The Riemannian geometry of physical systems of curves

Summary The properties of a physical system S_k where k ≠−1, of ∞²ⁿ⁻¹ trajectories C. in a Riemannian space V_n are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system S_k, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S ₀ in a Riemannian space V_n, is obtained. Finally, contact transformations and the transformation theory of a physical system S_k where k ≠−1, are considered in detail. To Enrico Bompiani on his scientific Jubilee Kasner,Differential geometric aspecte of dynamics, The Princeton Colloquium Lectures, 1909. Published by the ? American Mathematical Society, Providence, Rhode Island, 1913, and reprinted 1934.     

Boundary control and canonical realizations of a two-velocity dynamical system

The paper is devoted to the problems of controllability and realization for dynamical systems with various types of interacting waves that propagate with different velocities. One-velocity and a two-velocity dynamical systems are significantly different from the physical point of view. One can reconstruct a one-velocity system by its transfer function. For a two-velocity system a unique reconstruction is impossible. A procedure is proposed that allows us to construct by a transfer function of a two-velocity system a one-velocity system (a model) with the same transfer function. We give a “dynamical” interpretation for the triangular Krein factorization and for the corresponding construction of a triangular integral. For a transformation operator that connects a two-velocity system and its one-velocity model, a representation is given in terms of projectors on the accessible sets. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1994, pp. 18–44.     

On the Lipschitz stability of complex systems of differential equations

Mathematical models of real physical processes and phenomena can be described by the so-called complex systems of differential equations. Among them are systems that decompose into interacting subsystems and intrasystemic connections. They have significant nonlinearities. In this connection the application of the existing methods of study of the dynamics of the behavior of such systems is quite difficult. We propose some approaches to the study of Lipschitz stability connected with the technique of applying Lyapunov functions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 12–16.     

A hierarchy of axiom systemZF

In this paper we develop a sequenceZ ₀, ...,Z _α,... of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo's system Z and the union of all systems in the sequence is justZF. And we prove that for ordinal number α>1, there exists a sequence of ℵ_a+1 axiom systems between systemsZ _α andZ _α+1 such that these systems satisfy the above condition (1).     

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L ₁. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations, we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models (parallel case) in three dimensions.     

Quantum Uncertainty and the Spectra of Symmetric Operators

In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=〈φ,S φ〉, through a function ΔS _t of t, i.e., ΔS[φ]≥ΔS _t , for all physical states φ with 〈φ,S φ〉=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function ΔS _t . We also discuss potential applications in quantum and classical information theory.      

Universal minimal topological dynamical systems

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy < log a, admit a generating partition with no more than a sets. In Symbolic Dynamics terminology, these results can be phrased— ℕ^ℤ is a universal system in the category of aperiodic systems, and [a]^ℤ is a universal system for aperiodic ergodic systems with entropy < log a. Weiss ([We89], 1989) presented a Minimal system, on a Compact space (a subshift of ) which is universal for aperiodic systems. In this work we present a joint generalization of both results: given ɛ, there exists a minimal subshift of [a]^ℤ, universal for aperiodic ergodic systems with entropy < log a − ɛ.     

On the Exel Crossed Product of Topological Covering Maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕ is a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems.     

The pauli principle,stability, and bound states in systems of identical pseudorelativistic particles

Based on analyzing the properties of the Hamiltonian of a pseudorelativistic system Z_n of n identical particles, we establish that for actual (short-range) interaction potentials, there exists an infinite sequence of integers n_s, s = 1, 2, …, such that the system is stable and that sup _s n_s+1 n_s⁻¹ < + ∞. For a stable system Z_n, we show that the Hamiltonian of relative motion of such a system has a nonempty discrete spectrum for certain fixed values of the total particle momentum. We obtain these results taking the permutation symmetry (Pauli exclusion principle) fully into account for both fermion and boson systems for any value of the particle spin. Similar results previously proved for pseudorelativistic systems did not take permutation symmetry into account and hence had no physical meaning. For nonrelativistic systems, these results (except the estimate for n_s+1 n_s⁻¹ ) were obtained taking permutation symmetry into account but under certain assumptions whose validity for actual systems has not yet been established. Our main theorem also holds for nonrelativistic systems, which is a substantial improvement of the existing result. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 116–129, October, 2008.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.     

Geometric minimality conditions for systems of exponentials inL p [−π, π]

A proof is given of the stability theorem for minimal systems of exponentialse(Λ) = {e ^iλx }λ∈Λ inL _p [−π, π], where Λ ⊂ ℂ is a discrete subset. Geometric minimality conditions for such systems are obtained. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 773–777, November, 1995. I wish to express gratitude to A. A. Shkalikov, who posed the problem and paid constant attention to this work.     

Numeration systems,fractals and stochastic processes

A numeration system Ω is a compactification of the set of real numbers keeping the actions of addition and positive multiplication in a natural way. That is, Ω is a compact metrizable space with #Ω≥2 to which ℝ acts additively andG acts multiplicatively satisfying the distributive law, whereG is a nontrivial closed multiplicative subgroup of ℝ₊. Moreover, the additive action is minimal and uniquely ergodic with 0-topological entropy, while the multiplication by λ has |log λ|-topological entropy attained uniquely by the unique invariant probability measure under the additive action. We construct Ω as above as a colored tiling space corresponding to a weighted substitution. This framework contains especially the substitution dynamical systems and β-transformation systems with periodic expansion of 1, both of which have discreteG. It also contains systems withG=ℝ₊. We study α-homogeneous cocycles on it with respect to the addition. They are interesting from the point of view of fractal functions or sets as well as self-similar processes. We obtain the zeta-functions of Ω with respect to the multiplication.     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

Divergent Fourier series of integrable functions with respect to orthonormal systems

There exist two orthonormal systems such that the Fourier series of each functionƒ ∈ L[0, 1],ƒ ≠ 0, with respect to at least one of these systems diverges on a set of positive measure. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 717–724, May, 1998.     

Switching Sets Regolari E Cubiche Rigate Di P G (5, q ) Piani Di Tralsazione Ad Essi Associati G (5, q ) Piani Di Tralsazione Ad Essi Associati

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ. The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ⊂ Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ − Φ) ∪ Φ′. The planes Π, of orderq ³, are a generalization of the finite Hall planes.     

The reconstruction theorem for endomorphisms

The reconstruction theorem deals with dynamical systems which are given by a map ψ : M → M together with a read out function 𝒻 : M → ℝ. Restricting to the cases where ψ is a diffeomorphism, it states that for generic (ψ, 𝒻 ) there is a bijection between elements x ∈ M and corresponding sequences (𝒻(x), 𝒻 (ψ(x)), . . . , 𝒻 (ψ ^{k -1}(x))) of k successive observations, at least for k sufficiently big. This statement turns out to be wrong in cases where ψ is an endomorphism. In the present paper we derive a version of this theorem for endomorphisms (and which is equivalent to the original theorem in the case of diffeomorphisms). It justifies, also for dynamical systems given by endomorphisms, the algorithms for estimating dimensions and entropies of attractors from obervations. Received: 20 June 2002     

The Spectrum of Tetrahedral Quadruple Systems

An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair (S, T){(S, \mathcal{T})} where S is a finite set of v elements and T{\mathcal{T}} is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of T{\mathcal{T}} . When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4.     

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.