Nondegenerate Monge–Ampère Structures in Dimension 6

We define a nondegenerate Monge–Ampère structure on a six-dimensional manifold as a pair (, ), such that is a symplectic form and is a 3-differential form which satisfies =0 and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi–Yau structure and we study its integrability from the point of view of Monge–Ampère operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge–Ampère equation in dimension 4. We study from this point of view the example of the Stenzel metric on T*S ³.     

The Monge–Ampère equation: Various forms and numerical solution

We present three novel forms of the Monge–Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge–Ampère problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge–Ampère equation is a sum of such distributions.     

Complex solutions of Monge–Ampère equations

We describe a method to reduce partial differential equations of Monge–Ampère type in four variables to complex partial differential equations in two variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge–Ampère equations and the Plebanski equations.     

Information Processing in the Brain as Optimal Entropy Transport: A Theoretical Approach

We consider brain activity from an information theoretic perspective. We analyze the information processing in the brain, considering the optimality of Shannon entropy transport using the Monge–Kantorovich framework. It is proposed that some of these processes satisfy an optimal transport of informational entropy condition. This optimality condition allows us to derive an equation of the Monge–Ampère type for the information flow that accounts for the branching structure of neurons via the linearization of this equation. Based on this fact, we discuss a version of Murray’s law in this context.     

Type II hidden symmetries of the second heavenly equation

A Type II hidden symmetry of the non-linear second heavenly equation in gravitational physics is identified. Its provenance from other partial differential equations is studied. Two reductions of the second heavenly equation produce the Monge–Ampère equation in similarity variables and new analytic solutions are possible.     

Minimal switching voltage for magnetization reversals in asymmetric nanorings

Micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation are presented to study spin-polarized current induced magnetization switching in asymmetric nanoring-shaped magnetic tunnel junctions. The results show that in a nanoring with an intermediate eccentric distance S, the critical switching voltage V_C reaches the minimum whose magnitude is less than half of the voltage for a symmetric nanoring. In addition to the spin-transfer torque, the current induced Ampère field is found to play a crucial part in the switching process. Analysis is given to explain the existence of such an energy valley that leads to the minimal voltage.     

Transient radiation and conduction heat transfer inside a plane-parallel participating gray medium with boundaries having different reflecting characteristics

A hybrid ray-tracing method is developed for the solution to the radiative transfer in a plane-parallel participating medium having one specular surface and another diffuse surface. By this method, radiative transfer coefficients (RTCs) for specular–diffuse (S–D) surfaces are deduced. The medium surfaces are considered to be semitransparent. The effects of convection–radiation parameter, conduction–radiation parameter and refractive index on transient coupled heat transfer are investigated. Results show that the temperature curves of the medium having S–D surfaces is higher than those of the medium having S–S surfaces (two specular surfaces); the total heat flux at steady state for the S–D surfaces is lower than that for the S–S surfaces.     

Mössbauer effect study of antiferromagnetic Fe–Mn–Si alloys

Antiferromagnetic Fe–30Mn–Si alloys containing 2.0–8.7 at.% Si are known to exhibit several attractive physical properties at Néel temperatures which render them candidate materials for functional alloys applications. The Néel transitions and anomalous transport phenomena have been studied extensively in a wide temperature range. In the present work, the hyperfine interactions are studied by Mössbauer spectroscopy measured at temperatures 95–623 K. It is found that the Mössbauer spectra are singlets at temperatures above the Néel temperature and doublets below the Néel temperature. The alloys have a small hyperfine field around the Fe nuclei below the Néel temperature and the hyperfine field increases linearly with increasing silicon concentration. This can be explained by the presence of a localised net magnetic moment on the Fe nuclei which is induced by the silicon atoms. A decrease in isomer shift with increasing silicon concentration is observed and this can be accounted for by the change in the occupation of the Fe 3d shell. There is a small quadrupole splitting, it increases with increasing silicon concentration, and is consistent with the lattice shrinking and magnetostriction.     

Algorithmic Reduction of Poincaré–Dulac Normal Forms and Lie Algebraic Structure

The Poincaré–Dulac normal form of a given resonant system is in general nonunique; given a specific normal form, one would like to further reduce it to a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The algorithm can be applied under some condition, nongeneric but often met in applications. When applicable, it is only necessary to solve linear equations, and is more powerful than the one proposed in previous work by the same author [Lett. Math. Phys. 42 (1999), 103–114; and Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 461–514].     

The difference between Newtonian and relativistic forces

This paper discusses a new turn in the 148-year old electrodynamic force law controversy between the 1822 Ampère force law of the Newtonian electrodynamics and Grassmann's 1845 law which has become the electrodynamic force law of relativistic electromagnetism. Faced with the infallibility of Ampère's empirical law, defenders of relativity theory now argue that Ampère's law is equivalent to the relativistic law. This paper demonstrates that, far from being equivalent, the laws require two different mechanics of solid bodies, disagree on internally generated stresses, and predict different force distributions.     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

TDPAC calculations for mixed SDW and Zeeman hyperfine interactions

The time differential perturbed angular correlation curves have been calculated for the hyperfine field distribution arising from the superposition of an isotropic system of spin density waves and of an external magnetic field. The results of these calculations have been applied to theCr–Ta system. The agreement between the experiments and the calculations is fair. This shows that the spin density waves of chromium are not significantly polarized by a 12 kOe magnetic field. However a large negative shift of the Zeeman interaction is observed in the liquid nitrogen experiment: H/H=–0.18±0.02. The negative sign of the shift shows that the hyperfine field is antiparallel to the local magnetization.

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Résumé Les courbes de corrélation angulaire différentielle ont été calculées pour une distribution de champs hyperfins résultant de la superposition des interactions d'un noyau avec un système d'ondes de densité de spin d'une part et avec un champ magnétique extérieur d'autre part. Les résultats de ces calculs ont été appliqués à l'alliageCr–Ta. L'accord entre l'expérience et les calculs est bon ce qui montre que les ondes de densité de spin ne sont pas polarisées par un champ magnétique de 12 kOe. Cependant un déplacement de l'interaction de Zeeman de –0.18±0.02 observé à 77 K permet d'affirmér que le champ hyperfin est opposé à l'aimantation locale.

     

On Covariant Phase Space and the Variational Bicomplex

The notion of a phase space in classical mechanics is well known. The extension of this concept to field theory however, is a challenging endeavor, and over the years numerous proposals for such a generalization have appeared in the literature. In this paper We review a Hamiltonian formulation of Lagrangian field theory based on an extension to infinite dimensions of J.-M. Souriau's symplectic approach to mechanics. Following G. Zuckerman, we state our results in terms of the modern geometric theory of differential equations and the variational bicomplex. As an elementary example, we construct a phase space for the Monge–Ampere equation.     

Uniqueness of rotating charged black holes in five-dimensional minimal gauged supergravity

We study a five-dimensional spacetime admitting, in the presence of torsion, a non-degenerate conformal Killing–Yano 2-form which is closed with respect to both the usual exterior differentiation and the exterior differentiation with torsion. Furthermore, assuming that the torsion is closed and co-closed with respect to the exterior differentiation with torsion, we prove that such a spacetime is the only spacetime given by the Chong–Cvetič–Lü–Pope solution for stationary, rotating charged black holes with two independent angular momenta in five-dimensional minimal gauged supergravity.     

Analysis of semiconductor surfaces and interfaces

Recent development in the experimental and theoretical analysis of semiconductor surfaces is described. Special attention is given to the Secondary Ion Mass Spectroscopy technique and to its use in the ultrasensitive elemental analysis of semiconductors. Applications to III–V compounds are described.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.     

Schlesinger Transformations for Linearisable Equations

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every Itô algebra with a quotient identity has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy–Khinchin Theorem in the classical case. Two basic examples of noncommutative Itô finite group algebras are considered.     

Excitation of an ionospheric Alfvén resonator by a plasma wave discharge

We present a theoretical interpretation of the excitation of an ionospheric Alfvén resonator (IAR) by an electron beam propagated along the external magnetic field. The growth rate of generated Alfvén waves is found and the IAR excitation efficiency is estimated. The theoretical results are in qualitative agreement with the experimental data.Lobachevsky State University, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 3-4, pp. 312–317, March–April, 1995.     

On the integrability of symplectic Monge–Ampère equations

Let u$u$ be a function of n$n$ independent variables x¹,…,xⁿ$x^1,\dots ,x^n$, and let U=(_uij)$U=\left(u_{ij}\right)$ be the Hessian matrix of u$u$. The symplectic Monge–Ampère equation is defined as a linear relation among all possible minors of U$U$. Particular examples include the equation detU=1$detU=1$ governing improper affine spheres and the so-called heavenly equation, _u13u24−_u23u14=1$u_{13}{u}_{24}-u_{23}{u}_{14}=1$, describing self-dual Ricci-flat 4$4$-manifolds. In this paper we classify integrable symplectic Monge–Ampère equations in four dimensions (for n=3$n=3$ the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(_uij)=0$F\left(u_{ij}\right)=0$ in more than three dimensions is necessarily of the symplectic Monge–Ampère type.     

Two stage laser stabilization using external cavities: Application to a 10 μm CO2 laser locked to an OsO4 transition

A cw carbon dioxide laser, operating on the 10 m R(0) transition (28.832 THz), was stabilized by locking its frequency to an evacuated Fabry–Pérot cavity. The Fabry–Pérot cavity was in turn stabilized to the saturated absorption resonance of the Q(15) line of ¹⁸⁸OsO₄ , contained in a second Fabry–Pérot cavity. Cesium-clock referenced, frequency chain measurements of the resultant line frequency are presented together with detailed studies of the line shift sensitivities to various operating parameters. PACS 42.62.Eh; 42.62.F; 33.20.Ea