Concavity of the Lagrangian for quasi-periodic orbits

Percival introduced a “Lagrangian” for finding quasi-periodic orbits. For suitable area preserving mappings, we show that Percival's “Lagrangian” is strictly concave with respect to an appropriate affine structure on its domain. Consequently, the “Lagrangian” admits a unique maximum in the case of irrational frequencies. Supported by NSF grant MCS 79-02017     

On the reduction to a complete class in multiple decision problems (2)

Summary As a criterion for the reduction to a complete class of decision rule in case where actions, samples and states are finite in number, “regret-relief ratio” criterion and “incremental loss-gain ratio” criterion were introduced in 2-state of nature case [2]. In this paper, “generalized regret-relief ratio” criterion ink-state of nature case is introduced as an extension of “regret-relief ratio” criterion and its usefulness is shown with an example. The Institute of Statistical Mathematics     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Remarks on a principle of localization

We prove that the Brauer class of a crossed product is a sum of symbols iff its “local” components are. Analogously we show that a solution of the “Goldie rank conjecture” would follow from the “local” statements; an extension of a result of Cliff-Sehgal is an easy corollary.     

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

The semantics of knowledge attributions

The basic idea of conversational contextualism is that knowledge attributions are context sensitive in that a given knowledge attribution may be true if made in one context but false if made in another, owing to differences in the attributors’ conversational contexts. Moreover, the context sensitivity involved is traced back to the context sensitivity of the word “know,” which, in turn, is commonly modelled on the case either of genuine indexicals such as “I” or “here” or of comparative adjectives such as “tall” or “rich.” But contextualism faces various problems. I argue that in order to solve these problems we need to look for another account of the context sensitivity involved in knowledge attributions and I sketch an alternative proposal.     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

A problem with directional derivative in the theory of galvanomagnetic effects

We construct an asymptotics of the solution the Laplace equation in a “long” rectangle with the directional derivative given on its “long sides” and Dirichlet data on its “short sides.” By using the asymptotics, we calculate one of the integral characteristics, namely, the magnetoresistance. We obtain new formulas for the low-magnetic field magnetoresistance. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 520–532, April, 1999.     

On approximation of discontinuous solutions to the Buckley-Leverett equation

In this paper, the Lax-Wendroff and “cabaret” schemes for the Buckley-Leverett equation are studied. It is shown that these schemes represent unstable solutions. The choice of an unstable solution depends on the Courant number only. A finite-element version of the “cabaret” scheme is given.     

On the concept of decision aiding process: an operational perspective

The paper presents the concept of decision aiding process as an extension of the decision process. The aim of the paper is to analyse the type of activities occurring between a “client” and an “analyst” both engaged in a decision process. The decision aiding process is analysed both under a cognitive point of view and an operational point of view: i.e. considering the “products”, or cognitive artifacts the process will deliver at the end. Finally the decision aiding process is considered as a reasoning process for which the update and revision problems hold.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

“Almost quasistationary” approximation for the problem of solidification front stability

“Almost quasistationary” approximation is suggested for the investigation of the problem of solidification front stability. It is appropriate for the initial stage of the process when sizes of particles are sufficiently small. The cases of “sphere like” and “cylinder-like” nuclei are considered. Capillary forces are taken into account.     

The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS Model

An analysis of the RSS model in mathematical economics involves the study of an infinite-horizon variational problem in discrete time. Under the assumption that the felicity function is upper semicontinuous and “supported” at the value of the maximally-sustainable level of a production good, we report a generalization of results on the equivalence, existence and asymptotic convergence of optimal trajectories in this model. We consider two parametric specifications, and under the second, identify a “symmetry” condition on the zeroes of a “discrepancy function” underlying the objective function that proves to be necessary and sufficient for the asymptotic convergence of good programs. With a concave objective function, as is standard in the antecedent literature, we show that the symmetry condition reduces to an equivalent “non-interiority” condition.     

On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system. Research done in the University of Michigan-Ann Arbor, submission from Temasek Laboratories, National University of Singapore.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

KAM Stability for a three-body problem of the Solar system

A new (iso-energetic) KAM method is tested on a specific three-body problem “extracted” from the Solar system (Sun-Jupiter + asteroid 12 Victoria). Analytical results in agreement with the observed data are established. This paper is a concise presentation of [2]. Supported by the MIUR projects: “Dynamical Systems: Classical, Quantum, Stochastic” and “Variational Methods and Nonlinear Differential Equations” Received: February 3, 2004     

Orthogonal and bounded orthogonal spectral representations

The concept of an orthogonal spectral representation (OTSR) of a Hilbert spaceH relative to a spectral measureE(.) is introduced and it is shown that every Hilbert space admits an OTSR relative to a given spectral measure. Apart from the various results obtained about OTSRs, the principal result of Allan Brown (1974) is deduced as an easy consequence of this study. A new complete system of unitary invariants called the “equivalence of OTSRs”, is given for spectral measures. Two special types of OTSRs called “BOTSR” and “COBOTSR” are introduced and characterized respectively in terms of the “GCGS-property” and “CGS-property” of the associated spectral measure. Various complete systems of unitary invariants are given for spectral measures with the GCGS-property. Finally, the Wecken-Plesner-Rohlin theorem on hermitian operators with simple spectra is generalized to arbitrary spectral measures.     

Asymptotically optimal choice of the smoothing parameter in a nonparametric kernel estimator for a probability density

This paper presents a method of estimation of an “optimal” smoothing parameter (window width) in kernel estimators for a probability density. The obtained estimator is calculated directly from observations. By “optimal” smoothing parameters we mean those parameters which minimize the mean integral square error (MISE) or the integral square error (ISE) of approximation of an unknown density by the kernel estimator. It is shown that the asymptotic “optimality” properties of the proposed estimator correspond (with respect to the order) to those of the well-known cross-validation procedure [1, 2]. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 67–80, Perm, 1990.