new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

extrapolation of the dirichlet problem for elliptic equations with complex coefficients abstract in this paper we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p ellipticity specifically let be a chord arc domain in r n and the operator l i a i j x j b i x i be elliptic with b i x k x 1 for a small k let p 0 sup p 1 a is p elliptic we establish that if the l q dirichlet problem is solvable for l for some 1 q p 0 n 1 n 2 then the l p dirichlet problem is solvable for all p in the range q p 0 n 1 n 2 in particular if the matrix a is real or n 2 the l p dirichlet problem is solvable for p in the range q

gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

non self adjoint hamiltonians defined by riesz bases we discuss some features of non self adjoint hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a riesz basis of hilbert space among other things we give conditions under which these hamiltonians can be factorized in terms of generalized lowering and raising operators

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

an efficient local search framework for the minimum weighted vertex cover problem the minimum weighted vertex cover mwvc problem an extension of the classical minimum vertex cover mvc problem is an important np complete combinatorial optimization problem with a wide range of applications the objective of this paper is to design an efficient local search algorithm to solve the mwvc problem first the weighted edge strategy is proposed to define the dynamic scoring strategy so that our algorithm can find different possible optimal solutions second the weighted configuration checking wcc strategy is proposed to overcome the cycling problem in local search by combining the wcc strategy with the scoring strategy we design the vertex selection strategy to determine the vertex to be selected as a candidate solution component based on these strategies a novel local search framework namely diversion local search based on weighted configuration checking dlswcc is presented dlswcc is evaluated against several state of the art algorithms on various benchmark instances experimental results show that dlswcc outperforms its competitors in terms of both solution quality and computational efficiency in most classical instances specifically dlswcc can obtain 22 new upper bounds of 71 moderate scale problem instances 5 new upper bounds of 15 large scale problem instances and 56 new upper bounds of 56 massive graph instances

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

some remarks on quasi hermitian operators a quasi hermitian operator is an operator that is similar to its adjoint in some sense via a metric operator i e a strictly positive self adjoint operator whereas those metric operators are in general assumed to be bounded we analyze the structure generated by unbounded metric operators in a hilbert space following our previous work we introduce several generalizations of the notion of similarity between operators then we explore systematically the various types of quasi hermitian operators bounded or not finally we discuss their application in the so called pseudo hermitian quantum mechanics

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

iterated tabu search and variable neighborhood descent for packing unequal circles into a circular container this paper presents an iterated tabu search and variable neighborhood descent its vnd algorithm for packing unequal circles into a circular container pucc the algorithm adapts the tabu search procedure of iterated tabu search algorithms and proposes a tabu search and variable neighborhood descent ts vnd procedure we observe there are strong complementarities between the small circles and the large vacant places and propose the insert neighborhood to match up the small circles with the large vacant places although the insert neighborhood is inefficient and time consuming it is an important supplement to the classic swap neighborhood as it could arrange the small circles properly predicated on these features we employ the insert neighborhood only at chosen local minima of the swap neighborhood that shows promise for an improvement the traditional tabu search procedure is then transformed into a hybrid procedure composed of two alternative parts namely variable neighborhood descent and tabu search respectively besides this reformed procedure its vnd also incorporates other new features such as an adaptive evaluation function a novel method for accelerating the neighborhood exploration and the collision accidents criterion for evaluating how intensively the area near the current solution has been explored the computational results on three well established benchmark sets show that the proposed algorithm not only has a good discovery capability but also can provide good results within a reasonable time for a total of 84 benchmark instances the proposed algorithm improves the best known results on 23 instances matches 60 and only misses one

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

the bipartite quadratic assignment problem and extensions abstract we introduce a new binary quadratic program that subsumes the well known quadratic assignment problem this problem is shown to be np hard when some associated parameters are restricted to simple values but solvable in polynomial time when some other parameter values are restricted three different neighborhood structures are introduced a best solution in all these neighborhoods can be identified in polynomial time even though two of these neighborhoods are of exponential size different local search algorithms and their enhancements using tabu search are developed using these neighborhoods experimental analysis show that two hybrid algorithms obtained by combining these neighborhoods in specific ways yield results that are superior to the algorithms that use these neighborhoods separately extensive computational results and statistical analysis of the resulting data are presented

adaptive memory tabu search for binary quadratic programs recent studies have demonstrated the effectiveness of applying adaptive memory tabu search procedures to combinatorial optimization problems in this paper we describe the development and use of such an approach to solve binary quadratic programs computational experience is reported showing that the approach optimally solves the most difficult problems reported in the literature for challenging problems of limited size which are capable of being approached by exact procedures we find optimal solutions considerably faster than the best reported exact method moreover we demonstrate that our approach is significantly more efficient and yields better solutions than the best heuristic method reported to date finally we give outcomes for larger problems that are considerably more challenging than any currently reported in the literature

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

interpolation theory and refinement of nested hilbert spaces using the method of quadratic interpolation between hilbert spaces we show that a nested hilbert space with countable index set admits arbitrarily many proper refinements by nested hilbert spaces

effective metaheuristic algorithms for the minimum differential dispersion problem abstract this paper presents tabu search and memetic search algorithms for solving the minimum differential dispersion problem the tabu search algorithm employs a neighborhood decomposition candidate list strategy and a rarely used solution based tabu memory unlike the typical attribute based tabu list the solution based tabu strategy leads to a more highly focused intensification process and avoids tuning the tabu tenure while employing coordinated hash functions that accelerate the determination of tabu status the memetic search algorithm incorporates the tabu search procedure within it and makes use of a crossover operator that generates solution assignments by an evaluation mechanism that includes both quality and distance criteria experimental results on a benchmark testbed of 250 problems reveal that our tabu search algorithm is capable of discovering better solutions for 179 71 6 of the problem instances while our memetic search algorithm finds better solutions for 157 62 8 of the instances collectively yielding better solutions for 181 72 4 of the test problems than recently reported state of the art algorithms

a hybrid metaheuristic approach to solving the ubqp problem this paper presents a hybrid metaheuristic approach hma for solving the unconstrained binary quadratic programming ubqp problem by incorporating a tabu search procedure into the framework of evolutionary algorithms the proposed approach exhibits several distinguishing features including a diversification based combination operator and a distance and quality based replacement criterion for pool updating the proposed algorithm is able to easily obtain the best known solutions for 31 large random instances up to 7000 variables which no previous algorithm has done and find new best solutions for three of nine instances derived from the set partitioning problem demonstrating the efficacy of our proposed algorithm in terms of both solution quality and computational efficiency furthermore some key elements and properties of the hma algorithm are also analyzed

square roots of elliptic operators consider an elliptic sesquilinear form defined on v v by j u v ajk u xk t6v xj ak u xkv ju t6v xj auvdx where v is a closed subspace of h1 which contains c0 is a bounded lipschitz domain in rn ajk ak j a l and re ajk k j 0 for all cn with 1 let l be the operator with largest domain satisfying j u v lu v for all v then l i is a maximal accretive operator in l2 for a sufficiently large real number it is proved that l i 12 is a bounded operator from v to l2 provided mild regularity of the coefficients is assumed in addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense then the corresponding square root operators also depend differentiably on t the latter result is new even when the forms j are hermitian

wave packet dynamics in slowly modulated photonic graphene abstract mathematical analysis on electromagnetic waves in photonic graphene a photonic topological material which has a honeycomb structure is one of the most important current research topics by modulating the honeycomb structure numerous topological phenomena have been observed recently the electromagnetic waves in such media are generally described by the 2 dimensional wave equation it has been shown that the corresponding elliptic operator with a honeycomb material weight has dirac points in its dispersion surfaces in this paper we study the time evolution of the wave packets spectrally concentrated at such dirac points in a modulated honeycomb material weight we prove that such wave packet dynamics is governed by the dirac equation with a varying mass in a large but finite time our analysis provides mathematical insights to those topological phenomena in photonic graphene

approximating the cut norm via grothendieck s inequality the cut norm a c of a real matrix a a ij i in r j in s is the maximum over all i subset r j subset s of the quantity sum i in i j in j a ij this concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems here we show that the problem of approximating the cut norm of a given real matrix is max snp hard and we provide an efficient approximation algorithm this algorithm finds for a given matrix a a ij i in r j in s two subsets i subset r and j subset s such that sum i in i j in j a ij geq rho a c where rho 0 is an absolute constant satisfying rho 0 56 the algorithm combines semidefinite programming with a rounding technique based on grothendieck s inequality we present three known proofs of grothendieck s inequality with the necessary modifications which emphasize their algorithmic aspects these proofs contain rounding techniques which go beyond the random hyperplane rounding of goemans and williamson j acm 42 1995 pp 1115 1145 allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

heat kernels of second order complex elliptic operators and applications we study the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on rn we obtain gaussian bounds without further assumption ifn 2 and when the principal part has holder continuous coefficients ifn 3 a thorough study of the boundedness properties of the green operator is made in dimension 1 we construct the fundamental solution of these operators in dimension 2 boundedness results of the maximal accretive square roots onlpsobolev spaces and holder spaces are obtained whenn 2 under holder continuity assumption on the coefficients boundedh functional calculi onlp 1 p are also discussed

characterizations of domains of fractional powers for non negative operators abstract it is known that a closed linear operator a defined densely on a banach space x is non negative if and only if a generates a bounded analytic times resolvent family for some 0 2 in this paper by using such resolvent families as well as the komatsu representations we characterize systematically the domains of fractional powers of non negative operators on banach spaces gaps between the subset 0 d a and the whole space x are discussed as well

average value of solutions for the bipartite boolean quadratic programs and rounding algorithms we consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem bbqp with m n variables a closed form formula is developed to compute the average objective function value a of all solutions in o m n time however computing the median objective function value of the solutions is shown to be np hard also we show that any solution with objective function value no worse than a dominates at least 2 m n 2 solutions and this bound is the best possible further we show that such a solution can be identified in o m n time and hence the domination ratio of this algorithm is at least 1 4 we then show that for any fixed natural numbers a and b such that a b 1 no polynomial time approximation algorithm exists for bbqp with domination ratio larger than 1 2 1 m n unless p np it is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than a one of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value a new integer programming formulation of bbqp is also given and computational results with our rounding algorithms are reported

square roots of elliptic second order divergence operators on strongly lipschitz domains l p theory we study l p estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form l mathrm div a nabla on lipschitz domains subject to dirichlet or to neumann boundary conditions pursuing our work 4 where we considered operators on mathbb r n we obtain among other things l 1 2 f p le c nabla f p for all 1 p infty if l is real symmetric and the domain bounded which is new for 1 p 2 we also obtain similar results for perturbations of constant coefficients operators our methods rely on a singular integral representation calderon zygmund theory and quadratic estimates a feature of this study is the use of a commutator between the resolvent of the laplacian dirichlet and neumann and partial derivatives which carries the geometry of the boundary

path relinking for unconstrained binary quadratic programming abstract this paper presents two path relinking algorithms to solve the unconstrained binary quadratic programming ubqp problem one is based on a greedy strategy to generate the relinking path from the initial solution to the guiding solution and the other operates in a random way we show extensive computational results on five sets of benchmarks including 31 large random ubqp instances and 103 structured instances derived from the maxcut problem comparisons with several state of the art algorithms demonstrate the efficacy of our proposed algorithms in terms of both solution quality and computational efficiency it is noteworthy that both algorithms are able to improve the previous best known results for almost 40 percent of the 103 maxcut instances

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

layer potentials and boundary value problems for elliptic equations with complex l coefficients satisfying the small carleson measure norm condition abstract we consider divergence form elliptic equations l u a u 0 in the half space r n 1 x t r n 0 whose coefficient matrix a is complex elliptic bounded and measurable in addition we suppose that a satisfies some additional regularity in the direction transverse to the boundary namely that the discrepancy a x t a x 0 satisfies a carleson measure condition of fefferman kenig pipher type with small carleson norm under these conditions we establish a full range of boundedness results for double and single layer potentials in l p hardy sobolev bmo and holder spaces furthermore we prove solvability of the dirichlet problem for l with data in l p r n bmo r n and c r n and solvability of the neumann and regularity problems with data in the spaces l p r n h p r n and l 1 p r n h 1 p r n respectively with the appropriate restrictions on indices assuming invertibility of layer potentials for the t independent operator l 0 a 0

maximal regularity for non autonomous evolution equations we consider the maximal regularity problem for non autonomous evolution equations open image in new window 1 each operator a t is associated with a sesquilinear form mathop varvec mathfrak a t cdot cdot on a hilbert space h we assume that these forms all have the same domain and satisfy some regularity assumption with respect to t e g piecewise alpha holder continuous for some open image in new window we prove maximal l p regularity for all u 0 in the real interpolation space open image in new window the particular case where p 2 improves previously known results and gives a positive answer to a question of lions equations differentielles operationnelles et problemes aux limites springer berlin 1961 on the set of allowed initial data u 0

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

inapproximability results for maximum edge biclique minimum linear arrangement and sparsest cut we consider the minimum linear arrangement problem and the uniform sparsest cut problem so far these two notorious np hard graph problems have resisted all attempts to prove inapproximability results we show that they have no polynomial time approximation scheme unless np complete problems can be solved in randomized subexponential time furthermore we show that the same techniques can be used for the maximum edge biclique problem for which we obtain a hardness factor similar to previous results but under a more standard assumption

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

integrating tabu search and vlsn search to develop enhanced algorithms a case study using bipartite boolean quadratic programs abstract the bipartite boolean quadratic programming problem bbqp is a generalization of the well studied boolean quadratic programming problem the model has a variety of real life applications however empirical studies of the model are not available in the literature except in a few isolated instances in this paper we develop efficient heuristic algorithms based on tabu search very large scale neighborhood vlsn search and a hybrid algorithm that integrates the two the computational study establishes that effective integration of simple tabu search with vlsn search results in superior outcomes and suggests the value of such an integration in other settings complexity analysis and implementation details are provided along with conclusions drawn from experimental analysis in addition we obtain solutions better than the best previously known for almost all medium and large size benchmark instances

partial inner product spaces metric operators and generalized hermiticity motivated by the recent developments of pseudo hermitian quantum mechanics we analyze the structure of unbounded metric operators in a hilbert space it turns out that such operators generate a canonical lattice of hilbert spaces that is the simplest case of a partial inner product space pip space next we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties then we apply some of the previous results to operators on a particular pip space namely a scale of hilbert spaces generated by a metric operator finally we reformulate the notion of pseudo hermitian operators in the preceding formalism

a levenberg marquardt method based on sobolev gradients abstract we extend the theory of sobolev gradients to include variable metric methods such as newton s method and the levenberg marquardt method as gradient descent iterations associated with stepwise variable inner products in particular we obtain existence uniqueness and asymptotic convergence results for a gradient flow based on a variable inner product