Mean-Field Critical Behavior for the Contact Process

The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d>4:()– _c as _c and ()( _c–)^–1 as _c, where () and () are the spread probability and the susceptibility of the infection respectively, and _c is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.     

Central extensions and semi-infinite wedge representations of Krichever-Novikov algebras for more than two points

For the generalized Krichever-Novikov algebras of meromorphic vector fields and their induced modules of weight a different basis is given. With respect to this basis the module structure is generalized graded. Local central extensions of these algebras and their representations on the space of semi-infinite wedge product of forms of weight are studied. In this generalization, one again obtains c = -2(6² -6 +1) as the value for the central charge.     

Mathematical properties of gap-functions in strongly coupled superconductors

We consider the Dyson equation associated with the BCS superconducting state from a mathematical point of view. The Dyson equation gives rise to a modified gap equation that is similar to the BCS gap equation, but with a different kernel. We first show that for strong coupling (such that the McMillan parameter ||1) both the real and imaginary parts of the solution (E) of the modified gap equation alternate in sign as function of the excitation energyE, the periods being 4₀ for positive and 4₀/3 for negative . (₀ is the frequency of an Einstein spectrum of phonons). A closed, algebraic approximation to (E) is 2||₀log[cotan(E/ )]. Finally, the poles of the kernel of the integral equation are located in the complex-E plane. For the new-type, oscillatory solution of the modified gap equation the analogue of the causal (zero-temperature) Green's function is shown to have different analytic properties from those of the smooth Eliashberg solution of BCS theory.     

Theory of fluctuations in limit-cycles of second-order bifurcation type

The fluctuations in limit cycles of second-order bifurcation (transition from a stable to an unstable focus) are investigated near the bifurcation point _c, being an external control parameter. Two different methods are applied: a time- and space-dependent Fokker-Planck equation obtained from an ^1/2-expansion of the master equation ( being the volume) and a time- and space-dependent Langevin equation. Both methods give the same results. It is shown that the dependence of the radial correlation on ² = | – _c| and the time-behaviour of the phase correlation (ensemble dephasing) are determined by the dimensionality of space.     

Differential equations in the spectral parameter,Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrödinger operators whose eigenfunctions (, ) satisfy a differential equation in the spectral parameter of the formB(, )=(x). We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.     

A contact process with a single inhomogeneous site

The one-dimensional basic contact process is a Markov process for which particles give birth on vacant nearest neighbor sites at rate >0 and particles die at rate one. We introduce a one-dimensional contact process with a single inhomogeneous site: the evolution is as above except that a particle located at the origin does not die. Let _c be the critical value of the basic contact process. We show that for _c the upper invariant measures of the inhomogeneous contact process and the basic contact process coincide except at a finite number of sites. The behavior at = _c is much more intersting: the upper invariant measure of the inhomogeneous contact process concentrates on configurations with infinitely many particles, while it is known that the critical basic contact process dies out. So a single inhomogeneity may provoke a perturbation unbounded in space. As a byproduct of our analysis we prove that the connectivity probabilities of the critical basic contact process are not summable. We also give a biological interpretation of this model.     

Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media

The amplification of light signals (angular frequency _S in some isotropic media (D₂O, fused silica, and Schott type SF10 glasses) by noncollinear phase-matched parametric four-photon interaction ₁+_2S+₁ is studied theoretically. Computer simulations are carried out for fundamental and second-harmonic pump pulses of a mode-locked Nd: glass laser. Degenerate interaction (wavelength ₁=₂=1054nm or 527 nm) and nondegenerate interaction (₁=1054nm, ₂=527 nm are considered. Characteristic phase-matching parameters and gain parameters versus wavelength are determined. Limitations by spectral bandwidth, optical absorption, optical damage, self-phase modulation, self-focusing and stimulated Raman scattering are analysed.     

Determination of the scattering amplitude

The problem to determine the elastic scattering amplitude from the differential cross-section by the unitarity equation is reexamined. We prove that the solution is unique and can be determined by a convergent iteration if the parameter =sin of Newton and Martin is bounded by <₂0.86. The method is based on a fixed point theorem for holomorphic mappings in a complex Banach space.     

On the Asymptotic Density in a One-Dimensional Self-Organized Critical Forest-Fire Model

Consider the following forest-fire model where the possible locations of trees are the sites of . Each site has two possible states: vacant or occupied. Vacant sites become occupied at rate 1. At each site ignition (by lightning) occurs at ignition rate , the parameter of the model. When a site is ignited, its occupied cluster becomes vacant instantaneously. In the literature similar models have been studied for discrete time. The most interesting behaviour occurs when the ignition rate approaches 0. It has been stated by Drossel, Clar and Schwabl (1993) that then (in our notation) the density of vacant sites (at stationarity) is of order 1/ log (1/). Their argument uses a scaling ansatz and is not rigorous. We give a rigorous and mathematically more natural proof for our version of the model, and point out how it can be modified for the model studied by Drossel et al. Our proof shows that regardless of the initial configuration, already after time of order log (1/) the density is of the above mentioned order 1/ log (1/). We also obtain bounds on the cluster size distribution, showing that the scaling ansatz of Drossel et al. needs correction.Also at Vrije Universiteit Amsterdam.Acknowledgement We thank Jeff Steif for stimulating discussions about these and related problems.     

Loss in low-finesse Ti:LiNbO3 optical waveguide resonators

We propose to measure the contrast of Fabry-Perot resonances in low-finesse (end face polished) integrated optical resonators to determine an upper limit of the waveguide attenuation coefficient. The method is discussed; its absolute accuracy increases with decreasing loss. As example, experimental results are presented for 3 and 10 m wide Ti: LiNbO₃ strip guides; the attenuation coefficients are 0.46 dB/cm (EH₁₁) in the narrower guide at =0.63 m, respectively 0.19 dB/cm (EH₁₁) and 0.09 dB/cm (HE₁₁) in the 10 m wide guide at =1.15 m.     

Analytic structure in λ of theS-matrix for strongly singular potentials

The asymptotic behaviour in the -plane of solutions of the Schrödinger equation for scattering on singular potentials is investigated. The asymptotic behaviour of the Jost functions and theS-matrix is obtained. Furthermore, the general analytic form in the -plane of the Jost functions and theS-matrix is established. Some properties of the distribution of poles of theS-matrix are proved.On leave of absence from the Institute Ruder Bokovi, Zagreb, and the Zagreb University, Yugoslavia.     

New proofs of the existence of the Feigenbaum functions

A new proof of the existence of analytic, unimodal solutions of the Cvitanovi-Feigenbaum functional equation g(x) = –g(g(–x)),g(x) 1 - const|x|^r at 0, valid for all in (0, 1), is given, and the existence of the Eckmann-Wittwer functions [8] is recovered. The method also provides the existence of solutions for certain given values ofr, and in particular, forr=2, a proof requiring no computer.     

Luminescence of Higher Plants upon Excitation by Laser Radiation with Different Energy and Time Parameters

The luminescence response of the leaves of higher plants exposed to laser radiation with different energy and time parameters has been investigated at a wavelength of 685 nm. The plants were irradiated by laser radiation at 1060, 632, and 530 nm. Radiation at = 632 nm was used as the control. It is shown that radiation at = 1060 nm suppresses photosynthesis, whereas that at = 530 nm stimulates it.     

Calculation and optimization of parameters of radiating distributed feedback structures

A simple method is suggested for calculation of reflection, radiation and transmission coefficients for the distributed feedback structure in the second diffraction order. The method is based on a slight difference between coefficients of reflectionR and radiationI of the surface wave for = (where is the light wavelength corresponding to a precise resonance for the grating length I) and those for =_l (where _l is the light wavelength corresponding to the resonance for the finite grating length). The simplicity of the method makes it possible to use it for optimization of the distributed feedback structure by a number of parameters. The technique can be used in the case of thin-film and diffused waveguides for both TE and TM modes.     

Antibound states for a class of one-dimensional Schrödinger operators

Let (x) be the Dirac's delta,q(x)L ¹ (R)L ² (R) be a real valued function, and , R; we will consider the following class of one-dimensional formal Schrödinger operators on . It is known that to the formal operator may be associated a selfadjoint operatorH(,) onL ²(R). Ifq is of finite range, for >0 and || is small enough, we prove thatH(,) has an antibound state; that is the resolvent ofH(,) has a pole on the negative real axis on the second Riemann sheet.Work done while the author was supported by an undergraduate fellowship of the (Italian) National Research Council (CNR).     

Realization of Integrated Narrow Bandpass Filters in the Infrared Region

We demonstrated a high-efficiency technique for the integration of narrow bandpass filters (NBPFs). A filter array integrated with 16 filters on a single substrate has been fabricated by only twice depositing processes. The bandpass of the filters can be tuned widely and range from 2.295 to 2.967 m in this case. The relative full width at half maximum (FWHM) / of each filter is less than 0.7%. And the intensity of the bandpass is up to 41.3%.     

Nano-structured oriented carbon films grown by PLD and CVD methods

PLD and CVD methods, plasma or bias assisted, have been used to prepare thin films of nano-clustered graphite. The experimental conditions (vz. deposition apparatus, substrate temperature, working pressure, inert sustaining gases, parent species, and applied voltage) have been changed with the final aim of obtaining small graphene particles with the basal planes oriented along the growth vector and perpendicular to the n-Si100 substrate. Pulsed laser ablation (Nd:YAG, 2nd harmonic: =532 nm, h=2.33 eV, =7 ns, =10 Hz, 7 J/cm²), assisted by an RF-plasma, of a pyrolytic graphite target gave good results for nano-structure formation, provided high substrate temperature and high inert gas pressure are maintained. CVD methods, in the presence of high substrate temperature and a DC bias, showed a good attitude to drive a longitudinal growth of graphene layers and nano-wires from a reactive gas flow of Argon/Hydrogen and Methane. The morphology of the films grown at different conditions have been characterised by scanning electron microscopy (SEM). Film quality and nano-particle dimensions have been estimated by Raman spectroscopy. PACS 81.07.-b; 81.15.-z; 78.30.-j; 68.37.-d     

The ground state energy of Sdhrödinger operators

We studye()=inf spec(-+V) and examine whene()<0 for all 0. We prove thatc ²e()–d ² for suitableV and all small ||.Research partially funded under NSF grant number DMS-9101716.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Evaporation characteristics of a barium oxide film on a molybdenum backing in relation to temperature and coverage

The evaporation characteristics of a BaO-Mo₁₁₂ system, obtained by deposition of BaO on a heated ribbon, were investigated. The evaporation rate from such a system was found to be higher than in the case of deposition on an unheated ribbon, but the film was still active in a wide temperature range. Only a part of the film evaporates at each temperature and the part increases with temperature increase. At T = 1400 °K a film with < 1 evaporates completely, whereas a film with > 1 does not evaporate completely. Probably some of the barium oxide dissociates and is not detected by the receiver ribbon. The heat of evaporation of the film depends on the temperature; /T 10^–3 eV/deg.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 99–103, October, 1971.In conclusion the authors express their thanks to diploma students who assisted in the measurements: Nsu Hsing-Chou, A. I. Solov'eva, D. L. Natadze, and E. S. Ovchinnikova.