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.

On Unique Representations of Certain Dynamical Systems Produced by Continuous-Time Recurrent Neural Networks

2002 ◽  
Vol 14 (12) ◽  
pp. 2981-2996 ◽  
Author(s):  
Masahiro Kimura
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
State Space ◽  
Recurrent Neural Networks ◽  
Continuous Time ◽  
Feedforward Neural Networks ◽  
Approximation Problem ◽  
Affine Map ◽  
System A ◽  
Affine Maps

This article extends previous mathematical studies on elucidating the redundancy for describing functions by feedforward neural networks (FNNs) to the elucidation of redundancy for describing dynamical systems (DSs) by continuous-time recurrent neural networks (RNNs). In order to approximate a DS on Rn using an RNN with n visible units, an n—dimensional affine neural dynamical system (A-NDS) can be used as the DS actually produced by the above RNN under an affine map from its visible state-space Rn to its hidden state-space. Therefore, we consider the problem of clarifying the redundancy for describing A-NDSs by RNNs and affine maps. We clarify to what extent a pair of an RNN and an affine map is uniquely determined by its corresponding A-NDS and also give a nonredundant sufficient search set for the DS approximation problem based on A-NDS.

scholarly journals QUANTUM FIELD THEORY ON CURVED BACKGROUNDS — A PRIMER

2013 ◽  
Vol 28 (17) ◽  
pp. 1330023 ◽  
Author(s):  
MARCO BENINI ◽  
CLAUDIO DAPPIAGGI ◽  
THOMAS-PAUL HACK
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Degrees Of Freedom ◽  
Algebraic Approach ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
System A ◽  
Step Procedure ◽  
Free Fields

Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

SAMPLING DYNAMICAL SYSTEMS

1999 ◽  
Vol 3 (4) ◽  
pp. 602-609 ◽  
Author(s):  
Melvin J. Hinich
Keyword(s):  
Dynamical Systems ◽  
Sampling Rate ◽  
Identification Problem ◽  
Frequency Component ◽  
Linear Dynamical Systems ◽  
System A ◽  
Fixed Sample ◽  
Single Input ◽  
Exogenous Time Series

Linear dynamical systems are widely used in many different fields from engineering to economics. One simple but important class of such systems is called the single-input transfer function model. Suppose that all variables of the system are sampled for a period using a fixed sample rate. The central issue of this paper is the determination of the smallest sampling rate that will yield a sample that will allow the investigator to identify the discrete-time representation of the system. A critical sampling rate exists that will identify the model. This rate, called the Nyquist rate, is twice the highest frequency component of the system. Sampling at a lower rate will result in an identification problem that is serious. The standard assumptions made about the model and the unobserved innovation errors in the model protect the investigators from the identification problem and resulting biases of undersampling. The critical assumption that is needed to identify an undersampled system is that at least one of the exogenous time series be white noise.

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