a variational analysis of the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface we will give existence results by using variational methods in a noncoercive case a key tool in our analysis is a new moser trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u1 and u2 2012 wiley periodicals inc

lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

positive semidefinite matrix completion universal rigidity and the strong arnold property this paper addresses the following three topics positive semidenite psd matrix completions universal rigidity of frameworks and the strong arnold property sap we show some strong connections among these topics using semidenite programming as unifying theme our main contribution is a sufcient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix such partial matrices are an essential tool in the study of the gram dimension gd g of a graph g a recently studied graph parameter related to the low psd matrix completion problem additionally we derive an elementary proof of connelly s sucient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sucient conditions we also give a geometric characterization of psd matrices satisfying the strong arnold property in terms of nondegeneracy of an associated semidenite program which we use to establish some links between the gram dimension gd and the colin de verdi ere type graph parameter

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

characterizing generic global rigidity a d dimensional framework is a graph and a map from its vertices to ed such a framework is globally rigid if it is the only framework in ed with the same graph and edge lengths up to rigid motions for which underlying graphs is a generic framework globally rigid we answer this question by proving a conjecture by connelly that his sucient condition is also necessary the condition comes from considering the geometry of the length squared mapping essentially the graph is generically locally rigid i the rank of is maximal and it is generically globally rigid i the rank of the gauss map on the image of is maximal we also show that this condition is eciently checkable with a randomized algorithm

global completability with applications to self consistent quantum tomography let p 1 ldots p n in mathbb r d be unknown vectors and let omega subseteq 1 ldots n 2 assume that the inner products p i t p j are fixed for all i j in omega do these inner product constraints up to simultaneous rotation of all vectors determine p 1 ldots p n uniquely here we derive a necessary and sufficient condition for the uniqueness of p 1 ldots p n i e global completability which is applicable to a large class of practically relevant sets omega moreover given omega we show that the condition for global completability is universal in the sense that for almost all vectors p 1 ldots p n in mathbb r d the completability of p 1 ldots p n only depends on omega and not on the specific values of p i t p j for i j in omega this work was motivated by practical considerations namely matrix factorization techniques and self consistent quantum tomography

solving sat and sat modulo theories from an abstract davis putnam logemann loveland procedure to dpll t we first introduce abstract dpll a rule based formulation of the davis putnam logemann loveland dpll procedure for propositional satisfiability this abstract framework allows one to cleanly express practical dpll algorithms and to formally reason about them in a simple way its properties such as soundness completeness or termination immediately carry over to the modern dpll implementations with features such as backjumping or clause learning we then extend the framework to satisfiability modulo background theories smt and use it to model several variants of the so called lazy approach for smt in particular we use it to introduce a few variants of a new efficient and modular approach for smt based on a general dpll x engine whose parameter x can be instantiated with a specialized solver solvert for a given theory t thus producing a dpll t system we describe the high level design of dpll x and its cooperation with solvert discuss the role of theory propagation and describe different dpll t strategies for some theories arising in industrial applications our extensive experimental evidence summarized in this article shows that dpll t systems can significantly outperform the other state of the art tools frequently even in orders of magnitude and have better scaling properties

finding a tree structure in a resolution proof is np complete the resolution tree problem consists of deciding whether a given sequence like resolution refutation admits a tree structure this paper shows the np completeness of both the resolution tree problem and a natural generalization of the resolution tree problem that does not involve resolution

existence and multiplicity result for the singular toda system abstract we consider the toda system on a compact surface g u 1 2 1 h 1 e u 1 h 1 e u 1 d v g 1 2 h 2 e u 2 h 2 e u 2 d v g 1 4 j 1 j 1 j p j 1 u 2 2 2 h 2 e u 2 h 2 e u 2 d v g 1 1 h 1 e u 1 h 1 e u 1 d v g 1 4 j 1 j 2 j p j 1 where h i are smooth positive functions i are positive real parameters p j are given points on and i j are numbers greater than 1 we give existence and multiplicity results using variational and morse theoretical methods it is the first existence result when some of the i j s are allowed to be negative

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

invariant conformal metrics on ⁿ in this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its schouten tensor also we study conformal metrics on the sphere which are invariant by a k parameter subgroup of conformal diffeomorphisms of the sphere giving a bound on its maximum dimension moreover we classify conformal metrics on the sphere whose eigenvalues of the shouten tensor are all constant we call them isoparametric conformal metrics and we use a clas

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

moser trudinger inequalities for singular liouville systems in this paper we prove a moser trudinger inequality for the euler lagrange functional of general singular liouville systems on a compact surface we characterize the values of the parameters which yield coercivity for the functional hence the existence of energy minimizing solutions for the system and we give necessary conditions for boundedness from below we also provide a sharp inequality under assuming the coefficients of the system to be non positive outside the diagonal the proofs use a concentration compactness alternative poho aev type identities and blow up analysis

the np hardness of finding a directed acyclic graph for regular resolution let r be a resolution refutation given as a sequence of clauses without explicit description of the underlying dag then it is np complete to decide whether r is a regular resolution refutation

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

approximate cone factorizations and lifts of polytopes in this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix this provides a robust generalization of the famous result of yannakakis that polyhedral lifts of a polytope are controlled by exact nonnegative factorizations of its slack matrix our approximations behave well under polarity and have efficient representations using second order cones we establish a direct relationship between the quality of the factorization and the quality of the approximations and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron

compressibility of positive semidefinite factorizations and quantum models we investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products this can either be regarded as compression of positive semidefinite factorizations of nonnegative matrices or if the matrices are subject to additional normalization constraints as compression of quantum models we derive both lower and upper bounds on compressibility applications are broad and range from the analysis of experimental data to bounding the one way quantum communication complexity of boolean functions

existence and non existence results for the su 3 singular toda system on compact surfaces we consider the su 3 singular toda system on a compact surface g u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 4 m 1m 1m pm 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 4 m 1m 2m pm 1 where hi are smooth positive functions on i r pm and im 1 we give both existence and non existence results under some conditions on the parameters i and im existence results are obtained using variational methods which involve a geometric inequality of new type non existence results are obtained using blow up analysis and localized poho aev type identities

bounding the set of quantum correlations we introduce a hierarchy of conditions necessarily satisfied by any distribution p ensuremath alpha ensuremath beta representing the probabilities for two separate observers to obtain outcomes ensuremath alpha and ensuremath beta when making local measurements on a shared quantum state each condition in this hierarchy is formulated as a semidefinite program among other applications our approach can be used to obtain upper bounds on the quantum violation of an arbitrary bell inequality it yields for instance tight bounds for the violations of the collins et al inequalities

positive semidefinite rank let m in mathbb r p times q m rp q be a nonnegative matrix the positive semidefinite rank psd rank of m is the smallest integer k for which there exist positive semidefinite matrices a i b j ai bj of size k times k k k such that m ij mathrm trace a i b j mij trace aibj the psd rank has many appealing geometric interpretations including semidefinite representations of polyhedra and information theoretic applications in this paper we develop and survey the main mathematical properties of psd rank including its geometry relationships with other rank notions and computational and algorithmic aspects

volume comparison and the σk yamabe problem abstract in this paper we study the problem of finding a conformal metric with the property that the k th elementary symmetric polynomial of the eigenvalues of its weyl schouten tensor is constant a new conformal invariant involving maximal volumes is defined and this invariant is then used in several cases to prove existence of a solution and compactness of the space of solutions provided the conformal class admits an admissible metric in particular the problem is completely solved in dimension four and in dimension three if the manifold is not simply connected

a general existence result for the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 u1 4 j 1m 1 j pj 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 u2 4 j 1m 2 j pj 1 which is motivated by the study of models in non abelian chern simons theory here h1 h2h1 h2 are smooth positive functions 1 2 1 2 two positive parameters pipi points of the surface and 1 i 2 j 1 i 2 j non negative numbers we prove a general existence result using variational methods the same analysis applies to the following mean field equation u 1 heu heudvg 1 2 he u he udvg 1 which arises in fluid dynamics

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

liouville s formula in arbitrary planar domains abstract this work extends to an arbitrary planar domain the global version of liouville s formula for a general solution of the equation u k e 2 u that equation is lifted to the universal covering space where a solution is globally determined by a holomorphic function g which is projected into a multivalued representation of the original solution applying the method to a punctured disk results from chou wan are unified and an integral formula for the solutions is derived

equivariant semidefinite lifts and sum of squares hierarchies a central question in optimization is to maximize or minimize a linear function over a given polytope p to solve such a problem in practice one needs a concise description of the polytope p in this paper we are interested in representations of p using the positive semidefinite cone a positive semidefinite lift psd lift of a polytope p is a representation of p as the projection of an affine slice of the positive semidefinite cone mathbf s d such a representation allows linear optimization problems over p to be written as semidefinite programs of size d such representations can be beneficial in practice when d is much smaller than the number of facets of the polytope p in this paper we are concerned with so called equivariant psd lifts also known as symmetric psd lifts which respect the symmetries of the polytope p we present a representation theoretic framework to study equivariant psd lifts of a certain class of symmetric polytopes known as orbitopes our main result

the liouville equation in an annulus abstract we give the solutions to the liouville equation in an annulus a of r 2 that satisfy a certain neumann condition on each component of a as a consequence we classify all the metrics of constant curvature in a that have constant geodesic curvature on a

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

unrestricted resolution versus n resolution abstract an n resolution proof is a resolution proof in which the resolution rule is restricted one clause to which it is applied must only consist of negative literals n resolution is complete schoning 1987 p 109 stickel 1985 p 86 we construct an infinite family of propositional formulas and show these formulas have unrestricted resolution proofs whose length is polynomially bounded in their size all n resolution proofs of these formulas are of superpolynomial length

self consistent tomography of the state measurement gram matrix state and measurement tomography make assumptions about the experimental states or measurements these assumptions are often not justified because state preparation and measurement errors are unavoidable in practice here we describe how the gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so called globally completable this is for instance the case if the unknown measurements are known to be projective and nondegenerate the computed gram matrix determines the states and the measurement operators uniquely up to simultaneous rotations in the space of hermitian matrices we prove the reliability of the proposed method in the limit of a large number of independent measurement repetitions

the liouville equation in a half plane abstract we classify the solutions of the equation u a e u 0 in the half plane r 2 that satisfy the neumann boundary condition u t c e u 2 on r 2 an analogous problem in the once punctured disc d r 2 is also solved

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone we investigate the completely positive semidefinite cone mathcal cs n a new matrix cone consisting of all n times n matrices that admit a gram representation by positive semidefinite matrices of any size in particular we study relationships between this cone and the completely positive and the doubly nonnegative cone and between its dual cone and trace positive noncommutative polynomials we use this new cone to model quantum analogues of the classical independence and chromatic graph parameters alpha g and chi g which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0 1 scalars in the programs defining the classical parameters we can formulate these quantum parameters as conic linear programs over the cone mathcal cs n using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials

sparser johnson lindenstrauss transforms we give two different and simple constructions for dimensionality reduction in e2 via linear mappings that are sparse only an o e fraction of entries in each column of our embedding matrices are non zero to achieve distortion 1 e with high probability while still achieving the asymptotically optimal number of rows these are the first constructions to provide subconstant sparsity for all values of parameters improving upon previous works of achlioptas 2003 and dasgupta et al 2010 such distributions can be used to speed up applications where e2 dimensionality reduction is used

non negative matrix factorization revisited uniqueness and algorithm for symmetric decomposition non negative matrix factorization nmf has found numerous applications due to its ability to provide interpretable decompositions perhaps surprisingly existing results regarding its uniqueness properties are rather limited and there is much room for improvement in terms of algorithms as well uniqueness aspects of nmf are revisited here from a geometrical point of view both symmetric and asymmetric nmf are considered the former being tantamount to element wise non negative square root factorization of positive semidefinite matrices new uniqueness results are derived e g it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second order cone checking this condition is shown to be np complete yet this and other results offer insights on the role of latent sparsity in this context on the computational side a new algorithm for symmetric nmf is proposed which is very different from existing ones it alternates between procrustes rotation and projection onto the non negative orthant to find a non negative matrix close to the span of the dominant subspace simulation results show promising performance with respect to the state of art finally the new algorithm is applied to a clustering problem for co authorship data yielding meaningful and interpretable results

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

on the automatizability of resolution and related propositional proof systems a propositional proof system is automatizable if there is an algorithm that given a tautology produces a proof in time polynomial in the size of its smallest proof this notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound this new notion is called weak automatizability among other characterizations we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time after studying the robustness of the definition we prove the equivalence between i resolution is weakly automatizable ii res k is weakly automatizable and iii res k has feasible interpolation when k 1 in order to prove this result we show that res 2 has polynomial size proofs of the reflection principle of resolution which is a version of consistency we also show that res k for every k 1 proves its consistency in polynomial size while resolution does not in fact we show that resolution proofs of its own consistency require almost exponential size this gives a better lower bound for the monotone interpolation of res 2 and a separation from resolution as a byproduct our techniques also give us a way to obtain a large class of examples that have small resolution refutations but require relatively large width this answers a question of alekhnovich and razborov related to whether resolution is automatizable in quasipolynomial time

dimensionality reduction of sdps through sketching abstract we show how to sketch semidefinite programs sdps using positive maps in order to reduce their dimension more precisely we use johnson lindenstrauss transforms to pro duce a smaller sdp whose solution preserves feasibility or approximates the value of the original problem with high probability these techniques allow to improve both complexity and storage space requirements they apply to problems in which the schatten 1 norm of the matrices specifying the sdp and also of a solution to the problem is constant in the problem size furthermore we provide some results which clarify the limitations of positive linear sketches in this setting

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

on prescribing geodesic curvature on d2 abstract some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

the yamabe problem for higher order curvatures let m be a compact riemannian manifold of dimension n 2 the k curvature for k 1 2 n is defined as the k th elementary symmetric polynomial of the eigenvalues of the schouten tenser the k yamabe problem is to prove the existence of a conformal metric whose k curvature is a constant when k 1 it reduces to the well known yamabe problem under the assumption that the metric is admissible the existence of solutions is known for the case k 2 n 4 for locally conformally flat manifolds and for the cases k n 2 in this paper we prove the solvability of the k yamabe problem in the remaining cases k n 2 under the hypothesis that the problem is variational this includes all of the cases k 2 as well as the locally conformally flat case

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

symmetry results for elliptic schrodinger systems on half spaces abstract in this paper we prove some symmetry and monotonicity results for classical positive solutions for elliptic schrodinger systems on half spaces with the neumann type condition our methods employ the alexandrov serrin method of moving planes combined with hopf s lemma at a corner

scalar curvature rigidity of geodesic balls in sn in this paper we prove a scalar curvature rigidity result for geodesic balls in s n this result contrasts sharply with the counterexamples to min oo s conjecture constructed in deformations of the hemisphere that increase scalar curvature invent math 185 2011 175 197

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

blow up analysis for solutions to neumann boundary value problem abstract in this paper we will analyze the blow up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2 dimensional surface m g with boundary we will provide some accurate point wise estimates for the profile of these sequences

a double mean field equation related to a curvature prescription problem abstract we study a double mean field type pde related to a prescribed curvature problem on compacts surfaces with boundary 0 1 u 2 k e u k e u 1 in u 2 h e u 2 h e u 2 1 on here and are real parameters k h are smooth positive functions on and respectively and is the outward unit normal vector to we provide a general blow up analysis then a moser trudinger inequality which gives energy minimizing solutions for some range of parameters finally we provide existence of min max solutions for a wider range of parameters which is dense in the plane if is not simply connected

hard examples for resolution exponential lower bounds are proved for the length of resolution refutations of sets of disjunctions constructed from expander graphs using the method of tseitin since these sets of clauses encode biconditionals they have short polynomial length refutations in a standard axiomatic formulation of propositional calculus

on the relative complexity of resolution refinements and cutting planes proof systems an exponential lower bound for the size of tree like cutting planes refutations of a certain family of conjunctive normal form cnf formulas with polynomial size resolution refutations is proved this implies an exponential separation between the tree like versions and the dag like versions of resolution and cutting planes in both cases only superpolynomial separations were known a urquhart bull symbolic logic 1 1995 pp 425 467 j johannsen inform process lett 67 1998 pp 37 41 p clote and a setzer in proof complexity and feasible arithmetics amer math soc providence ri 1998 pp 93 117 in order to prove these separations the lower bounds on the depth of monotone circuits of raz and mckenzie in combinatorica 19 1999 pp 403 435 are extended to monotone real circuits an exponential separation is also proved between tree like resolution and several refinements of resolution negative resolution and regular resolution actually this last separation also provides a separation between tree like resolution and ordered resolution and thus the corresponding superpolynomial separation of a urquhart bull symbolic logic 1 1995 pp 425 467 is extended finally an exponential separation between ordered resolution and unrestricted resolution also negative resolution is proved only a superpolynomial separation between ordered and unrestricted resolution was previously known a goerdt ann math artificial intelligence 6 1992 pp 169 184

on the complexity of resolution with bounded conjunctions we analyze size and space complexity of res k a family of propositional proof systems introduced by krajicek in fund math 170 1 3 2001 123 which extend resolution by allowing disjunctions of conjunctions of up to k 1 literals we show that the treelike res k proof systems form a strict hierarchy with respect to proof size and also with respect to space moreover resolution while simulating treelike res k is almost exponentially separated from treelike res k to study space complexity for general res k we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for resolution and to extend them to res k

regular resolution versus unrestricted resolution a resolution proof of an unsatisfiable propositional formula is called regular if and only if no variable is eliminated with the resolution rule twice on any branch of the proof tree representing the resolution proof an infinite family of unsatisfiable propositional formulas is constructed and the following shown these formulas have polynomial size unrestricted resolution proofs whereas all regular resolution proofs of these formulas are of superpolynomial length

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

classification of singular radial solutions to the σk yamabe equation on annular domains the study of the kth elementary symmetric function of the weyl schouten curvature tensor of a riemannian metric the so called k curvature has produced many fruitful results in conformal geometry in recent years in these studies in conformal geometry the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic pde important advances have been made in recent years in the understanding of the analytic behavior of solutions of the pde however the singular behavior of these solutions which is important in describing many important questions in conformal geometry is little understood this note classifies all possible radial solutions in particular the singular solutions of the k yamabe equation which describes conformal metrics whose k curvature equals a constant although the analysis involved is of elementary nature these results should provide useful guidance in studying the behavior of singular solutions in the general situation

robust self testing of the singlet in this paper we introduce a general framework to study the concept of robust self testing which can be used to self test maximally entangled pairs of qubits epr pairs and local measurement operators the result is based only on probabilities obtained from the experiment with tolerance to experimental errors in particular we show that if the results of an experiment approach the cirel son bound or approximate the mayers yao type correlations then the experiment must contain an approximate epr pair more specifically there exist local bases in which the physical state is close to an epr pair possibly encoded in a larger environment or ancilla moreover in these bases the measurements are close to the qubit operators used to achieve the cirel son bound or the mayers yao results 2012 iop publishing ltd