Spectral and Nonlinear Properties of the Sum of Boolean Functions

Boolean functions are the mathematical basis of modern cryptographic algorithms. However, in practice, a set of interrelated Boolean functions is often used to construct a cryptographic algorithm. This circumstance makes the task of research of cryptographic quality, in particular, the distance of the nonlinearity of the sum of few Boolean functions important. The nonlinearity distance of a Boolean function is determined by the maximum value of its Walsh-Hadamard transform coefficients. In this paper, we proposed a formula that is the equivalent of the summation of Boolean functions in the Walsh-Hadamard transform domain. The application of this formula, as well as the Walsh-Hadamard spectral classification made it possible to determine the structure of WalshHadamard transform coefficients, and the distance of the nonlinearity when summing the Boolean functions lengths N 8 and N 16 , indicating valuable practical application for information protection

Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results

We continue our investigation of the nonlinear SUSY for complex potentials started in the Part I (math-ph/0610024) and prove the theorems characterizing its structure in the case of non-diagonalizable Hamiltonians. This part provides the mathematical basis of previous studies. The classes of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians are specified and the asymptotics of formal eigenfunctions and associated functions are derived. Several results on the normalizability of associated functions at infinities are rigorously proved. Finally the Index Theorem on relation between Jordan structures of intertwined Hamiltonians depending of the behavior of elements of canonical basis of supercharge kernel at infinity is proven.Comment: 31 pp., comments on PT symmetry and few relevant refs are adde

Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed

Energy State of a Plastically Deformed Surface Layer

AbstractThe paper reports the results of experimental research on the substantiation of the criterion for steel strengthening effectiveness established on a basis of an energetic approach to the consideration of the mechanism for the surface layer formation with dynamic methods of plastic forming. Using the analogy between the processes of energy absorption of the crystal lattice under mechanical loading and under heating, the work demonstrates that the maximum specific energy which can absorb the crystal lattice corresponds to the value equal to the difference between the heat content (enthalpy) of the material in the solid state, at the melting temperature and enthalpy H_TS at 2930К. The proposed method and experimental device allowing to estimate the stored energy in the plastically deformed surface layer as the difference between the work expended in plastic deformation of the material and the quantity of the released heat. It was established that the energy growth limit in the local plastically deformed volume of a surface occurs at making of 11-13 acts of the action force; the further increase in acts influences the energy state of the surface as it becomes stabilized

Программные связи и обеспечение устойчивости движения электромеханического манипулятора

For the theoretical study of the dynamics of manipulation robots, define design parameters and control laws, you must have a current mechanical models that accurately describe the properties of real robots. The choice of the computational model in each case is determined by the kinematic scheme of the manipulator, mechanical properties (inertial, elastic, dissipative, and the like) parts and assemblies, type and characteristics of the drives, as well as the required accuracy of the calculation. The objective of the control is to ensure the motion of the mechanical system under some requirements that make up its program. Program motion of the system can be performed by the application to the system of control of forces, the system settings change in the process, building of special control devices (controllers) or a combination of these. The original objectives of the control theory are inverse problems of classical dynamics. From the mathematical point of view, calculation model manipulation robot is a system of differential equations. This model may include equations describing the phenomena non-mechanical nature, for example, electrical processes in the circuits of the motors of the actuators. In this article the author examines the issues of ensuring conditions of the asymptotic stability software movement mechanical and electromechanical systems with holonomic and nonholonomic constraints. For example, the three-tier model controllable electromechanical manipulator conditions of the asymptotic stability of a given movement. The described approaches to ensuring the asymptotic stability of electromechanical systems can be used in the study of stability of motion proprietary mechanical systems, mechanics of controlled motion in the solution of management tasks manipulators, transport and space systems.Для теоретического изучения динамики манипуляционных роботов, определения конструктивных параметров и законов управления необходимо иметь расчётные механические модели, с достаточной точностью описывающие свойства реальных роботов. Выбор расчётной модели в каждом конкретном случае определяется кинематической схемой манипулятора, механическими свойствами (инерционными, упругими, диссипативными и т.п.) его деталей и узлов, типом и характеристиками приводов, а также необходимойточностью производимых расчётов. Задачей управления является обеспечение движения механической системы согласно некоторым требованиям, которые составляют её программу. Программное движение системы может быть осуществлено приложением к системе управляющих сил, изменением параметров системы в процессе движения, построением специальных управляющих устройств (регуляторов) или сочетанием этих возможностей. Исходными задачами теории управления являются обратные задачи классической динамики. С математической точки зрения расчётная модель манипуляционного робота представляет собой систему дифференциальных уравнений. Эта модель может содержать уравнения, описывающие также явления немеханической природы, например электрические процессы в цепях электродвигателей приводов. В данной статье автором исследуются вопросы обеспечения условий асимптотической устойчивости программного движения механических и электромеханических систем с голономными и неголономными связями. На примере модели трёхзвенного управляемого электромеханического манипулятора обеспечиваются условия асимптотической устойчивости заданного движения. Описываемые подходы к обеспечению условий асимптотической устойчивости электромеханических систем могут быть использованы при исследовании устойчивости движения несвободных механических систем, в механике управляемого движения, при решении задач управления роботами-манипуляторами, транспортными и космическими системами

Investigation of Microstructure of Oxygen-Containing Copper

The results of the study of fine structure of continuous casted bars of oxygen-containing copper produced by casting in the belt water-cooled crystallizer are presented. The character of fractures of samples of copper cast bars in different directions depending on technological parameters of continuous casting was investigated. For determination of chemical composition of samples of cast bars and detection of presence of possible impurities in copper the microspectral analysis was carried out. It is shown that removing of sources of gasing of copper melt leads to decreasing of volume fraction of eutectic Cu-Cu2O, discontinuity flaws and pores in the structure of cast bars.     Keywords: Copper, Continuos casting, Rolling, Rolled wire, Contirod method, Microstructure, Fracture, Microspectral analysis

Construction Method for Infinite Families of Bent Sequences

Bent-sequences is one of the most important classes of Boolean functions, which are widely used in modern cryptographic algorithms, and telecommunication systems that are based on CDMA and OFDM standards. The problem of synthesis of bent-sequences of large lengths is actual and widely discussed. However, in view of the high complexity and unpredictability of the class of bent-sequences, the creation of methods for their synthesis faces significant difficulties. In this paper, a recursive method for constructing infinite families of bent-sequences, based on easily synthesized bent-sequences of small length, has been developed. As the basis of this method, the operations of interleaving of elements and strings, which are widely used in the theory of synthesis of perfect binary arrays, are applied. Effective reproduction rules for bent-sequences in the time domain based on the operation of rearrangement of segments, a rotor, and dimensional changes are proposed. The method developed allows rapid acquisition of a lot of bentsequences of any predefined length. Moreover, the obtained bent-sequences belong to different classes according to Agievich classification, which is important from the cryptographic point of view

Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics

When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials and potentials with finite number of bound states. After the survey of the results existing in the subject the algebraic and analytic properties of hidden-symmetry differential operators are rigorously elaborated in the Theorems and illuminated by several examples

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.Comment: 13 pages, LaTeX with AmsFont