Fluctuating Number of Energy Levels in Mixed-Type Lemon Billiards

In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B=0.1953,0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry–Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as E, in agreement with the theoretical prediction by Steiner.

Prony's method modification of multi frequency signals spectral analysis.

The issues of digital spectral analysis of signals were considered. The possibility of accurate representation of a complex discrete sequence in the form of a linear combination of multiple harmonics without attenuation has been proved and implemented. A finite algorithm modifying the Prony's method was proposed, with a guaranteed arrangement of the poles of the corresponding autoregressive model on the circle of a unit radius. A scheme of economization of the order of the model in conditions of data redundancy in relation to the estimated limited number of harmonic components of the signal was developed.

On the Sensitivity of the Gamma of the Quantum Cryptographic System AKM2017 to Changes in the Session Key

The quantum cryptographic system AKM2017 is considered. The results of the analysis of the dependence of the degree of difference between the encryption and decryption gamut on the degree of difference between the corresponding session keys are presented. The equality of these degrees of distinction is revealed and substantiated. For an arbitrarily fixed encryption session key, the distribution of session decryption keys by classes is revealed and described, depending on the value of the degree of difference between the encryption and decryption gamuts. One class is made up of all session decryption keys, leading to the same value of the degree of difference between the encryption and decryption gamuts. A geometric interpretation of the specified distribution by classes is given in the form of placement along circles (class is a circle) on the surface of a sphere of unit radius centered at the origin of the Euclidean rectangular coordinate system in a three-dimensional linear space over the field of real numbers. The stated results can be used to solve the problems of optimizing the values of the parameters of accuracy and reliability of the functioning of variants of practical implementations of the quantum cryptographic system AKM2017, for example, when setting up a session decryption key, which makes it possible to guarantee a predetermined small value of the mathematical expectation of the number of incorrectly decrypted binary plain text.

MATHEMATICAL MODEL AND AREA OF EXISTENCE OF THE FAMILY CRANKS AND MOBILE MECHANISMS WITH THE MAXIMUM TRANSMISSION ANGLE

The article examines crank-rocker mechanisms. Such mechanisms are used in transport and technological machines. The article is devoted to the search for a new family of crank-rocker mechanisms. A mathematical model of a new family of crank-rocker mechanisms is obtained. In this family, the maximum transmission angle reaches 90 when the crank angle is 75. Thus, the new family of crank-rocker mechanisms differs from the known families by the position of the mechanism in which the maximum of the transmission angle function takes place. It is shown that, with a certain ratio of link lengths, the new family corresponds to the known families KKM-5 and KKM-7. The area of existence of a new family of crank-rocker mechanisms is established. This area is bounded by the arc of the circle of the unit radius and the curve. The mentioned curve is based on a joint solution of a mathematical model of a new family of mechanisms and the famous Kolchin straight line. The dependence for the minimum transmission angle is obtained. A formula for determining the angle of the rocker arm span is proposed. A graphical interpretation of the mentioned dependencies and formulas is constructed. The scope of existence of a new family of crank-rocker mechanisms and graphical interpretations are used in determining the geometric parameters of mechanisms. These mechanisms are part of a new family of mechanisms.

Analytical solutions for three-dimensional Coulomb transition matrix at negative energy and integer values of interaction parameter

With the use of a stereographic projection of momentum space into a four-dimensional sphere of unit radius, the possibility of analytical solution of the three-dimensional two-body Lippmann–Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has also been applied to the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

Фундаментальный анализ сингулярных и резонансных явлений в колебательных полиадах молекулы дифторсилилена

The singular structure of the lower vibrational states of the difluorosilylene molecule (up to four quanta of total excitation) was studied by expanding the energies of each state in the series of high-order Rayleigh-Schrödinger perturbation theory and analyzing their implicit multivalued properties using the fourth degree Padé-Hermite approximants. The quartic potential energy surface in dimensionless normal coordinates was calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of the values of multivalued approximants reproduces the variational solution with high accuracy, while other values, starting from the fourth polyad, in many cases coincide with the energies of other states of the polyad. The Fermi and Darling-Dennison resonances are analyzed on the basis of the coincidence of the singular complex branch points of the approximants for interacting states inside or near a circle of unit radius on the complex plane. It was found that a pair of states can have several coinciding branch points of solutions, including those inside the unit circle. It is concluded that this approach is an effective method for determining the polyad structure of vibrational states. The calculation parameters are selected, which are necessary for the reproducibility of key results. The calculations were carried out using a software package in the Fortran language using a package of arithmetic calculations with a long mantissa of real numbers (200 digits).

Деформационное уширение и тонкая структура спектральных линий в оптических спектрах диэлектрических кристаллов, содержащих редкоземельные ионы

The procedure of calculation of the spectral line shape in optical spectra of rare-earth ions in crystals with the inclusion of random deformations of an elastically anisotropic crystal lattice caused by point defects is developed. The distribution function of components of the random strain tensor in the case of a low defect concentration is obtained as the generalized six-dimensional Lorentz distribution. The distribution function parameters are represented by the integral functional of the strain tensor components on a sphere of unit radius containing an isotropic point defect in its center. The numerical calculations of the strain tensors induced by point defects and the parameters of the distribution functions of random strains in LiLuF_4 and LaAlO_3 crystals have been performed. The calculated envelope with the doublet structure corresponding to the Γ_2(^3 H _4) → Γ_34(^3 H _5) singlet–doublet transition in the absorption spectrum of Pr^3+ ions in the LiLuF_4 crystal agrees well with the data of the measurements.

Drapeable unit arcs fit in the unit 30° sector

We show that a 30° circular sector of unit radius contains an isometric copy of every drapeable unit arc, and we describe the family of drapeable exit arcs of unit length in the sector. The conjecture that this sector is a cover for the family of all unit arcs remains unresolved.

Fatigue Life of Construction Elements in a Complex Stress State

This paper presents a method of assessing fatigue strength for a body in conditions of cyclical tension and torsion. The theoretical calculations have been conducted using the criterion of averaged structural microdamage resulting from local slips. The description of microdamages of such a body, in the view of the slip concept, was done by using a half-sphere with a unit radius, on the surface of which the location of all local physical planes and slip systems was determined employing three angles. A computer method was used to determine the slip boundaries in a complex stress state, analyzing the slip condition for all combinations of angles. Based on the calculated values of the microdamages' intensity function, the number of loading cycles until the moment of fracture initiation was estimated. Experimental verification of the suggested criterion was conducted using cylindrical smooth specimens, made of C45 steel. The tests of fatigue strength were made under conditions of a constant amplitude of zero-start pulsating loads.