scholarly journals Classes of bent functions identified by specific normal forms and generated using Boolean differential equations

2011 ◽  
Vol 24 (3) ◽  
pp. 357-383 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Normal Forms ◽  
Direct Calculation ◽  
Bent Functions ◽  
Second Step ◽  
Complexity Classes ◽  
The Relationship

This paper aims at the identification of classes of bent functions in order to allow their construction without searching or sieving. In order to reach this aim, we studied first the relationship between bent functions and complexity classes defined by the Specific Normal Forms of all Boolean functions. As result of this exploration we found classes of bent functions which are embedded in different complexity classes defined by the Specific Normal Form. In the second step to reach our global aim, we utilized the found classes of bent functions in order to express bent functions in terms of derivative operations of the Boolean Differential Calculus. In detail, we studied bent functions of two and four variables. This exploration leads finally to Boolean differential equations that will allow the direct calculation of all bent functions of two and four variables. A given generalization allows to calculate subsets of bent functions for each even number of Boolean variables.

scholarly journals Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions

2015 ◽  
Vol 28 (1) ◽  
pp. 51-76 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Short Introduction ◽  
The Core ◽  
Boolean Equation ◽  
General Difference ◽  
Derivatives Of

The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

COMPUTATION OF SIMPLEST NORMAL FORMS OF DIFFERENTIAL EQUATIONS ASSOCIATED WITH A DOUBLE-ZERO EIGENVALUE

2001 ◽  
Vol 11 (05) ◽  
pp. 1307-1330 ◽  
Author(s):  
Y. YUAN ◽  
P. YU
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Computational Efficiency ◽  
Normal Forms ◽  
Computer Systems ◽  
Computer Programs ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Explicit Formulae ◽  
User Friendly

In this paper a method is presented for computing the simplest normal form of differential equations associated with the singularity of a double zero eigenvalue. Based on a conventional normal form of the system, explicit formulae for both generic and nongeneric cases are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The recursive algebraic formulae have been implemented on computer systems using Maple. The user-friendly programs can be executed without any interaction. Examples are given to demonstrate the computational efficiency of the method and computer programs.

ON THE NONEXISTENCE of BENT FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1431-1438 ◽  
Author(s):  
YIN ZHANG ◽  
MEICHENG LIU ◽  
DONGDAI LIN
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Normal Forms ◽  
Bent Functions

In this paper, we study the nonexistence of bent functions in the class of Boolean functions without monomials of degree less than d in their algebraic normal forms (ANF). We prove that n-variable Boolean functions in such class are not bent when there are not more than n + d - 3 monomials in their ANFs. We also show that an n-variable Boolean function is not bent if it has no monomial of degree less than ⌈3n/8 + 3/4⌉ in its ANF.

Uniqueness and non-uniqueness of normal forms for vector fields

1988 ◽  
Vol 108 (1-2) ◽  
pp. 27-33 ◽  
Author(s):  
Alberto Baider ◽  
Richard Churchill
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Resonance Problem ◽  
Main Result Theorem ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Result Theorem ◽  
The Relationship

SynopsisThe use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

AN EFFICIENT METHOD FOR COMPUTING THE SIMPLEST NORMAL FORMS OF VECTOR FIELDS

2003 ◽  
Vol 13 (01) ◽  
pp. 19-46 ◽  
Author(s):  
P. YU ◽  
Y. YUAN
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Efficient Method ◽  
Vector Fields ◽  
Normal Forms ◽  
Nonlinear Transformation ◽  
New Method ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Computation Efficiency

A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the "simplest" formula which reduces the computation demand to minimum. At each order of the normal form computation, the formula generates a set of algebraic equations for computing the normal form and nonlinear transformation. Moreover, the new recursive method is not required for solving large matrix equations, instead it solves linear algebraic equations one by one. Thus the new method is computationally efficient. In addition, unlike the conventional normal form theory which uses separate nonlinear transformations at each order, this approach uses a consistent nonlinear transformation through all order computations. This enables one to obtain a convenient, one step transformation between the original system and the simplest normal form. The new method can treat general differential equations which are not necessarily assumed in a conventional normal form. The method is applied to consider Hopf and Bogdanov–Takens singularities, with examples to show the computation efficiency. Maple programs have been developed to provide an "automatic" procedure for applications.

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

2020 ◽  
Vol 30 (02) ◽  
pp. 2050028 ◽  
Author(s):  
Chuncheng Wang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Transmission Line ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Line Equation ◽  
Bifurcation Problems ◽  
Functional Differential

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.

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.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals CagA-PositiveHelicobacter pyloriInfection and Reduced Sperm Motility, Vitality, and Normal Morphology

2013 ◽  
Vol 35 ◽  
pp. 229-234 ◽  
Author(s):  
E. Moretti ◽  
G. Collodel ◽  
L. Mazzi ◽  
M. S. Campagna ◽  
N. Figura
Keyword(s):  
Sperm Motility ◽  
Normal Forms ◽  
Molecular Mimicry ◽  
Sperm Quality ◽  
Semen Analysis ◽  
Normal Morphology ◽  
Partial Linear ◽  
Caga Status ◽  
The Relationship ◽  
Autoimmune Phenomena

Helicobacter pylori(HP) infection, particularly when caused by strains expressing CagA, may be considered a concomitant cause of male and female reduced fertility. This study explored, in 87 HP-infected males, the relationship between infection by CagA-positive HP strains and sperm parameters. HP infection and CagA status were determined by ELISA and Western blotting; semen analysis was performed following WHO guidelines. The amino acid sequence of human enzymes involved in glycolysis and oxidative metabolism were “blasted” with peptides expressed by HP J99. Thirty-seven patients (42.5%) were seropositive for CagA. Sperm motility (18% versus 32%; ), sperm vitality (35% versus 48%; ) and the percentage of sperm with normal forms (18% versus 22%; ) in the CagA-positive group were significantly reduced versus those in the CagA-negative group. All the considered enzymes showed partial linear homology with HP peptides, but four enzymes aligned with four different segments of the samecagisland protein. We hypothesize a relationship between infection by strains expressing CagA and decreased sperm quality. Potentially increased systemic levels of inflammatory cytokines that occur in infection by CagA-positive strains and autoimmune phenomena that involve molecular mimicry could explain the pathogenetic mechanism of alterations observed.

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.

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