Normal Forms and Cohomology Classes at Single Resonances

2020 ◽  
pp. 24-30
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Normal Form ◽  
Periodic Orbit ◽  
Normal Forms ◽  
Double Resonance ◽  
Action Space ◽  
Unperturbed System ◽  
Coordinate Change ◽  
Easy Consequence ◽  
Anisotropic Norm ◽  
Dirichlet Theorem

This chapter discusses the single-resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single-resonance regime. The classical partial averaging theory indicates that after a coordinate change, the system has the normal form away from punctures. In order to state the normal form, one needs an anisotropic norm adapted to the perturbative nature of the system. The chapter also uses the idea of Lochak to cover the action space with double resonances. A double resonance corresponds to a periodic orbit of the unperturbed system. Finally, the chapter looks at a lemma which is an easy consequence of the Dirichlet theorem.

Double Resonance: Geometric Description

2020 ◽  
pp. 31-38
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Mechanical System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Double Resonance ◽  
Energy Reduction ◽  
Geometric Description ◽  
Coordinate Change ◽  
Energy Surfaces

This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with perturbation. The system is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The chapter then formulates the non-degeneracy conditions and theorems about their genericity. It also considers the normally hyperbolic invariant cylinders, and sketches the proof using local and global maps. The periodic orbits obtained in Theorem 4.4 correspond to the fixed points of compositions of local and global maps, when restricted to the suitable energy surfaces.

Forcing equivalence between kissing cylinders

2020 ◽  
pp. 133-144
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Normal Form ◽  
Variational Problem ◽  
Mechanical System ◽  
Homology Class ◽  
Double Resonance ◽  
Fast Time ◽  
Coordinate Change ◽  
System A ◽  
Time Periodic ◽  
Slow System

This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.

Derivation of the slow mechanical system

2020 ◽  
pp. 162-181
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Normal Form ◽  
Degrees Of Freedom ◽  
Double Resonance ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Coordinate Change ◽  
System P ◽  
Resonant Normal Form ◽  
Coordinate Changes ◽  
Slow System

This chapter proves various normal form results and formulates the coordinate changes that are used to derive the slow system at the double resonance. The discussions here apply to arbitrary degrees of freedom. The results also apply to the proof of the main theorem by restricting to the case n = 2. First, the chapter reduces the system near an n-resonance to a normal form. After that, it performs a coordinate change on the extended phase space, and an energy reduction to reveal the slow system. The chapter then describes a resonant normal form, before explaining the affine coordinate change and the rescaling, revealing the slow system. Finally, it discusses variational properties of these coordinate changes.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

scholarly journals Normal form approach in the motion planning of space robots: a case study

2021 ◽  
Author(s):  
Krzysztof Tchoń ◽  
Katarzyna Zadarnowska
Keyword(s):  
Normal Form ◽  
Motion Planning ◽  
Normal Forms ◽  
Inverse Dynamics ◽  
Planning Problem ◽  
Inverse Dynamics Problem ◽  
Velocity Level ◽  
Form Approach ◽  
Space Approach

AbstractWe examine applicability of normal forms of non-holonomic robotic systems to the problem of motion planning. A case study is analyzed of a planar, free-floating space robot consisting of a mobile base equipped with an on-board manipulator. It is assumed that during the robot’s motion its conserved angular momentum is zero. The motion planning problem is first solved at velocity level, and then torques at the joints are found as a solution of an inverse dynamics problem. A novelty of this paper lies in using the chained normal form of the robot’s dynamics and corresponding feedback transformations for motion planning at the velocity level. Two basic cases are studied, depending on the position of mounting point of the on-board manipulator. Comprehensive computational results are presented, and compared with the results provided by the Endogenous Configuration Space Approach. Advantages and limitations of applying normal forms for robot motion planning are discussed.

NORMAL FORMS FOR FUZZY LOGIC — AN APPLICATION OF KOLMOGOROV’S THEOREM

1996 ◽  
Vol 04 (04) ◽  
pp. 331-349 ◽  
Author(s):  
VLADIK KREINOVICH ◽  
HUNG T. NGUYEN ◽  
DAVID A. SPRECHER
Keyword(s):  
Neural Networks ◽  
Fuzzy Logic ◽  
Normal Form ◽  
Normal Forms ◽  
Approximation Property ◽  
Universal Approximation ◽  
Approximation Result ◽  
Logical Operations ◽  
Kolmogorov's Theorem

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

scholarly journals On the structure of least common multiple matrices from some class of matrices

2018 ◽  
Vol 10 (1) ◽  
pp. 179-184
Author(s):  
A.M. Romaniv
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Smith Normal Form ◽  
Least Common Multiple ◽  
The Matrix ◽  
Singular Matrices ◽  
Class Of Matrices ◽  
Smith Normal Forms ◽  
Invertible Matrices ◽  
Principal Ideal Domains

For non-singular matrices with some restrictions, we establish the relationships between Smith normal forms and transforming matrices (a invertible matrices that transform the matrix to its Smith normal form) of two matrices with corresponding matrices of their least common right multiple over a commutative principal ideal domains. Thus, for such a class of matrices, given answer to the well-known task of M. Newman. Moreover, for such matrices, received a new method for finding their least common right multiple which is based on the search for its Smith normal form and transforming matrices.

scholarly journals A combinatory account of internal structure

2011 ◽  
Vol 76 (3) ◽  
pp. 807-826 ◽  
Author(s):  
Barry Jay ◽  
Thomas Given-Wilson
Keyword(s):  
Normal Form ◽  
Internal Structure ◽  
Pattern Matching ◽  
Normal Forms ◽  
Expressive Power ◽  
Combinatory Logic ◽  
Natural Numbers ◽  
Type Information ◽  
Computable Functions

AbstractTraditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

2004 ◽  
Vol 14 (09) ◽  
pp. 3337-3345 ◽  
Author(s):  
JIANPING PENG ◽  
DUO WANG
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Sufficient Condition ◽  
First Order ◽  
Recursive Formulas ◽  
Unique Normal Forms

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

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