Semilinear boundary value problems for degenerate pseudodifferential operators in spaces of Sobolev type

Boundary value problems for degenerate semilinear elliptic pseudodifferential operators are considered. Using the apparatus of the theory of pseudodifferential operators, function spaces introduced in [4–8], and the Rabinowitz construction [12] based on the Borsuk theorem (see [11]), we prove the existence of solutions of the problem in suitable function spaces. To the memory of Leonid Romanovich Volevich     

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

Classical Yang-Mills (Y.M.) equations with static external sources are formulated as a Hamiltonian system with gauge symmetry in A₀ = 0 gauge. Using the concept of a “momentum mapping” (J. Marsden and A. Weinstein, Rep. Math. Phys.5 (1974), 121) on symplectic manifolds with symmetry, an analogue of centrifugal potential of a mass point in a spherically symmetric potential is derived. This gives rise to an effective potentialV_eff, whose critical points are rigorously proved to be in one-to-one correspondence with static Y.M. solutions. V_eff additionally depends on the prescribed external source ?, which is as a constant of motion analogous to angular moment of the mass point. Thus bifurcation of static solutions is caused by bifurcation of critical points of V_eff under variation of the external parameter ?. Some closing remarks on dynamics and stability on gauge orbit space are added.     

On quantization of free fields in stationary space-times

In Section 1 we analyse the structure of the infinite-dimensional Hamiltonian system described by the Klein-Gordon equation (free real scalar field) in stationary space-times with closed space sections; we give an existence and uniqueness theorem for the Lichnerowicz distribution kernelG ₁ together with its proper Fourier expansion, and we construct the Hilbert spaces of frequency-part solutions defined by means ofG ₁.In Section 2 an analysis, a theorem and a construction similar to the above are formulated for thefree real field spin 1, massm>0, in one kind of static space-times.In this letter, only results are given. For detailed proofs and further results, see reference [9], [10] and [11].     

Lorentzian framed structures in general relativity

We study a class of four-dimensional real Lorentzian manifolds generated by a global operator (satisfying a cubic equation). We characterize empty, Einstein, conformally flat spaces and construct mathematical models of two physical space-times. Locally, the solutions presented have been identified with known solutions. Kruskal space-time is generated by using the warped product technique [11].     

Non-uniqueness of solutions of Percival's Euler-Lagrange equation

Percival [5,6] introduced a Langrangian and an Euler-Lagrange equation for finding quasi-periodic orbits. In [3], we studied area preserving twist homeomorphisms of the annulus, using Percival's formalism. We showed that Percival's Lagrangian has a maximum on a suitable function space, and that a point where it takes its maximum is a solution of Percival's Euler-Lagrange equation. Moreover, in the rigorous interpretation of Percival's formalism which we gave in [3], the solutions of Percival's Euler-Lagrange equation correspond bijectively to a certain class of minimal sets. (We will prove this in Sect. 2.) In [4], we showed that Percival's Lagrangian takes its maximum at only one point. In this paper, we show that there existC area preserving twist diffeomorphisms of the annulus, for which there exists at least one solution of Percival's Euler-Lagrange equation where Percival's Lagrangian does not take its maximum. In other words, solutions of Percival's Euler-Lagrange equation need not be unique.Supported by NSF Grant No. MCS 79-02017     

Bright and dark solitons in a quasi-1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters

We investigate the exact bright and dark solitary wave solutions of an effective 1D Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature Pérez-García et al. (2004) [50], Serkin et al. (2007) [51], Gurses (2007) [52] and Kundu (2009) [53], we point out that the effective 1D equation resulting from the Gross-Pitaevskii (GP) equation can be transformed into the standard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schrödinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton bound state, as the atomic scattering length and gain/loss term are varied. However, when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.     

Static and time-dependent solutions of Einstein-Maxwell-Yukawa fields

An exact solution of Einstein-Maxwell-Yukawa field equations has been obtained in a space-time with a static metric. A critical analysis reveals that the results previously obtained by Patel [9], Singh [10], and Taub [11] are particular cases of our solution. The singular behavior of the solutions has also been discussed in this paper. Further, extending the technique developed by Janis et al. [12], for static fields, to the case of nonstatic fields, an exact time-dependent axially symmetric solution of EMY fields has been obtained. Our solution in the nonstatic case is nonsingular in the sense of Bonnor [15] and presents a generalization of the results obtained by Misra [7] to the case when a zero-mass scalar field coexists with a source free electromagnetic field.     

The orbital stability of the cnoidal waves of the Korteweg-de Vries equation

The cnoidal wave solution of the integrable Korteweg-de Vries equation is the most basic of its periodic solutions. Following earlier work where the linear stability of these solutions was established, we prove in this Letter that cnoidal waves are (nonlinearly) orbitally stable with respect to so-called subharmonic perturbations: perturbations that are periodic with period any integer multiple of the cnoidal-wave period. Our method of proof combines the construction of an appropriate Lyapunov function with the seminal results of Grillakis, Shatah and Strauss (1987, 1990) [17] and [18]. The integrability of the Korteweg-de Vries equation is used in that we need the presence of at least one extra conserved quantity in addition to those expected from the Lie point symmetries of the equation.     

Spherically symmetric static solutions of the Einstein-Maxwell equations

We report a new formalism to obtain solutions of Einstein-Maxwell’s equations for static spheres assuming the matter content to be a charged perfect fluid of null-conductivity. Defining three new variablesu=4πεr ²,ν=4πpr ^{2 2} andw=(4π/3)(ρ+ε)r ² whereε, ρ andε denote respectively energy densities of the electric, matter and free gravitational fields whereasp is the fluid pressure, Einstein’s field equations are rewritten in an elegant form. The solutions given by Bonnor [1], Nduka [2], Cooperstock and De la Cruz [3], Mehra [4], Tikekar [5,6], Xingxiang [7], Patino and Rago [8] are all shown to possess simple relations betweenu, v, andw whereas Pant and Sah’s [9] solution for which all the three functions,u, v, andw are constants is a trivial case of the present formalism, We have presented six new solutions with ε = 2ρ. For the first three solutionsw andu are constants withv as a variable whereas the remaining three solutions satisfy the equation of state for isothermal gas;v =kw =-ku where (i)k is an arbitrary constant but not equal to 1 or 1/3 (ii)k = 1 and (iii)k = 1/3. We also obtained a generalization of Cooperstock and De la Cruz’s [3] solution which is regular for 2ρ > ε but singular for 2ρ ≤ ε.     

Numerical method for hydrodynamic modulation equations describing Bloch oscillations in semiconductor superlattices

We present a finite difference method to solve a new type of nonlocal hydrodynamic equations that arise in the theory of spatially inhomogeneous Bloch oscillations in semiconductor superlattices. The hydrodynamic equations describe the evolution of the electron density, electric field and the complex amplitude of the Bloch oscillations for the electron current density and the mean energy density. These equations contain averages over the Bloch phase which are integrals of the unknown electric field and are derived by singular perturbation methods. Among the solutions of the hydrodynamic equations, at a 70 K lattice temperature, there are spatially inhomogeneous Bloch oscillations coexisting with moving electric field domains and Gunn-type oscillations of the current. At higher temperature (300 K) only Bloch oscillations remain. These novel solutions are found for restitution coefficients in a narrow interval below their critical values and disappear for larger values. We use an efficient numerical method based on an implicit second-order finite difference scheme for both the electric field equation (of drift-diffusion type) and the parabolic equation for the complex amplitude. Double integrals appearing in the nonlocal hydrodynamic equations are calculated by means of expansions in modified Bessel functions. We use numerical simulations to ascertain the convergence of the method. If the complex amplitude equation is solved using a first order scheme for restitution coefficients near their critical values, a spurious convection arises that annihilates the complex amplitude in the part of the superlattice that is closer to the cathode. This numerical artifact disappears if the space step is appropriately reduced or we use the second-order numerical scheme.     

A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J³Y as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first-order field theories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395], to the case of second-order field theories, and we apply our theory to the Camassa–Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser–Veselov rigid body algorithm to the setting of nonlinear PDEs with second-order Lagrangians.     

Accuracy and reproducibility of automated white matter hyperintensities segmentation with lesion segmentation tool: A European multi-site 3T study

Brain vascular damage accumulate in aging and often manifest as white matter hyperintensities (WMHs) on MRI. Despite increased interest in automated methods to segment WMHs, a gold standard has not been achieved and their longitudinal reproducibility has been poorly investigated. The aim of present work is to evaluate accuracy and reproducibility of two freely available segmentation algorithms. A harmonized MRI protocol was implemented in 3T-scanners across 13 European sites, each scanning five volunteers twice (test-retest) using 2D-FLAIR. Automated segmentation was performed using Lesion segmentation tool algorithms (LST): the Lesion growth algorithm (LGA) in SPM8 and 12 and the Lesion prediction algorithm (LPA). To assess reproducibility, we applied the LST longitudinal pipeline to the LGA and LPA outputs for both the test and retest scans. We evaluated volumetric and spatial accuracy comparing LGA and LPA with manual tracing, and for reproducibility the test versus retest. Median volume difference between automated WMH and manual segmentations (mL) was −0.22[IQR = 0.50] for LGA-SPM8, −0.12[0.57] for LGA-SPM12, −0.09[0.53] for LPA, while the spatial accuracy (Dice Coefficient) was 0.29[0.31], 0.33[0.26] and 0.41[0.23], respectively. The reproducibility analysis showed a median reproducibility error of 20%[IQR = 41] for LGA-SPM8, 14% [31] for LGA-SPM12 and 10% [27] with the LPA cross-sectional pipeline. Applying the LST longitudinal pipeline, the reproducibility errors were considerably reduced (LGA: 0%[IQR = 0], p < 0.001; LPA: 0% [3], p < 0.001) compared to those derived using the cross-sectional algorithms. The DC using the longitudinal pipeline was excellent (median = 1) for LGA [IQR = 0] and LPA [0.02]. LST algorithms showed moderate accuracy and good reproducibility. Therefore, it can be used as a reliable cross-sectional and longitudinal tool in multi-site studies.     

Equivalence between the Osserman condition and the Rakić duality principle in dimension 4

We show that four-dimensional Riemannian manifolds which satisfy the Raki? duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant). Thus, since it was proved in Raki? (1999) [9] that Osserman manifolds satisfy the Raki? duality principle, both conditions are equivalent.     

Physics of novel site controlled InGaAs quantum dots on (1 1 1) oriented substrates

Recent work has shown that site-controlled dots (QD) grown on (1 1 1)B GaAs substrates, pre-patterned with tetrahedral pyramidal recesses (Baier et al., 2006) [1], (Pelucchi et al., 2007) [2], (Zhu et al., 2007) [3] are suitable for the generation of single and entangled photons (Young et al., 2009) [4]. We recently introduced InGaAs/GaAs site controlled QD structures which demonstrated record breaking spectral purity, and we showed that increasing the indium concentration of the active region allows easy tunability of the emission wavelength (Mereni et al., 2009) [5], [6]. We present here the first theoretical analysis of the emission energies and optical properties of this system as a function of QD height and In concentration. We model the dots using an 8 band k.p theory chosen to provide the best convergence and performance for structures oriented specifically along the (1 1 1) crystallographic direction.     

Stationary two-black-hole configurations: A non-existence proof

Based on the solution of a boundary problem for disconnected (Killing) horizons and the resulting violation of characteristic black hole properties, we present a non-existence proof for equilibrium configurations consisting of two aligned rotating black holes. Our discussion is principally aimed at developing the ideas of the proof and summarizing the results of two preceding papers (Neugebauer and Hennig, 2009 [2], Hennig and Neugebauer, 2011 [3]). From a mathematical point of view, this paper is a further example (Meinel et al., (2008) [29]) for the application of the inverse (“scattering”) method to a non-linear elliptic differential equation.     

Complex relativity and real solutions. IV. Perturbations of vacuum Kerr-Schild spaces

In this paper the theory of integrable double Kerr-Schild (IDKS) spaces is examined. The vacuum field equations are shown to reduce to the single equation of Plebaski and Robinson [20]. These metrics are given essentially in terms of one potentialH. First-order perturbations ofH lead to metric (gravitational) perturbations of vacuum algebraically degenerate spaces in a direct manner and give results in agreement with those of Cohen and Kegeles [6, 7, 8], Stewart [9], Teukolsky [5], Torres del Castillo [12, 13], and others. Higher-order perturbations ofH are also obtained with the view that, in the limit, these solutions should yield (new) exact vacuum solutions. The success of this construction lies in the (complex) geometric structure of IDKS spaces. This structure induces a natural splitting of the field equations which allows a potentialization of the perturbation (as well as the vacuum metric itself). It also allows massless spin 1/2 and 1 fields to be examined on the IDKS background in a similar manner.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

Pathway model,superstatistics, Tsallis statistics,and a generalized measure of entropy

The pathway model of Mathai [A pathway to matrix-variate gamma and normal densities, Linear Algebra Appl. 396 (2005) 317–328] is shown to be inferable from the maximization of a certain generalized entropy measure. This entropy is a variant of the generalized entropy of order α$\alpha$, considered in Mathai and Rathie [Basic Concepts in Information Theory and Statistics: Axiomatic Foundations and Applications, Wiley Halsted, New York and Wiley Eastern, New Delhi, 1975], and it is also associated with Shannon, Boltzmann–Gibbs, Rényi, Tsallis, and Havrda–Charvát entropies. The generalized entropy measure introduced here is also shown to have interesting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure, expected value, and information content in a scheme. Particular cases of the pathway model are shown to be Tsallis statistics [C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479–487] and superstatistics introduced by Beck and Cohen [Superstatistics, Physica A 322 (2003) 267–275]. The pathway model's connection to fractional calculus is illustrated by considering a fractional reaction equation.     

Growth of a single layer gold stripe and investigation of the preferable growth direction on reconstructed Au(1 1 1) surface using STM

We have investigated the growth of nanometer-scale gold stripes on reconstructed Au(1 1 1) surface using scanning tunneling microscopy (STM). The experiment was carried out under the conditions of ultrahigh vacuum and room temperature. The stripes were grown by the scanning motion of the STM tip over the area containing more than one step edge with the tunnel resistance less than several tens of mega ohms (MΩs). Unlike the previous reports [J.C. Heyraud, J.J. Metoris, Surf. Sci. 100 (1989) 519; V.M. Hallmark, S. Chiang, J.F. Rabolt, J.D. Swalen, R.J. Wilson, Phys. Rev. Lett. 59 (1987) 2879], we found, by directly comparing the direction of the stripes and the orientation of the underlying lattice, that the gold stripes grow preferentially along [1,−1,0] direction and its threefold symmetric directions at (1 1 1) surface of fcc structure. We also found that the scanning direction of the STM tip does not affect the direction of the stripe growth although the growth rate is suppressed remarkably when the scanning direction is close to [1,1,−2] direction of Au(1 1 1) surface.