average case intractability vs worst case intractability we show that not all sets in np or other levels of the polynomial time hierarchy have efficient average case algorithms unless the arthur merlin classes ma and am can be derandomized to np and various subclasses of p poly collapse to p furthermore other complexity classes like p pp and pspace are shown to be intractable on average unless they are easy in the worst case

parallel adaptive simplical re meshing for deforming domain cfd computations deforming domains occur in many fields of computational fluid dynamics cfd such as interface tracking simulation of pumps and engines and fluid structure interaction the deformation of the domain presents a challenge to the integrity of the computational mesh substantial motion of the domain boundaries requires vertex motion and changes in mesh connectivity for cases of simple boundary motion or structured meshes predetermined changes to the mesh structure can be sufficient however without a priori knowledge of how the domain will change a more robust solution is required the present work offers a parallelized solution for simplical meshes that is well suited to extremely complex geometry the mesh continuously evolves without user intervention or the use of target meshes varying length scales imposed by evolving boundary curvatures and narrow gaps are resolved with a fast length scale algorithm the set of algorithms are incorporated into an object oriented code structure that permits broad application to a range of cfd problems the robustness and versatility of the algorithm is demonstrated in several examples representing motion of internal and external boundaries where the boundary motion may or may not be known a priori

an algorithm for coarsening unstructured meshes we develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary unstructured nonuniform triangular mesh using these hierarchical basis functions we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations we show that such iterative methods perform as well as those developed for the usual case of structured locally refined meshes in particular we show that the generalized condition numbers for such iterative methods are of order j 2 where j is the number of hierarchical basis levels

on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

relativized arthur merlin versus merlin arthur games arthur merlin games were introduced recently by babai in order to capture the intuitive notion of efficient probabilistic proof systems considered as complexity classes they are extensions of np it turned out that one exchange of messages between the two players is sufficient to simulate a constant number of interactions thus at most two new complexity classes survive at the constant levels of this new hierarchy am and ma depending on who starts the communication it is known that ma subseteq am in this paper we answer an open problem of babai we construct an oracle c such that am c sigma 2 p c ne emptyset since ma c subseteq sigma 2 p c it follows that for some oracle c mac amc our prooftechnique is a modification of the technique used by baker and selman to show that 2 p and 2 p can be separated by some oracle this result can be interpreted as an evidence that with one exchange of messages the proof system is stronger when arthur starts the communication

error estimates for the raviart thomas interpolation under the maximum angle condition the classical error analysis for the raviart thomas interpolation on triangular elements requires the so called regularity of the elements or equivalently the minimum angle condition however in the lowest order case optimal order error estimates have been obtained in g acosta and r g duran siam j numer anal 37 2000 pp 18 36 replacing the regularity hypothesis by the maximum angle condition which was known to be sufficient to prove estimates for the standard lagrange interpolation in this paper we prove error estimates on triangular elements for the raviart thomas interpolation of any order under the maximum angle condition also we show how our arguments can be extended to the three dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in g acosta and r g duran siam j numer anal 37 2000 pp 18 36

darboux transformation and explicit solutions for two integrable equations a new n fold darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by qiao z j qiao phys lett a 192 1994 316 322 by the darboux transformation explicit soliton and multi soliton solutions for the two equations are obtained in particular soliton and complexiton solutions are shown through some figures

circuit lower bounds for merlin arthur classes we show that for each k 0 mathsf ma 1 mathsf ma with 1 bit of advice does not have circuits of size n k this implies the first superlinear circuit lower bounds for the promise versions of the classes mathsf ma mathsf am and mathsf zpp parallel mathsf np we extend our main result in several ways for each k we give an explicit language in mathsf ma cap mathsf coma 1 which does not have circuits of size n k we also adapt our lower bound to the average case setting i e we show that mathsf ma 1 cannot be solved on more than 1 2 1 n k fraction of inputs of length n by circuits of size n k furthermore we prove that mathsf ma does not have arithmetic circuits of size n k for any k as a corollary to our main result we obtain that derandomization of mathsf ma o 1 implies the existence of pseudorandom generators computable using o 1 bits of advice

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

parallel space time hp adaptive discretization scheme for parabolic problems abstract we introduce an integration scheme for parabolic problems our parallelizable method uses adaptive h p finite elements in space and finite differences in time the strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition we verified the performance of our method against two different benchmarks in both two dimensional model problem on an l shaped domain and three dimensional pennes bioheat equation settings results show a significant speedup in computational time when compared with the sequential version of the algorithm moreover we develop a mathematical framework to analyze similar schemes which include h p spatial adaptivity our framework describes error propagation rigorously and as such allows to analyze convergence properties of these mixed methods

tetrahedral mesh generation using delaunay refinement with non standard quality measures this paper studies the practical performance of delaunay refinement tetrahedral mesh generation algorithms by using non standard quality measures to drive refinement we show that sliver tetrahedra can be eliminated from constrained delaunay tetrahedralizations solely by refinement despite the fact that quality guarantees cannot be proven the algorithm can consistently generate meshes with dihedral angles between 18circ and 154 using a fairer quality measure targeting every type of bad tetrahedron dihedral angles between 14 and 154 can be obtained the number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius to shortest edge ratio we also study the use of mesh improvement techniques on delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20 regardless of the quality measure used to drive refinement the algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth copyright 2011 john wiley sons ltd

on pseudorandomness and resource bounded measure abstract in this paper we extend a key result of nisan and wigderson j comput system sci 49 1994 149 167 to the nondeterministic setting for all 0 we show that if there is a language in e dtime 2 o n that is hard to approximate by nondeterministic circuits of size 2 n then there is a pseudorandom generator that can be used to derandomize bp np in symbols bp np np by applying this extension we are able to answer some open questions in lutz theory comput systems 30 1997 429 442 regarding the derandomization of the classes bp p k and bp p k under plausible measure theoretic assumptions as a consequence if p 2 does not have p measure 0 then am coam is low for p 2 thus in this case the graph isomorphism problem is low for p 2 by using the nisan wigderson design of a pseudorandom generator we unconditionally show the inclusion ma zpp np and that ma coma is low for zpp np

anisotropic error bounds of lagrange interpolation with any order in two and three dimensions abstract in this paper using the newton s formula of lagrange interpolation we present a new proof of the anisotropic error bounds for lagrange interpolation of any order on the triangle rectangle tetrahedron and cube in a unified way

an approach to refining three dimensional tetrahedral meshes based on delaunay transformations a technique for refining three dimensional tetrahedral meshes is proposed in this paper the proposed technique is capable of treating arbitrary unstructured tetrahedral meshes convex or non convex with multiple regions resulting in high quality constrained delaunay triangulations the tetrahedra generated are of high quality nearly equilateral sliver tetrahedra which present a real problem to many algorithms are not produced with the new method the key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the voronoi delaunay theory and a reposition of nodes technique the computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra making it ideal for direct employment in a fully automatic finite element analysis system for 3 d adaptive mesh refinement application to some test problems is presented to show the effectiveness and applicability of the new method

local remeshing for large amplitude grid deformations fluid structure interaction fsi modelling can involve large deformations in the fluid domain which could lead to degenerating mesh quality and numerical inaccuracies or instabilities if allowed to amplify unchecked complete remeshing of the entire domain during the solution process is computationally expensive and can require interpolation of solution variables between meshes as an alternative we investigate a local remeshing algorithm with two emphases a the identification and remedy of flat degenerate tetrahedra and b the avoidance of node motion and hence associated interpolation errors initially possible topological changes are examined using a dynamic programming algorithm to maximise the minimum local element quality through edge reconnection in the 3d situation it was found that reconnection improvements tend to be limited to long edges and those with few three or four element neighbours the remaining degenerate elements are classified into one of four types using three proposed metrics the minimum edge to edge distance ee the minimum node to edge distance ne and the shortest edge length se and removed according to the best manner for their type optimised thresholds for identifying and classifying elements for removal were found to be ee 0 18 ne 0 21 se 0 2

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

on the interpolation error estimates for q_1 quadrilateral finite elements in this paper we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over frac 1 j p 1 for 1 leq p leq infty on the reference element where j is the jacobian of the nonaffine mapping this allows us to introduce weak geometric conditions depending on p leading to interpolation error estimates in the w 1 p norm for any p in 1 infty which can be regarded as a generalization of the regular decomposition property rdp condition introduced in g acosta and r g duran siam j numer anal 38 2000 pp 1073 1088 for p 2 and new rdp conditions nrdp for p neq2 we avoid the use of the reference family elements which allows us to extend the results to a larger class of elements and to introduce the nrdp condition in a more unified way as far as we know the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 leq p leq infty

high order unstructured curved mesh generation using the winslow equations we propose a method to generate high order unstructured curved meshes using the classical winslow equations we start with an initial straight sided mesh in a reference domain and fix the position of the nodes on the boundary on the true curved geometry in the interior of the domain we solve the winslow equations using a new continuous galerkin finite element discretization this formulation appears to produce high quality curved elements which are highly resistant to inversion in addition the corresponding nonlinear equations can be solved efficiently using picard iterations even for highly stretched boundary layer meshes compared to several previously proposed techniques such as optimization and approaches based on elasticity analogies this can significantly reduce the computational cost while producing curved elements of similar quality we show a number of examples in both two and three space dimensions including complex geometries and stretched boundary layers and demonstrate the high quality of the generated meshes and the performance of the nonlinear solver

a fictitious energy approach for shape optimization this paper deals with shape optimization of continuous structures as in early works on shape optimization coordinates of boundary nodes of the fe domain are directly chosen as design variables convergence problems and problems with jagged shapes are eliminated by a new regularization technique an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design the energy constraint defines a feasible design space whose size can be varied by one parameter the upper energy limit by construction the proposed regularization is applicable to a wide range of problems although in this paper the application is restricted to linear elastostatic problems

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

acute type refinements of tetrahedral partitions of polyhedral domains we present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron the resulting family of partitions is of acute type thus all the tetrahedra satisfy the maximum angle condition both these properties are highly desirable in finite element analysis

quality mesh generation in higher dimensions we consider the problem of triangulating a d dimensional region our mesh generation algorithm called qmg is a quadtree based algorithm that can triangulate any polyhedral region including nonconvex regions with holes furthermore our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles finally our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by qmg is bounded above by a factor times the number produced by any competing algorithm where the factor depends on the aspect ratio bound satisfied by the competing algorithm the qmg algorithm has been implemented in c and is used as a mesh generator for the finite element method

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

quasi periodic solution of the 2 1 dimensional boussinesq burgers soliton equation abstract a 2 1 dimensional bossinesq burgers soliton equation is proposed which has a affinitive connection with the boussinesq burgers soliton hierarchy through a natural nonlinearization of the boussinesq burgers s eigenvalue problems a finite dimensional hamiltonian system is obtained and is proved to be completely integrable in liouville sense the abel jacobi coordinates are constructed to straighten out the hamiltonian flows from which the quasi periodic solutions of the 2 1 dimensional boussinesq burgers equation are derived by resorting to the riemann theta functions

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

the mini mixed finite element for the stokes problem an experimental investigation abstract o h 3 2 superconvergence in pressure and velocity has been experimentally investigated for the two dimensional stokes problem discretized with the mini mixed finite element even though the classic mixed finite element theory for the mini element guarantees linear convergence for the total error recent theoretical results indicate that superconvergence of order o h 3 2 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured three directional triangular meshes the numerical experiments presented here suggest a more general validity of o h 3 2 superconvergence possibly to automatically generated and unstructured triangulations in addition the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise linear part of the computed velocity finding that the former is generally closer to the exact velocity whereas the latter conserves mass better

quasi periodic solution of a new 2 1 dimensional coupled soliton equation a new 2 1 dimensional integrable soliton equation is proposed which has a close connection with the levi soliton hierarchy through the nonlinearization of the levi eigenvalue problems we obtain a finite dimensional integrable system the abel jacobi coordinates are constructed to straighten out the hamiltonian flows by which the solutions of both the 1 1 and 2 1 levi equations are obtained through linear superpositions an inversion procedure gives the quasi periodic solution in the original coordinates in terms of the riemann theta functions

qualitative measures for initial meshes this paper reviews geometric measures used to assess the shape of finite elements in two and three dimensional meshes measures have been normalized and made scale invariant whenever possible this paper also introduces a universal similarity region that enhances comparisons of triangles and their measures as a byproduct the usr provides a dynamic way to compare improved triangular meshes copyright 2000 john wiley sons ltd

a pseudorandom oracle characterization of bpp it is known from work of bennett and gill siam j comput 10 1981 pp 96 113 and of ambos spies in proc 1st structure in complexity theory conference 1986 pp 23 34 that the following conditions are equivalent i l in text bpp ii for almost all oracles a l in text p a it is shown here that the following conditions are also equivalent to i and ii iii the set of oracles a for which l in text p a has pspace measure 1 iv for every pspace random oracle a l in text p a it follows from this characterization and its proof that almost every a in text espace is leqslant t text p hard for text bpp a succinctly the main content of the proof is that pseudorandom generators exist relative to every pseudorandom oracle

restricted flows of soliton hierarchies coupled kdv and harry dym case restricted flows of soliton hierarchies associated with the energy dependent schrodinger spectral problem are determined explicitly a remarkable connection with separable potentials is used for proving complete integrability of the restricted flows a previously unknown lagrangian and hamiltonian formulation of the neumann system is found whole families of generalizations of the neumann and garnier systems are given

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

error estimates for 3 d narrow finite elements we obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three dimensional narrow elements first for the lagrange interpolation we prove optimal error estimates both in order and regularity in l p for p 2 for p 2 it is known that this result is not true applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than h 2 these results are valid both for tetrahedral and rectangular elements second for the case of rectangular elements we obtain optimal in order and regularity error estimates for an average interpolation valid for functions in w 1 s p with 1 p and 0 s 1

a complexity theory based on boolean algebra a projection of a boolean function is a function obtained by substituting for each of its variables a variable the negation of a variable or a constant reducibilities among computational problems under this relation of projection are considered it is shown that much of what is of everyday relevance in turing machine based complexity theory can be replicated easily and naturally in this elementary framework finer distinctions about the computational relationships among natural problems can be made than in previous formulations and some negative results are proved

on mesh geometry and stiffness matrix conditioning for general finite element spaces the performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation pde models it is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix in this paper for typical second order elliptic problems some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes the derivation of such relations for general high order elements is based on a new trace formula for the element stiffness matrix it is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces these results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing

on the equivalence of regularity criteria for triangular and tetrahedral finite element partitions in this note we examine several regularity criteria for families of simplicial finite element partitions in r d email protected 2 3 these are usually required in numerical analysis and computer implementations we prove the equivalence of four different definitions of regularity often proposed in the literature the first one uses the volume of simplices the others involve the inscribed and circumscribed ball conditions and the minimal angle condition

quasi periodic solutions for some 2 1 dimensional integrable models generated by the jaulent miodek hierarchy some 2 1 dimensional integrable models including the modified kadomtsev petviashvili equation generated by the jaulent miodek hierarchy are investigated with the help of the jaulent miodek eigenvalue problem these 2 1 dimensional integrable models are separated into compatible hamiltonian systems of ordinary differential equations using the generating function flow method the involutivity and the functional independence of the integrals are proved the abel jacobi coordinates are introduced from which the quasi periodic solutions for these 2 1 dimensional integrable models are derived by resorting to the riemann theta functions pacs number 0230j

on the generalization of the synge křižek maximum angle condition for d simplices abstract in this note we present a generalization of the maximum angle condition proposed by j l synge in 1957 and m k i ek in 1992 for triangular and tetrahedral elements respectively for the case of higher dimensional simplicial finite elements its relations to the other angle type conditions commonly used in interpolation theory and in finite element analysis are discussed

a local radial basis function method for advection diffusion reaction equations on complexly shaped domains abstract time dependent advection diffusion reaction and diffusion reaction equations are used as models in biology chemistry physics and engineering as representative examples we focus on a chemotaxis model and a turing system from biology and apply a local radial basis function method to numerically approximate the solutions the numerical method can efficiently approximate large scale problems in complexly shaped domains

new finite dimensional integrable systems and explicit solutions of hirota satsuma coupled kortweg de vries equation the nonlinearization approach is extended to the hirota satsuma coupled kortweg de vries kdv equation associated with a 4 4 matrix spectral problem from which two new finite dimensional integrable hamiltonian systems are obtained in the liouville sense it is shown that the solutions of this coupled kdv equation are reduced to solving a compatible system of ordinary differential equations the reductions of the two hamiltonian systems and their separability are discussed an interesting connection between the two reduced hamiltonian systems in the case of one parameter and known two dimensional integrable systems is revealed as application some explicit solutions of the hirota satsuma coupled kdv equation are derived

algebraic mesh quality metrics quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics the theory based on the jacobian and related matrices provides a means of constructing classifying and evaluating mesh quality metrics the jacobian matrix is factored into geometrically meaningful parts a nodally invariant jacobian matrix can be defined for simplicial elements using a weight matrix derived from the jacobian matrix of an ideal reference element scale and orientation invariant algebraic mesh quality metrics are defined the singular value decomposition is used to study relationships between metrics equivalence of the element condition number and mean ratio metrics is proved the condition number is shown to measure the distance of an element to the set of degenerate elements algebraic measures for skew length ratio shape volume and orientation are defined abstractly with specific examples given two combined metrics shape volume and shape volume orientation are algebraically defined and examples of such metrics are given algebraic mesh quality metrics are extended to nonsimplicial elements a series of numerical tests verifies the theoretical properties of the metrics defined

optimization of the multigrid convergence rate on semi structured meshes by local fourier analysis in this paper a local fourier analysis for multigrid methods on tetrahedral grids is presented different smoothers for the discretization of the laplace operator by linear finite elements on such grids are analyzed a four color smoother is presented as an efficient choice for regular tetrahedral grids whereas line and plane relaxations are needed for poorly shaped tetrahedra a novel partitioning of the fourier space is proposed to analyze the four color smoother numerical test calculations validate the theoretical predictions a multigrid method is constructed in a block wise form by using different smoothers and different numbers of pre and post smoothing steps in each tetrahedron of the coarsest grid of the domain some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm

local nonobtuse tetrahedral refinements around an edge in this note we show how to generate and conformly refine nonobtuse tetrahedral meshes locally around and towards an edge so that all dihedral angles of all resulting tetrahedra remain nonobtuse the proposed technique can be used e g for a numerical treatment of solution singularities and also for various mesh adaptivity procedures near the reentrant corners of cylindric type 3d domains

error estimates for raviart thomas interpolation of any order on anisotropic tetrahedra we prove optimal order error estimates for the raviart thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition namely the so called three dimensional maximum angle condition and the regular vertex property for tetrahedra our techniques are different from those used in previous papers on the subject and the results obtained are more general in several aspects first intermediate regularity is allowed that is for the raviart thomas interpolation of degree k 0 we prove error estimates of order j 1 when the vector field being approximated has components in w j 1 p for triangles or tetrahedra where 0 j k and 1 p these results are new even in the two dimensional case indeed the estimate was known only in the case j k on the other hand in the three dimensional case results under the maximum angle condition were known only for k 0

low end uniform hardness versus randomness tradeoffs for am impagliazzo and wigderson proceedings of the 39th annual ieee symposium on foundations of computer science ieee computer society washington dc 1998 pp 734 743 proved a hardness versus randomness tradeoff for bpp in the uniform setting which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions in slightly weaker settings gutfreund shaltiel and ta shma comput complexity 12 2003 pp 85 130 proved a uniform hardness versus randomness tradeoff for am but that result worked only on the high end of possible hardness assumptions in this work we give uniform hardness versus randomness tradeoffs for am that are near optimal for the full range of possible hardness assumptions following gutfreund shaltiel and ta shma we do this by constructing a hitting set generator hsg for am with resilient reconstruction our construction is a recursive variant of the miltersen vinodchandran hsg comput complexity 14 2005 pp 256 279 the only known hsg construction with this required property the main new idea is to have the reconstruction procedure operate implicitly and locally on superpolynomially large objects using tools from pcps low degree testing self correction together with a novel use of extractors that are built from reed muller codes for a sort of locally computable error reduction as a consequence we obtain gap theorems for am and am cap coam that state roughly that either am or am cap coam protocols running in time t n can simulate all of exp arthur merlin games are powerful or else all of am or am cap coam can be simulated in nondeterministic time s n arthur merlin games can be derandomized for a near optimal relationship between t n and s n as in gutfreund shatiel and ta shma the case of am cap coam yields a particularly clean theorem that is of special interest due to the wide array of cryptographic and other problems that lie in this class

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

a geometrical approach to mesh smoothing we study the problem of smoothing finite element meshes of triangles and tetrahedra where vertices are recursively moved to improve the overall quality of the elements with respect to a given shape quality metric we propose a geometric approach to solving the local optimization problem level sets of the given metric are used to characterize the set of optimal point s we also introduce a new mesh quality metric for tetrahedra

the maximum angle condition is not necessary for convergence of the finite element method we show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems this condition is only sufficient in fact finite element approximations may converge even though some dihedral angles of simplicial elements tend to

silver exudation a silver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge silvers are notoriously common in 3 dimensional delaunay triangulations even for well spaced point sets we show that if the delaunay triangulation has the ratio property introduced in miller et al 1995 then there is an assignment of weights so the weighted delaunay traingulation contains no silvers we also give an algorithm to compute such a weight assignment

pp is closed under truth table reductions beigel reingold and spielman j comput system sci 50 191 202 1995 showed that pp is closed under intersection and a variety of special cases of polynomial time truth table closure we extend their techniques to show that pp is closed under general polynomial time truth table reductions we also show that pp is closed under constant round truth table reductions

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

lawson s triangulation is nearly optimal for controlling error bounds one way of evaluating a triangulation method is to study how well it controls error bounds and stability in computations lawson s method which maximizes the smallest angle over all triangulations controls error bounds for linear interpolates as well as for certain derivative estimates in particular no other triangulation improves these bounds by more than a factor of 2 although lawson s triangulation method does not minimize the largest edge over all triangulations it nearly does this edge is no longer than 2 sqrt 3 times the longest edge in any other triangulation thus lawson s triangulation method controls error bounds in polynomial interpolation schemes

an immersed finite element method based on a locally anisotropic remeshing for the incompressible stokes problem abstract in the present paper we study a finite element method for the incompressible stokes problem with a boundary immersed in the domain on which essential constraints are imposed such type of methods may be useful to tackle problems with complex geometries interfaces such as multiphase flow and fluid structure interaction the method we study herein consists in locally refining elements crossed by the immersed boundary such that newly added elements called subelements fit the immersed boundary in this sense this approach is of a fitted type but with an original mesh given independently of the location of the immersed boundary we use such a subdivision technique to build a new finite element basis which enables us to represent accurately the immersed boundary and to impose strongly dirichlet boundary conditions on it however the subdivision process may imply the generation of anisotropic elements which for the incompressible stokes problem may result in the loss of inf sup stability even for well known stable element schemes we therefore use a finite element approximation which appears stable also on anisotropic elements we perform numerical tests to check stability of the chosen finite elements several numerical experiments are finally presented to illustrate the capabilities of the method the method is presented for two dimensional problems

a zero one law for rp and derandomization of am if np is not small we show that if rp does not have p measure zero then zpp exp as corollaries we obtain a zero one law for rp in exp and that both probabilistic classes zpp and rp have the same measure in exp we also prove that if np does not have p measure zero then np am

on the shape of tetrahedra from bisection we present a procedure for bisecting a tetrahedron t successively into an infinite sequence of tetrahedral meshes j 0 91 g2 which has the following properties 1 each mesh sfn is conforming 2 there are a finite number of classes of similar tetrahedra in all the 3n n 0 3 for any tetrahedron t7 in s7n jj t cl1 t where ti is a tetrahedron shape measure and cl is a constant 4 3 j c2 1 2 ni3j t where d t denotes the diameter of tetrahedron t1 and c2 is a constant estimates of cl and c2 are provided properties 2 and 3 extend similar results of stynes and adler and of rosenberg and stenger respectively for the 2 d case the diameter bound in property 4 is better than one given by kearfott

fast centroidal voronoi delaunay triangulation for unstructured mesh generation a fast unstructured mesh generation algorithm based on conforming centroidal voronoi delaunay triangulation cfcvdt algorithm ju 2007 is proposed in this paper in the new algorithm the constrained delaunay triangulation cdt algorithm is used only for the generation of the initial mesh the mesh quality shall be continuously improved by updating the positions of vertices and flipping edges in a number of iterations since the most time consuming procedure in cfcvdt algorithm is the cdt in each iteration which has been successfully avoided in this new algorithm the efficiency has been significantly improved furthermore the meshes generated by this algorithm have similar high quality features as that generated by cfcvdt when complex interfaces are involved our algorithm can keep the mesh conforming to the interfaces very efficiently by using various density functions this algorithm can produce high quality non uniform meshes for potentially many applications a fast high quality triangular mesh generation algorithm fcfcvdt is proposed based on the cfcvdt algorithm the fcfcvdt algorithm significantly improves the efficiency of cfcvdt algorithm at the same time maintains the mesh quality it is capable of generating high quality body interface fitted meshes for complicate domains interfaces

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses

inverse mapping and distortion measures for quadrilaterals with curved boundaries by using the theory of geodesics in differential geometry inverse relations of the mapping for 8 node quadrilaterals with curved boundaries are derived and expressed in terms of the element geodesic co ordinates defined at the local origin the parameters in the expressions of the geodesic co ordinates in terms of the isoparametric co ordinates are suggested to be the distortion measures of the element the mathematical meanings of these parameters can be discussed and explained by using the coefficient vectors in the equations defining the isoparametric co ordinate transformation this method of defining the element distortion measures is consistent and general and can be applied to any other two or three dimensional isoparametric finite elements