Effective Łojasiewicz inequalities in semialgebraic geometry

The main result of this paper can be stated as follows: letV ⁿ be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by someD > 1. LetF, G[X₁,, X_n] be polynomials with degF, degG D inducing onV continuous semialgebraic functionsf, g:VR. Assume that the zeros off are contained in the zeros ofg. Then the following effective ojasiewicz inequality is true: there exists an universal constantc ₁ and a positive constantc ₂ (depending onV, f,g) such that for allxV. This result is generalized to arbitrary given compact semialgebraic setsV and arbitrary continuous functionsf,g:V . An effective global ojasiewicz inequality on the minimal distance of solutions of polynomial inequalities systems and an effective Finiteness Theorem (with admissible complexity bounds) for open and closed semialgebraic sets are derived.     

Influence of Saffman's lift force on the motion of a particle in a Couette layer

The article is concerned with the study of the effect of E. S. Asmolov's corrections to Saffman's lift force for the wall vicinity and a nonzero ratio of Reynolds numbers. It is shown in what way these corrections change the particle paths in a Couette layer and the conditions of deposition.Notation x=X/D, y=Y/D dimensionless longitudinal and transverse coordinates - u=U _p /U , =V _p /U dimensionless projections of particle velocity on the longitudinal and transverse axes - =tU /D dimensionless time - U ²/(18D) Stokes number - = _g / _p , coefficient of the gas kinematic viscosity - particle diameter - /D - _g , _p densities of the gas and particle material - du/d - dv/d - P _s Saffman's force - C coefficient in the formula for Saffman's force - yRe _d ^1/2 - A v _r Re _d ^1/2 - 3.08 - Re V _r / - Re _k ²/)U _g /Y - A Re/Re _k ^1/2 - Re _d U D/ - V _r ((U _g –U _p )²+V _p ² )^1/2 Indices g refers to gas parameters - p refers to the parameters of particles - 0 at the time momentt=0 - S Saffman's force - k Reynolds number based on the velocity gradient - based on velocity - r relative velocity - x projection on thex axis     

A dilatometric method to measure the thermal diffusivity of nonmetallic liquids

A new method to measure the thermal diffusivity of liquids is presented. It requires determination of the time dependence of the thermal expansion of the liquid when it is subjected to a heat source at the top of the cell containing the liquid. The high accuracy of the method (about 3%) is due to an essential reduction of convective currents and also to the absence of temperature detectors, which generally introduce unwanted perturbations on the thermal Field.Nomenclature Thermal conductivity - c Specific heat - Density - c = specific heat x density - h Newton coefficient - Thermal diffusivity - T, 0 Temperature - tV Electric signal - Calibration coefficient - _exp, _th Volume change of the liquid     

Nucleate boiling

The study deals with the effect of the surface conditions on the nucleate boiling curve. A relation is proposed which describes the complete nucleate boiling curve.Notation q thermal flux - q* thermal flux at which the liquid boils after one-phase convection - q_c thermal flux during one-phase convection - q_cr1, q_cr2 first and the second critical thermal flux - T saturation temperature - T superheat of the heating surface relative to the saturation temperature - T* superheat prior to boiling of the liquid after one-phase convection - T_cr1 superheat during the first boiling crisis - T_cr3min minimum superheat at which the third boiling crisis can occur - P pressure - P_cr critical pressure - heat transfer coefficient during nucleate boiling - R_cr radius of a critical vapor forming nucleus - coefficient of surface tension - r latent heat of evaporation - thermal conductivity of the liquid - kinematic viscosity of the liquid - , densities of the liquid and the vapor - g gravitational constant - k Boltzmann constant - N Avogadro number - h Planck's constant Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 394–401, March, 1981.     

Thermal conductivity and heat capacity of solid silver bromide (AgBr) under pressure

The thermal conductivity, , and the heat capacity per unit volume, c _p , have been measured for solid silver bromide (AgBr) using the transient hot-wire method. Measurements were made at temperatures in the range 100–400 K and at pressures up to 2 GPa. c _p was found to be independent of temperature and pressure over these ranges. of AgBr was found to be similar to that of AgCl, which was measured previously. For AgBr, only acoustic phonons needed to be taken into account up to 340 K, but optic phonons probably carried some heat at higher temperatures. The Leibfried-Schlömann (LS) formula could describe the ratio (AgCl)/(AgBr), but not the ratio (1 GPa)/(0) for either substance. An empirical modification of the LS formula could describe the latter ratios but not the former. Further theoretical developments are required for understanding of (P) for even such relatively simple substances as AgCl and AgBr.     

Thermal conductivity of hydrocarbons in the naphthene group under high pressures

The thermal conductivity of hydrocarbons in the naphthene group has been experimentally determined. An equation is now proposed for calculating the thermal conductivity over the given temperature and pressure ranges.Notation thermal conductivity - ₂₀ and ₃₀ values of the thermal conductivity at 20 and 30°C, respectively - t₀,P₀ thermal conductivity at t₀, p₀ - _t p thermal conductivity at temperature t and under pressure P - change in thermal conductivity - P pressure - P_melt melting pressure - P₀ atmospheric pressure - t₀ 20°C temperature - T, t temperature - T_cr critical temperature - temperature coefficient of thermal conductivity - ₂₀ temperature coefficient of density - density - ₂₀ density at 20°C - _cr critical density - M molar mass - =T/T_cr referred temperature - v specific volume - v₀ specific volume at 20°C - v change in specific volume - _{3 0} a coefficient - B (t) a function of the temperature - S a quadratic functional - W_i, weight of the i-th experimental point - _i error of the i-th experimental value of thermal conductivity - B _{y, =0.6} value of B (t) at T = 0.6T_cr - B = B (t)/B_{, =0.6} referred value of coefficient B (t) Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 491–499, September, 1981.     

On the solution of nonstationary heat-conduction problems with variable heat-transfer coefficient

The article describes an exact method for calculating the temperature field in solids when they are heated in a medium with a variable heat-transfer coefficient and a nonuniform initial temperature distribution.Notation temperature - L thickness of plate - x space coordinate - a thermal diffusivity - thermal conductivity - heat-transfer coefficient - t time - X=x/L dimensionless coordinate - Fo=at/L² Fourier number - Bi(Fo)=(Fo)L/ Biot number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 5, pp. 921–924, May, 1971.     

Thermal boundary resistance in superfluid4He nearT λ

New measurements of the boundary resistivity in superfluid⁴He (2ppb³He) nearT are reported as a function of ¦¦ ( =T/T (Q) — 1) and of heat flux Q in a cell with parallel polished copper surfaces. Here we call T(Q) the temperature where the superfluid state abruptly disappears. In this design, the sidewall gaps between the copper pieces and the stainless steel spacer were eliminated. In contrast to several previous experiments but in agreement with those of Li and Lip a, no largeQ-dependent boundary resistivity anomaly was detected. However, as ¦¦ 0 the small weakly divergent resistivity was observed and its dependence onQ over the experimental range 1 <Q < 80 W/cm² was found to be very small. These new results are compared with previous experiments and predictions. An explanation of the previously observed anomalous transport phenomena is presented in terms of a heat flow through the sidewall gaps in these cells, and its limitation by a critical flow value _c. This phenomenological model can be fit satisfactorily to the observations. In the appendix we calculate _c from mutual friction.     

The thermal conductivity of AISI 304L stainless steel

A compilation and critical analysis of the thermal conductivity () of AISI 304 stainless steel (SS) between 100 and 1707 K has been given in the literature. The author represented his recommended values of by an inflection in the A versus temperature relationship between 300 and 500 K. Because a physical mechanism had not been identified that would produce such a temperature dependence in of 304 SS, interest was generated in the possible existence of an as yet undiscovered phenomenon that might cause such an inflection. Consequently, experimental verification of the inflection was sought. The present paper presents recent measurements of , the electrical resistivity, and the absolute Seebeck coefficient of 304L SS from 300 to 1000 K and of the thermal diffusivity () from 297 to 423 K. The values computed from the a measurements were within ± 1.6% of the directly measured An inflection was not observed in the temperature dependence of between 300 and 500 K. After careful evaluation and because a physical mechanism still has not been identified which would produce such an inflection, the authors conclude that the inflection in the vs T relationship reported in the literature was caused by the data analysis technique.     

The transverse acoustic impedance of dilute solutions of3He in superfluid4He

The transverse acoustic impedanceZ=R–iX of dilute solutions of³He in superfluid⁴He has been measured at a frequency (/2) of 20.5 MHz at temperaturesT from 30 mK to the transition at T. The³He concentrations studied werec=0.014, 0.031, 0.053, 0.060, and 0.092 below 1 K, thoughc decreased slightly near the point. The impedance was found from the temperature dependence of the quality factor and the resonant frequency of anAT-cut quartz crystal resonator immersed in the liquid. Below 1 K,Z is due to the Fermi gas of³He quasiparticles, and in the collisionless limit, 1 ( is a relaxation time),R remains constant whileX goes to zero. Measurements ofR(c, T) andX(c, T) were analyzed to determine the momentum accommodation coefficient (c, T) and (c, T). The relaxation times were in good agreement with previous work, while (c, T) was independent ofc, but increased from 0.29±0.03 below 0.1 K to 1.0±0.1 above 0.8 K. Various mechanisms are suggested to explain this. Between 1.0 and 1.5 K the³He quasiparticles and the thermally excited rotons are in the hydrodynamic region, 1. Values of the total viscosity (c, T) were obtained and analyzed to give the³He gas viscosity and the³He-³He and roton-³He scattering rates, both of which were energy-dependent. The superfluid healing length a was also measured. Near the point we founda=(0.1±0.03)^–2/3 nm, where =1–T/T, proportional to the phase coherence length . Our data are consistent with the hypothesis that _s/T is a universal constant for superfluid dilute solutions, where _s is the superfluid density. Between 1.0 and 1.8 K we found thata(c, T) was comparable to measurements in³He-⁴He films.     

Temperature dependence of electrical resistivity of deformed and undeformed V-rich V3Si single crystals

The electrical resistivity (T) of V-rich V₃Si single crystals (T _c-11.4 K) was measured from 4.2 to 300 K along the directions of [1 0 0] and [1 1 1] before and after plastic deformation at 1573 K. Anisotropy of (T) was observed although V₃Si has the cubic A15 structure. Plastic deformation does not affect the normal-state (T) behaviour but changes the normal-superconducting transition width T_c. At low temperatures (T _c<T 40 K), (T) varies approximately as T ⁿ where n-2.5 and this behaviour does not contradict the (0)- phase-diagram plot proposed by Gurvitch, where is the electron-phonon coupling constant and (0) is the residual resistivity.     

Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden

Zusammenfassung Messungen des Anwachsens von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkav gekrümmten Wänden (Görtler-Taylor-Wirbel) ergaben drei deutlich getrennte Bereiche: Es traten zunächst Wirbel mit der Wellenläge 0,9 auf (=Grenzschichtdicke, =Höhe einer Zelle, die zwei gegensinnig drehende Wirbel enthält). Je nach Größe der mit der Verdrängungsdicke ₁ der Grenzschicht gebildeten Reynolds-Zahl erschienen dann kurze Zeit später Wirbel mit 2,5, wenn war. Im Bereiche dagegen traten stattdessen bei den hier durchgeführten Versuchen immer Wirbel mit der Wellenlänge 6,5 auf. Bei werden die ersten Tollmien-Schlichting-Wellen mit der Wellenlänge _TS 6· angefacht. In ihren wandnahen Bereichen der Wellentäler könnten sich dann die oben genannten Längswirbel der Wellenlänge 6,5· ausbilden, die die zwei-in eine dreidimensionale Störung allseits gleicher Größenordnung verwandeln können.

---

The development of longitudinal vortices in boundary layers growing with time along concave walls

Summary Measurements of the growth of longitudinal vortices in boundary layers growing with time along concave walls (Görtler-Taylor vortices) rendered three distinctly separated regions. First, vortices with a wave-length 0.9 appeared (-boundary layer thicness, =height of a cell containing two counterrotating vortices). Then, depending on the Reynolds number R _{a 1}/v ₁=displacement thickness), vortices with 2.5 appeared shortly afterwards, provided . In the region , however, the wave-length was 6.5. For the first Tollmien-Schlichting waves with _TS 6 were excited. In the wave-throughs close to the wall the abovementioned longitudinal vortices with wave length 6.5 may then be formed. This might transform the two-dimensional into a three-dimensional flow of equal order of magnitude in all directions.

---

Zeichenerklärungen R _a Innenradius - Re _a Reynolds-Zahl gebildet mit dem InnenradiusR _a - Reynolds-Zahl gebildet mit der Verdrängungsdicke ₁ - kritische Taylor-Zahl - h Standhöhe der Flüssigkeit im Zylinder - t Zeit - z Anzahl - Steigungswinkel der Geraden - Grenzschichtdicke - ₁ Verdrängungsdicke - Wellenlänge (enthält ein gegensinnig rotierendes Längswirbelpaar) - v kinematische Zähigkeit - Winkelgeschwindigkeit Indizes K Knickpunkt der Geradensteigung - L unterhalb des Knickpunktes der Geradensteigung - TS Tollmien-Schlichting - e Einsatz der Wirbelentstehung     

Measurements of Critical Current of Superfluid 4He through An Array of Submicron Apertures very near Tλ

Far from the lambda transition the critical flow of superfluid ⁴ He through a small orifice is determined by thermal nucleation of quantized vortices. Between 300 mK and 2 K linearly decreasing critical flow velocity has been observed earlier. As the temperature approaches T the size of the vortex core increases and becomes comparable to that of the orifice. We report here measurements of the critical mass current in this temperature range. An array of 24 3×0.17 m holes in parallel with a macroscopic parallel path and flexible-diaphragm Helmholtz resonator have been used. The temperature range explored was from 80 mK to 20 K below T. Preliminary analysis of the data shows that for a reduced temperature t=(T–T)/T1·10 ^–4 the critical current scales approximately as t ^1.25 . Closer to T the critical phase difference across the array becomes comparable to 2 and the results have to be analyzed in terms of Josephson effect. The superfluid density has been measured at the same time as the critical current.     

Computational thermography: Numerical modeling of the thermal regimes of building structures

We consider numerical methods of simulating thermal regimes of building structures that make it possible to create optimum structures as regards power consumption by using more accurate calculations than those available in existing construction specifications and regulations. Possible means of reducing energy expenditures for formation of an optimum microclimate in living quarters are described.Notation R thermal resistance to heat transfer - _i thermal conductivity - c _i heat capacity - _i moisture content - i number of a layer - S thermal inertia of the material - density of the substance - frequency of harmonic vibrations - t time - Fo Fourier number - thermal diffusivity - t time step - x spatial step - Bi Biot number - h _c coefficient of convective heat transfer - k thermal conductivity - T ambient temperature - T _w wall temperature - Nu Nusselt number - Ra _l Rayleigh number - Gr _l Grashof number - Pr Prandtl number - g free fall acceleration - coefficient of thermal volumetric expansion of air - l characteristic length - coefficient of kinematic viscosity of air - determining temperature - thermal conductivity of the material - q heat flux - s area of the heat transfer surface - perimeter of the heat transfer surface - T free stream velocity - air viscosity Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute of the Academy of Sciences of Belarus, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66 No. 6, pp. 733–738, June, 1994.     

Transport properties of helium near the liquid-vapor critical point. II. Thermal conductivity of a3He-4He mixture

Measurements of the thermal conductivity are reported for an 80%³He-20%⁴He mixture above the critical point along several isotherms and near-critical isochores, using the same techniques and apparatus described for a study of³He. Using again the assumption that the observed conductivity can be decomposed into a sum of a regular and a singular contribution _reg and _sing, it is shown that along two near-critical isochores, _sing diverges. In particular, along the isochore showing the largest at the phase transition, the divergence is nearly the same as for³He and can be roughly characterized by a simple power law (T-T_c)^– with 0.58. This observation is contrary to predictions that foresee _sing0 asT _c is approached. The relaxation times characterizing the attainment of steady state conditions after switching the heat flux on and off show a similar behaviour as a function of reduced temperature as do those for pure³He. This result might indicate a substantial coupling between concentration and entropy diffusion. In the Appendix, the correlation length for³He nearT _c is calculated from heat conductivity, viscosity, and specific heat data.Work supported by Grant DMR 8024056 of the National Science Foundation.     

High-frequency dielectric properties of Mg-Al-Si,Ca-Al-Si and Y-Al-Si oxynitride glasses

Recently developed coaxial line techniques [1] have been used to determine, at room temperature, the values of the real () and imaginary (') parts of the dielectric constants for some Mg-Al-Si, Ca-Al-Si and Y-Al-Si oxynitride glasses over the frequency range 500 MHz to 5 GHz. The frequency dependencies of and ' are consistent with the universal law of dielectric response in that (-_t8)^(n–1) and '^(n–1) for all glass compositions; the high experimental value of the exponent (n=1.0±0.1) suggests the limiting form of lattice loss [2] situation. In this frequency range, as previously reported [3] at longer wavelengths, the addition of nitrogen increases the dielectric constant, (); in both the oxide and oxynitride glasses is also influenced by the cation, being increased with cation type in the order magnesium, yttrium, calcium as at lower frequencies.     

Transport in the low-temperature limit of a Landau Fermi liquid

The Boltzmann equation for Landau quasiparticles is solved for T 0 by a specialization of a method discussed by Sykes and Brooker. The quasiparticle distribution function is expanded in Legendre polynomials, assuming a boundary condition which imposes axial symmetry, and even-order terms are assumed to relax together with relaxation time _e , odd-order terms with relaxation time _o . By letting wavelength , with finite, one obtains a first-sound solution, and by lettingT 0, and then , one obtains a zero-sound solution. When these solutions are used to calculate the pressure, it is found that the first-sound solution is consistent with hydrodynamics, exhibiting viscosity = _s , while the zero-sound velocityc ₁=[^–1(B₁+4/3_s)]^1/2, so that phenomenologically zero-sound propagates like a longitudinal elastic wave in a glass. A higher zero-sound mode is also predicted, but is heavily damped. The heat flux is calculated and found to obey Vernotte's equation, which contains an intertial term, added to Fourier's law, that becomes significant asT 0.     

Experimental study of the boundary between single-phase convection and boiling in underheated cryogenic liquids

The effect of pressure and underheating on the position of the boundary between heat-transfer regimes in liquid helium and hydrogen is investigated.Notation q heat flux - p pressure - =T_s–T underheating - T_s saturation temperature - T temperature of liquid - T=T_wa – T T_s=T_wa – T_s - T_wa temperature of heat-emitting surface - A,a, B, b, C constants - m, n indices - Nu Nusselt number - Ra Rayleigh number - thermal conductivity - coefficient of cubical expansion - kinematic viscosity - g acceleration - standard deviation Indices 01 conditions of convection-boiling transition - 02 conditions of boiling-convection transition Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 1, pp. 5–11, January, 1982.     

Spatial Evolution of High Frequency Phonons in Superfluid 4He from a Pulse-Heated Metal Film

An almost monochromatic spectrum of high frequency (/k _B 10 K) phonons in superfluid ⁴ He is created by a short ( 0.1s) pulse of Joule-heating in a metal film submerged in the liquid at saturated vapour pressure (svp). These phonons have lifetimes that tend to infinity as T 0, and are the ones effective in quantum evaporation experiments. Most of these high frequency (hf) phonons are not injected into the liquid ⁴ He across the metal—liquid interface, but are created in the liquid by energy-increasing interactions which begin with the injected phonons of much lower energy (/k _B 1 K). These hf phonons are created up to 5 millimetres in front of the heater, hence the time of flight from a heater to a detector only gives an approximate value (lower bound) of their energies. Here we present measurements at svp of phonon energy fluxes in liquid ⁴ He at various distances from a pulse-heated metal film. Analysis of these signals gives an improved determination of the hf phonon spectrum (peaked at 10.20 ± 0.05 K with HWHM 0.2 K on the high energy side).     

Electron-phonon coupling as a mechanism for high-temperature superconductivity

We have solved the Eliashberg gap equations which are valid for arbitrary phonon frequency, (_ph), electron-phonon coupling constant, (), and screened Coulomb interaction, ( ^*). We have used values of (_ph),, and ( ^*) appropriate to the cuprate superconductors, and calculated the density of states, the pair potential., and the value of the gap at T=0 K. Using the linearized Eliashberg equations in the matrix representation, we have calculatedT _c and 2/k T _c . We have found that we can account for the highT _c 's in the cuprates with reasonable values for, ^*, and _ph.