Group codes on certain algebraic curves with many rational points

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq ² over _q , whereq = 2q ₀ ² andq ₀ = 2 ⁿ , such that dimension + minimal distance q ² + 1 – q ₀ (q – 1). The codes are ideals in the group algebra _q [S], whereS is a Sylow-2-subgroup of orderq ² of the Suzuki-group of orderq ² (q ² + 1)(q – 1). The curves used for construction have in relation to their genera the maximal number of GF _q -rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.Supported by Deutsche Forschungsgemeinschaft while visiting Essen University     

Propagation of a shock wave in a locally anisotropic medium containing channels with elevated thermal diffusivity

We consider the interaction between a shock wave and thin extended channels with elevated a thermal diffusivity. It is shown that a thermal perturbation ahead of the shock wave front leads to the effect of a warm layer, i.e., to the formation of a large-scale self-similarly growing precursor.Notation T temperature - p pressure - density - velocity - e specific internal energy - thermal conductivity coefficient - effective specific heat ratio - q heat flux density - t time - x, y Cartesian coordinates Institute of the Dynamics of the Geospheres, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 643–650, June, 1994.     

Interactions of oxides in heating and combustion

Special features of the component interaction in two groups of SHS systems are investigated: I) oxideoxidizer+oxide-reducer and II) oxide mixtures in which the reaction of formation of a complex oxide is not redox. It is shown that the components of mixtures I first transform into the components of the corresponding mixture II and only after that into the final products. Mass transfer of the reagents in all steps involves eutectic melts as well as gaseous intermediate products whose formation is caused by impurities. Basic macrosteps of the interaction processes of Ba and Sr oxides with WO₃ and MoO₃ are proposed and confirmed experimentally.Institute of the Chemistry of Solids and the Processing of Mineral Raw Material, Siberian Branch of the Russian Academy of Sciences, Novosibirsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No.5, pp.594–597, November, 1993.     

Finding irreducible and primitive polynomials

In the paper some new fast constructions of irreducible and primitive polynomials are presented. For instance, it is shown, that for anyQ large enough one can design a finite field _q withq=Q + o(Q) elements in polynomial time (log Q) ^o (1).     

Pressure-induced non-bulk superconductivity in Eu1.2Mo6(S1−x Se x )8

High-pressure study has been made both resistively and magnetically on Eu_1.2Mo₆(S_1–x Se _x )₈. Superconductivity was detected at a lower pressure in samples forx0 than forx=0 in spite of the larger unit cell volume of the former. The superconductivity observed remains non-bulk up to 20 kbar. The maximum onset temperature of the superconductivity transition decreases withx.On leave from the Physics Institute, Chinese Academy of Sciences, Peking, China.     

Analysis and optimal control of discrete time‐varying systems Via Hahn polynomials

Abstract

Recently, the problem of analysis and optimal control of discrete time‐invariant systems has been extensively studied using finite series expansion of discrete orthogonal polynomials. This paper is to extend the applicable scope of discrete orthogonal polynomials to discrete time‐varying systems. The finite set of Hahn polynomials {q_ik)], i=0, 1, …, N} is chosen as the finite series expansion basis due to its general form and useful properties. First, for treating the product of two discrete‐time functions by Hahn series expansion, a new algorithm is derived to compute the Hahn series expansion coefficients of products qi(k)q_j (k), i, j=0, 1, …, N. These Hahn coefficients are then used to establish a product operational matrix for relating the Hahn coefficient vector of a product function to those of its component functions. This product operational matrix, along with the relations for connecting the Hahn coefficient vectors of a discrete function x(k) and its time‐shifted x(k+1), is finally applied to derive computational algorithms for solving the problems of analysis and optimal control of discrete time‐varying systems via finite Hahn series. Computed results are provided to illustrate the applicability of the proposed algorithms.     

Investigation into threshold stress intensity factors of materials under cyclic loading. Report 2. Prediction of fatigue limits and fatigue crack growth

On the basis of the results described in Report 1 [Probl. Prochn., No. 4, 5–16 (1998)], the author considers the possibility of using threshold stress intensity factors for the prediction of fatigue limits for titanium alloys by the formulas of linear elastic fracture mechanics and presents a substantiation of the method for predicting fatigue crack growth in structural elements. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 5–11, September–October, 1998.     

Mathematical model of least cost planning of regional energy supply

A number of simple mathematical models for least cost planning of energy supply are considered. Optimum plans have been found and ranking of energy sources is suggested.Institute of Mathematics, Academy of Sciences of Belarus; Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 1, pp. 149–155, January–February, 1995.     

Influence of thermal conductivity of a material on an unsteady heat removal parameter

In preparing [1] the authors additionally studied the influence of the thermophysical properties of a material on its unsteady breakdown. They found that the unsteady heat removal parameter d₀ depends logarithmically on the thermal conductivity.Institute for High Temperatures, Academy of Sciences of the USSR, Moscow. Institute for Development of Materials, Ukrainian Academy of Sciences, Kiev. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 62, No. 4, pp. 546–551, April, 1992.     

A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 3. Antiplane deformation

Within the framework of the approach proposed by L. P. Mazurak, L. T. Berezhnyts'kyi, and P. S. Kachur [“Method for determination of elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 1. Mathematical foundations,”Fiz.-Khim. Mekh. Mater.,33, No. 6, 21–31 (1997)], we construct a new method for the determination of elastic equilibrium of cylindrical bodies with noncanonical curvilinear foreign elastic inclusions under conditions of longitudinal shear. Unlike the method of perturbation of the form of a boundary, this method imposes no restrictions on the form of inclusions. The method is based on a procedure of determination of contour integrals of the Cauchy type by using the Faber polynomials. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 21–26, March–April, 1999.     

A linear representation of the diffusion of solvents into insoluble polymers

The amount,q, of a solvent taken up by a sample of an insoluble polymer during a time,t, can be represented accurately by the relationq=t/(at + b), which has a very simple linear equivalent: (t/q)=at+b, wherea andb are constants. The same behaviour is also shown by the exchange of ions in an ion-exchange resin and in the case of a swell process, whereq may be measured as a volume increase. The kinetic relationq=k(Q−q) ², where Q=a⁻¹ andk=Q ⁻²b⁻¹, is compared with Fick's first law of diffusion and the dependence of the diffusion coefficient and the concentration gradient onq is discussed.     

Behaviour of the complex dielectric function and energy loss spectrum of copper in the [110] direction in the energy range 2 < ħω ⩽ 4 Rydberg

The complex wave vector- and frequency-dependent dielectric function, (q,) = ₁ (q,) + ₂ (q,), for copper is calculated using realistic band energies and wave functions. The Bloch states _bk (r) are obtained from the modified augmented plane wave method (MAPW) with the Chodorow potential. Results are presented for the principal direction ofq, [110]. About 70 bands and 60 plane waves at 10k-points in the 1/48th of the Brillouin zone are considered. An additional peak was found at the energy loss spectrum of copper whose centre is situated near 2.1 Rydberg, its centre moves gradually to the high energy side with increasing |q|. According to our knowledge, it is the first time this result has been calculated theoretically or even measured experimentally in the [110] direction for copper.     

Pulling of glass microcapillaries: Theory and experiment

An approximate analytical solution is obtained which predicts a microcapillary form during pulling which is close to that observed in experiments.Institute of Mechanics and Biomechanics, Bulgarian Academy of Sciences, Sofia. Insitute for Problems of Mechanics, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 63, No. 6, pp. 722–727, December, 1992.Indices 0, p, and 1 denote R, h, and T referring to the initial section (x=0), heating zone boundary (x=l), and fiber section on the receiving device (x=L).     

Analysis of the stress-strain state of a rectangular orthotropic plate under the action of biaxial tension and shear

We propose a method for the analysis of the stress-strain state of rectangular orthotropic plates under the conditions of biaxial tension and shear. According to this method, we specify a field of displacements in the plate in the form of the product of a matrix of differential operators by an unknown vector function or a column of three scalar functions, each of which admits a representation in the form of the sum of the products of an unknown function of one variable by a trigonometric function of the other variable. The number of independent coefficients of each function is determined from the boundary conditions by the method of Fourier series. We also performed calculations of the stressed state in the plate depending on the ratio of its sides and the type of loading. ^aKarpenko Physicomechanical Institute, National Academy of Sciences of Ukraine, Lviv.^bInstitute for Applied Problems in Mathematics and Mechanics, National Academy of Sciences of Ukraine, Lviv.^cLutsk Industrial Institute, Lutsk, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 47 – 55, January – February, 1998.     

Structure of helium at 4 and 8 K from neutron diffraction

We determine by neutron diffraction the static structure factorS(q) of fluid helium at 4 K and 8 K at number densityn = 19.5 nm^–3 for wavenumbers 0.5 < q < 7 nm^–1.S(q) decreases very rapidly from its thermodynamic valueS(0) atq = 0 to aboutS(0)/2 atq 5 nm^–1 confirming the behavior seen in X-ray diffraction experiments. We combine the experimental data with previous larger-q results and compare them with classical model calculations that indicate long range correlations 1/r ⁴ between the helium particles. By means of computer simulations we show that the structure is incompatible with any Lennard-Jones interaction.     

Phonon dispersion in He II at long wavelengths and low temperatures

Phonon dispersion in liquid He II at long wavelengths and low temperatures is calculated using a generalized polarization potential approach and shown to yield results in good agreement with both superthermal phonon propagation and neutron scattering experiments at pressures up to 24 bar. It is shown that the long-wavelength dispersion relation is of the form _q =s ₀ q(1+₂ q ²+₃ q ³+...) suggested by Feenberg and by Kemoklidze and Pitaevskii, but that the validity of such an expansion is restricted to wave vectorsq0.1Å ^–1. For larger values ofq, one can force a polynomial fit, containing only odd powers ofq, to the dispersion curve and other parameters of interest, but the number of terms which must be kept is so large as to render such an approach unwieldy.Work supported in part by the National Science Foundation Grant NSF DMR72-03026.     

Noisy interpolation of sparse polynomials in finite fields

We consider a polynomial analogue of the hidden number problem introduced by Boneh and Venkatesan, namely the sparse polynomial noisy interpolation problem of recovering an unknown polynomial f(X) ∈ [X] with at most w non-zero terms from approximate values of f(t) at polynomially many points t ∈ selected uniformly at random. We extend the polynomial time algorithm of the first author for polynomials f(X) of sufficiently small degree to polynomials of almost arbitrary degree. Our result is based on a combination of some number theory tools such as bounds of exponential sums and the number of solutions of congruences with the lattice reduction technique. The new idea is motivated by Waring's problem and uses a recent bound on exponential sums of Cochrane, Pinner, and Rosenhouse.     

Decomposition of algebras overF q (X 1,...,X m )

LetA be a finite dimensional associative algebra over the fieldF whereF is a finite (algebraic) extension of the function fieldF _q(X ₁,...,X _m). Here F_q denotes the finite field ofq elements (q=p^l for a primep). We address the problem of computing the Jacobson radical Rad (A) ofA and the problem of computing the minimal ideals of the radical-free part (Wedderburn decomposition). The algebraA is given by structure constants overF andF is given by structure constants overF _q(X ₁,...,X _m). We give algorithms to find these structural components ofA. Our methods run in polynomial time ifm is constant, in particular in the casem=1. The radical algorithm is deterministic. Our method for computing the Wedderburn decomposition ofA uses randomization (for factoring univariate polynomials overF _q).Research partially supported by Hungarian National Foundation for Scientific Research (OTKA), Grants 2581, F4116 and EC Cooperative Action IC 1000 (ALTEC).     

Primitive polynomial remainder sequences in elimination theory

Primitive polynomial remainder sequences (pprs) are more than a tool for computing gcd's; the content computations in the course of computing the pprs of two multivariate polynomialsf ₁ andf ₂ provide information on the common zeros off ₁ andf ₂. Because of this additional property, primitive polynomial remainder sequences can be used for solving systems of algebraic equations.This work has been supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project no. P6763, the Austrian Ministry of Science, project ESPRIT BRA 3125 MEDLAR, and the U.S. Army Research Office through the ACSyAM branch of the Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027.     

Effects of nitriding on titanium alloy strength and plasticity

Summary Studies have been made on the effects of temperature and time on nitriding and from the pressure of the nitrogen on the strength and plasticity of pseudo- and + titanium alloys. Factors are observed that provide for retaining a high level of mechanical characteristics.Karpenko Physicomechanics Institute, Ukrainian Academy of Sciences, L'vov. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 1, pp. 81–84, January–February, 1993.