Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

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.     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

Gravitation and universal Fermi coupling in general relativity

The generally covariant Lagrangian densityG = + 2K _matter of the Hamiltonian principle in general relativity, formulated by Einstein and Hilbert, can be interpreted as a functional of the potentialsg _ikand of the gravitational and matter fields. In this general relativistic interpretation, the Riemann-Christoffel form _kl ⁱ = _kl ⁱ for the coefficients _kl ⁱ of the affine connections is postulated a priori. Alternatively, we can interpret the LagrangianG as a functional of , g_ik, and the coefficients _kl ⁱ . Then the _kl ⁱ are determined by the Palatini equations. From these equations and from the symmetry _kl ⁱ = _lk ⁱ for all matter fields with /=0 the Christoffel symbols again result. However, for Dirac's bispinor fields, / becomes dependent on the Dirac current, essentially with a coupling factor Khc. In this case, the Palatini equations define a new transport rule for the spinor fields, according to which a second universal interaction results for the Dirac spinors, besides Einstein's gravitation. The generally covariant Dirac wave equations become the general relativistic nonlinear Heisenberg wave equations, and the second universal interaction is given by a Fermi-like interaction term of the V-A type. The geometrically induced Fermi constant is, however, very small and of the order 10^–81erg cm³     

Grassmann-Cartan algebra and Klein-Gordon equations

Given a Riemannian structure (M, g), a hypothesis is investigated that if= _p=0 ⁿ _p (M) is submitted to the differential condition (g++)=0, =mc/—which implies that each component of fulfills the Klein-Gordon equation (- ²) _p =0, ought to be interpreted as a natural complex of the bosonic fields. Then it is found that the complex admits the interpretation in the sense of first quantization with (M) being a convex set of states, with the structure of a Hilbert space over . The definite spin states of bosons are then pure states which are not conserved by the temporal evolution.     

Замечание К теории по степенного возникновения подви жных слоев в плазме инертных газов

[1] t _B , t _B . , t _B , . .

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A note on the theory of the successive production of moving striations in the plasma of inert gases

Approximate expressions are derived on the basis of Pekárek's theoretical paper [1] for the period of the maximum t_B of a wave packet produced by the passage of a wave of stratification before the aperture of a photomultiplier, and for its time width in the half-height t_B. The relaxation time of a wave of stratification, following from the theory [1], can thus be calculated by means of the experimentally measured velocity of motion of the maximum of a wave packet u and its width t_B. The calculation is supplemented by numerical data on the magnitude of errors committed by using approximate expressions.

     

Amplitude and energy coefficients of reflected and refracted light

The tensor of reflected and refracted light was first introduced into the principal axes and then the latter were transformed into thep ands direction of the electric field strength. The amplitude coefficient of thep-components of refracted light has different forms depending on which complex unit vector is unit with respect to the square or the absolute value. The energy coefficient has the same value in both cases.

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, p s . p- , . .

     

Anderson localization for the almost Mathieu equation: A nonperturbative proof

We prove that for any diophantine rotation angle and a.e. phase the almost Mathieu operator (H()) _n = _n–1 + _n+1 +cos(2(+n)) _n has pure point spectrum with exponentially decaying eigenfunctions for 15. We also prove the existence of some pure point spectrum for any 5.4.     

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 .     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.     

On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping , analytic and -close to the identity, there exists an analytic autonomous Hamiltonian system, H such that its time-one mapping _H differs from by a quantity exponentially small in 1/. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows exactly, namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H, or equivalently of the rescaled Hamiltonian K=^-1H, which differs fromK, but turns out to be ⁵ close to it. Special attention is devoted to numerical integration for scattering problems.     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl ²(), (l _Y, )(n) = (l+1)+(l–)+ cos(n+) (n) have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.     

Spectral superbroadening of self-focused picosecond laser pulses in D2O

Intense picosecond light pulses of a mode-locked Nd: glass laser at _L =1054 nm (fundamental wavelength) and _SH =527 nm (second harmonic wavelength) are passed through a sample of D₂O under self-focusing conditions. Spectrally structured superbroadened, spatially bell-shaped emission in the forward direction is obtained. Primary generation processes are pump-pulse-degenerate stimulated parametric four-photon interaction (₁ + _{1 3} + ₄) and stimulated Raman scattering (_{1 R} + ), which occur concurrently (₁= _L or _SH angular pump frequency, #x03C9; _R first Stokes frequency, #x03C9; signal frequency, #x03C9;₃ signal frequency, #x03C9;₄ idler frequency). The parametric four-photon interaction occurs under collinear non-phase-matched conditions and under longitudinally phase-matched, transversally non-phase-matched (erenkov-like) conditions. Subsequent interaction processes are pump-pulse-nondegenerate four-photon interaction of the type ₁ + _{R 3} + ₄, coherent antiStokes Raman scattering (CARS, ₁ + ₁ – _{4 3}), inverse Raman scattering ( _A + ₁ + ), and cascading light up-conversion of the type ₁ + _(i) – _{R (i+1)}.     

Double quantum transitions in paramagnetic resonance

The theory of double quantum transitions of the M=±2 type, with regard to inhomogeneously broadened spin systems is studied in this paper with the approximation ²T₂T₃ 1. We suppose that the inhomogeneous broadening is formed by an inhomogeneous crystal field. The obtained results describe the magnitude of absorption as a function of the h.f. power and also describe the shape of the absorption curve. It is demonstrated that in inhomogeneously broadened spin systems the absorption curve of double quantum transitions has the form of the difference of two different Lorentz's curves and that at the saturation ( ²T₂T₁ 1) the absorption increases with the cube of the h.f. field intensity. The shape of the curves is expressed by means of phenomenological relaxation constants of the system.

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M=±2 ²T₂T₃ 1. , - . . , ( ²T₂T₁ 1) . .

     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

The contrast gain in a guest-host nematic display due to a periodic boundary condition

The mean square tilt angle of a nematic slab with finite anchoring energy and periodic boundary conditions has been theoretically investigated, as a function of the slab geometry and of the reduced extrapolation length. If the anchoring strength is free-surfacelike, the contrast is affected by a loss 10% at room temperature if the ratio between the anchoring pitch and the cell thickness is 0.5.Glossary anchoring pitch - h cell thickness - /h - ( = x/, = y/h) reduced coordinates - (, ) local tilt angle - elastic constant - w_a anchoring energy anisotropy - b=/w _a de Gennes-Kleman extrapolation length - B=b/h reduced extrapolation length - T _NI nematic-isotropic transition temperature - :=(T/T _NI ) – 1 reduced temperature - easy axis direction - _MAX - _± ² mean square tilt angle along the boundary - () absorbance coefficients of the p-dye - r /: dichroic ratio - c contrast - G contrast gain - S order parameter     

High-resolution laser spectroscopy of theQ v(0) transitions in solid parahydrogen

Our recent high-resolution laser spectroscopy of theQ _v(0) (=0,J=00) transitions in solid parahydrogen is discussed. The systems studied include the fundamental vibrational bands of impurity D₂ and HD, the first and second overtones of parahydrogen, and the charge-induced spectrum of-ray irradiated parahydrogen. Additionally, Stark and stimulated Raman-gain spectroscopies are applied to the solid. The linewidths are as sharp as 2 MHz HWHM, which is highly unusual for a solid. Our spectra demonstrate a variety of physical phenomena, particularly thek = 0 selection rule, as well asJ = 1/J = 0 pair intermolecular interactions.     

Microwave relaxation and phenomenological damping in thin films

The relation between relaxation timeT, frequency swept resonance linewidth , and phenomenological damping is given by =2/T=(_x+_y), where _x,y = (H ₀+(N _x,y –N _z ) 4M _s ).N _x,y,z are sample demagnetizing factors,H ₀ is the effectivez-directed static field, 4M _s is the saturation induction, and is the gyromagnetic ratio. This fairly simple but general relation shows that the numerical relation between damping and relaxation at a given frequency can be quite different for in-plane and normally magnetized thin films. For thesame loss processes, so thatT andT are equal, is larger than . For permalloy films at 1 GHz, =15 . In addition, the conventional field swept linewidth, H=/, is simply related to only forN _x =N _y . Both and H are geometry dependent and do not provide an intrinsic measure of the relaxation. These results are confirmed by both resonance and transient response experiments. The large values of for large angle switching may also be partially explained by this analysis because the relevant magnetization motion is due to a demagnetizing field normal to the film plane.Visiting scientist on leave fromRaytheon Company, U.S.A. Supported by the Japan Society for the Promotion of Science.