Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
P. Biran and K. Cieliebak @cite_12 @cite_22 generalized Seidel's result in various ways. In the case @math , for every @math , Corollary follows from their results. Further references about results on the topology of Lagrangian embeddings are provided in @cite_12 @cite_22 .
@cite_22 @cite_12
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Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed. In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ n to the more general case of subcritical Stein manifolds. We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion (O ( _ X n ) , ) where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an (O ( ( _ X ) n ) ) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as an important class of structured random JL embeddings, are formed by randomizing the column signs of a circulant matrix generated by a random vector. With the help of recent decoupling techniques and matrix-valued Bernstein inequalities, we obtain a new bound @math for Gaussian circulant JL embeddings. Moreover, by using the Laplace transform technique (also called Bernstein's trick), we extend the result to subgaussian case. The bounds in this paper offer a small improvement over the current best bounds for Gaussian circulant JL embeddings for certain parameter regimes and are derived using more direct methods. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g. We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1 12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1 (22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c d , for a constant c > 0. The exponent c d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2 d log 3 2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically. In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes). Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data. 1 Preliminaries.- 1.1 Introduction.- 1.2 Measures and Functions.- 1.3 Weak Topologies.- 1.4 Convergence of Measures.- 1.5 Complements.- 1.6 Notes.- I Markov Chains and Ergodicity.- 2 Markov Chains and Ergodic Theorems.- 2.1 Introduction.- 2.2 Basic Notation and Definitions.- 2.3 Ergodic Theorems.- 2.4 The Ergodicity Property.- 2.5 Pathwise Results.- 2.6 Notes.- 3 Countable Markov Chains.- 3.1 Introduction.- 3.2 Classification of States and Class Properties.- 3.3 Limit Theorems.- 3.4 Notes.- 4 Harris Markov Chains.- 4.1 Introduction.- 4.2 Basic Definitions and Properties.- 4.3 Characterization of Harris recurrence.- 4.4 Sufficient Conditions for P.H.R.- 4.5 Harris and Doeblin Decompositions.- 4.6 Notes.- 5 Markov Chains in Metric Spaces.- 5.1 Introduction.- 5.2 The Limit in Ergodic Theorems.- 5.3 Yosida's Ergodic Decomposition.- 5.4 Pathwise Results.- 5.5 Proofs.- 5.6 Notes.- 6 Classification of Markov Chains via Occupation Measures.- 6.1 Introduction.- 6.2 A Classification.- 6.3 On the Birkhoff Individual Ergodic Theorem.- 6.4 Notes.- II Further Ergodicity Properties.- 7 Feller Markov Chains.- 7.1 Introduction.- 7.2 Weak-and Strong-Feller Markov Chains.- 7.3 Quasi Feller Chains.- 7.4 Notes.- 8 The Poisson Equation.- 8.1 Introduction.- 8.2 The Poisson Equation.- 8.3 Canonical Pairs.- 8.4 The Cesaro-Averages Approach.- 8.5 The Abelian Approach.- 8.6 Notes.- 9 Strong and Uniform Ergodicity.- 9.1 Introduction.- 9.2 Strong and Uniform Ergodicity.- 9.3 Weak and Weak Uniform Ergodicity.- 9.4 Notes.- III Existence and Approximation of Invariant Probability Measures.- 10 Existence of Invariant Probability Measures.- 10.1 Introduction and Statement of the Problems.- 10.2 Notation and Definitions.- 10.3 Existence Results.- 10.4 Markov Chains in Locally Compact Separable Metric Spaces.- 10.5 Other Existence Results in Locally Compact Separable Metric Spaces.- 10.6 Technical Preliminaries.- 10.7 Proofs.- 10.8 Notes.- 11 Existence and Uniqueness of Fixed Points for Markov Operators.- 11.1 Introduction and Statement of the Problems.- 11.2 Notation and Definitions.- 11.3 Existence Results.- 11.4 Proofs.- 11.5 Notes.- 12 Approximation Procedures for Invariant Probability Measures.- 12.1 Introduction.- 12.2 Statement of the Problem and Preliminaries.- 12.3 An Approximation Scheme.- 12.4 A Moment Approach for a Special Class of Markov Chains.- 12.5 Notes. The notion of a basic embedding appeared in research motivated by Kolmogorov–Arnold's solution of Hilbert's 13th problem. Let K,X,Y be topological spaces. An embedding K⊂X×Y is called basic if for every continuous function f:K→R there exist continuous functions g:X→R, h:Y→R such that f(x,y)=g(x)+h(y) for any point (x,y)∈K. Let Ti be an i-od. Theorem. There exists only a finite number of prohibited' subgraphs for basic embeddings into R×Tn. Consequently, for a finite graph K there is an algorithm for checking whether K is basically embeddable into R×Tn. Our theorem is a generalization of Skopenkov's description of graphs basically embeddable into R2, and our proofs is a (non-trivial) extension of that one. In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. Abstract We make explicit the geometric content of Mel’nikov’s method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of families of Lagrangian submanifolds N e ± , with N 0 + = N 0 − and constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits of a given Hamiltonian system are detected by the zeros of the Mel’nikov 1-form. This 1-form admits an integral expression which is non-convergent in general. We discuss different solutions to this convergence problem. We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu’s functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem. @S0163-1829 98!03335-9# Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114. We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing (M )-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough (M ). A closely related problem is to show that two independent Poisson processes on ( R ) are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems. Abstract This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in Dvoretzky's theorem from order of ɛ 2 to order of ɛ (except for log factors). The second is a joint generalization of (Milman's version of) Dvoretzky's theorem and (a recent generalization by Klartag and Mendelson of) the Johnson–Lindenstrauss Lemma. Let (L ) be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold ((X , ) ). We assume that the first homology group (H_1 (L ; A) ) with coefficients in a commutative ring (A ) injects into the group (H_1 (X ; A) ) and that (X ) contains no Maslov zero pseudo-holomorphic disc with boundary on (L ). Then, we prove that for every generic choice of a tame almost-complex structure (J ) on (X ), every relative homology class (d H_2 (X , L ; Z ) ) and adequate number of incidence conditions in (L ) or (X ), the weighted number of (J )-holomorphic discs with boundary on (L ), homologous to (d ), and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of (J ), provided that at least one incidence condition lies in (L ). These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring (A ). We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree