searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

fast hamiltonicity checking via bases of perfect matchings for an even integer t 2 the matching connectivity matrix ht is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices an entry ht m1 m2 is 1 if m1 and m2 form a hamiltonian cycle and 0 otherwise motivated by applications for the hamiltonicity problem we show that ht has rank exactly 2t 2 1 over gf 2 the upper bound is established by an explicit factorization of ht as the product of two submatrices the matchings labeling columns and rows respectively of the submatrices therefore form a basis xt of ht the lower bound follows because the 2t 2 1 2t 2 1 submatrix with rows and columns labeled by xt can be seen to have full rank we obtain several algorithmic results based on the rank of ht and the particular structure of the matchings in xt first we present a 1 888n no 1 time monte carlo algorithm that solves the hamiltonicity problem in directed bipartite graphs second we give a monte carlo algorithm that solves the problem in 2 s 2 pwno 1 time when provided with a path decomposition of width pw for the input graph moreover we show that this algorithm is best possible under the strong exponential time hypothesis in the sense that an algorithm with running time 2 s2 e pwno 1 for any e 0 would imply the breakthrough result of a 2 e n time algorithm for cnf sat for some e 0

singularities of electromagnetic fields in polyhedral domains in this paper we investigate the singular solutions of time harmonic maxwell equations in a domain which has edges and polyhedral corners it is now well known that in the presence of non convex edges the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the laplace operator 4 5 we show how this type of result can be derived from the classical mellin analysis and how this analysis leads to sharper results concerning the singular parts which belong to h 1 for the singular functions we exhibit simple and explicit formulas based on generalized dirichlet and neumann singularities for the laplace operator these formulas are more explicit than the results announced in our note 10

the vertex separation and search number of a graph abstract we relate two concepts in graph theory and algorithmic complexity namely the search number and the vertex separation of a graph let s g denote the search number and vs g denote the vertex separation of a connected undirected graph g we show that vs g s g vs g 2 and we give a simple transformation from g to g such that vs g s g we characterize those trees having a given vertex separation and describe the smallest such trees we also note that there exist trees for which the difference between search number and vertex separation is indeed 2 we give algorithms that for any tree t compute vs t in linear time and compute an optimal layout with respect to vertex separation in time o n log n vertex separation has previously been related to progressive black white pebble demand and has been shown to be identical to a variant of search number node search number and to path width which has been related to gate matrix layout cost all these properties are known to be computationally intractable for fixed k an o n log 2 n algorithm is known which decides whether a graph has path width at most k

weighted sobolev spaces and regularity for polyhedral domains we prove a regularity result for the poisson problem u f u p g on a polyhedral domain p r3 using the babuska kondratiev spaces kam p these are weighted sobolev spaces in which the weight is given by the distance to the set of edges 4 33 in particular we show that there is no loss of kam regularity for solutions of strongly elliptic systems with smooth coefficients we also establish a trace theorem for the restriction to the boundary of the functions in kam p

set partitioning via inclusion exclusion given a set n with n elements and a family mathcal f of subsets we show how to partition n into k such subsets in 2 n n o 1 time we also consider variations of this problem where the subsets may overlap or are weighted and we solve the decision counting summation and optimization versions of these problems our algorithms are based on the principle of inclusion exclusion and the zeta transform in effect we get exact algorithms in 2 n n o 1 time for several well studied partition problems including domatic number chromatic number maximum k cut bin packing list coloring and the chromatic polynomial we also have applications to bayesian learning with decision graphs and to model based data clustering if only polynomial space is available our algorithms run in time 3 n n o 1 if membership in mathcal f can be decided in polynomial time we solve chromatic number in o 2 2461 n time and domatic number in o 2 8718 n time finally we present a family of polynomial space approximation algorithms that find a number between chi g and lceil 1 epsilon chi g rceil in time o 1 2209 n 2 2461 e epsilon n

remarks on mixed finite element methods for problems with rough coefficients this paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients the so lution of such problems will in general be rough and it is well known that the usual ritz or displacement finite element method will be inaccurate in general the purpose of the paper is to help clarify the issue of whether the use of mixed variational principles leads to finite element schemes i e to mixed methods that are more accurate than the ritz or displacement method for such problems for one dimensional problems it is well known that certain mixed methods are more accurate and robust than the ritz method for problems with rough coefficients our results for two dimensional problems are mostly of a negative character through an examination of examples we show that certain standard mixed methods fail to provide accurate approximations for problems with rough coefficients except in some special situations this paper is concerned with the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients the solu tions of such problems will in general be rough and it is well known that the usual ritz or displacement finite element method based on piecewise linear approximating functions is inaccurate in general the purpose of the paper is to help clarify the issue of whether the use of mixed variational principles leads to finite element schemes i e to mixed methods that are more accurate than the ritz or displacement method for such problems mixed variational principles arise naturally in the mathematical formulation of many physical problems for example the laws of linear elasticity may be described in terms of a displacement variational formulation involving only displacements or in terms of a mixed variational formulation involving both stresses and displace ments whose equations express the stress strain relation and the balance of forces in purely mathematical terms one can obtain a mixed formulation from a displacement formulation by introducing new variables for some of the derivatives or certain linear combinations of derivatives of the unknown func tion a mixed approximation method is obtained by basing a finite element

the multidirectional neumann problem in ℝ4 the neumann problem as formulated in lipschitz domains with l p boundary data is solved for harmonic functions in any compact polyhedral domain of 4 that has a connected 3 manifold boundary energy estimates on the boundary are derived from new polyhedral rellich formulas together with a whitney type decomposition of the polyhedron into similar lipschitz domains the classical layer potentials are thereby shown to be semi fredholm to settle the onto question a method of continuity is devised that uses the classical 3 manifold theory of e e moise in order to untwist the polyhedral boundary into a lipschitz boundary it is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary in this way the flattening arguments of b e j dahlberg and c e kenig for the h 1 at neumann problem can be extended to polyhedral domains in 4 a compact polyhedral domain in 6 of m l curtis and e c zeeman based on a construction of m h a newman shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra

intractability of clique width parameterizations we show that edge dominating set hamiltonian cycle and graph coloring are w 1 hard parameterized by clique width it was an open problem explicitly mentioned in several papers whether any of these problems is fixed parameter tractable when parameterized by the clique width that is solvable in time g k cdot n o 1 on n vertex graphs of clique width k where g is some function of k only our results imply that the running time o n f k of many clique width based algorithms is essentially the best we can hope for up to a widely believed assumption from parameterized complexity namely fpt neq w 1

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains the symbolic calculus of pseudodifferential operators of mellin type is applied to study layer potentials on a plane domain q whose boundary 9q is a curvilinear polygon a singularity type is a zero of the determinant of the matrix of symbols of the mellin operators and can be used to calculate the bad values of p for which the system is not fredholm on lp 9q using the method of layer potentials we study the singularity types of the system of elastostatics lu uau au vdivu 0 in a plane domain q whose boundary 9k2 is a curvilinear polygon here u 0 and u a oo when a oo the system is the stokes system of hydrostatics for the traction double layer potential we show that all singularity types in the strip 0 re z i lie in the interval 1 1 so that the system of integral equations is a fredholm operator of index 0 on lp aq for all p 2 p oo the explicit dependence of the singularity types on a and the interior angles 0 of aq is calculated the singularity type of each corner is independent of a if the corner is nonconvex introduction recently there has been considerable interest in using layer potentials to solve lp boundary value problems for elliptic operators and systems on a lipschitz domain q in r for the systems of elastostatics dkv and hydrostatics fkv dahlberg fabes kenig and verchota have used rellich type identities to prove that the double layer potential integral equations yield a fredholm operator of index 0 on l 2 q for p 2 only limited information is available on the boundary integral equations for general lipschitz domains in rn the general problem of the notion of symbol on the boundary of a general lipschitz domain is still very much open in this paper we treat a very special case a curvilinear polygonal domain in r in this 2 dimensional case a precise symbolic calculus of pseudodifferential operators of mellin type is available we show that certain double layer boundary integral equations yield operators which for all p 2 p 00 are received by the editors december 14 1988 1980 mathematics subject classification 1985 revision primary 35j25 45e05 research partially supported by the italian cnr 1990 american mathematical society 0002 9947 90 1 00 25 per page

data reduction for graph coloring problems this paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances we are interested in how well polynomial time data reduction can provably shrink instances of coloring problems in terms of the chosen parameter it is well known that deciding 3 colorability is already np complete hence parameterizing by the requested number of colors is not fruitful instead we pick up on a research thread initiated by cai dam 2003 who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial time solvable for example parameterizing by the number k of vertex deletions needed to make the graph chordal we obtain various upper and lower bounds for kernels of such parameterizations of q coloring complementing email protected s study of the time complexity with respect to these parameters our results show that the existence of polynomial kernels for q coloring parameterized by the vertex deletion distance to a graph class f is strongly related to the existence of a function f q which bounds the number of vertices which are needed to preserve the no answer to an instance of q list coloring on f

spectra for solvable lie algebras of bundle endomorphisms the aim of the paper is to investigate spectral properties of the lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space under natural hypotheses such lie algebras are solvable being in general infinite dimensional the spectral theory of finite dimensional solvable lie algebras of operators is extended to this natural class of infinite dimensional solvable lie algebras the discussion uses the language of continuous fields of c algebras the flag manifolds in c algebraic framework are naturally involved here they providing the basic method for obtaining flags of vector bundles

regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3 part i countably normed spaces on polyhedral domains this is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains and it concentrates on exploring the structure of these spaces such as the embeddings of weighted sobolev spaces the relation between weighted sobolev spaces and weighted continuous function spaces and the relations between the weighted sobolev spaces and countably weighted sobolev spaces in cartesian coordinates and in the spherical and cylindrical coordinates these well defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in 3 which are essential for the design of effective computation and the analysis of the h p version of the finite element method for solving elliptic problems in three dimensional nonsmooth domains arising from mechanics and engineering

irreducible representations of infinite dimensional lie algebras abstract some results concerning the operatorially irreducible topologically irreducible operatorially and topologically irreducible representations of a class of infinite dimensional lie algebras by bounded operators in a complex banach space are obtained

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

invariant subspaces of operator lie algebras and lie algebras with compact adjoint action it is proved that a lie algebra of compact operators with a non zero volterra ideal is reducible has a nontrivial invariant subspace a number of other criteria of reducibility for collections of operators is obtained the results are applied to the structure theory of lie algebras of compact operators and normed lie algebras with compact adjoint action

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

an asymptotic formula for the commutators in what follows we shall describe some asymptotic formulas for the commutators defined by the values of the analytic functional calculus of some commutingn tuple of bounded operators in a complex banach space

computing the chromatic number using graph decompositions via matrix rank abstract computing the smallest number q such that the vertices of a given graph can be properly q colored known as the chromatic number is one of the oldest and most fundamental problems in combinatorial optimization the q coloring problem has been studied intensively using the framework of parameterized algorithmics resulting in a very good understanding of the best possible algorithms for several parameterizations based on the structure of the graph for example algorithms are known to solve the problem on graphs of treewidth tw in time o q tw while a running time of o q e tw is impossible assuming the strong exponential time hypothesis seth while there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators almost nothing is known about parameterizations based on edge separators we fill this gap by studying q coloring parameterized by cutwidth and parameterized by pathwidth in bounded degree graphs our research uncovers interesting new ways to exploit small edge separators we present two algorithms for q coloring parameterized by cutwidth ctw a deterministic one that runs in time o 2 ctw where is the square matrix multiplication exponent and a randomized one with runtime o 2 ctw in sharp contrast to earlier work the running time is independent of q the dependence on cutwidth is optimal we prove that even 3 coloring cannot be solved in o 2 e ctw time assuming seth our algorithms rely on a new rank bound for a matrix that describes compatible colorings combined with a simple communication protocol for evaluating a product of two polynomials this also yields an o d 2 1 pw time randomized algorithm for q coloring on graphs of pathwidth pw and maximum degree d such a runtime was first obtained by bjorklund but only for graphs with few proper colorings we also prove that this result is optimal in the sense that no o d 2 1 e pw time algorithm exists assuming seth

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara