Strong Poincaré recurrence theorem in MV-algebras

The classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

Diagnostics of stochastic resonance using Poincaré recurrence time distribution

In the paper we calculate the distribution density of Poincaré recurrence times for a one-dimensional nonhyperbolic cubic map subjected to white noise and a harmonic signal. It is established that for small vicinities of recurrence the distribution density is not described by an exponential law and is periodically modulated with the external signal frequency. It is shown that the Fourier spectrum of the distribution density exhibits a well-distinguishable peak at the external signal frequency. The peak amplitude achieves its maximal value in the regime of stochastic resonance (SR), that can be used for detecting the SR effect.     

Functional inequalities and subordination: stability of Nash and Poincaré inequalities

We show that certain functional inequalities, e.g. Nash-type and Poincaré-type inequalities, for infinitesimal generators of C ₀ semigroups are preserved under subordination in the sense of Bochner. Our result improves earlier results by Bendikov and Maheux (Trans Am Math Soc 359:3085–3097, 2007, Theorem 1.3) for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.     

Sporadic randomness,Maxwell's Demon and the Poincaré recurrence times

In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure.     

On the direct proof of the Poincaré theorem on invariant tori

We present the direct proof of the Poincaré theorem on invariant tori.     

A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.     

On extension and refinement of the Poincaré inequality

The aim of this paper is to analyze the heat semigroup ${(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}$ generated by the usual Laplacian operator Δ on ${\mathbb{R}^{d}}$ equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup.     

Hilbert function,generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.     

The Poincaré index and the χy-characteristic of Hirzebruch

In the paper, we discuss several methods for computing the homology of contravariant and covariant versions of the classical De Rham complex on analytic spaces. Our approach is based on the theory of holomorphic and regular meromorphic differential forms, and is applicable in different settings depending on concrete types of varieties. Among other things, we describe how to compute by elementary calculations the homological index of vector fields and differential forms given on Cohen–Macaulay curves, graded normal surfaces, complete intersections and some others. In these situations, making use of ideas of X. Gómez-Mont, we derive explicit expressions for the local topological index of Poincaré and its generalizations. Furthermore, applying similar methods to the study of certain other complexes, we investigate some challenges, relating to the computation of classical topological–analytical invariants, such as the Euler characteristic of the Milnor fibre of an isolated singularity, the multiplicity of the discriminant of the versal deformation, the dimension of torsion and cotorsion modules, and so on.     

Two-party graphs and monotonicity properties of the Poincaré mapping

The local differential of a system of nonlinear differential equations with a T-periodic right-hand side is representable as a directed sign interaction graph. Within the class of balanced graphs, where all paths between two fixed vertices have the same signs, it is possible to estimate the sign structure of the differential of the global Poincaré mapping (a shift in time T). In this case all vertices of a strongly connected graph naturally break into two sets (two parties). As appeared, the influence of variables within one party is positive, while that of variables from different parties is negative. Even having simplified the structure of a local two-party graph (by eliminating its edges), one can still exactly describe the sign structure of the differential of the Poincaré mapping. The obtained results are applicable in the mathematical competition theory.     

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.     

On the interlacing of the zeros of Poincaré series

The Ramanujan Journal - Rankin proved that the Poincaré series for $$mathbf{SL}(2,{{mathbb {Z}}})$$ that are not cusp forms have all their zeros on the unit circle in the standard...     

Integration over the Space of Functions and Poincaré Series Revisited

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.     

Value distribution of Painlevé transcendents of the first and the second kind

We examine value distribution properties of the first and the second Painlevé transcendents. For every transcendental meromorphic solution ϕ(z) (resp. ψ(z)) of the first (resp. second) Painlevé equation, the deficiency δ(g,ϕ) (resp. δ(g, ψ)) of a small functiong(z) does not exceed 1/2. Furthermore, for ϕ(z), the ramification index satisfies ϑ(gφ)≤5/12.