convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction diffusion problems the numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems the quasi optimal order error estimates are proved in the e weighted h 1 norm valid uniformly up to a logarithmic factor in the singular perturbation parameter by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

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

a priori error estimates of regularized elliptic problems approximations of the dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth generalized functions via convolution in this work we show a priori rates of convergence of this approximation process in standard sobolev norms with minimal regularity assumptions on the approximation of the dirac delta distribution the application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp h 1 and l 2 error estimates for the corresponding regularized problem as an application we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non regularized counterpart provided that the support of the dirac delta approximation is set to a multiple of the mesh size at a fraction of the implementation complexity numerical experiments are provided to support our theories

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

piecewise polynomial interpolation in muckenhoupt weighted sobolev spaces and applications we develop a constructive piecewise polynomial approximation theory in weighted sobolev spaces with muckenhoupt weights for any polynomial degree the main ingredients to derive optimal error estimates for an averaged taylor polynomial are a suitable weighted poincare inequality a cancellation property and a simple induction argument we also construct a quasi interpolation operator built on local averages over stars which is well defined for functions in l 1 l1 we derive optimal error estimates for any polynomial degree on simplicial shape regular meshes on rectangular meshes these estimates are valid under the condition that neighboring elements have comparable size which yields optimal anisotropic error estimates over n n rectangular domains the interpolation theory extends to cases when the error and function regularity require different weights we conclude with three applications nonuniform elliptic boundary value problems elliptic problems with singular sources and fractional powers of elliptic operators

on the finite element method for elliptic problems with degenerate and singular coefficients we consider dirichlet boundary value problems for second order elliptic equations over polygonal domains the coefficients of the equations under consideration degenerate at an inner point of the domain or behave singularly in the neighborhood of that point this behavior may cause singularities in the solution the solvability of the problems is proved in weighted sobolev spaces and their approximation by finite elements is studied this study includes regularity results graded meshes and inverse estimates applications of the theory to some problems appearing in quantum mechanics are given numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi uniform meshes

the query complexity of finding local minima in the lattice in this paper we study the query complexity of finding local minimum points of a boolean function this task occurs frequently in exact learning algorithms for many natural classes such as monotone dnf o log n term dnf unate dnf and decision trees on the negative side we prove that any possibly randomized algorithm that produces a local minimum of a function f chosen from a sufficiently rich concept class using a membership oracle for f must ask n2 membership queries in the worst case in particular this lower bound applies to the class of decision trees a simple algorithm is known that achieves this lower bound on the positive side we show that for the class o log n term dnf finding local minimum points requires only n log n membership queries and more generally tn membership queries for t term dnf with tn this efficient procedure improves the time and query complexity of known learning algorithms for the class o log n term dnf 2001 elsevier science

algebraicity of local holomorphisms between real algebraic submanifolds of complex spaces we prove that a germ of a holomorphic map f between c n and c n sending one real algebraic submanifold mc n into another m c n is algebraic provided mcontains no complex analytic discs and m is generic and minimal we also propose an algorithm for finding complex analytic discs in a real submanifold

properties of discrete delta functions and local convergence of the immersed boundary method many problems involving internal interfaces can be formulated as partial differential equations with singular source terms numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions we study the convergence properties of such discretizations for constant coefficient elliptic problems using the immersed boundary method as an example we show how the order of the differential operator order of the finite difference discretization and properties of the discrete delta function all influence the local convergence behavior in particular we show how a recently introduced property of discrete delta functions the smoothing order is important in the determination of local convergence rates

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

a singularly perturbed problem with two parameters in two dimensions on graded meshes abstract a numerical approximation of a convection reaction diffusion problem by standard bilinear finite elements is considered using duran lombardi and duran shishkin type meshes we prove first order error estimates in an energy norm numerical examples confirm our theoretical results and show smaller errors compared to the well known shishkin mesh

a posteriori error estimates for the stokes problem with singular sources abstract we propose a posteriori error estimators for classical low order inf sup stable and stabilized finite element approximations of the stokes problem with singular sources in two and three dimensional lipschitz but not necessarily convex polytopal domains the designed error estimators are proven to be reliable and locally efficient on the basis of these estimators we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform

uniform error estimates for certain narrow lagrange finite elements error estimates of dupont and scott are used to derive uniform error estimates for lagrange finite elements in jrn n 2 under the following conditions 1 the elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis and 2 the space of approximating functions ft in each element t must include the space of polynomials of degree 1 n 2 if n is odd this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded for example if n 3 the usual requirement with shape regularity is that each ft contain all first degree polynomials the result of this paper requires that each ft contain all quadratic polynomials and consequently does not apply to linear courant elements in tetrahedrons or trilinear tensor elements in rectangular boxes counterexamples in these two cases are included

piecemeal graph exploration by a mobile robot we study how a mobile robot can learn an unknown environment in a piecemeal manner the robot s goal is to learn a complete map of its environment while satisfying the constraint that it must return every so often to its starting position for refueling say the environment is modeled as an arbitrary undirected graph which is initially unknown to the robot we assume that the robot can distinguish vertices and edges that it has already explored we present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph g v e in which the robot explores every vertex and edge in the graph by traversing at most o e v1 o 1 edges this nearly linear algorithm improves on the best previous algorithm in which the robot traverses at most o e v2 edges we also give an application of piecemeal learning to the problem of searching a graph for a treasure

a general reflection principle in c2 abstract a real analytic hypersurface m through 0 in cn is said to have the reflection property if any holomorphic mapping defined on one side of m not totally degenerate at 0 and mapping m into another real analytic hypersurface in cn extends holomorphically to a full neighborhood of 0 in cn the main result of this paper is that a real analytic hypersurface in c2 has the reflection property if and only if it is not levi flat

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

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

learning of event recording automata in regular inference a regular language is inferred from answers to a finite set of membership queries each of which asks whether the language contains a certain word one of the most well known regular inference algorithms is the l algorithm due to dana angluin however there are almost no extensions of these algorithms to the setting of timed systems we extend angluin s algorithm for on line learning of regular languages to the setting of timed systems since timed automata can freely use an arbitrary number of clocks we restrict our attention to systems that can be described by deterministic event recording automata deras we present three algorithms tl s g tl n s g and tl s for inference of deras in tl s g and tl n s g we further restrict event recording automata to be event deterministic in the sense that each state has at most one outgoing transition per action learning such an automaton becomes significantly more tractable the algorithm tl n s g builds on tl s g by attempts to construct a smaller in number of locations automaton finally tl s is a learning algorithm for a full class of deterministic event recording automata which infers a so called simple dera which is similar in spirit to the region graph

l p convergence of the immersed boundary method for stationary stokes problems in this paper we analyze the convergence of the immersed boundary method as applied to a static stokes flow problem using estimates obtained in y liu and y mori siam j numer anal 50 2012 pp 2986 3015 we consider a problem in which a d dimensional structure is immersed in an n dimensional domain and prove error estimates for both the pressure and the velocity field in the l p 1 leq p leq infty norm one interesting consequence of our analysis is that the asymptotic error rates in the l 1 norm do not depend on either d or n and in the l p p 1 norm they only depend on n d the resulting estimates are checked numerically for optimality

error estimates in weighted sobolev norms for finite element immersed interface methods abstract when solving elliptic partial differential equations in a region containing immersed interfaces possibly evolving in time it is often desirable to approximate the problem using an independent background discretisation not aligned with the interface itself optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface at the cost of a higher complexity in the implementation a much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions as done in immersed boundary methods these methods are known to have inferior convergence properties depending on the global regularity of the solution across the interface when compared to enriched methods in this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon restricted to a small neighbourhood of the interface in particular we show that optimal approximations can be constructed in a natural and inexpensive way simply by reformulating the problem in a distributionally consistent way and by resorting to weighted norms when computing the global error of the approximation

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

supercloseness on graded meshes for q1 finite element approximation of a reaction diffusion equation in this paper we analyze the standard piece wise bilinear finite element approximation of a model reaction diffusion problem we prove supercloseness results when appropriate graded meshes are used the meshes are those introduced in duran and lombardi 2005 8 but with a stronger restriction on the graduation parameter as a consequence we obtain almost optimal error estimates in the l 2 norm thus completing the error analysis given in duran and lombardi 2005 8

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

a fem for an optimal control problem of fractional powers of elliptic operators we study solution techniques for a linear quadratic optimal control problem involving fractional powers of elliptic operators these fractional operators can be realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension thus we consider an equivalent formulation with a nonuniformly elliptic operator as the state equation the rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation we discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes for the control problem we analyze two approaches one that is semidiscrete based on the so called variational approach where the control is not discretized and the other one that is fully discrete via the discretization of the control by piecewise constant functions for both approaches we derive a priori error estimates with respect to degrees of freedom numerica

map learning with uninterpreted sensors and effectors this dissertation presents a set of methods by which a learning agent called a critter can learn a sequence of increasingly abstract and powerful interfaces to control a robot whose sensorimotor apparatus and environment are initially unknown the result of the learning is a rich hierarchical model of the robot s world its sensorimotor apparatus and environment the learning methods rely on generic properties of the robot s world such as almost everywhere smooth effects of actions on sensory features at the lowest level of the hierarchy the critter analyzes the effects of its actions in order to define control signals one for each of the robot s degrees of freedom it uses a generate and test approach to define sensory features that capture important aspects of the environment it uses linear regression to learn action models that characterize context dependent effects of the control signals on the learned features it uses these models to define high level control laws for finding and following paths defined using constraints on the learned features the critter abstracts these control laws which interact with the continuous environment to a finite set of actions that implement discrete state transitions at this point the critter has abstracted the robot s world to a finite state machine and can use existing methods to learn its structure

delta function approximations in level set methods by distance function extension in a k tornberg b engquist numerical approximations of singular source terms in differential equations j comput phys 200 2004 462 488 it was shown for simple examples that the then m

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

inference of finite automata using homing sequences abstract we present new algorithms for inferring an unknown finite state automaton from its input output behavior even in the absence of a means of resetting the machine to a start state a key technique used is inference of a homing sequence for the unknown automaton our inference procedures experiment with the unknown machine and from time to time require a teacher to supply counterexamples to incorrect conjectures about the structure of the unknown automaton in this setting we describe a learning algorithm that with probability 1 outputs a correct description of the unknown machine in time polynomial in the automaton s size the length of the longest counterexample and log 1 we present an analogous algorithm that makes use of a diversity based representation of the finite state system our algorithms are the first which are provably effective for these problems in the absence of a reset we also present probabilistic algorithms for permutation automata which do not require a teacher to supply counterexamples for inferring a permutation automaton of diversity d we improve the best previous time bound by roughly a factor of d3 log d

on cr mappings between algebraic cauchy riemann manifolds and separate algebraicity for holomorphic functions we prove the algebraicity of smooth cr mappings between algebraic cauchy riemann manifolds a generalization of separate algebraicity principle is established

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

a robust numerical method for a control problem involving singularly perturbed equations we consider an unconstrained linear quadratic optimal control problem governed by a singularly perturbed convection reaction diffusion equation we discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by duran and lombardi in du an and lombardi 2005 2006 we prove convergence of this scheme in addition when the state equation is a singularly perturbed reaction diffusion equation we derive quasi optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes we present several numerical experiments when the state equation is both a reaction diffusion and a convection reaction diffusion equation these numerical experiments reveal a competitive performance of the proposed solution technique

a high order mixed polygonal finite element for incompressible stokes flow analysis abstract this paper presents a new high order mixed scheme of polygonal finite element method pfem for incompressible flow computation the system of polygonal basis functions with bubble enriched functions is applied to represent the fluid velocity field while the pressure field is approximated by the set of polygonal basis functions the new element is named as minipe from a practical point of view the present element naturally satisfies the well known inf sup condition the numerical tests of the incompressible fluid flow problems governed by stokes equation systems are presented to verify the performance quality of proposed method with respect to the stability and convergence rate

a posteriori error estimation for stabilized mixed approximations of the stokes equations the practical implementation of the lowest order pl p0 linear velocity constant pressure finite element method for the steady state incompressible navier stokes equations is addressed in this work three different types of a posteriori error indicator are introduced and each is shown to be equivalent to the discretization error our numerical results show that these indicators can be used to drive an adaptive refinement process which is specially tailored to create grids which conform to the requirements of the local stabilization it is also shown that the indicators provide an effective method for detecting local singularities in the flow solution

poincare automorphisms for nondegenerate cr quadrics where a cele re lr and p ic 2 in 1962 tanaka 11 proved the analogous result for arbi trary nondegenerate hyperquadrics in i e l z w e l e x l e l m w z z where is a nondegenerate hermit ian form in nondegenerate hyperquadrics serve as quadrat ic models of hypersurfaces in c 1 with nondegenerate levi form nondegenerate quadrics in r k are the quadratic models of surfaces with nondegenerate in sense of baouendi jacobowi t tr6ves 1 and beloshapka 2 vector valued levi form

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

error estimates of triangular finite elements under a weak angle condition in this note by analyzing the interpolation operator of girault and raviart given in v girault p a raviart finite element methods for navier stokes equations theory and algorithms in springer series in computational mathematics springer verlag berlin 1986 over triangular meshes we prove optimal interpolation error estimates for lagrange triangular finite elements of arbitrary order under the maximal angle condition in a unified and simple way the key estimate is only an application of the bramble hilbert lemma

discontinuous galerkin method for hyperbolic equations involving delta singularities negative order norm error estimates and applications in this paper we develop and analyze discontinuous galerkin dg methods to solve hyperbolic equations involving delta singularities negative order norm error estimates for the accuracy of dg approximations to delta singularities are investigated we first consider linear hyperbolic conservation laws in one space dimension with singular initial data we prove that by using piecewise k th degree polynomials at time t the error in the h k 2 norm over the whole domain is k 1 2 th order and the error in the h k 1 mathbb r backslash mathcal r t norm is 2k 1 th order where mathcal r t is the pollution region due to the initial singularity with the width of order mathcal o h 1 2 log 1 h and h is the maximum cell length as an application of the negative order norm error estimates we convolve the numerical solution with a suitable kernel which is a linear combination of b splines to obtain l 2 error estimate of 2k 1 th order for the post processed solution moreover we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying duhamel s principle numerical examples including an acoustic equation and the nonlinear rendez vous algorithms are given to demonstrate the good performance of dg methods for solving hyperbolic equations involving delta singularities

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

the poisson and stokes problems on weighted spaces in lipschitz domains and under singular forcing abstract we show the well posedness of the poisson and stokes problems on weighted spaces over general lipschitz domains for a particular range of p we consider those weights in the muckenhoupt class a p that have no singularities in a neighborhood of the boundary of the domain

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

the blob projection method for immersed boundary problems a new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two dimensional domain is presented the method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled these are described by a collection of regularized point forces and the velocity field they induce is computed directly on a regular cartesian grid via a smoothed dipole potential comparisons between this method and the immersed boundary method of peskin and mcqueen are presented the results show that the method proposed here preserves volumes better and has a higher order of convergence

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

on regularizations of the dirac delta distribution in this article we consider regularizations of the dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations pdes we study the convergence of a sequence of distributions s h to a singular term s as a parameter h associated with the support size of s h shrinks to zero we characterize this convergence in both the weak topology of distributions and a weighted sobolev norm these notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature this framework allows different regularizations to be compared the convergence of solutions of pdes with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted sobolev norms we also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems

an efficient query learning algorithm for ordered binary decision diagrams in this paper we propose a new algorithm that exactly learns ordered binary decision diagrams obdds with a given variable ordering via equivalence and membership queries our algorithm uses at most n equivalence queries and at most 2n log2m 3n membership queries where n is the number of nodes in the target reduced obdd and m is the number of variables the upper bound on the number of membership queries is smaller by a factor of o m compared with that for the previous best known algorithm proposed by r gavalda and d guijarro learning ordered binary decision diagrams proceedings of the 6th international workshop on algorithmic learning theory 1995 pp 228 238

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

accuracy of three dimensional analysis of regularized singularities summary in computational mechanics the quadrature of discontinuous and singular functions is often required to avoid specialized quadrature procedures discontinuous and singular fields can be regularized however regularization changes the algebraic structure of the solving equations and this can lead to high errors we show how to acquire accurate and consistent results when regularization is carried out a three dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method the accuracy obtained via gaussian quadrature is compared with that obtained by means of cubpack adaptive quadrature fortran tool the use of regularized functions with non compact and compact support is investigated through an error evaluation procedure based on the use of their fourier transform the proposed procedure leads to the remarkable conclusion that regularized delta functions with non compact support exhibit superior performance copyright 2014 john wiley sons ltd

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

a pde approach to space time fractional parabolic problems we study solution techniques for parabolic equations with fractional diffusion and caputo fractional time derivative the latter being discretized and analyzed in a general hilbert space setting the spatial fractional diffusion is realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension we write our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition we propose and analyze an implicit fully discrete scheme first degree tensor product finite elements in space and an implicit finite difference discretization in time we prove stability and error estimates for this scheme

finite element approximation of elliptic problems with dirac measure terms in weighted spaces applications to one and three dimensional coupled problems in this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving dirac measures using weighted sobolev spaces and weighted discrete norms our approach handles both the cases where the measure is simply a right hand side and where it represents an additional term i e solution dependent in the formulation of the problem the main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains where the coupling terms are seen as dirac measures concentrated on the fractures we first establish a decomposition lemma which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces then we consider the stability of the galerkin approximation with finite elements in weighted norms with uniform and graded meshes we introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf sup conditions in weighted norms then we focus on the convergence of the finite element method due to the lack of regularity the convergence rates are suboptimal for uniform meshes we show that using graded meshes optimal rates are recovered our theoretical results are supported by several numerical experiments finally we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three dimensional media with one dimensional fractures that are found in the analysis of microvascular flows

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

discretization of multipole sources in a finite difference setting for wave propagation problems abstract seismic sources are commonly idealized as point sources due to their small spatial extent relative to seismic wavelengths the acoustic isotropic point radiator is inadequate as a model of seismic wave generation for seismic sources that are known to exhibit directivity therefore accurate modeling of seismic wavefields must include source representations generating anisotropic radiation patterns such seismic sources can be modeled as multipoles that is a time dependent linear combination of spatial derivatives of the spatial delta function since the solutions of linear hyperbolic systems with point source right hand sides are necessarily singular standard results on convergence of grid based numerical methods finite difference or finite element do not imply convergence of numerical solutions we present a method for discretizing multipole sources in a finite difference setting an extension of the moment matching conditions developed for the dirac delta function in other applications along with numerical evidence demonstrating the accuracy of these approximations using this analysis we develop a weak convergence theory for the discretization of a family of symmetric hyperbolic systems of first order partial differential equations with singular source terms solved via staggered grid finite difference methods we show that grid independent space time averages of the numerical solutions converge to the same averages of the continuum solution and provide an estimate for the error in terms of moment matching and truncation error conditions numerical experiments confirm this result but also suggest a stronger one optimal convergence rates appear to be achieved point wise in space away from the source

an adaptive stabilized finite element scheme for a water quality model abstract residual type a posteriori error estimators are introduced in this paper for an advection diffusion reaction problem with a dirac delta source term the error is measured in an adequately weighted w 1 p norm these estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

a regularized xfem framework for embedded cohesive interfaces this work focuses on xfem modeling of elastodamaging interfaces characterized by regularized kinematics after regularization of the displacement discontinuity a process zone of finite width with continuous strain fields emerges the stress fields are independently energy deduced modeled via specific constitutive laws and subsequently introduced within the virtual work principle the methodology can address cohesive interfaces with vanishing and finite thickness in a unified way two dimensional results for a strain localization problem and several delamination tests are discussed in detail and compared with previous numerical experimental results