Solution spectrum of nonlinear diffusion equations

The stationary version of the nonlinear diffusion equation–c/t +D c = ₁c – ₂c² can be solved with the ansatz c = _p ^{= 1/} A_p(coshkx) ^–p , inducing a band structure with regard to the ratio ₁/₂. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c = skp = 1/ A _pq (coshkx)^–p [(cosht) ^–q–1 sinht + b(cosht)^–q] represents a solution spectrum of the time-dependent nonlinear equation, and the stationary version can be found from the asymptotic behavior of the expansion. The solutions can be associated with reactive processes propagating along molecular chains, and their applicability to biophysical processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheroids.     

Reactive pulsed laser deposition and laser induced crystallization of SnO2 transparent conducting thin films

Transparent conducting SnO₂ thin films with a thickness between 1000–2000 Å were deposited on glass, quartz and silicon substrates using standard pulsed laser deposition techniques with two different targets (Sri and SnO₂) and with three different laser wavelengths (1.06, 0.532 and 0.266 ) from a Q-switched Nd: YAG laser. Tin dioxide films with optical transmission over most of the visible spectrum exceeding 80% were obtained using a Sn target and a background oxygen pressure of 20 Pa. The electrical resistivity () depended strongly on the substrate temperature during deposition, with the lowest values of of about 10^–2 -cm obtained when the substrate was maintained at 400°C during deposition. Using SnO₂ targets, predominantly amorphous phase SnO₂ films were deposited on Si substrates and then transformed into polycrystalline Sn₃O₄ by laser induced crystallization ( = 1.06 m). Whereas these later films were essentially non-conducting as deposited ( > 400 -cm), the electrical resistivity was permanently reduced after laser induced crystallization by a factor greater than 1000 to a value of approximately 4 × 10^–1 -cm.     

ОПРЕДЕЛЕНИЕ ВРЕМЕНИ ЖИЗНИ μ-МЕЗОНОВ ПО АНОМАЛЬНОМУ ПОГЛ0Щ ЕНИЮ МЕТОДОМ ЯДЕРНЫХ ЭМУЛЬСИЙ

- ;- . ( ) . , - .     

Spectral properties of certain composition operators arising in statistical mechanics

By applying the theory of linear positive operators in a Banach space we derive spectral properties of certain composition operators in the Banach spaceA () of holomorphic functions over some domain . Examples of such operators are provided by the so called generalized transfer matrices of classical one-dimensional lattice systems.     

Inverse Problem for a General Bounded Domain inR3 with Piecewise Smooth Mixed BoundaryConditions

We study the influence of a finite container on an ideal gas. The trace of theheat kernel (t) = _{= 1}exp(–t), where {} _{= 1}are the eigenvalues of the negative Laplacian – ² = – ³ _{= 1}(/x )² in the (x ¹, x ², x ³)-space,is studied for a general bounded domain with a smooth bounding surface S, where afinite number of Dirichlet, Neumann, and Robin boundary conditions on thepiecewise smooth parts S _i (i = 1, ..., n) of S are considered such that S =U _{i = 1} S _i . Some geometrical properties of (the volume, the surface area, the meancurvature, and the Gaussian curvature) are determined. Furthermore,thermodynamic quantities, particularly the energy, for an ideal gas enclosed inthe general bounded domain with Dirichlet, Neumann, and Robin conditionsare examined with the help of the asymptotic expansions of (t) for short timet. We show that these thermodynamic quantities depend on some geometricproperties of .     

The electronic structure of Rhodium: Angle-resolved studies of photoelectron and secondary electron emission

We report electron spectroscopic studies of the Rh(111) surface, with the aim to obtain bulk band-structure information. We have measured normal photoemision using tunable synchrotron radiation in the range of photon energies between 11 eV and 55 eV, and angle-dependent photoemission along the LUX and LKL azimuths using the He resonance lines (=21.2 eV, 40.8 eV). To complement these data, we studied angleresolved secondary electron emission after excitation with electrons and photons. We derive parts of the one-electron energy dispersionE(k) along L, and determine the energies of several bulk critical points (in eV):E(> ₇₊/₈₊)=–2.75±0.10,E(> ₈₊=–0.85±0.10,E(> _7–=16.1±0.5,E(> _6–/> _8–)=20.5±0.5,E(X ₇₊)=–5.0±0.1,E(L ₆₊)=–5.6±0.5,E(L ₆₊/L ₄₊₊₅₊)=–2.65±0.10,E(L ₆₊)=9.0±0.5 eV. Our results are compared to several available band structure calculations.     

Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in R d

Let A=A ₀+v(x) where A ₀ is a second-order uniformly elliptic self-adjoint operator in R ^d and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A ₀ are Hölder continuous, we study the asymptotic behaviour of the counting function N(A,) () with the remainder estimates depending on the regularity hypotheses. Our strongest regularity hypotheses involve Lipschitz continuity and give the remainder estimate N(A,)O({}^–), where may take an arbitrary value strictly smaller than the best possible value known in the smooth case. In particular, our results are obtained without any hypothesis on critical points of the potential.     

Levinson's Theorem for the Schrödinger Equation in One Dimension

Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx ^–2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn ₊(n _–) is related to the phase shift ₊ (0)[_– (0)] of the scattering states with the same parity at zero momentum as ₊ (0)+ /2 =n ₊ and _– (0) =n _– for the noncritical case, and ₊ (0) =n ₊ and_– (0) – /2 =n _– for the critical case.     

Percolation cluster numbers by real-space renormalization

The average numbern _s (p) of percolation clusters withs sites is calculated for the triangular lattice using real-space renormalization. Fors up to 2,000 the whole range of concentrationsp was analyzed;n _s varied over sixty decades. We found logn _s –s forp belowp _c =0.5, andn _s s ^–, =2.35 atp=p _c . For smallp one hasn _s (p · ) ^s . Nearp _c we found the scaling fromn _s s ^– f((p _c –p) ·s ) with=0.53. Presumably for the first time renormalization methods were used to calculate percolation properties not only nearp _c but also far away from the critical point.Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Fine Structures of the Permittivity and Characteristic Losses of Electrons in Fluorite

Experimental-computational spectra of the permittivity and characteristic losses –Im^–1 for energies in the range 5–21 eV at a temperature of 4.2 K and theoretical spectra of and –Im^–1 of a fluorite crystal are resolved into elementary transition bands. The parameters of transition bands (energies of their maxima E _i, band halfwidths H _i and areas S _i, and oscillator forces f _i) are determined. A correlation of the spectral bands of and –Im^–1is established, and their specific features are elucidated.     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

830 nm emission in sodium vapor

We demonstrate that the intriguing 830 nm coherent emission, which is observed when sodium vapor is pumped with a high-power pulsed laser tuned near the 3S4D two-photon transition, is due to an axially phase-matched six-wave mixing process. This conclusion is based upon the observation of emission near 584 nm, which is coupled to the 830 nm emission in the six-wave mixing process: ₁+₂=2_L–_4D4P –_4P3D . In addition, we have observed coherent emission near 1.16 m, which is due to an analogous process involving cascade through the 4S (as opposed to the 3D) state. We calculate the wavelengths of all photons involved in these processes using the standard formulas of parametric wave-mixing theory, and show that they can be predicted to within experimental uncertainties. Finally we report observations of significant blue shifts of the 830 nm and 1.16 m emissions in a mixed sodium-potassium vapor. These shifts can be readily understood by considering the effect of the potassium on the frequency-dependent refractive index of the vapor. Due to these results, other recent interpretations of the 830 nm emission as stimulated excimer emission on the Na₂ 1³ _g ⁺ 1³ _u ⁺ band must now be rejected.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

Derivation of a quasiclassical scattering-phase approximation from the phase equations

A new radial phase equation is obtained for particles in a spherically symmetric potential field; the equation allows the quasiclassical asymptotes of the scattering phases _l() to be investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 64–69, August, 1978.     

Particle polarizations in decays of5HeΛ and4HΛhypernuclei

Nucleon polarizations are examined in the hypernucleus decays⁵He, ⁴He + p + _–,⁴H, ³H + p + _–, and⁴H, ³He + n + _–. The effect of the strong interaction in the final state is taken into account. A significant role is predicted for the resonance interaction in the p-⁴He system in the formation of the proton polarization in the⁵He decay. It is also shown that in the⁴H, ³H + p + _– decay the proton-polarization components should attain significant magnitudes, while in the charge-exchange decay⁴H, ³He + n + _–, the neutron polarization is expected to be not too large.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 100–103, October, 1988.     

On the invariant sets of a family of quadratic maps

The Julia setB for the mappingz (z–)² is considered, where is a complex parameter. For 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onB are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenB and -chains ± (± (± ...), which is explored for –1/42 and for with ||1/4, with the aid of the Böttcher equation. ThenB is shown to be a Hölder continuous curve for ||<1/4.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-8002731     

Über dieKα-Satelliten von Fluor in Lithiumfluorid

Using a focussing crystal spectrometer with photoelectric registration theK-emission spectrum of fluorine in solid lithium fluoride (=18.3 Å) was studied. If the spectrum is excited by Co or CuL-radiation or bremsstrahlung of a W-target the satellitesK ₃ andK ₄ appear with considerable intensity, whereas they appear only faintly ifL-radiation of Fe is used for excitation. In agreement with these observations the calculated threshold energies of the satellites are near the photon energy of FeL _{1, 2}. The measured energies of the components agree with values obtained from the energies of the singly and doubly ionized F^– ion as calculated by means of optical data. The results show that the satellitesK ₃ andK ₄ in the spectrum of fluorine in solid LiF correspond to transitions in the doubly ionized F^–-ion.     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

On the asymptotics of occurrence times of rare events for stochastic spin systems

We consider translation-invariant attractive spin systems. LetT _,x ^v be the first time that the average spin inside the hypercube reaches the valuex when the process is started from an invariant measure with density smaller thanx. We obtain sufficient conditions for (1) ¦¦^–1 logT _,x ^v (x) in distribution as ¦¦ , and ¦¦^–1 logT _,x ^v (x) as ¦¦ , where (x):= –lim ¦¦^–1 log {(average spin inside ) x. And (2)T _,x ^v /ET _,x ^v converges to a unit mean exponential random variable as ¦¦ . Both (1) and (2) are proven under some type of rapid convergence to equilibrium. (1) is also proven without extra conditions for Ising models with ferromagnetic pair interactions evolving according to an attractive reversible dynamics; in this case is a thermodynamic function. We discuss also the case of finite systems with boundary conditions and what can be said about the state of the system at the timeT _,x ^v .On leave from São Paulo University.     

One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ¹ It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural parameter characterizing equilibrium states is the spectral density matrix function (SDMF)F(), [– , ). Time evolution of a space profile of local equilibrium parameters is described by a space-time SDMFF(t;x, ) t, xR ¹. The hydrodynamic equation forF(t; x, ) which we derive in this paper means that the normal mode profiles indexed by are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, ) and a familyP ,>0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points close to the value ^–1 x in the stateP is approximately described by the SDMFF(x, ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in. Given nonzerotR ¹, we consider the states P _–1 ,>0, describing the system at the time moments ^–1 t during its harmonic time evolution. We check that the covariance at lattice points close to ^–1 x in the state P _–1 is approximately described by a SDMFF(t;x, ) and establish the connection betweenF(t; x, ) andF(x,).