Bounding the Trellis State Complexity of Algebraic Geometric Codes

Let be an algebraic geometric code of dimension k and length n constructed on a curve over F_q. Let be the state complexity of and the Wolf upper bound on . We introduce a numerical function R that depends on the gonality sequence of and show that where g is the genus of . As a matter of fact, R(2g–2)g–(₂–2) with ₂ being the gonality of over F_q, and thus in particular we have that The authors were partially supported respectively by the Grants VA02002 (Junta de Castilla y León), Proc. 300681/97-6 (CNPq-Brazil) and SB2000-0225 (Secretaria de Estado de Educación y Universidades del Ministerio de Educación, Cultura y Deportes de España)     

On the bit security of the Diffie-Hellman key

Let _p be a finite field of p elements, where p is prime. The bit security of the Diffie-Hellman function over subgroups of ^*_p and of an elliptic curve over _p, is considered. It is shown that if the Decision Diffie-Hellman problem is hard in these groups, then the two most significant bits of the Diffie-Hellman function are secure. Under the weaker assumption of the computational (rather than decisional) hardness of the Diffie-Hellman problems, only about (log p)^1/2 bits are known to be secure.     

Codes over <Emphasis Type="Bold">Σ</Emphasis><Subscript>2</Subscript><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> and Jacobi forms over the Quaternions

We introduce codes over the ring We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring _2m and Jacobi forms over the half-space of quaternions. This motivates us to construct an algebra homomorphism from a certain invariant polynomial ring, where the complete weight enumerators belong, to the ring of Jacobi forms over the quaternions. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.This research was partially supported by KOSEF R01-2003-00011596-0Some of the results of this paper were presented at the IEEE Information Theory Workshop, April, 2003     

On the Bounds of the Bilinear Complexity of Multiplication in Some Finite Fields

From the existence of a tower of algebraic function fields, we improve upper bounds on the bilinear complexity of multiplication in all the extensions of the finite fields and where p is a prime 5. In particular, we improve asymptotic upper bounds on this complexity for prime finite fields of characteristic p>5.     

Cryptographic implications of Hess' generalized GHS attack

A finite field K is said to be weak for elliptic curve cryptography if all instances of the discrete logarithm problem for all elliptic curves over K can be solved in significantly less time than it takes Pollard's rho method to solve the hardest instances. By considering the GHS Weil descent attack, it was previously shown that characteristic two finite fields are weak. In this paper, we examine characteristic two finite fields for weakness under Hess' generalization of the GHS attack. We show that the fields are potentially partially weak in the sense that any instance of the discrete logarithm problem for half of all elliptic curves over , namely those curves E for which is divisible by 4, can likely be solved in significantly less time than it takes Pollard's rho method to solve the hardest instances. We also show that the fields are partially weak, that the fields are potentially weak, and that the fields are potentially partially weak. Finally, we argue that the other fields where N is not divisible by 3, 5, 6, 7 or 8, are not weak under Hess' generalized GHS attack.     

The adjacency graph of a real algebraic surface

The paper deals with the question of recognizing the mutual positions of the connected components of a non-singular real projective surface S in the real projective 3-space. We present an algorithm that answers this question through the computation of the adjacency graph of the surface; it also allows to decide whether each connected component is contractible or not. The algorithm, combined with a previous one returning as an output the topology of the surface, computes a set of data invariant up to ambient-homeomorphism which, though not sufficient to determine the pair , give information about the nature of the surface as an embedded object. This research was partially performed with the contribution of M.I.U.R. and of Eurocontract HPRN-CT-2001-00271     

Stability characteristics of suspension flow through wavy-walled channels

Summary. A numerical study of the linear temporal stability characteristics of particulate suspension flow through a converging-diverging symmetric wavy-walled channel is considered. The basic flow is a superposition of plane channel flow of particulate suspension and periodic flow components arising due to the small amplitude sinusoidal waviness of the channel walls. The disturbance equations are derived within the framework of Floquet theory and solved using the spectral collocation method. The effects of small amplitude sinusoidal waviness of the channel walls and those of the presence of particles on the initial growth of the disturbances are examined. Two-dimensional stability calculations for particulate suspensions indicate the presence of fast growing unstable modes that arise due to the waviness of the walls. Neutral stability calculations are performed in the disturbances wavenumber-Reynolds number (_s–Re) plane, for the wavy channel with representative values of wavenumber () and the wall amplitude to semi-channel height ratio () for different values of volume fraction density of the particles (C). It is observed that the critical Reynolds number for transition decreases with increase of and C. However, the flow can be modulated by suitable wall excitation which in turn can stabilize the flow.     

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.     

Error estimates for the Fourier-finite-element approximation of the Lamé system in nonsmooth axisymmetric domains

This paper is concerned with the effective implementation of the Fourier-finite-element method, which combines the approximating Fourier and the finite-element methods, for treating the Dirichlet problem for the Lamé equations in axisymmetric domains with conical vertices and reentrant edges. The partial Fourier decomposition reduces the three-dimensional boundary value problem to an infinite sequence of decoupled two-dimensional boundary value problems on the plane meridian domain The asymptotic behavior of the solutions of the reduced problems near angular points of _a is described by suitable singular functions and treated numerically by linear finite elements on locally graded meshes. For it is proved that the rate of convergence of the combined approximations in is of the order where h and N are the parameters of the finite-element- and Fourier-approximation, respectively, with h0 and N.Acknowledgements This work was done while the author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, under the Junior Associateship Scheme. The visit was sponsored by the Swedish International Development Cooperation Agency (SIDA). The author gratefully acknowledges this support.Junior Associate of the Abdus Salam ICTP.     

Linear response of a viscous liquid sheet to oscillatory external pressure perturbation in the long wave approximation. Varicose excitation

Summary. In this paper, a plane viscous liquid sheet is considered in contact with two passive media, from which two external perturbations of small intensity, given by local pressure distributions, act on the sheet. Of the two possible ways of action of the sources of perturbation only the symmetric one is pursued here, for which the sheet reacts in the so called varicose mode. The action of the sources is taken either as instantaneous, or as persisting infinitely long. During the persistent action the sources are considered to move with constant velocity along the sheet and the perturbation is changing harmonically with circular frequency _e. In the paper, the linear response of the viscous sheet to the external perturbation is computed and analyzed. The starting point of the computation is a general formula for the exact linear response of the sheet as an inverse Fourier-Laplace transform, the derivation of which is reproduced in Appendix A of this paper. The formula shows the response to depend on several characteristic numbers, of which one is intrinsic to the unperturbed sheet, whereas the others also imply the properties of the source. The intrinsic number can be taken as in which the sheet thickness h, the density of the liquid, its kinematic viscosity , and it surface tension with respect to the two ambient media enter. It is related by to the Ohnesorge number Oh. First the concept of the long wave approximation used is discussed in the context: it implies the neglect of all the hard branches _n(k),n=1,..., of spectral modes of the sheet, for which On the remaining two soft branches _±(k) of the spectrum, for which lim_k0_±(k)=0, the asymptotic expansion of (k) fork0 is considered and the whole branches are approximated by the (extrapolation of the) lowest order of the expansion. Finally, the whole Fourier integrand of the signal is substituted by its lowest order asymptotic expression aroundk=0. Within this approximation the response of the sheet to an instantaneous point perturbation is computed in its rest frame of reference. It is a long time asymptotic form of the exact response to the instantaneous point perturbation, the kernel of the general signal evolution of the sheet. The computed asymptotic form of this kernel bears a strong formal resemblance to the evolution kernel of the heat equation. The analogy is, however, not physical: whereas for ²<4 the long wave modes of the two soft branches of the sheet spectrum are purely absorptive, like the modes of the heat equation, for ²>4 their modes are dispersive and dissipative waves, with their peculiar propagation properties. By adequate application of the superposition principle the responses to instantaneous line perturbation and to harmonically persistent perturbations by moving point and line sources are computed from the asymptotic evolution kernel. Because of the specific form of this kernel, the long time surface deflection responses to the harmonic excitations do not depend separately on the additional characteristic numbers _eandW=||/ (where _eand || are considered in units of 4/h² and 2/h, respectively), but only on their ratio _e/W². The properties of these signals in dependence on and _e/W², as well as their implications for technical processes involving liquid sheets, like courtain coating, are discussed in detail.AcknowledgementThe support of this work by the German Research Foundation (DFG) is gratefully acknowledged.     

Structural properties and enumeration of quasi cyclic codes

Given any finite fieldF _q , an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF _q ^N which is invariant underT ⁿ for some integern, 0 <n N, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theF _q []-submodules ofF _q ^N where the product(gl)· is naturally defined as ₀ + ₁T ⁿ +...+ _m T ^mn if()= ₀ + ₁ +...+ _{m m} .In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicF _q []-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.     

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.     

Nonlinear viscoelasticity and relaxation phenomena of polymer solids

Summary The specific Helmholtz free energy of nonlinear viscoelastic isotropic polymer solids is assumed to be a separable, symmetric function along three orthogonal principal strain directions, i.e.=( ₁,q _n ⁽¹⁾)+( ₂,q _n ⁽²⁾)+( ₃,q _n ⁽³⁾), where the _r (r=1, 2, 3) are three principal stretch ratios and theq _n ( ^r ) are a set of internal state variables. It is shown that, by using this postulated form of the free energy function, one can readily characterize the mechanical response of viscoelastic polymer solids.

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Nichtlineare Viskoelastizität und Relaxationserscheinungen in festen Polymeren

Zusammenfassung Die spezifische Helmholtzsche freie Energie nichtlinear-viskoelastischer, isotroper, fester Polymere wird als separable, symmetrische Funktion der Form=( ₁,q _n ⁽¹⁾)+( ₂,q _n ⁽²⁾)+( ₃,q _n ⁽³⁾) angenommen mit den drei Hauptdehnungsverhältnissen _r (r=1, 2, 3) und inneren Zustandsvariablenq _n ( ^r ). Es wird gezeigt, daß dann das mechanische Verhalten der viskoelastischen festen Polymere schnell charakterisiert werden kann.

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This represents the results of one phase of research being carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Effect of gamma irradiation and silver doping on YBCO superconductor

The effect of -irradiation and silver doping on the properties of YBCO superconductors has been studied by electrical resistance measurements and x-ray diffraction techniques. The zero-resistance superconducting transition temperature of (YBCO)_1–xAg _x systematically increase by about 2 K with an increase of Ag content up tox=0.04. Further increase in Ag content decreases the transition temperature. The normal-state resistance decreases up to a factor of 3.4 with an increase of Ag content. No significant change in the transition width has been observed due to silver addition up tox=0.30. Undoped YBCO and the Ag-doped specimens of (YBCO)_1.96Ag_0.04 were irradiated with-rays up to an integrated-dose of 77 MR. The normal-state resistance of the specimens gradually increases and the transition temperature slightly decreases with increase of-dose. The intensities of the x-ray diffraction peaks of the monolayers of the sample powders decrease with -irradiation without noticeable change of diffraction angles or peak widths. These observations indicate that the high-T _c supconductor YBCO is quite sensitive to-irradiation but drastic lattice expansion does not occur up to a-dose of 77 MR.     

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     

Statistical analysis of high-order moments in a turbulent boundary layer with large density gradients

Summary. An experimental study of the effects of large density variations on the turbulent transfers in a boundary layer is carried out in this paper. Density differences were generated by tangential injection of air or helium in a mixture of these two components. In order to study the fine structure of wall turbulence in the presence of significant density variations, a statistical analysis of the experimental data, obtained in a wind tunnel, is carried out. The results show that the relaxation of the skewness factor of u(S_u) is carried out more quickly in the external layer than close to the wall, for the air injection as well as for the helium injection. S_u grows close to the injection slot in an appreciable way, and this increase definitely is stronger for the air injection than for the helium injection. This growth of the skewness factor close to the injection slot can be explained by the increase in the longitudinal convective flux of turbulent energy in this zone. For the distributions of the flatness factor F_u the results show that there is no significant effect of the density gradient on the intermittent structure of the instantaneous longitudinal velocity in the developed zone, x/5, where is the boundary-layer thickness. Differences remain, of course, between the profiles of F_u for density gradients in the relaxation zone, where the longitudinal gradients are significant. It should, however, be noted that has a smaller effect on the flatness factor compared to the skewness factor. The statistical analysis carried out in this study shows that the helium injection in the boundary layer generates more violent ejections than the air injection. This result is confirmed by the significant contribution of the ejections to turbulent mass flux.     

The dispersion of3He quasiparticles in He II from neutron scattering

In an inelastic neutron scattering (INS) experiment on³He-⁴He mixtures one observes, besides the photon-roton mode which is barely modified by the admixture of³He, an additional excitation at lower energies which is interpreted as quasi-particle-hole excitations of a nearly free Fermi gas. We reanalyse INS data ofx ₃=1% and 4.5% mixtures at various pressures to extract the mean energy of the fermions. In the momentum range 9<q<17 nm^–1 (above 2k _F ) follows very closely the relation =A ₂ q ²+A ₄ q ⁴ at all concentrations, pressures and temperatures observed. In a 4.5% mixture (T _F 0.3 K), measurements were performed for temperatures in the range 0.07<T<0.9 K. We find bothA ₂ andA ₄ to be strongly temperature dependent. For the interpretation of thermodynamical properties, the single particle energy _k is parametrized as _k =_o+1/(2ms*) ·k ₂ · (1+k ²). Neglecting interactions between fermions, we calculate from the free-particle _k the scattering functionS(q, ) and the mean value of the fermion peak energy _q = S ₃(q, )d/S ₃(q, )d. We find that follows closely _q , deviating at most by 10%. A comparison to the measuredA ₂ andA ₄ directly yieldsms* (x ₃,p, T) and (x ₃,p, T). In the limitx ₃=0,p=0 andT=0, the density and concentration dependence of the inertial mass is in excellent agreement with values found by Sherlock and Edwards. The temperature dependence of the specific heat data from Greywall and Owers-Bradleyet al. are well represented by our model atT<0,5 K.     

Heat and mass transfer of a multiatomic gas in a cylindrical capillary with arbitrary Knudsen numbers

Processes of heat and mass transfer of a multiatomic gas in a cylindrical channel of circular cross section with arbitrary Knudsen numbers are considered on the basis of a model kinetic equation, taking account of the excitation of rotational and vibrational degrees of freedom of the molecules.Notation Kn Knudsen number - f, f^tr total and translational Eucken factors - R_o capillary radius - m molecular mass - k Boltzmann's constant - n, T numerical density and temperature of gas - v_i i-th component of the molecular velocity - h_ij perturbation function - E_i ^(r), e_j ^(v) energy of the i-th rotational and j-th vibrational levels - E_o ^(r), E_o ^(v) equilibrium values of the rotational and vibrational energy - P_i ^(r), P_i ^(v) probability of rotational and vibrational states of energy E _i ^r and E _j ^v - , logarithmic pressure and temperature gradients - T_o mean gas temperature - R rarefaction parameter of gas - C _V ^r , C _V ^v contributions of rotational and vibrational degrees of freedom of the molecule to the specific heat at constant volume - U macroscopic gas velocity - q^(t), q^(r), q^(v) components of the heat flux density due to translational, rotational, and vibrational degrees of freedom of the molecules - P, pressure and dynamic viscosity of the gas - l free path length of molecules - up velocity of Poiseuille flow - u_T rate of thermal creep - cross-sectional area of capillary - I_n, I_q numerical and heat fluxes averaged over the channel cross section - universal index characterizing the thermomolecular pressure difference - ^t, ^r, ^v thermal conductivities due to translational, rotational, and vibrational degrees of freedom of the molecules - mass density of the gas - D_rr, D_vv diffusion coefficients of rotationally and vibrationally excited molecules among the unexcited molecules - Z_r rotational collisional number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 71–82, July, 1984.     

Optimization of the designs of small-size helium cryostats

Using a computer program that simulates thermophysical processes occurring in small helium cryostats, the surface emissivities of actual cryostats are verified. The efficiency of the heat transfer from the walls of the reservoir suspension tube to the helium vapors is determined. The program makes it possible to optimize the designs of cryostats according to the specific conditions of their use.Notation Stefan-Boltzmann, constant - _bg, _ho, _sh, _res, _det emissivities of the background surrounding the cryostat, housing surface of the cryostat, radiation shield, helium reservoir, and radiation detector - T _bg,T _ho,T _sh,T _res temperatures of the background, cryostat housing, shield, and reservoir - _re1, _re2, _re3 reduced emissivities of the shield, housing, and reservoir - F _sh,F _ho,F _res surface areas of shield, housing, and reservoir - F ₁,F ₂,F ₃ areas of the entrance window in the cryostat housing, of the filter on the radiation shield, and of the inlet hole of the matching optics - ₁, ₂ emissivities of the filter and entrance window - ₁ solid angle at which the background radiation is absorbed by optical filter - ₂ solid angle at which radiation is transferred by matching optics to the radiation detector - ₁, ₂, ₃ transmission of radiation by way of the entrance window, filter, and matching optics - S cross-sectional area of the helium reservoir suspension tube - (T) temperature dependence of the thermal conductivity of the tube material - n ₁,n ₂,n ₃ number of mechanical couplings between the housing and the shield, electric wires, and mechanical couplings between the shield and reservoir - S ₁,S ₂,S ₃ cross-sectional areas of the mechanical coupling between the housing and shield, of the electric wire, and of the mechanical coupling between the shield and reservoir - l ₁,l ₂,l ₃,l ₄ lengths of mechanical couplings and electric wires over the body-shield and the shield-reservoir portions - ₁(T), ₂(T), ₃(T) temperature dependences of the thermal conductivities of the materials of mechanical couplings between the housing and shield, of electric wires, and mechanical couplings between the shield and reservoir - specific resistance of the material of electric wires - I working electric current - d inner diameter of the helium reservoir suspension tube - L ₁,L ₂,L ₃ lengths of the portions of the tube in the first, second, and third zones participating in heat exchange with helium vapors - U ₁,U ₂,U ₃ coefficients of heat conduction from the tube walls to helium vapors in the first, second, and third zones - T ₁, T ₂, T ₃ temperature differences in the corresponding zones of the tube - G ₁ weighted quantity of nitrogen evaporated per unit time - r ₁ latent heat of nitrogen vapor generation - G weighted quantity of helium evaporated per unit time - r latent heat of helium vapor generation - T _h temperature of helium vapors - c _p heat capacity of gaseous helium - G weighted quantity of helium evaporated per unit time due to heat supply only along the helium reservoir suspension tube All-Russian Scientific Center S. I. Vavilov State Optical Institute, St. Petersburg, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 760–766, September–October, 1995.     

Flow of a generalized second grade fluid between heated plates

Summary We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.

List of symbols

Alphanumeric A ₁,A ₂ Kinematical tensor - b Body force - C Dimensionless parameter related to the pressure gradient - h Separation between the plates - L Velocity gradient - m Power-law index - M Constant appearing in the Reynolds viscosity model - p Pressure field - Modified pressure field - q Heat flux vector - r Radiant heating - T Cauchy's stress tensor - l Unit tensor - v Velocity vector - V Characteristic velocity - x Axis along the plate - y Axis perpendicular to the plate Greek ₁, ₂ Normal stress coefficient - Specific internal energy - Dimensionless parameter related to the viscous dissipation - Conservative body force field - Specific entropy - Thermal conductivity - Coefficient of viscosity - ₀ Reference viscosity - Second invariant of the stretching tensor - Temperature - ₁ Temperature of the lower plate - ₂ Temperature of the upper plate - Density - Specific Helmholtz free energy Operators div Divergence - grad Gradient - tr Trace