Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.     

Quantum modeling of nonlinear dynamics of stock prices: Bohmian approach

We use quantum mechanical methods to model the price dynamics in the financial market mathematically. We propose describing behavioral financial factors using the pilot-wave (Bohmian) model of quantum mechanics. The real price trajectories are determined (via the financial analogue of the second Newton law) by two financial potentials: the classical-like potential V (q) (“hard” market conditions) and the quantumlike potential U(q) (behavioral market conditions). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 405–415, August, 2007.     

Bogoliubov equations and functional mechanics

The functional classical mechanics based on the probability approach, where a particle is described not by a trajectory in the phase space but by a probability distribution, was recently proposed for solving the irreversibility problem, i.e., the problem of matching the time reversibility of microscopic dynamics equations and the irreversibility of macrosystem dynamics. In the framework of functional mechanics, we derive Bogoliubov-Boltzmann-type equations for finitely many particles. We show that a closed equation for a one-particle distribution function can be rigorously derived in functional mechanics without any additional assumptions required in the Bogoliubov method. We consider the possibility of using diffusion processes and the Fokker-Planck-Kolmogorov equation to describe isolated particles.     

Random walks in random environment: What a single trajectory tells

We present a procedure that determines the law of a random walk in an iid random environment as a function of a single “typical” trajectory. We indicate when the trajectory characterizes the law of the environment, and we say how this law can be determined. We then show how independent trajectories having the distribution of the original walk can be generated as functions of the single observed trajectory.     

Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem

One of modern approaches to the problem of coordination of classical mechanics and statistical physics — functional mechanics is considered. Deviations from classical trajectories are calculated and evolution of themoments of distribution function is constructed. The relation between the received results and absence of the Poincaré-Zermelo paradox in functional mechanics is discussed. Destruction of periodicity of movement in functional mechanics is shown and decrement of attenuation for classical invariants of movement on a trajectory of functional mechanical averages is calculated.     

Determinantal singularities and newton polyhedra

Topological invariants of determinantal singularities are studied in terms of Newton polyhedra. The approach is based on the notion of a toric resolution of a determinantal singularity. Computations are carried out in the more general setting of “elimination theory in the context of Newton polyhedra.”     

Determinantal singularities and newton polyhedra

Topological invariants of determinantal singularities are studied in terms of Newton polyhedra. The approach is based on the notion of a toric resolution of a determinantal singularity. Computations are carried out in the more general setting of “elimination theory in the context of Newton polyhedra.”     

“Locus” and “Trace” in Cabrigéomètre: relationships between geometric and functional aspects in a study of transformations

The present text describes and characterises the tools “Locus” and “Trace” of Cabri-géomètre II, in relations to a study of geometric transformation, more precisely, the passage from the notion of transformation of figures to the notion of applications¹ that map points on the plane onto the plane itself. In particular it discusses how the conception of image of a figure under a transformation can evolve—through interaction in a “milieu” organised around Cabri-géomètre—such that students move from views of figure-images as undecomposible entities to see them as sets of image-points. Moreover, the study allowed the identification that the notion of trajectory (in a dynamic interpretation) has an important role in this conceptually difficult passage and that dynamic geometry environment renovate this notion.     

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

The notion of Loschmidt echo (also called “quantum fidelity”) has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of “quantum chaos”, quantum computation and decoherence. In this paper, we study this quantity semiclassically (as ), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called “Ehrenfest time” , we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the “return probability” (again on reference states being the coherent states) by the same method, as a function of t and . Submitted: April 27, 2006; Accepted: May 11, 2006     

Hyperbolic Calculus

The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universal) Clifford algebra forIR ^1,0 [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus” analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical string motions.     

New global distributions in number theory and their applications

Consideration of an example corresponding to the partition of integers results in revision of certain physical concepts. A substantial term is added to the Bose–Einstein distribution. The notion of “fractional dimension” is introduced. Some effects considered earlier as pure quantum effects are explained from the classical standpoint. A distribution depending on three fixed points is given.     

Derivation of the Boltzmann equation and entropy production in functional mechanics

A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation of classical mechanics. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which are rational numbers. Hence, one can speak about the accuracy of the initial probability density function in the Liouville equation. We assume that the initial data for the microscopic density functions are assigned by the macroscopic one (so one can say about a kind of hierarchy and subordination of the microscale to the macroscale) and derive the Boltzmann equation, which leads to the entropy production.     

A Newton-like method for nonlinear system of equations

Tanabe (1988) proposed a variation of the classical Newton method for solving nonlinear systems of equations, the so-called Centered Newton method. His idea was based on a deviation of the Newton direction towards a variety called “Central Variety”. In this paper we prove that the Centered Newton method is locally convergent and we present a globally convergent method based on the centered direction used by Tanabe. We show the effectiveness of our proposal for solving nonlinear system of equations and compare with the Newton method with line search.     

A global optimization method for the design of space trajectories

The problem of optimally designing a trajectory for a space mission is considered in this paper. Actual mission design is a complex, multi-disciplinary and multi-objective activity with relevant economic implications. In this paper we will consider some simplified models proposed by the European Space Agency as test problems for global optimization (GTOP database). We show that many trajectory optimization problems can be quite efficiently solved by means of relatively simple global optimization techniques relying on standard methods for local optimization. We show in this paper that our approach has been able to find trajectories which in many cases outperform those already known. We also conjecture that this problem displays a “funnel structure” similar, in some sense, to that of molecular optimization problems.     

Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional. The author passed away in 2006 prior to publication of the article.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS Model

An analysis of the RSS model in mathematical economics involves the study of an infinite-horizon variational problem in discrete time. Under the assumption that the felicity function is upper semicontinuous and “supported” at the value of the maximally-sustainable level of a production good, we report a generalization of results on the equivalence, existence and asymptotic convergence of optimal trajectories in this model. We consider two parametric specifications, and under the second, identify a “symmetry” condition on the zeroes of a “discrepancy function” underlying the objective function that proves to be necessary and sufficient for the asymptotic convergence of good programs. With a concave objective function, as is standard in the antecedent literature, we show that the symmetry condition reduces to an equivalent “non-interiority” condition.