It is well known that the projection of a matrix A onto a Krylov subspace span { h, Ah, Ah, . . . , Ak−1h } results in a matrix of Hessenberg form. We show that the projection of the same matrix A onto an extended Krylov subspace, which is of the form span { A−krh, . . . , A−2h, A−1h,h, Ah, Ah . . . , A`h } , is a matrix of so-called extended Hessenberg form which can be characterized uniquely ...

We consider a perturbation of the Anosov-type system, which leads to the appearance of a hierarchical set of islands-around-islands. We demonstrate by simulation that the boundaries of the islands are sticky to trajectories. This phenomenon leads to the distribution of Poincaré recurrences with power-like tails in contrast to the exponential distribution in the Anosov-type systems.

Wave packets in a system governed by a Hamiltonian with a generic nonlinear spectrum typically exhibit both full and fractional revivals. It is shown that the latter can be eliminated by inducing suitable geometric phases in the states, by varying the parameters in the Hamiltonian cyclically with a period T . Further, with the introduction of this natural time step T , the occurrence of near re...

with initial terms u0 ux = ••• = ur_2 = 0, ur_1 1. Then (u) is called a unit sequence with coefficients al5 a25 -.., ar. For a positive integer AT, the primitive period of (u) modulo AT, denoted by K(M), is the least positive integer m such that un + m = un (mod AT) for all nonnegative integers n greater than or equal to some fixed integer n0. It is known that the primitive period modulo M of a...

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We ...

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs ( iφ , iφ2 ), where i is a natural number and φ is the golden ratio 1 2 (1 + √ 5). It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if φ is adjoined to the rationals.

The recurrence of states is a fundamental behaviour of dynamical systems. A modern technique of nonlinear data analysis, the recurrence plot, visualises and analyses the recurrence structure and allows us to detect transitions in the system’s dynamics by using recurrence quantification analysis (RQA). In the last decade, the RQA has become popular in many scientific fields. However, a sufficien...

We prove the variational principle for dimensions for Poincaré recurrences, in the case of invariant sets of dynamical systems with continuous time. To achieve this goal we show that these dimensions can be expressed as roots of a non–homogeneous Bowen equation.

QUERY: Authors present an important application of the of the extreme value theory to Poincaré recurrences in dynamical systems, which can provide estimations of some metrics of the ergodic chaotic attractors, namely the phase-space local dimension and local recurrence. Authors apply the technique to relevant daily fields: the sea level pressure, surface air temperature and precipitation rate, ...

This paper explores the performance of a family of algorithms for computing the Walsh-Hadamard transform, a useful computation in signal and image processing. Recent empirical work has shown that the family of algorithms exhibit a wide range of performance and that it is non-trivial to determine which algorithm is optimial on a given computer. This paper provides a theoretical basis for the per...