On the persistence of elliptic lower-dimensional tori in Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann’s non-degeneracy condition, which is essential for the proof of our result.     

The partial malliavin calculus and its application to non-linear filtering

In this article, the Malliavin calculus is used to derive regularity properties of the conditional distribution of one ltd process given a second Ito process. The relation between the processes involved is the usual one assumed in the study of filtering theory. The non-degeneracy which we require is stated in terms of Malliavins covariance matrix in Theorem (3.15). More practical conditions are given in Lemma (3.19) for general Tto processes and in Lemma (3.29) for diffusions. Finally, in Theorem (4.6) a “localized” version of these results is given for diffusions.     

Homotopy fiber sequences induced by 2-functors

This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors.     

Homogenization of divergence-form operators with lower-order terms in random media

The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non-degeneracy and boundedness. Received: 27 August 1999 / Revised version: 27 October 2000 / Published online: 26 April 2001     

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky-MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.     

Effectiveness for infinite variable words and the Dual Ramsey Theorem

We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA₀ over RCA₀. We show that neither VW(2,2) nor OVW(2,2) is provable in WKL₀. These results give partial answers to questions posed by Friedman and Simpson [3].The first authors research is partially supported by an NSF VIGRE Grant at Indiana University. The second authors research is partially supported by an NSF Postdoctoral Fellowship.     

Characterization of Stochastic Viability of Any Nonsmooth Set Involving Its Generalized Contingent Curvature

Abstract

Thanks to the Stroock and Varadhan “Support Theorem” and under convenient regularity assumptions, stochastic viability problems are equivalent to invariance problems for control systems (also called tychastic viability), as it has been singled out by Doss in 1977 for instance. By the way, it is in this framework of invariance under control systems that problems of stochastic viability in mathematical finance are studied. The Invariance Theorem for control systems characterizes invariance through first‐order tangential and/or normal conditions whereas the stochastic invariance theorem characterizes invariance under second‐order tangential conditions. Doss's Theorem states that these first‐order normal conditions are equivalent to second‐order normal conditions that we expect for invariance under stochastic differential equations for smooth subsets. We extend this result to any subset by defining in an adequate way the concept of contingent curvature of a set and contingent epi‐Hessian of a function, related to the contingent curvature of its epigraph. This allows us to go one step further by characterizing functions the epigraphs of which are invariant under systems of stochastic differential equations. We shall show that they are (generalized) solutions to either a system of first‐order Hamilton‐Jacobi equations or to an equivalent system of second‐order Hamilton‐Jacobi equations.     

The great theorem of A.A. Markoff and Jean Bernoulli sequences

A proof of Markoff’s Great Theorem on the Lagrange spectrum using continued fractions is sketched. Markoff’s periods and Jean Bernoulli sequence ¹ are used to obtain a simple algorithm for the computation of the Lagrange spectrum below 3.     

Eigenvalues and colorings of digraphs

Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.     

Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Brooks’ Theorem via the Alon-Tarsi Theorem

We give a proof of Brooks’ Theorem and its choosability extension based on the Alon-Tarsi Theorem; this also shows that Brooks’ Theorem remains valid in a more general game coloring setting.     

Generalized Serre relations for Lie algebras associated with positive unit forms

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’s Theorem [J.-P. Serre, Complex Semisimple Lie Algebras (G.A. Jones, Trans.), Springer-Verlag, New York, 1987] gives then a representation of the given Lie algebra in generators and relations in terms of the Cartan matrix.In this work, we generalize Serre’s Theorem to give an explicit representation in generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of linearly independent roots which generate all roots as linear combinations with integral coefficients.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

关于Borg定理

Borg定理是判定周期系数二阶线性微分方程稳定的一个重要定理,这个定理在一定意义下是不可改进的.本文利用判别式的一个新形式,在弱的限制下,得到判定周期系数二阶线性微分方程稳定的定理,所得结论改进了Borg定理的判定结果.     

Sliding puzzles and rotating puzzles on graphs

Sliding puzzles on graphs are generalizations of the Fifteen Puzzle. Wilson has shown that the sliding puzzle on a 2-connected graph always generates all even permutations of the tiles on the vertices of the graph, unless the graph is isomorphic to a cycle or the graph θ₀ [R.M. Wilson, Graph puzzles, homotopy, and the alternating group, J. Combin. Theory Ser. B 16 (1974) 86–96]. In a rotating puzzle on a graph, tiles are allowed to be rotated on some of the cycles of the graph. It was shown by Scherphuis that all even permutations of the tiles are also obtainable for the rotating puzzle on a 2-edge-connected graph, except for a few cases. In this paper, Scherphuis’ Theorem is generalized to every connected graph, and Wilson’s Theorem is derived from the generalized Scherphuis’ Theorem, which will give a uniform treatise for these two families of puzzles and reveal the structural relation of the graphs of the two puzzles.     

Existence of nontrivial homoclinic orbits for fourth-order difference equations

We obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth-order difference equations by using Mountain Pass Theorem, a weak convergence argument and a discrete version of Lieb’s lemma.     

Natural density of rectangular unimodular integer matrices

In this paper, we compute the natural density of the set of k×n integer matrices that can be extended to an invertible n×n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form . Connections with Cesàro’s Theorem on the density of coprime integers and Quillen-Suslin’s Theorem are also presented.     

Comparison theorems for noncanonical third order nonlinear differential equations

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives. We prove comparison theorems on property A between linear and nonlinear equations. Some integral criteria ensuring property A for nonlinear equations are also given. Our assumptions on the nonlinearity of f are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.      

Multivariate process capability via Löwner ordering

Probability bounds can be derived for distributions whose covariance matrices are ordered with respect to Löwner partial ordering, a relation that is based on whether the difference between two matrices is positive definite. One example is Anderson’s Theorem. This paper develops a probability bound that follows from Anderson’s Theorem that is useful in the assessment of multivariate process capability. A statistical hypothesis test is also derived that allows one to test the null hypothesis that a given process is capable versus the alternative hypothesis that it is not capable on the basis of a sample of observed quality characteristic vectors from the process. It is argued that the proposed methodology is viable outside the multivariate normal model, where the p-value for the test can be computed using the bootstrap. The methods are demonstrated using example data, and the performance of the bootstrap approach is studied empirically using computer simulations.