylov iteration such accuracy lossless truncation is impossible nonlinearity is but one of many causes of this sort of solution coefficient equiconvergence when bandwidth truncation is possible though our second theme is to show that a modest amount of floating point operations and memory can be saved by an unsymmetric truncation in which the number of elements retained to the left of the main diagonal is roughly double the number kept to the right our second improvement is to replace the m term spectral series for q x by its m 2 m 2 chebyshev pade rational approximation this sometimes allow one to halve the matrix bandwidth reducing the linear algebra costs by a factor of four

schur weyl reciprocity for ariki koike algebras let g glm1 glmr be a levi subalgebra of glm with m ri 1mi and v the natural representation of the quantum group uq g we construct a representation of the ariki koike algebra hn r on the n fold tensor space of v commuting with the action of uq g and prove the schur weyl reciprocity for the actions of uq g and hn r on it

modal preconditioning of galerkin spectral methods fourier galerkin discretizations of pdes can often be preconditioned by a matrix that is the union of a dense m m matrix plus a large matrix that is either diagonal or banded with small bandwidth this was suggested forty years ago by l m delves for our two dimensional example the preconditioned iteration converges geometrically fast to machine precision in less than ten iterations if the residual of the partial differential equation is evaluated by fast fourier transform fft the cost scales almost linearly in the number of unknowns on a 6144 6144 grid the computation needed less than an hour in matlab on a laptop the dense block requires a total degree ordering the fft evaluation of the residual requires speedometer ordering of the fourier coefficient matrix we show that this dual bookkeeping is essential but not difficult

the complexity of promise problems with applications to public key cryptography a promise problem is a formulation of partial decision problem complexity issues about promise problems arise from considerations about cracking problems for public key cryptosystems using a notion of turing reducibility between promise problems this paper disproves a conjecture made by even and yacobi 1980 that would imply nonexistence of public key cryptosystems with np hard cracking problems in its place a new conjecture is raised having the same consequence in addition the new conjecture implies that np complete sets cannot be accepted by turing machines that have at most one accepting computation for each input word

graph isomorphism is in spp we show that graph isomorphism is in the complexity class spp and hence it is in p in fact in modkp for each k 2 these inclusions for graph isomorphism were not known prior to membership in spp we derive this result as a corollary of a more general result we show that a generic problem find group has an fpspp algorithm this general result has other consequences for example it follows that the hidden subgroup problem for permutation groups studied in the context of quantum algorithms has an fpspp algorithm also some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism e g coset intersection are in spp and thus in modkp for each k 2

computational complexity of computing a partial solution for the graph automorphism problems it is known that a nontrivial automorphism on a given graph is computed by using any oracle that computes a pair of vertices u v such that u is mapped to v by some nontrivial automorphism in this paper we consider a weaker oracle acting as follows for a given graph the oracle returns a pair v b of a vertex v and a bit email protected 0 1 with the promise that if it returns v 0 then the vertex v is fixed by some nontrivial automorphism but if it returns v 1 then the vertex v is moved by some nontrivial automorphism provided that the given graph has a nontrivial automorphism we here note that the oracle may return an arbitrary pair as its answer in case that the given graph has no nontrivial automorphism we then show a stronger result that such an oracle is still powerful enough to compute a nontrivial automorphism we also show that a similar result holds for rightga a ga complete problem we further investigate the computational complexity of computing a partial solution for prefixga which is known to be gi complete for this problem we show that when we consider any oracle similar to one mentioned above the oracle does not help us to solve prefixga unless gi t p ga

the modular representation theory of q schur algebras we developed some basic theory of characteristic zero modular representations of q schur algebras we described a basis of the q schur algebra in terms of the relative norm which was first introduced by p hoefsmit and l scott and studied the product of two such basis elements we also defined the defect group of a primitive idempotent in a q schur algebra and showed that such a defect group is just the vertex of the corresponding indecomposable zf module

solvable black box group problems are low for pp in this paper we show that several algorithmic problems for abelian black box groups and solvable black box groups are low for pp in particular we prove that membership testing group intersection order verification and group isomorphism for abelian groups are in spp and group factorization coset intersection and double coset membership for abelian groups are in lwpp similarly we show that membership testing order verification and group isomorphism for solvable groups are in zppspp we also show that group intersection group factorization coset intersection and double coset membership for solvable groups are low for pp

on the isomorphism of graphs having some eigenvalues of moderate multiplicity abstract graph isomorphism is one of the most prominent problems in computer science which is not known to be np complete or polynomial time solvable babai grigoryev and mount stoc 1982 showed that for graphs which only have eigenvalues of bounded multiplicity except at most one eigenvalue this problem is in p evdokimov and ponomarenko combinatorica 1999 extended the result of babai et al to graphs for which all eigenvalues have multiplicity o log n log log n where n is the number of vertices in this paper we present an elementary proof that if for two graphs the sum of squares of the multiplicities corresponding to the eigenvalues with unbounded multiplicity is o log n log log n then isomorphism testing can be solved in polynomial time although our result is completely covered by evdokimov and ponomarenko and hence it is not new our proof only uses elementary methods and is completely different from their considerably more demanding approach

a frobenius formula for the characters of the hecke algebras this paper uses the theory of quantum groups and the quantum yang baxter equation as a guide in order to produce a method of computing the irreducible characters of the hecke algebra this approach is motivated by an observation of m jimbo giving a representation of the hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group u q mathfrak s l n by rewriting the solutions of the quantum yang baxter equation for u q mathfrak s l n in a different form one can avoid the quantum group completely and produce a frobenius formula for the characters of the hecke algebra by elementary methods using this formula we derive a combinatorial rule for computing the irreducible characters of the hecke algebra this combinatorial rule is aq extension of the murnaghan nakayama for computing the irreducible characters of the symmetric group along the way one finds connections apparently unexplored between the irreducible characters of the hecke algebra and hall littlewood symmetric functions and kronecker products of symmetric groups

a fast and well conditioned spectral method a spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions the method leads to matrices that are al

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

on tensor spaces over hecke algebras of type bn abstract let w b n be the weyl group of type b n and h b n be the associated iwahori hecke algebra in this paper we study the n tensor space v n where dim v 2 m with natural actions introduced in r m green hyperoctahedral schur algebras j algebra 192 1997 418 438 of w b n and of h b n for each composition 1 m of n let e be the corresponding initial basis element of v n see 3 8 for definition we show that if d is a distinguished right coset representative of s in w b n then the action of the natural basis element t d on e coincides with the permutation action of d up to a scalar as an application we prove that the n tensor space decomposes at the integral level into a direct sum of some permutation modules over hecke algebra h b n with respect to certain standard parabolic subalgebras

uncertainty quantification for complex systems with very high dimensional response using grassmann manifold variations abstract this paper addresses uncertainty quantification uq for problems where scalar or low dimensional vector response quantities are insufficient and instead full field very high dimensional responses are of interest to do so an adaptive stochastic simulation based methodology is introduced that refines the probability space based on grassmann manifold variations the proposed method has a multi element character discretizing the probability space into simplex elements using a delaunay triangulation for every simplex the high dimensional solutions corresponding to its vertices sample points are projected onto the grassmann manifold the pairwise distances between these points are calculated using appropriately defined metrics and the elements with large total distance are sub sampled and refined as a result regions of the probability space that produce significant changes in the full field solution are accurately resolved an added benefit is that an approximation of the solution within each element can be obtained by interpolation on the grassmann manifold the method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory

the isomorphism problem for k trees is complete for logspace we show that for k constant k tree isomorphism can be decided in logarithmic space by giving an o klogn space canonical labeling algorithm the algorithm computes a unique tree decomposition uses colors to fully encode the structure of the original graph in the decomposition tree and invokes lindell s tree canonization algorithm as a consequence the isomorphism the automorphism as well as the canonization problem for k trees are all complete for deterministic logspace completeness for logspace holds even for simple structural properties of k trees we also show that a variant of our canonical labeling algorithm runs in time o k 1 n where n is the number of vertices yielding the fastest known fpt algorithm for k tree isomorphism

computing functions with parallel queries to np the class 2p of languages polynomial time truth table reducible to sets in np has a wide range of different characterizations we consider several functional versions of 2p based on these characterizations we show that in this way the three function classes fllognp fplognp and fp np are obtained in contrast to the language case the function classes seem to all be different we give evidence in support of this fact by showing that fllognp coincides with any of the other classes then l p and that the equality of the classes fplognp and fp np would imply that the number of nondeterministic bits needed for the computation of any problem in np can be reduced by a polylogarithmic factor and that the problem can be computed deterministically with a subexponential time bound of order 2no 1 log log n