A sum formula related to ellipsoids with applications to lattice point theory

In this article we consider finite and infinitep-dimensional sums over functionsf, where the argument off is represented by a positive definite quadratic form. We develop a sum formula like theEuler-Maclaurin orPoisson sum formula. Applications to exponential sums and lattice point problems are given.     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

A new approach to the Morse-Conley theory and some applications

Summary We present a new approach to the Morse theory which is based on a generalization of the Conley index to non locally compact spaces. The variant of the Morse theory which we obtain seems suitable for the applications to nonlinear functionals analysis. Some applications are given here; they mainly concern the study of periodic solutions of second order Hamiltonian systems. Other applications are in some quoted papers.     

Lyapunov exponents of hyperbolic attractors

 Let μ ₊ be the SBR measure on a hyperbolic attractor Ω of a C ² Axiom A diffeomorphism (M,f) and v the volume measure on M. As is known, μ ₊ -almost every is Lyapunov regular and the Lyapunov characteristic exponents of (f,Df) at x are constants $\lambda^{(i)}(\mu_+,f),1\leq i\leq s$. In this paper we prove that $v$-almost every $x$ in the basin of attraction $W^s(\Omega)$ is positively regular and the Lyapunov characteristic exponents of $(f,Df)$ at $x$ are the constants . Similar results are also obtained for nonuniformly completely hyperbolic attractors. Received: 20 September 2001     

Transformation invariance of Lyapunov exponents

Lyapunov exponents represent important quantities to characterize the properties of dynamical systems. We show that the Lyapunov exponents of two different dynamical systems that can be converted to each other by a transformation of variables are identical. Moreover, we derive sufficient conditions on the transformation for this invariance property to hold. In particular, it turns out that the transformation need not necessarily be globally invertible.     

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.     

A multiset hook length formula and some applications

A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov–Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.     

A generalization of the Lipschitz summation formula and some applications

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.

     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

A renorming in some Banach spaces with applications to fixed point theory

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(⋅)} which satisfy some special conditions. We define an equivalent norm ?⋅? on X such that if C is a convex bounded closed subset of (X,?⋅?) which is τ-relatively sequentially compact, then every nonexpansive mapping T:C→C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier-Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G=T, we recover P.K. Lin's renorming in the sequence space ?₁. Moreover, we give new norms in ?₁ with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L₁(μ) can be renormed to have the FPP.     

某一类控制系统中发生的指数偏差现象

我们考虑在某类控制系统中,以Lyapunov指数作为观测量,远离给定遍历测度的那些周期测度.对于这类周期测度的数量关于周期的指数增长率,我们将用测度熵在某个集合上的上界给出它的一个上限控制.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Local Baire classification of Lyapunov exponents

One considers two different definitions of the Baire class of a functional at a point. These definitions are in agreement with the common definition of the Baire class. The semicontinuity of a functional at a point is associated with its inclusion into the first Baire class at that point in the sense of the said definitions for Lyapunov exponents of a homogeneous nth-order system. In particular, it is shown that for the two smallest exponents, the inclusion into the first Baire class at a point is equivalent to semicontinuity in the sense of one of the two definitions and continuity in the sense of the other. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 56–70, 2007.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

A robust, locally interpretable algorithm for Lyapunov exponents

An enhanced version of the well known Wolf algorithm for the estimation of the Lyapunov characteristic exponents (LCEs) is proposed. It permits interpretation of the local behavior of non-linear flows. The new variant allows for reliable calculation of the non-uniformity-factors (NUFs). The NUFs can be interpreted as standard deviations of the LCEs. Since the latter can also be estimated by the Wolf algorithm, however, without local information on the flow, the new version ensures local interpretability and therefore allows the calculation of the NUFs. The local contributions to the LCEs which we call “local LCEs” can at least be calculated up to three dimensions. Application of the modified method to a hyperchaotic flow in four dimensions shows that an extension to many dimensions is possible and promises new insight into so far not fully understood high-dimensional non-linear systems.     

On applications of the Taylor formula in some quasispaces

We consider some metric spaces with quasimetric (quasispaces) comprising uniformly regular (equiregular) Carnot — Carathéodory quasispaces whose quasimetric is induced by C ^ϒ−1-smooth vector fields of formal degree not higher than ϒ. For these spaces, some analogues of the Campbell — Hausdorff formula are derived, which allows us to prove a theorem on a nilpotent tangent cone, a theorem on isomorphism of various nilpotent tangent cones defined at a common point, and a local approximation theorem.