a fast algorithm for optimum height limited alphabetic binary trees in this paper an o nl log n time algorithm is presented for construction of an optimal alphabetic binary tree with height restricted to l this algorithm is an alphabetic version of the package merge algorithm and yields an o nl log n time algorithm for the alphabetic huffman coding problem the alphabetic package merge algorithm is quite simple to describe but appears hard to prove correct garey siam j comput 3 1974 pp 101 110 gives an o n 3 log n time algorithm for the height limited alphabetic binary tree problem itai siam j comput 5 1976 pp 9 18 and wessner inform process lett 4 1976 pp 90 94 independently reduce this time to o n 2l for the alphabetic problem in siam j comput 16 1987 pp 1115 1123 a rather complex o n 3 2 l log 1 2 n time hybrid algorithm is given for length limited huffman coding the package merge algorithm discussed in this paper first appeared in tech report 88 01 ics dept univ of california irvine ca but without proof of correctness

quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

a balanced finite element method for singularly perturbed reaction diffusion problems consider the singularly perturbed linear reaction diffusion problem varepsilon 2 delta u bu f in omega subset mathbb r d u 0 on partial omega where d ge 1 the domain ome

maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions a semilinear second order parabolic equation is considered in a regular and a singularly perturbed regime for this equation we give computable a posteriori error estimates in the maximum norm semidiscrete and fully discrete versions of the backward euler crank nicolson and discontinuous galerkin dg r methods are addressed for their full discretizations we employ elliptic reconstructions that are respectively piecewise constant piecewise linear and piecewise quadratic for r 1 in time we also use certain bounds for the green s function of the parabolic operator

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

unary prefixed huffman coding for a group of quantized generalized gaussian sources we propose a coding algorithm called unary prefixed huffman uph coding for a family of infinite sources called quantized generalized gaussian gg sources compared with the existing codes for these gg sources the uph algorithm provides a more adaptive approach and its coding efficiency is upper bounded by entropy 2

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

quantum cohomology and the periodic toda lattice we describe a relation between the periodic one dimensional toda lattice and the quantum cohomology of the periodic flag manifold an infinite dimensional kahler manifold this generalizes a result of givental and kim relating the open toda lattice and the quantum cohomology of the finite dimensional flag manifold we derive a simple and explicit differential operator formula for the necessary quantum products which applies both to the finite dimensional and to the infinite dimensional situations

the uniform saturation property for a singularly perturbed reaction diffusion equation although adaptive finite element methods for solving elliptic problems often work well in practice they are usually not proven to converge for poisson like problems an exception is given by the method of dorfler 8 that was later improved by morin nochetto and siebert 11 in this paper we extend these methods by constructing an adaptive finite element method for a singularly perturbed reaction diffusion equation that in energy norm converges uniformly in the size of the reaction term moreover in this algorithm the arising galerkin systems are solved only inexactly so that generally the number of arithmetic operations is equivalent to the number of triangles in the final partition

conditions for optimality of the huffman algorithm a new general formulation of huffman tree construction is presented which has broad application recall that the huffman algorithm forms a tree in which every node has some associated weight by specifying at every step of the construction which nodes are to be combined to form a new node with a new combined weight we characterize a wide class of weight combination functions the quasilinear functions for which the huffman algorithm produces optimal trees under correspondingly wide classes of cost criteria in addition known results about huffman tree construction and related concepts from information theory and from the theory of convex functions are tied together suggestions for possible future applications are given

maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction diffusion problem a singularly perturbed semilinear reaction diffusion equation posed in the unit square is discretized on arbitrary nonuniform tensor product meshes we establish a second order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter no mesh aspect ratio assumption is made numerical results are presented that support our theoretical estimate

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

quantum cohomology of flag manifolds abstract we study two aspects of quantum schubert calculus a presentation of the small quantum cohomology ring of partial flag manifolds and a quantum giambelli formula our proof gives a relationship between universal schubert polynomials as defined by fulton and quantum schubert polynomials as defined by fomin gelfand and postnikov and later extended by ciocan fontanine intersection theory on hyperquot schemes is an essential element of the proof

a short note on the redundancy of degreea it is known that there always exists an instantaneous uniquely decodable code such that the redundancy of the discrete source is bounded between 0 and 1 johnsen 5 provided sharper bounds on the redundancy by making assumptions on the most likely source letter in this paper we have obtained bounds on the redundancy of degree by applying johnsen s approach

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

pointwise approximation of corner singularities for a singularly perturbed reaction diffusion equation in an l shaped domain a singularly perturbed reaction diffusion equation is posed in a two dimensional l shaped domain subject to a continuous dirchlet boundary condition its solutions are in the holder space c 2 3 and typically exhibit boundary layers and corner singularities the problem is discretized on a tensor product shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3 2 we establish almost second order convergence of our numerical method in the discrete maximum norm uniformly in the small diffusion parameter numerical results are presented that support our theoretical error estimate

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

a note on the convergence of linear finite elements let u be the solution of the dirichlet problem for laplace s equation in a smooth domain omega and let u h be its piecewise linear finite element approximation if the family of partitions is quasi uniform it is known that left u u h right l p leqq ch 2 u w 2 p quad text for 1 p infty and left u u h right l infty leqq ch 2 log frac 1 h u w 2 infty and moreover the log 1 h factor cannot be removed from the last estimate the objective of this paper is to show that if u in w 2 infty the error is of optimal order in the bounded mean oscillation bmo norm that is left u u h right text bmo leqq ch 2 u w 2 infty this norm is very close to the uniform norm and moreover the space bmo plays the role of l infty in many applications in real analysis the result is obtained by a duality argument as in the l p case making use of an appropriate regularity result

optimal prefix codes for sources with two sided geometric distributions a complete characterization of optimal prefix codes for off centered two sided geometric distributions of the integers is presented these distributions are often encountered in lossless image compression applications as probabilistic models for image prediction residuals the family of optimal codes described is an extension of the golomb codes which are optimal for one sided geometric distributions the new family of codes allows for encoding of prediction residuals at a complexity similar to that of golomb codes without recourse to the heuristic approximations frequently used when modifying a code designed for nonnegative integers so as to apply to the encoding of any integer optimal decision rules for choosing among a lower complexity subset of the optimal codes given the distribution parameters are also investigated and the relative redundancy of the subset with respect to the full family of optimal codes is bounded

existence of optimal prefix codes for infinite source alphabets it is proven that for every random variable with a countably infinite set of outcomes and finite entropy there exists an optimal prefix code which can be constructed from huffman codes for truncated versions of the random variable and that the average lengths of any sequence of huffman codes for the truncated versions converge to that of the optimal code also it is shown that every optimal infinite code achieves kraft s inequality with equality

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

optimal prefix codes for infinite alphabets with nonlinear costs let p p i be a measure of strictly positive probabilities on the set of nonnegative integers although the countable number of inputs prevents usage of the huffman algorithm there are nontrivial p for which known methods find a source code that is optimal in the sense of minimizing expected codeword length for some applications however a source code should instead minimize one of a family of nonlinear objective functions beta exponential means those of the form loga sigmaip i an i where n i is the length of the ith codeword and a is a positive constant applications of such minimizations include a novel problem of maximizing the chance of message receipt in single shot communications a 1 this paper introduces methods for finding codes optimal for such exponential means one method applies to geometric distributions while another applies to distributions with lighter tails the latter algorithm is applied to poisson distributions and both are extended to alphabetic codes as well as to minimizing maximum pointwise redundancy the aforementioned application of minimizing the chance of buffer overflow is also considered

eine aussage zurl stabilitat und zur genauen konvergenzordnung derh 0 0 projektionen in this paper we settle a question concerning the necessity of the logh factor appearing in error estimates of linear finite element solutions let 2 be a bounded convex polygon and t h a regular family of subdivisions of into triangles be furthers 0 h h 0 1 the corresponding linear finite element spaces andp h h 0 1 s 0 h theh 0 1 projections with respect to the uniform norm asymptotically estimates of the form parallel u p h u parallel infty leqq c logh dist u s 0 h u in c bar omega cap h 0 1 omega hold as pointed out by schatz 1980 for functionsu with bounded second derivatives this implies parallel u p h u parallel infty leqq ch 2 log h u 2 infty a result which was proved by nitsche 1975 and scott 1976 with different methods it was an open question if the factor logh is necessary or only due to the methods of proof in this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if logh is replaced by a termo logh

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

on the redundancy of optimal codes with limited word length some limitations are given on the redundancy of d ary codes with maximal codeword length l an upper bound that improves on a previous result of e n gilbert 1971 is given it is then shown that the redundancy of these constrained codes is very close to that of the unconstrained huffman codes when the number of codewords n is such that nd sup 1 l becomes negligible a tight bound is given on the redundancy when only the most likely probabilities are known in the binary case a tight lower bound is given on the redundancy when only the least likely probability is known

lagrangian empirical design of variable rate vector quantizers consistency and convergence rates the lagrangian formulation of variable rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized lloyd algorithms for quantizer design the lagrangian formulation is demonstrated to provide a convenient framework for analyzing the empirical design of variable rate vector quantizers in particular the consistency of empirical design based on minimizing the lagrangian performance over a stationary and ergodic training sequence is shown for sources with finite second moment the finite sample performance is also studied for independent training data and sources with bounded support

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

optimal prefix codes for pairs of geometrically distributed random variables optimal prefix codes are studied for pairs of independent integer valued symbols emitted by a source with a geometric probability distribution of parameter q 0 1 covering the range q 1 2 the described codes produce the expected reduction in redundancy with respect to the 1 d case while maintaining low complexity coding operations

a combinatorial approach to golomb forests optimal binary prefix free codes for infinite sources with geometrically distributed frequencies e g p pi 1 p i 0 0p1 were first implicitly suggested by golomb over 30 years ago in the context of run length encodings ten years later gallager and van voorhis exhibited such optimal codes for all values of p these codes were derived by using the hu2man encoding algorithm to build optimal codes for finite sources and then showing that the finite codes converge in a very specific sense to the infinite one in this note we present a new combinatorial approach to solve the same problem one that does not use the hu2man algorithm but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence one consequence of this new approach is a complete characterization of all of the optimal codes in particular it shows that for all p 0p1 except for an easily describable countable set there is a unique optimal code but for each p in this countable set there are an uncountable number of optimal codes another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one a final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees

about adaptive coding on countable alphabets max stable envelope classes in this paper we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets this task is generally not achievable without restricting the class of sources over which universality is desired building on our prior work we propose natural families of sources characterized by a common dominating envelope we particularly emphasize the notion of adaptivity which is the ability to perform as well as an oracle knowing the envelope without actually knowing it this is closely related to the notion of hierarchical universal source coding but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible namely by including max stable heavy tailed envelopes which are excellent models in many applications such as natural language modeling we derive a minimax lower bound on the redundancy of any code on such envelope classes including an oracle that knows the envelope we then propose a constructive code that does not use knowledge of the envelope the code is computationally efficient and is structured to use an expanding threshold for auto censoring etac and we therefore dub it the etac code we prove that the etac code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length and can be therefore qualified as a near adaptive code over families of heavy tailed envelopes for finite and light tailed envelopes the penalty is even less and the same code follows closely previous results that explicitly made the light tailed assumption our technical results are founded on methods from regular variation theory and concentration of measure

how accurate are finite elements on anisotropic triangulations in the maximum norm abstract in kopteva 2014 a counterexample of an anisotropic triangulation was given on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate this example was given in the context of a singularly perturbed reaction diffusion equation in this paper we present further examples of unanticipated pointwise convergence behaviour of lagrange finite elements on anisotropic triangulations in particular we show that linear finite elements may exhibit lower than expected orders of convergence for the laplace equation as well as for certain singular equations and their accuracy depends not only on the linear interpolation error but also on the mesh topology furthermore we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher order finite elements are employed on anisotropic meshes a theoretical justification will be given for some of the observed numerical phenomena

a dichotomous search for a geometric random variable we are given a two state system that starts in state 0 ends in state 1 and makes a single transition from state 0 to state 1 during n periods if the system is in state 0 it moves to state 1 in the next period with a known positive time independent probability once it reaches state 1 it remains there by observing the state of the system at some intermediate period we can learn whether this transition occurred earlier or not an optimal search strategy minimizes the expected number of observations needed to locate the exact transition time in this paper we show how to compute efficiently and how to approximate the optimal strategy applications to the problem arise in the areas of quality control and maintenance of communication and supply lines

fully localized a posteriori error estimators and barrier sets for contact problems we derive novel pointwise a posteriori error estimators for elliptic obstacle problems which except for obstacle resolution completely vanish within the full contact set localization we then construct a posteriori barrier sets for free boundaries under a natural stability or nondegeneracy condition we illustrate localization properties as well as reliability and efficiency for both solutions and free boundaries via several simulations in 2 and 3 dimensions

minimum delay codes huffman s algorithm finds a prefix free binary code on a weighted alphabet which minimizes the expected length of the code string for a single symbol a definition is given for the expected delay for a prefix free binary code on a weighted alphabet it is the expected time between a request to transmit the symbol and the completion of that transmission assuming a channel with fixed capacity where requests are queued an o n 5 time o n 3 space convex hull algorithm is given that finds a code of minimal expected delay where n is the size of the alphabet it is conjectured that the algorithm has substantially lower time and space complexities in the worst case and still lower in the average case the heart of the proof of polynomial time complexity is the convex hull theorem which states that under certain conditions a binary tree that minimizes one penalty measure can be changed to a binary tree that minimizes a second penalty measure in a carefully controlled sequence of changes called elementa

on the cost of worst case coding length constraints we investigate the redundancy that arises from adding a worst case length constraint to uniquely decodable fixed to variable codes over achievable huffman 1952 codes this is in contrast to the traditional metric of the redundancy over the entropy we show that the cost for adding constraints on the worst case coding length is small and that the resulting bound is related to the fibonacci numbers

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

fully computable robust a posteriori error bounds for singularly perturbed reaction diffusion problems a procedure for the construction of robust upper bounds for the error in the finite element approximation of singularly perturbed reaction diffusion problems was presented in ainsworth and babuska siam j numer anal 36 2 331 353 1999 which entailed the solution of an infinite dimensional local boundary value problem it is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article we view the present work as fulfilling that promise and as completing the investigation begun in ainsworth and babuska siam j numer anal 36 2 331 353 1999 by removing the obligation to solve a local problem exactly the resulting new estimator is indeed fully computable and the first to provide fully computable robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method

quantum sheaf cohomology and duality of flag manifolds we study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds realized as the ope ring of a 2 twisted two dimensional theories with 0 2 supersymmetry quantum sheaf cohomology generalizes the notion of quantum cohomology complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models glsms and for nonabelian glsms describing grassmannians in this paper we continue to explore the quantum sheaf cohomology of nonabelian theories we first propose a method to compute the generating relations for 0 2 glsms with 2 2 locus we apply this method to derive the quantum sheaf cohomology of products of grassmannians and flag manifolds the dual deformation associated with the biholomorphic duality gives rise to an explicit ir duality of two a 2 twisted 0 2 gauge theories

decay rates of adaptive finite elements with dorfler marking we investigate the decay rate for an adaptive finite element discretization of a second order linear symmetric elliptic pde we allow for any kind of estimator that is locally equivalent to the standard residual estimator this includes in particular hierarchical estimators estimators based on the solution of local problems estimators based on local averaging equilibrated residual estimators the zz estimator etc the adaptive method selects elements for refinement with dorfler marking and performs a minimal refinement in that no interior node property is needed based on the local equivalence to the residual estimator we prove an error reduction property in combination with minimal dorfler marking this yields an optimal decay rate in terms of degrees of freedom

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

a diary on information theory foreword on the mathematical notion of information games of chance and probability theory notes on the teaching of probability theory variations on a theme by fibonacci the mathematical theory of trees

source coding for quasiarithmetic penalties whereas huffman coding finds a prefix code minimizing mean codeword length for a given finite item probability distribution quasiarithmetic or quasilinear coding problems have the goal of minimizing a generalized mean of the form rho 1 sigmaipirho li where li denotes the length of the ith codeword p i denotes the corresponding probability and rho is a monotonically increasing cost function such problems proposed by campbell have a number of diverse applications several cost functions are shown here to yield quasiarithmetic problems with simple redundancy bounds in terms of a generalized entropy a related property also shown here involves the existence of optimal codes for well behaved cost functions optimal codes always exist for possibly infinite alphabet sources having finite generalized entropy an algorithm is introduced for finding binary codes optimal for convex cost functions this algorithm which can be extended to other minimization utilities can be performed using quadratic time and linear space this reduces the computational complexity of a problem involving minimum delay in a queue allows combinations of previously considered problems to be optimized and greatly expands the set of problems solvable in quadratic time and linear space

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

on quantum cohomology rings of partial flag varieties 0 introduction the main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces sln c p where p is a parabolic subgroup the quantum cohomology ring of a smooth projective variety or more generally of a symplectic manifoldx has been introduced by string theorists see va and w1 roughly speaking it is a deformation of the usual cohomology ring with parameter space given byh x the multiplicative structure of quantum cohomology encodes the enumerative geometry of rational curves on x in the past few years the highly nontrivial task of giving a rigorous mathematical treatment of quantum cohomology has been accomplished both in the realm of algebraic and symplectic geometry in various degrees of generality this can be found in beh behm km lit1 lit2 mcs and rt as well as in the surveys fp and t if one restricts the parameter space to h 1 1 x one gets the small quantum cohomology ring this ring in the case of partial flag varieties is the object of the present paper in order to state our main results we first describe briefly the classical side of the story we interpret the homogeneous space f sln c p as the complex projective variety parametrizing flags of quotients of c of given ranks say nk n1 by a classical result of ehresmann e the integral cohomology of f can be described geometrically as the free abelian group generated by the schubert classes these are the poincare duals of fundamental classes of certain subvarieties w f one for each element of the subset s s n1 nk of the symmetric group sn consisting of permutations w with descents in n1 nk a description of the multiplicative structure is provided by yet another classical theorem due to borel bor which gives a presentation forh f z specifically let 1 1 1 n1 2 1 2 n2 n1 k 1 1 k 1 n nk be n independent variables define an to be the block diagonal matrix diag d1 d2 dk 1 where

optimal algorithms for universal random number generation from finite memory sources we study random number generators rngs both in the fixed to variable length fvr and the variable to fixed length vfr regimes in a universal setting in which the input is a finite memory source of arbitrary order and unknown parameters with arbitrary input and output finite alphabet sizes applying the method of types we characterize essentially unique optimal universal rngs that maximize the expected output respectively minimize the expected input length in the fvr respectively vfr case for the fvr case the rng studied is a generalization of elias s scheme while in the vfr case the general scheme is new we precisely characterize up to an additive constant the corresponding expected lengths which include second order terms similar to those encountered in universal data compression and universal simulation furthermore in the fvr case we consider also a twice universal setting in which the markov order k of the input source is also unknown

huffman coding with an infinite alphabet a new type of sufficient condition is provided for a probability distribution on the nonnegative integers to be given an optimal d ary prefix code by a huffman type algorithm in the justification of our algorithm we introduce two new essentially one concepts as the definition of the optimality of a prefix d ary code which are shown to be equivalent to that defined in the traditional way these new concepts of the optimality are meaningful even for the case where the shannon entropy h p diverges

pointwise a posteriori error control for discontinuous galerkin methods for elliptic problems an a posteriori error bound for the maximum pointwise error for the interior penalty discontinuous galerkin method for a standard elliptic model problem on polyhedral domains is presented the computational domain is not required to be lipschitz thus allowing for domains with cracks and other irregular polyhedral domains the proof is based on the direct use of green s functions and varies substantially from the approach used in previous proofs of similar l infty estimates for continuous finite element methods in the literature numerical experiments indicating the good behavior of the resulting a posteriori bounds within an adaptive algorithm are also presented

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings