The two-dimensionalO(N) nonlinear σ-model: Renormalisation and effective actions

We establish the existence of the Wilson Renormalised trajectory of theO(N) sigma model in perturbation theory in the effective charge. This yields a proof of perturbative renormalisability, and is also relevant in the small-field analysis of the rigorous renormalisation group construction of the continuum theory.Laboratoire associé au CNRS UA 280On leave from the Tata Institute of Fundamental Research, Bombay, 400 005, IndiaResearch funded in part by NSF grant DMS 8601978     

Spectral behavior of quasi periodic potentials

Spectral properties of Schrödinger operators of the typeH =–+V, where is the Laplacian,V a quasiperiodic potential and a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as is increased.Research supported in part by grant NSF MCS 7900813     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.     

Can particle creation by a black hole be described in terms of more familiar laboratory processes?

Particle creation by a black hole is described in terms of temperature corrections to the Casimir effect. The results of Levin, Polevoy, and Ritov for spectral and total Poynting vector for a fluctuating electromagnetic field in a plane vacuum gap between two arbitrary media with different temperatures in flat spacetime are applied to clarify the situation that exists between the horizon of a nonrotating black hole and spatial infinity. This helps to reveal the mechanism of particle creation. The Hawking radiation is born inside the bell formed by a potential barrier of a black hole in all the region [2M, ]. Its blackbody spectrum is due to the interaction of field fluctuations with the surface of the bell. The particles between the walls are virtual ones. They can become real after passing through the [3M, ] tail, appearing to an observer at future infinityJ ⁺ as real ones. The arguments for and against the present standpoint are discussed.     

ФИЗИЧЕСКИЕ ОСНОВЫ ИЗ МЕРЕНИЯ НАПРАВЛЕНИЯ ВОЗДУШНОГО ПОТОКА МЕ ТОДОМ НАКАЛЕННОЙ ПРО ВОЛОКИ ПРИ ВЫСОКНХ ПОДЗВУКО ВЫХ СКОРОСТЯХ

d - l - t _d R _max - J , ( ) - - , . . - ( ) - - R R _max —R. ,      

Anisotropy of the low temperature resistivity of hexagonal single crystalline spin glass systems

The impurity contribution to the resistivity in zero field (T) of dilute hexagonal single crystals of ZnMn, CdMn and MgMn has been studied in the mK range on samples cut parallel () and perpendicular () to thec-axis, using a SQUID technique for the measurements. Typical spin glass behavior is found in (T) as well as (T) for all alloys, with Kondo like logarithmic increases at higher temperatures and maxima atT _m at lower temperatures, indicating the influence of impurity interactions. The differences in the corresponding isotropic resistivity _poly(T) between the three systems can qualitatively be understood within the framework of a theoretical model by Larsen, describing (T) as a function of universal quantitiesT/T _K and _RKKY/T _K , where _RKKY is the RKKY-interaction strength andT _K the Kondo temperature. With respect to the two lattice directions studied, the behavior of (T and (T is anisotropic in the Kondo regime as well as in the range where ordering becomes important. While the anisotropy in the Kondo slope can be understood by an anisotropic unitarity limit, the understanding of the anisotropy in region where impurity interactions are important remains problematic.Dedicated to Prof. Dr. S. Methfessel on the occasion of his 60th birthday     

On the equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.     

Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

LetN, be von-Neumann-Algebras on a Hilbert space , a comon cyclic and separarting vector. Assume to be cyclic and separating also forN . Denote byJ , J _N the modular conjugations to (, ), and _N the associated modular operators. If and these data define in a canonical way a conformal quantum field theory in a cricle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentiageometrie und Quantenphysik     

Method of moments in analysis of x-ray diffraction lines

A general relation between the moments of the functionsf,g, andh, in the integral equationh(x)=f(y)g(x–y) dy, is derived. This enables any moment of the unknown functionf to be calculated from the moments of the functionsg andh. In particular, if certain assumptions are fulfilled, the moments of the components of the doublet can be calculated with advantage from the moments of the total profile. The statistical significance of the moment characteristics is also emphasized.

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f, g, h h(x)=f(y)g(x–) dy, f g h. , . .

     

ОПРЕДЕНИЕ ВНУТРЕННЕГО КВАДРУПОЛЬНОГО МОМЕНТА Ег162 ИЗ фОТОЭЛЕКТРИЧЕСКОГО ИЗМЕРЕНИЯ ИЗОТОПИЧЕСКИХ СМЕЩЕНИЙ ЛИНИЙ Ег I И Ег II

The isotopic shifts in Er I and Er II spectra were investigated by means of a recording spectrometer with a Fabry-Perot interferometer. The experimental technique of measuring the shifts of the spectral lines consists of using monoisotopic sources of light and a photoelectric record of alternating interference patterns corresponding to the different isotopic components. The values of the relative shifts of 30 lines of erbium and the data of the deformation parameters of the nuclei of Er¹⁶⁶ and Er¹⁶⁸ were used for determining the intrinsic quadrupole moment of Er¹⁶². It follows then from the theory of the isotopic effect in the spectra of elements with deformed nuclei, that the value of the intrinsic quadrupole moment isQ ₀(Er ¹⁶²)=(7·25±0·30)× ×10^–24 cm².

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Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

A note on the Ising model in high dimensions

We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh0.Part of this work was done while this author was visiting Rutgers University, supported by NSF grant DMR-86-12369 and Princeton University, support by NSF grant PHY-85-15288-A01Partially supported by a NSF grant to Cornell UniversityPartially supported by NSF grant DMR 86-12369Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell and by a NSF grant to Cornell University. This author was supported by the NSF grant DMR-86-12369 while visiting Rutgers University (when this work was started). On leave from São Paulo University     

Scattering states and bound states in λ℘(φ)2

By analyzing the Bethe-Salpeter equation for even ()₂ models we show that for weak coupling the mass spectrum is discrete and of finite multiplicity below 2m. Moreover on even states of energy less than 4(m–) we show that theS matrix is unitary. Herem is the physical mass and =()0 as 0. Our results rely essentially only on a simple assumption about the analyticity of the Bethe-Salpeter kernel which has been verified for weak coupling. For the interaction ⁴, (/m _o ² 1) we show that there are no even bound states of energy less than 4(m–).Work supported in part by NSF, Grant MPS 74-13252     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

Fractal drums and then-dimensional modified Weyl-Berry conjecture

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set R ⁿ (n1) with fractal boundary of interior Minkowski dimension (n–1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the counting functionN() (i.e. the number of positive eigenvalues less than ) as +, which is of the form ^/2 times a negative, bounded and left-continuous function of . This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn2. In addition, we also obtain explicit upper and lower bounds on the second term ofN().     

Zur Temperaturabhängigkeit einiger Kenngrössen der Verfestigungskurven von Zinkeinkristallen

Zusammenfassung Es werden Ergebnisse beschrieben über die Temperaturabhängigkeit: der kritischen Schubspannung ₀, des Verfestigungskoeffizienten _A , im BereichA der Verfestigungskurve; des Verfestigungskoeffizienten _B im BereichB, der Längea _A des BereichsA und der Schubspannung _A am Ende des BereichsA. Die kritische Schubspannung wächst im Temperaturbereich 225–344°K linear mit abnehmender Temperatur. Der Temperaturverlauf des Verfestigungskoeffizienten _A , stimmt überein mit älteren Messungen an h.k.p. Metallen. Ähnlich verhält sich der Verfestigungskoeffizient _B . Die Schubspannung am Ende des BereichsA wird durch thermisch aktivierte Vorgänge beeinflußt. Der BereichA verkürzt sich im Temperaturbereich 0–85°C linear mit zunehmender Temperatur. Temperaturwechselversuche zeigen, daß im BereichB thermisch irreversible Änderungen in der Versetzungsstruktur auftreten.

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: ₀, _A , _B , _{A A} . 226 344°K . _A . _B . . 0–85°C . , .

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Herrn Dr. F. Kroupa und Ing. B. esták bin ich für das Durchlesen des Manuskripts sowie wertvolle Hinweise zu Dank verpflichtet. Mein besonderer Dank für zahlreiche fruchtbare Diskusionen und Anregungen gilt Herrn P. Kratochvíl. Herrn Prof. Dr. A. Seeger danke ich ganz besonders für Kritik des Manuskripts und wertvolle Ratschläge.     

Currents and Super-Operator Product Expansionsof N = 4 Super-W Algebra

The currents and the super-operators product expansions of theN = 4 super-Walgebra are obtained. The generators of this extended conformal algebra consistof the stress-tensor superfield J and the primary superfields _±v ^q of integer orhalf-integer conformal weight and Cartan–Weylcharge q. The algorithm forderiving the N = 4 super-W algebra is given for particular cases. Explicit formsof the operator product algebra for different values of are given.