AN ALGEBRAIC APPROACH TO THE KOLMOGOROV-SINAI ENTROPY

1996 ◽  
Vol 08 (02) ◽  
pp. 167-184 ◽  
Author(s):  
R. ALICKI ◽  
J. ANDRIES ◽  
M. FANNES ◽  
P. TUYLS
Keyword(s):  
Dynamical Systems ◽  
Algebraic Approach ◽  
Entropy Formula ◽  
Starting Point ◽  
Algebraic Formalism

We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for defining the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained proof of the entropy formula for general automorphisms of n-dimensional tori.

Considerations on the Vibratory Motion of a Medical Electro-Mechanical Equipment

2015 ◽  
Vol 811 ◽  
pp. 110-116
Author(s):  
Cristian Dragomirescu ◽  
Victor Iliescu
Keyword(s):  
Dynamical Systems ◽  
Medical Equipment ◽  
Initial Conditions ◽  
Therapeutic Benefit ◽  
Mechanical Equipment ◽  
Mechanical Device ◽  
Starting Point ◽  
Vibratory Motion

The paper analyses the vibratory motion of an electro-mechanical device acting medical equipment. The study is performed using the specific methods of the theory of the dynamical systems. The induced vibrations, their variation in time and the evolution of the system, as a function of the initial conditions, are studied, examining the fulfilment of the constraints prescribed in the literature, in order to achieve a maximal therapeutic benefit. The suitable operation ranges are determined, which establishes the starting point for the design of such equipment.

scholarly journals Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Jinde Cao ◽  
Wei Jiang
Keyword(s):  
Dynamical Systems ◽  
Necessary Conditions ◽  
Algebraic Approach ◽  
Coefficient Matrix ◽  
The State ◽  
State Structure ◽  
Singular Coefficient ◽  
Control Delay ◽  
State Response ◽  
Sufficient And Necessary Conditions

This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. The factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously. The state structure of fractional singular dynamical systems with control delay is characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability for such systems are established based on the algebraic approach. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.

Extension of the Dynamics of Unstable Systems

1998 ◽  
Vol 01 (01) ◽  
pp. 127-165 ◽  
Author(s):  
I. Antoniou ◽  
Z. Suchanecki
Keyword(s):  
Dynamical Systems ◽  
Evolution Operator ◽  
Algebraic Approach ◽  
Extended Formulation ◽  
Probabilistic Prediction ◽  
Unstable Systems ◽  
Dual Pairs ◽  
Spectral Decompositions ◽  
Prediction And Control ◽  
And Control

The work of the Brussels–Austin groups over the last six years has demonstrated that for unstable systems, classical or quantum, there exist spectral decompositions of the evolution In terms of resonances and resonance states which appear as eigenvalues and eigenprojections of the evolution operator. These new spectral decompositions are non-trivial only for unstable systems and define an extension of the evolution to suitable dual pairs. Duality here is the states/observables duality and the spectral decompositions extend the algebraic approach to dynamical systems to an intrinsically probabilistic and irreversible formulation. The extended formulation allows for probabilistic prediction and control beyond the traditional local techniques.

scholarly journals On Some Symmetries of Quadratic Systems

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1300
Author(s):  
Maoan Han ◽  
Tatjana Petek ◽  
Valery G. Romanovski
Keyword(s):  
Dynamical Systems ◽  
Algebraic Approach ◽  
Symmetry Groups ◽  
Planar Case ◽  
Affine Map ◽  
Polynomial Dynamical Systems ◽  
System A ◽  
Polynomial Map ◽  
Real Quadratic ◽  
General Method

We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.

Algebraic Schrödinger Representation of Quantum Chromodynamics in Temporal Gauge and Resolution of Constraints

1997 ◽  
Vol 52 (3) ◽  
pp. 220-240
Author(s):  
H. Stumpf ◽  
W. Pfister
Keyword(s):  
Energy Equation ◽  
Electric Energy ◽  
Algebraic Approach ◽  
Functional Space ◽  
Generating Functional ◽  
Vector Potentials ◽  
Temporal Gauge ◽  
Functional States ◽  
Algebraic Representations ◽  
Algebraic Formalism

Abstract The algebraic formalism of QCD is expounded in order to demonstrate the resolution of Gauß constraints on the quantum level. In the algebraic approach energy eigenstates of QCD in temporal gauge are represented in an algebraic GNS-basis. The corresponding Hilbert space is mapped into a functional space of generating functional states. The image of the QCD-Heisenberg dynamics becomes a functional energy equation for these states. In the same manner the Gauß constraints are mapped into functional space. In functional space the Gauß constraints can be exactly resolved. The resolutions are defined by nonperturbative recurrence relations. The longitudinal color electric energy can be expressed by means of these resolvents, which leads to "dressed" color Coulomb forces in temporal gauge. Although present in the system, the longitudinal vector potentials do not affect its energy eigenvalues. This leads to a selfconsistent subsystem within the functional energy equation in temporal gauge which has to be identified with a functional energy equation in Coulomb gauge. In addition this procedure implies a clear conception for the incorporation of various algebraic representations into the formal Heisenberg dynamics and establishes the algebraic “Schrödinger” equation for QCD in functional space.

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