New class of Finsler metrics

A class of Finsler spaces is introduced which is determined by the metric functionF(x, y)=[ +kB B )y y ]^1/2, whereB =B x, y andk is a constant. Various properties of these spaces are developed. A particular choice ofB is shown to produce a geodesic equation which is equivalent to the Lorentz equation of motion for a charged particle. Some general arguments for the physical applicability of Finsler spaces are also given.     

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.     

Generalization ability and information gain of clock-model perceptrons

We study the generalization abilityg _Q ofQ-state Clock-model perceptrons for (i) Hebbian and for certain Non-Hebbian learning procedures, namely (ii) learning with maximal stability, (iii) zero stability and (iv) optimal generalization, for the case of random training sets. Among other results we find thatg _Q behaves quite different in the Hebbian and in the Non-Hebbian cases in the limitQ. E.g. in the Hebbian case for finite ,g _Q vanishes always 1/Q, whereas in the Non-Hebbian cases considered,g _Q converges forQ to a non-trivial continuous functiong (), which vanishes for <2, but increases rapidly for >2. This means that for (ii), (iii) and (iv), as a function of atQ=, there is a 2nd-order phase transition from a non-generalizing phase for 2 to a generalizing phase for >2. Different behaviour of the Hebbian and Non-Hebbian cases, respectively, is also observed for the information gain obtained through learning. For the particular case of AdaTron Learning, which is identical to case (ii), we find a geometrical formulation forg _Q (), which is applicable to more general models.     

Distribution of the error term for the number of lattice points inside a shifted circle

We investigate the fluctuations inN (R), the number of lattice pointsnZ ² inside a circle of radiusR centered at a fixed point [0, 1)². Assuming thatR is smoothly (e.g., uniformly) distributed on a segment 0RT, we prove that the random variable has a limit distribution asT (independent of the distribution ofR), which is absolutely continuous with respect to Lebesgue measure. The densityp (x) is an entire function ofx which decays, for realx, faster than exp(–|x|^4–). We also obtain a lower bound on the distribution function which shows thatP (–x) and 1–P (x) decay whenx not faster than exp(–x ⁴⁺). Numerical studies show that the profile of the densityp (x) can be very different for different . For instance, it can be both unimodal and bimodal. We show that , and the variance depends continuously on . However, the partial derivatives ofD are infinite at every rational point Q ², soD is analytic nowhere.     

On the equivalence of the relativistic theories of gravitation

Einstein's equations are rewritten in terms of a certain torsionless linear connection which differs, in general, from the Levi-Civita metric connection . The torsionless connection appears in a natural way as the canonical momentum of the gravitational field g . Einstein's equations have a simple interpretation in terms of the connection . The equivalence of the so-calledpurely metric, purely affine, andmetricaffine theories of gravitation is proved.This work has been written under the financial support of Gruppo Nazionale per la Fisica Matematica of the Italian National Research Council.     

Gauge invariance of the Euler-Lagrange expressions

We prove, for a Lagrangian density L(g_ij;A _i ⁱ ;A _i ^j ), that the gauge invariance of the Euler-Lagrange expressionsE ⁱ (L) implies the existence of a gauge-invariant scalar densityL ₁, such thatE ⁱ (L) =E ⁱ (L₁). We then prove the uniqueness of the Yang-Mills field equations.     

Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H _mn= _{m, n+1}+ _{m, n–1}+ _{m, n} [(n+1)]–[n]) where =(5–1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.On leave from the Central Research Institute for Physics, Budapest, Hungary.     

Efficient high-repetition-rate fs-laser based X-ray source

The generation of [_Fe]K line radiation has been studied on a compact, ultrashort, laser based hard X-ray source, particularly suited for high-repetition-rate laser systems (1 to >10 kHz). Sub-millijoule, 25 fs Ti:sapphire laser pulses are applied to generate hard X-ray radiation on a commercially available ferric audio cassette tape target. Spectroscopic investigations reveal strong iron K emission in addition to a weak bremsstrahlung continuum. Under single exposure and single pulse conditions the X-ray yield is low. Energy conversion efficiencies of about _K=2×10^-7 are observed. Prepulse and multiple shot exposure techniques, both successfully used with high energy/low repetition rate systems, are investigated for these low energy/high repetition rate conditions. Depending on the applied pulse energy and intensity, the X-ray yield can be enhanced by more than three orders of magnitude. Photon fluxes exceeding 2×10⁹Kphotons/s are generated with the presented tape target yielding conversion efficiencies of nearly _K=10^-5. PACS 42.65.-k; 52.38.Ph; 52.50.Jm     

Monotonicity and concavity in Coulomb systems

The eigenvalues ofH()=H ₀+H ^*, whereH ^* is an arbitrary Coulomb potential, decrease with increasing 0. Linear and parabolic bounds for the ground-state energy are presented. These bounds are applied to the biexciton and the exciton at a neutral donor.     

What determinates the chemical changes of the internal conversion coefficients for inner atomic shells?

Calculations of internal conversion coefficients (ICC) of the E1–E4 and M1–M4 transitions for nuclei in ions show that the relative changes _i / _i of the ICC _i for deep inner subshells can differ significantly from the relative changes _i/_i of the electron densities _i at the nucleus. For the K conversion _i/ _i are many times greater than _i/_i. Especially large deviations of _i/ _i are characteristic of transitions of high multipolarity; however, for the M1 transitions they can also be significant. Illustrations of various dependencies of _i/ _iare presented for the conversion in the regionZ-50.     

Characterising and controlling surface defects

The controlled production of surface defects by ion irradiation and nanoindentation techniques is discussed. The characterisation of novel types of defects is emphasized, among those: quasi two-dimensional dislocation dipoles following coalescence at the surface of the vacancies induced by the irradiation and mesas and winding-loops as dislocation configurations originated by nanoindentation. The relationship between the latter defects and the onset of crystal plasticity is examined.Received: 16 May 2004, Published online: 10 August 2004PACS: 61.72.Ff Direct observation of dislocations and other defects (etch pits, decoration, electron microscopy, X-ray topography, etc.) - 68.35.Gy Mechanical properties; surface strains - 61.72.Bb Theories and models of crystal defects     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

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.     

Cooling-rate dependence of the residual energy of a one-dimensional configurational glass model

For a one-dimensional configurational glass model we have performed molecular dynamical calculations. The Newtonian equation of motion was solved numerically including a damping term. The residual energye _res() as a function of the damping constant , exhibits a power law behavioure _res(),0.061; in an intermediate range of . This behaviour can be explained as the freezing of a certain type of two level systems with an excitation energy and a barrier heightB. The exponent is approximately equal to /B. This relationship is justified analytically.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Bounded perturbations of dynamics

If and are one-parameter automorphism groups of a von Neumann algebraM is said to be a bounded perturbation of if _t – _t 0 ast0. We give a complete characterization of the bounded perturbations of . In particular, we show that if can be implemented by a strongly continuous one-parameter group with self-adjoint generator (Hamiltonian)H, then can be implemented in the same way and the corresponding HamiltonianH can be chosen to be of the formH=VHV ^–1+h, whereV is a unitary ofM andh=h*M.On leave of absence from II. Institut für Theoretische Physik, Universität Hamburg, D-2000 Hamburg 50, Federal Republic of Germany     

Scaling factors associated withM-furcations of the 1−μ∥x∥z map

A numerical study is made of the scaling behavior associated withM-furcations (M=3, 4, 5) in the mapx _t+1 =1–x _t ^z (z>1). The scaling constants and are calculated as functions ofz, as well as the more general scaling functions andf(a).     

On P-representation for parametric amplifier and parametric frequency converter

The statistical properties of a parametric amplifier and a frequency converter are studied by means of quantum mechanical methods. The Schrödinger picture and the P-representation of the density matrix are used. Carrying out the Fourier transformation of the Liouville equation a partial differential equation for a generating function is obtained. The inverse Fourier transform of a solution of this equation is a time-dependent P-representationPN( ₁, ₂,t). For the parametric amplifier a relation is derived which enables us to compute the functionPA( ₁, ₂,t) = =1< ₁, ₂/ ₁> is shown thatPA is classical distribution ifPN( ₁, ₂,0) is a positive distribution, while the P-representationPN( ₁, ₂,t) need not exist as a distribution and the P-representationPN( ₁, ₂,t) for the parametric frequency converter is constant along classical trajectories.The author wishes to thank Dr. J. Peina for stimulating discussions.     

Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.     

Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes- (LANS-) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS- equations in the literature⁽¹⁾ for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS- equations, denoted the Prandtl- equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl- equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.⁽²⁾     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN