on the efficiency of pairing heaps and related data structures the pairing heap is well regarded as an efficient data structure for implementing priority queue operations it is included in the gnu c library strikingly simple in design the pairing heap data structure nonetheless seems difficult to analyze belonging to the genre of self adjusting data structures with its design originating as a self adjusting analogue of the fibonacci heap it has been previously conjectured that the pairing heap provides constrant amortized time decrease key operations and experimental studies have supported this conjecture this paper demonstrates contrary to conjecture that the pairing heap requires more than constant amortized time to perform decrease key operations moreover new experimental findings are presented that reveal detectable growth in the amortized cost of the decrease key operation second a unifying framework is developed that includes both pairing heaps and fibonacci heaps the parameter of interest in this framework is the storage capacity available in the nodes of the data structure for auxiliary balance information fields in this respect fibonacci heaps require log log n bits per node when n items are present this is shown to be asymptotically optimal for data structures that achieve the same asymptotic performance bounds as fibonacci heaps and fall within this framework

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

non extremal sextic moment problems abstract positive semidefiniteness recursiveness and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems also positive semidefiniteness combined with another necessary condition consistency is a sufficient condition in the case of extremal moment problems i e when the rank of the moment matrix denoted by r and the cardinality of the associated algebraic variety denoted by v are equal however these conditions are not sufficient for non extremal sextic or higher order truncated moment problems in this paper we settle three key instances of the non extremal i e r v sextic moment problem as follows when r 7 positive semidefiniteness consistency and the variety condition guarantee the existence of a 7 atomic representing measure when r 8 we construct two determining algorithms corresponding to the cases v 9 and v to accomplish this we generalize the rank reduction technique developed in previous work where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure

on the dynamic finger conjecture for splay trees part ii the proof the following result is shown on an n node splay tree the amortized cost of an access at distance d from the preceding access is o log d 1 in addition there is an o n initialization cost the accesses include searches insertions and deletions

poisson structures on the cotangent bundle of a lie group or a principle bundle and their reductions on a cotangent bundle t g of a lie group g one can describe the standard liouville form and the symplectic form d in terms of the right maurer cartan form and the left moment mapping of the right action of g on itself and also in terms of the left maurer cartan form and the right moment mapping and also the poisson structure can be written in related quantities this leads to a wide class of exact symplectic structures on t g and to poisson structures by replacing the canonical momenta of the right or left actions of g on itself by arbitrary ones followed by reduction to g cross a weyl chamber e g this method also works on principal bundles

the core variety of a multisequence in the truncated moment problem abstract let m i i z n i m 0 0 denote a real n dimensional multisequence of finite degree m the truncated moment problem concerns the existence of a positive borel measure supported in r n such that 0 1 i r n x i d i z n i m we associate to 2 d an algebraic variety in r n called the core variety v v the core variety contains the support of each representing measure we show that if v is nonempty then 2 d 1 has a representing measure moreover if v is a nonempty compact or determining set then 2 d has a representing measure we also use the core variety to exhibit a sequence with positive definite moment matrix and positive riesz functional which fails to have a representing measure

the multidimensional truncated moment problem atoms determinacy and core variety abstract this paper is about the moment problem on a finite dimensional vector space of continuous functions we investigate the structure of the convex cone of moment functionals supporting hyperplanes exposed faces inner points and treat various important special topics on moment functionals determinacy set of atoms of representing measures core variety

recursiveness and propagation for 2 variable weighted shifts abstract for 1 variable weighted shifts it is well known that given three positive numbers 0 1 2 we always find stampfli s subnormal weighted shift whose first three weights are 0 1 and 2 we call it a type abc completion or shift on the other hand given four positive numbers 0 1 2 3 it is not always possible to find a subnormal weighted shift whose first four weights are 0 1 2 and 3 in this paper we extend the notion of a type abc shift to the 2 variable weighted shifts using a localization technique from the theory of truncated moment problems we next study an inner propagation phenomena of a 2 variable weighted shift whose core is of type abc we further establish the necessary and sufficient conditions for the subnormality of one step extensions of type abc 2 variable weighted shifts finally we study a certain invariant property under powers for type abc 2 variable weighted shifts

dimensional stability in truncated moment problems abstract an approach to truncated moment problems is developed via the riesz functional and an assumed dimensional stability of its associated hilbert spaces although equivalent to a concept of flatness introduced by r curto and l fialkow the dimensional stability discussed in this paper has a different geometric aspect and leads to statements parallel to those of the quoted authors as well as to some newer ones obtained by simpler arguments a stability equation giving a local characterization of the dimensional stability is also presented

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

conditional quadrature method of moments for kinetic equations kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution moment methods are an important class of approximate models derived from kinetic equations but require closure to truncate the moment set in quadrature based moment methods qbmm closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments e g spatial fluxes can be related to the finite moment set in this work a novel moment inversion algorithm based on 1 d adaptive quadrature of conditional velocity moments is introduced and shown to always yield realizable distribution functions i e non negative quadrature weights this conditional quadrature method of moments cqmom can be used to compute exact n point quadratures for multi valued solutions also known as the multi variate truncated moment problem and provides optimal approximations of continuous distributions in order to control numerical errors arising in volume averaging and spatial transport an adaptive 1 d quadrature algorithm is formulated for use with cqmom the use of adaptive cqmom in the context of qbmm for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing i e collision less systems elastic and inelastic particle particle collisions and external forces i e fluid drag

positivity of riesz functionals and solutions of quadratic and quartic moment problems abstract we employ positivity of riesz functionals to establish representing measures or approximate representing measures for truncated multivariate moment sequences for a truncated moment sequence y we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set k r n if and only if the associated riesz functional l y is k positive for a determining set k we prove that if l y is strictly k positive then y admits a representing measure supported in k as a consequence we are able to solve the truncated k moment problem of degree k in the cases i n k 2 4 and k r 2 ii n 1 k 2 and k is defined by one quadratic equality or inequality in particular these results solve the truncated moment problem in the remaining open cases of hilbert s theorem on sums of squares

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

multivariate nonnegative quadratic mappings in this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain with nonnegativity defined by a prespecified conic order in particular we consider the set cone of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone and study when it can be represented by linear matrix inequalities we also discuss the applications of the results in robust optimization especially the robust quadratic matrix inequalities and the robust linear programming models in the latter application the implementational errors of the solution are taken into account and the problem is formulated as a semidefinite program

rank pairing heaps we introduce the rank pairing heap an implementation of heaps that combines the asymptotic efficiency of fibonacci heaps with much of the simplicity of pairing heaps other heap implementations that match the bounds of fibonacci heaps do so by maintaining a balance condition on the trees representing the heap in contrast to these structures but like pairing heaps our trees can evolve to have arbitrary unbalanced structure also like pairing heaps our structure requires at most one cut and no other restructuring per key decrease in the worst case the only changes that can cascade during a key decrease are changes in node ranks although our data structure is simple its analysis is not

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

the truncated moment problem via homogenization and flat extensions abstract this paper studies how to solve the truncated moment problem tmp via homogenization and flat extensions of moment matrices we first transform tmp to a homogeneous tmp htmp and then use semidefinite programming sdp techniques to solve htmp our main results are 1 a truncated moment sequence tms is the limit of a sequence of tms admitting measures on r n if and only if its homogenized tms htms admits a measure supported on the unit sphere in r n 1 2 an htms admits a measure if and only if the optimal values of a sequence of sdp problems are nonnegative 3 under some conditions that are almost necessary and sufficient by solving these sdp problems a representing measure for an htms can be explicitly constructed if one exists

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

the multidimensional truncated moment problem caratheodory numbers abstract let a be a finite dimensional subspace of c x r where x is a locally compact hausdorff space and a f 1 f m a basis of a a sequence s s j j 1 m is called a moment sequence if s j f j x d x j 1 m for some positive radon measure on x each moment sequence s has a finitely atomic representing measure the smallest possible number of atoms is called the caratheodory number c a s the largest number c a s among all moment sequences s is the caratheodory number c a in this paper the caratheodory numbers c a s and c a are studied in the case of differentiable functions methods from differential geometry are used the main emphasis is on real polynomials for a large class of spaces of polynomials in one variable the number c a is determined in the multivariate case we obtain some lower bounds and we use results on zeros of positive polynomials to derive upper bounds for the caratheodory numbers

limits of positive flat bivariate moment matrices the bivariate moment problem for a sequence 6 of degree 6 remains unsolved but we prove that if the associated 10 10 moment matrix m3 satisfies m3 0 and rank m3 6 then admits a sequence of approximate representing measures and 5 has a representing measure more generally let fd denote the closure of the positive flat moment matrices of degree 2d in n variables each matrix in fd admits computable approximate representing measures and in 2013 jiawang nie and the author began to study concrete conditions for membership in this class let 2d i i zn i 2d 0 0 denote a real n dimensional sequence of degree 2d if the corresponding moment matrix md md is the limit of a sequence of positive flat moment matrices m k d i e m k d 0 and rank m k d rank m k d 1 then i md 0 ii rank md d 1 dim r x1 xn d 1 and iii 2d 1 admits a representing measure we extend our earlier results by proving conversely that for n 2 if md satisfies certain positivity and rank conditions related to i iii then md is the limit of positive flat moment matrices

solution of the truncated moment problem with variety y x 3 we show that positivity consistency and the variety condition which are sufficient to solve the truncated moment problem on planar curves of degree 2 are not sufficient for curves of higher degree using new partly algorithmic conditions based on positive moment matrix extensions we present a concrete solution to the truncated moment problem on the curve y x 3 we also use moment matrix extensions to solve in a less concrete sense truncated moment problems on curves of the form y g x and yg x 1 g x leading to degree bounded weighted sum of squares representations for polynomials which are strictly positive on such curves

on a new relation between jacobi and homogeneous poisson manifolds we establish a new very close relationship which links jacobi structures and homogeneous poisson structures defined on the same manifold and study the characteristic foliations of the related structures several examples of this construction are also given

an analogue of the riesz haviland theorem for the truncated moment problem abstract let 2 n i i 2 n denote a d dimensional real multisequence let k denote a closed subset of r d and let p 2 n p r x 1 x d deg p 2 n corresponding to the riesz functional l l p 2 n r is defined by l a i x i a i i we say that l is k positive if whenever p p 2 n and p k 0 then l p 0 we prove that admits a k representing measure if and only if l admits a k positive linear extension l p 2 n 2 r this provides a generalization from the full moment problem to the truncated moment problem of the riesz haviland theorem we also show that a semialgebraic set solves the truncated moment problem in terms of natural degree bounded positivity conditions if and only if each polynomial strictly positive on that set admits a degree bounded weighted sum of squares representation