On primitive and free roots in a finite field

In this paper them-dimensional extension of the finite field of orderq is investigated from an algebraic point of view. Looking upon the additive group as a cyclic module over the principal ideal domain , we introduce a new family of polynomials over which are the additive analogues of the cyclotomic polynomials. Two methods to calculate these polynomials are proposed. In combination with algorithms to compute cyclotomic polynomials, we obtain, at least theoretically, a method to determine all elements in of a given additive and multiplicative order; especially the generators of both cyclic structures, namely the generators of primitive normal bases in over , are characterized as the set of roots of a certain polynomial over .     

Laminar thermal boundary layer on a continuous accelerated sheet extruded in an ambient fluid

Summary Exact boundary layer similarity solutions are developed for flow, friction and heat transfer on a continuously accelerated sheet extruded in an ambient fluid of a lower temperature.Melt-spinning, polymer and glass industries and the cooling of extruded metallic plates are practical applications of this problem.Results for skin-friction and heat-transfer coefficients are given. Larger acceleration is accompanied by larger skin-friction and heat-transfer coefficients. Rapid cooling of the sheet is accompanied by a larger Nusselt number.Nomenclature sheet width - c dimensionless constant - c _f local skin friction coefficient - F dimensionless transformed stream function - G dimensionless transformed temperature - local heat transfer coefficient - fluid thermal conductivity - length of deformation zone - m exponent of surface speed variation - q exponent of surface temperature variation - T dimensionless temperature - sheet surface temperature - solidification temperature - ambient temperature - sheet thickness - u velocity component along the sheet - u _s sheet surface velocity - wind up velocity - v velocity component normal to the sheet - x dimensionless coordinate along the sheet - y dimensionless coordinate normal to the sheet - Nu Nusselt number, - Pr Prandtl number, - Re Reynolds number, - =Re^–0.5 - dimensionless similarity coordinate - dynamic viscosity - kinematic viscosity - fluid mass density - sheet mass density - wall shear stress - dimensionless stream function With 3 Figures     

On the motion of a curved and twisted rod

Summary In this paper, the equations of motion of a curved and twisted rod are derived from the basic principles of dynamics. The set of equations gives the extensional, flexural and torsional motions of the rod. The coupling among these types of motion due to the curvature and tortuosity of the rod is shown explicitly in the case of a helical spring. By successive simplification of the equations, the equations of motion of a circular ring and those of a straight rod are obtained. In this respect, the derived equations can be considered as a generalization of the elementary theories of rod in extensional, torsional and flexural motion.The dispersion relation of a helical spring is calculated for the two lower frequency modes. It is shown that the frequency-wave length relationship is not monotonically decreasing as in the cases of uncoupled flexural or torsional motion. Finally, frequencies are calculated based on the approximate frequency expression ofLove to show that Love's frequency expression for a helical rod is accurate.

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Zur Bewegung eines gekrümmten und verdrillten Stabes

Zusammenfassung In dieser Arbeit werden die Bewegungsgleichungen eines gekrümmten, verdrillten Stabes von den Grundgleichungen der Dynamik hergeleitet. Dieses System von Gleichungen legt die Bewegung des Stabes durch Längsdehnung, Biegung und Torsion fest. Die Kopplung dieser Bewegunsarten, durch Krümmung und Verdrillung, wird für den Fall einer Spiralfeder explizit gezeigt. Durch Vereinfachung der Gleichungen werden die Bewegungsgleichungen des Kreisringes und die des geraden Stabes erhalten. In dieser Hinsicht können die hergeleiteten Gleichungen als Verallgemeinerung der elementaren Theorein der Bewegung eines Stabes betrachtet werden.Die Dispersionsgleichung der Spiralfeder wird für die beiden niedrigsten Frequenzen berechnet. Es wird gezeigt, daß das Verhältnis Frequenz-Wellenlänge nicht wie in den Fällen der ungekoppelten Biege-oder Torsionsbewegung monoton abnimmt. Abschließend werden die Frequenzen nach dem Loveschen Näherungsausdruck berechnet, um zu zeigen, daß die Genauigkeit dieses Ausdruckes für Spiralfedern gut ist.

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Nomenclature position vector of a point on axis of curved rod - position vector relating any pointB in a plane perpendicular to the axis of the rod to pointP where plane cuts axis of rod - R modulus of - linear momentum vector - density - axial force vector = - axial force vector at origin - transverse shear force vector - transverse shear force vector at origin - moment vector - moment vector at origin - s distance measured along axis of rod - t time - a radius of gyration of rod for circular sections - l total length of spring along axis - A cross-sectional area of rod - s/a — normalized coordinates measured along axis of rod - applied load vector - angular momentum vector - trihedral of space curve-unit vectors in tangential, normal and binormal directions - _t axial rotation - displacement vector - normalized displacement vector= - ₀,₀ curvature and tortuosity of rod - , normalized curvature and tortuosity of rod - , circular frequency and normalized frequency respectively - E, G Young's modulus and shear modulus - v Poisson's ratio - Wave number - pitch angle of helical spring - d wire diameter of helical spring - R coil radius of helical spring - N number of turns of helical spring With 5 Figures     

Polynomial-time construction of codes I: Linear codes with almost equal weights

We prove that there are infinite families (C_i)_i0 of codes over F_q with polynomial complexity of construction whose relative weights are as close to as we want and are such that

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).     

Eine Erweiterung der Analogie zwischen der Gasdynamik und Flachwasserströmungen auf rotierende Systeme mit konservativen Massenkräften

Zusammenfassung Die bekannte Analogie zwischen den Theorien reibungsfreier flacher Wasserströmung und der isentropen idealen Gasdynamik wird im zweidimensionalen instationären Fall durch Zulassung rotierender Bezugssysteme und konservativer Massenkraftfelder erweitert. Damit kann man in einem rotierenden Wasserbehälter mit geeignet gestaltetem Boden u. a. atmosphärische Probleme näherungsweise studieren.

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Summary The well known analogy of the theories of nonviscous shallow water flows and isentropic ideal gasdynamics is generalized in the two-dimensional nonstationary case to apply to rotating systems with conservative body forces. This enables one, in particular, to simulate atmospheric problems by use of a rotating water tank with suitable bottom shape.

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Liste der verwendeten Symbole spezifische Wärme des Gases bei konstantem Druck - Fr in (2) definierteFroude-Zahl - Gravitationsbeschleunigung - H ortsabhängige Ruhetiefe bei Drehung des Wassers als starrer Körper - Druck - atmosphärischer Druck auf der freien Wasseroberfläche - Bezugslänge - Zeit - absolute Temperatur - Geschwindigkeitskomponenten parallel zu den Achsen - kartesische Koordinaten - durch Abweichung von der Drehung des Wassers als starrer Körper verursachte orts-und zeitabhängige Änderung der Wassertiefe - Verhältnis der spezifischen Wärmen des Gases - zu gehörende Wellenlänge - Dichte - Winkelgeschwindigkeit Mit 1 Textabbildung1 Boden des Wasserbehälters 2 freie Wasseroberfläche II freie Wasseroberfläche bei einer Drehung des Wassers als starrer KörperEin oberer Querstrich gibt an, daß die betreffende Größe eine physikalische Dimension hat; Größen ohne einen solchen Querstrich sind dimensionsfrei.     

Multiband superconductivity: A mapping to the extended attractive Hubbard model

Following our recent work which shows that purely repulsive multiband systems effectively reduce to the extended attractive Hubbard model having the on-site repulsion along with nearest-neighbor and next-nearst-neighbor attraction for the cuprate structures, we study the latter model in the mean-field theory.T _c against band filling for various symmetries of pairing is numerically calculated. The results show that, for the region of and expected for the mapping from the original multiband model,T _c has maximum values of the order of 100 K, which is consistent with the experimental results. For certain values of parameters,s- and -wave pairing have similar values of T_c, which may suggest the possibility of a mixed phase of the two.The numerical calculations were done on the HITAC S3800 in the University of Tokyo Computer Centre. This work was in part supported by a Grant-in-Aid from the Ministry of Education, Science, and Culture, Japan.     

Tunneling with very long relaxation times in glasses,organic materials,and Nb-Ti-H (D)

The long-time (t=10–200 h) heat release from glasses, from organic materials, and from Nb-Ti-H (D) was measured at 30T70 mK. For Suprasil W glass, Dimethyl-Siloxan, Stycast 1266, Stycast 2850 FT, Vespel, and for Nb-Ti-H (D) with various Ti and D concentrations, we found . Typical values are = 0.05 nW/g for the organic materials and for Nb-Ti-H (D) and = 0.005 nW/g for the glass att=100 h after cooldown from room temperature. For charging temperaturesT _i <5 K, we find the predicted dependence (investigated for Suprasil W glass and for Nb-Ti-D). The observed time and temperature dependences agree with predictions of the conventional two-level tunneling model for amorphous materials even at these very long times. No heat release was observed for Teflon, graphite, and Al₂O₃.     

Nonlinear liquid oscillations in spherical systems under zero-gravity

Summary Nonlinear free oscillations of the interface of a concentric frictionless immiscible liquid system in a spherical container are investigated in a zero-gravity environment. The natural frequencies are determined for the axisymmetric and asymmetric oscillations of the interfacial surface with the diameter ratio and density ratio as parameters. It was found that for small outer- to inner liquid density ratio the oscillations exhibit softening, while for large density ratios it renders hardening oscillation. The asymmetric oscillations exhibit in the softening range softer and in the hardening range harder liquid oscillations. For a liquid layer around a rigid center sphere the oscillations of the free liquid surface yields softening behavior, where for thinner layers the softening effect is more pronounced.Nomenclature a radius of spherical container, or radius of rigid center sphere - b radius of undisturbed interfacial surface, or radius of undisturbed free liquid surface - k=a/b diameter ratio - pressure - pressure (dimensionless) - , , spherical coordinates - dimensionless radius - R _i main radii of curvaturei=1, 2 time - dimensionless time - v _i liquid velocity (j=1 spherical layer region,j=2 inner liquid sphere region) - V volume of the liquid - Y _nm tesseral surface harmonics - _i density of liquids - velocity potential - dimensionless velocity potential - interfacial surface- or free surface elevation - dimensionless interfacial surface- or free surface elevation - ₀ maximum elevation - circular frequency - circular frequency - _n0 axisymmetric natural frequency - _n1 asymmetric natural frequencym=1 - _nm ⁽⁰⁾ natural frequency of linearized liquid system - mean curvature - _nm Kronecker symbol With 10 Figures     

Load relaxation studies of germanium

A series of compressive load relaxation experiments were conducted on germanium single crystals in the temperature range 400 to 885° C. The curvature of the log-log data obtained from load relaxation tests changes from concave upward to concave downward as the test temperature increases at fixed stress level, or as the strain level increases at fixed temperature. At intermediate temperatures, 600° C, the transition from concave upward to concave downward curvature happens on a single relaxation curve. These observations are consistent with the two-branch rheological model proposed by Hart to explain the deformation behaviour of metals and were analysed in terms of this model. The transition from concave upward to concave downward curvature could be moved to higher temperature by doping germanium with gallium, which decreases the dislocation glide velocity relative to that in pure germanium. The transition could be shifted to lower temperature by compressing samples along [1 1] rather than [1 0] because the [1 1] orientation favours cross-slip while the [1 0] orientation does not. Dislocation dipoles and straight dislocations dominated the microstructure of samples which had concave upward log-log curves, while well-developed dislocation cell structures dominated the microstructure of samples which yielded concave downward curves. The observed changes in the curvature of the load relaxation curves and the dislocation structure both indicate the increased importance of dislocation climb with increasing temperature. When compared through the Orowan equation, the load relaxation results are in good agreement with published stress-dislocation velocity data.     

Superconducting critical currents in Nb-Cu-Nb sandwiches

A preparation method is presented for Nb-Cu-Nb sandwiches for proximity-effect measurements with a large junction area of about 150×150 µm and intermediate Cu layers of thicknesses between 0.2 and 1.5 µm. The thickness of the Nb layers is about 150 µm. These samples are used to study the dependence of the superconducting critical currentI _c through the Cu layer as a function of a magnetic field applied parallel and perpendicular to the layers. Furthermore, the effects of flux lines trapped in the Nb layers on the critical currentsI _c andI _c are investigated. The results are used to discuss the influence of proximity effects in microcomposite superconductors.     

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.     

Viscous response of an anchored spinning liquid column to various axial excitations

Summary For a solidly rotating viscous cylindrical liquid column of finite length the response to axial synchronous, counter- and one-sided excitation is determined for anchored contact lines at the disc-rim. For a rotating column additional responses of inertial waves (hyperbolic range) appear for < 2 ₀, while in the elliptic range < 2 ₀ the sloshing response occurs. The various responses for the free surface displacement have been numerically evaluated. Only in the one-sided exitation case all resonance peaks appear, while for synchronous excitation only the odd resonances and for counter-excitation only the even resonance peaks occur.Notation a radius of column - h length of liquid bridge - I _n modified Bessel function - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution - Weber number - z ₀ excitation amplitude - liquid density - surface tension - surface tension parameter - Ohnesorge number - liquid surface displacement - kinematic viscosity - ₀ rotational speed - dimensionless rotational speed - forcing frequency - dimensionless forcing frequency - dimensionless forcing frequency for non-viscous liquid - a= root of bi-cubic Eq.(33) - root of bi-cubic Eq.(33)     

Anisotropic flow and fracture in beryl [Be3Al2(SiO3)6]

Flow and fracture resulting from Vickers indentation testing on {0 0 0 1} and {10 0 } planar orientations have been examined. Flow characterized by indent shape differentiation was analysed to belong to the slip system with planes of the types { 10 0} and {11 0}. The ensuing fracture paths were resolved to propagate along {1 0 0} and {1 1 } cleavage planes whileK _c values obtained for them were 0.196 and 0.248 MPam^1/2, respectively.     

Damped response of an axially excited rotating liquid bridge in zero-gravity

Summary The response of a solidly rotating finite liquid bridge due to axial excitation exhibits for frictionless liquid at the resonances singularities. For the experimenter in a spacelabmission the actual resonance amplitude is of quite some importance. For this reason damping, that has to be measured in ground tests, has been introduced into the results of the response.Notation a radius of the liquid bridge - h length of the liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - r, ,z polar coordinates - t time - excitation amplitude - elliptic case - hyperbolic case - abbreviation - damping factor of liquid - (z, t) free surface displacement - =² – ² surface tension - surface tension - liquid density - ₀ rotational speed of liquid bridge - forcing frequency of axial excitation - natural frequency of liquid bridge With 2 Figures     

Size effects and ductile-brittle transition of polypropylene

The influence of specimen width on fracture parameters has been investigated. The range examined was sufficiently large to obtain ductile and brittle fractures. With reference to previously published work, the phenomenology has been analysed by combining BCS model and Carpinteri's brittleness number approach.Nomenclature a crack length - f(a/W) shape function according to ASTM specification [16] - F(a/W) shape function according to Tada Paris notation [21] - E elastic modulus - K _IC plane strain fracture toughness - K _IC ^f fictitious plane strain fracture toughness - K _IC2 plane stress fracture toughness - J _IC ^f J-integral at maximum load - L span - weight average molecular weight - number average molecular weight - polydispersity - P _M maximum load - P _F load of brittle fracture - p _P load of plastic collapse - s brittleness number - V machine cross speed - W specimen width - _y yield stress - strain rate     

Nonstationary process of heat transfer in a tube with longitudinal fins

An analytical solution to the problem of nonstationary thermal interaction of a flow of a heat-transfer agent and a thin-walled tube with longitudinal fins is constructed for variable parameters of heat transfer.Notation u, temperatures of the fins - ,w temperatures of the tube walls - temperature of the flow of the heat-transfer agent - _i ,i= coefficients of heat transfer from the ambient medium to the fins and the tube walls, respectively - _i ,i= temperature distributions for the ambient medium - coefficients of heat transfer from the flow of the heat-transfer agent to the tube walls - q _i density of the heat flux to the corresponding portions of the tube - heat capacity, thermal conductivity, density, and thickness of the fin and tube material - c _p , ,G, F heat capacity, density, and flow rate of the heat-transfer agent, cross-sectional area of the tube - dimensions of the tube Bauman Moscow State Technical University. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 673–680, June, 1994.     

Thermal expansion and bulk modulus of cobalt

The ten third-order elastic (TOE) constants of hexagonal cobalt, which has nearly the idealc/a value, have been evaluated using the nearest-neighbor central interaction model for hexagonal materials proposed by Ramji Rao and Srinivasan. These TOE constants have been employed to evaluate the low-temperature limit of the thermal expansion, the Anderson-Grüneisen (AG) parameter , and the second Grüneisen constantq of cobalt. has the value 2.08 for cobalt. The temperature variation of the volume Grüneisen gamma of cobalt has been calculated on this model. The high-temperature limit has the value 2.13, which agrees well with the experimental value obtained from thermal expansion and specific heat data of cobalt. Anderson's theory has been used to explain the temperature dependence of the bulk modulus of cobalt and compared with the experimental values obtained from the elastic constant data of Fisher and Dever. The calculatedB _s values agree with the experimental values to within 1%.     

Two-lifetime model calculations of the quantum interference dominated transverse magnetoresistance of magnesium

The magnetoresistance of extremely pure, strain-free magnesium is dominated by the transport properties of a narrow slab of coupled orbits in the geometry in whichJ [11 0] andH lies in the (11 0) plane. In samples whose quantum state lifetime _s exceeds 10^–10 sec the conductivity of the coupled orbit system is controlled by _s as well as the large-angle scattering lifetime ₁. The magnetoresistance then exhibits complicated behavior, oscillating with |H| and varying rapidly with field orientation. The behavior withH aligned to within about 0.001° of [10 0] is due to manifold multiply connected open orbits, whereas forH tilted away from [10 0] by more than this the behavior is due to extended, multiply connected closed orbits. Detailed models for both these regimes are developed, taking into account the interference effects resulting from the feedback paths. Within these models the transport properties are calculated exactly, taking into account to all orders the amplitude propagation of wave packets within the network. Results are compared to experimental data and previous calculations, and the limitations of the model are discussed.Work supported in part by NSF grant DMR 7910533.     

Stresses in a cylindrical shell supported by a core of a different material

Summary The stress problem of a thin cylindrical shell supported by an elastic core of a different material and subjected to arbitrary loading on its curved surface is considered. The problem is solved by applying the three-dimensional theory of elasticity to the core and using membrane or bending solutions for the shell. Equilibrium and compatibility equations are satisfied at the junction of the shell and the core. It is pointed out that the procedure can easily be extended to the case of a hollow core with or without another shell of another material in it. Numerical results are presented to illustrate the effectiveness of even a weak core in reducing the shell stresses.

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Zusammenfassung Gegenstand der Untersuchung ist eine dünne Kreiszylinderschale, die durch einen elastischen Kern aus einem anderen Werkstoff gestützt ist und eine beliebige Belastung trägt. Die Lösung verbindet die strenge, dreidimensionale Theorie des zylindrischen Kerns mit der Membran- oder Biegetheorie der Schale. An der Grenze zwischen beiden Teilen müssen die Verschiebungen und gewisse Spannungskomponenten stetig übergehen. Es wird darauf hingewiesen, daß die Lösung leicht auf den Fall ausgedehnt werden kann, daß der Kern ein Hohlzylinder ist, der möglicherweise auf der Innenseite mit einer zweiten Zylinderschale verbunden ist. Zahlenergebnisse zeigen, daß selbst ein verhältnismäsig nachgiebiger Kern einen großen (und günstigen) Einfluß auf die Spannungen in der Schale ausübt.

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Principal Symbols a Radius of the middle surface of the shell - t Thickness of the shell - =1–t/2a - u _c,v _c,w _c Displacements respectively in the axial, circumferential and radial directions of a point in the core - X(x), (), (r/a) 3×3 square matrices - ,m Parameters - l Length of the cylinder - c A vector containing constantsc ₁,c ₂ andc ₃ - =r/a - =m+4(1–v _e) - E _c,v _e Elastic constants for the core material - Stresses at a point in the core - D _c - A vector containing _rx , _r and _r - (r/a) A 3×3 matrix - Displacements at the surfacer=a of the core - A vector containing - Amplitudes of displacements - A vector containing - =(x, ,a) - _ij Constants - A A square matrix containing constants _ij - Stress resultants in the shell as defined in reference [3] - p _x,p ,P _r Components of applied loading per unit area of shell's middle surface - () - ()· - u, v, w Displacements of a point on the middle surface of the shell - E _s,v _s Elastic constants for the shell material - D _s - K - k - p _xmn,p _mn,p _rmn Amplitudes of loadsp _x,p , p_r - u _mn, v_mn,w _mn Amplitudes of displacementsu, v, With 1 Figure