Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

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.     

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.While writing this paper, the authors were partially supported by grants MTM2004-03101 and TIN2004-07933-C03-02 of the Spanish Ministry of Education and Science, including FEDER funds of the European Union.     

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.     

Junctions of singularly degenerating domains with different limit dimensions 1

The asymptotic series for solutions of the mixed boundary-value problem for the Poisson equation in a domain, which is a junction of singularly degenerating domains, are constructed. In this paper, which is the first part of the publication, the three-dimensional problem (“wheel hub with spokes”) and the analogous two-dimensional problems are considered. The methods of matched and compound asymptotic expansions are used. It is shown that a special self-adjoint extension of the operator of the limit problem in the “hub” supplied by the straight-line segments (“limits of spokes”) can be chosen as an asymptotical model of the problem in question; the extension parameters are to be some integral characteristics of the boundary-layer problems. Bibliography: 39 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 18, pp. 3–78, 1995.     

Conditional lexicographic orders in constraint satisfaction problems

The lexicographically-ordered CSP (“lexicographic CSP” or “LO-CSP” for short) combines a simple representation of preferences with the feasibility constraints of ordinary CSPs. Preferences are defined by a total ordering across all assignments, such that a change in assignment to a given variable is more important than any change in assignment to any less important variable. In this paper, we show how this representation can be extended to handle conditional preferences in two ways. In the first, for each conditional preference relation, the parents have higher priority than the children in the original lexicographic ordering. In the second, the relation between parents and children need not correspond to the importance ordering of variables. In this case, by obviating the “overwhelming advantage” effect with respect to the original variables and values, the representational capacity is significantly enhanced. For problems of the first type, any of the algorithms originally devised for ordinary LO-CSPs can also be used when some of the domain orderings are dependent on assignments to “parent” variables. For problems of the second type, algorithms based on lexical orders can be used if the representation is augmented by variables and constraints that link preference orders to assignments. In addition, the branch-and-bound algorithm originally devised for ordinary LO-CSPs can be extended to handle CSPs with conditional domain orderings.     

Estimation of the Defect Status on Visual Field Longitudinal Data

In this paper, we show the use of Multivariate Time Series models, Markov Random Fields and Bayesian methodologies to solve an applied ophthalmological problem related to the study of glaucoma. Glaucoma is a very serious and widely extended eye disease characterized by a gradual decrease in the intensity of the patient’s sight. It is not, however, homogeneous over all the visual field, and starts at one or several sites and gradually spreads to nearby sites. Measurement of the patient’s “seeing threshold” at different points in the visual field is an important diagnostic tool for glaucoma and other diseases. It results in a map with 52 numerical values, each of which represents the level of intensity perceived by the patient at that site, and ranges from 0 (complete blindness) to 35 (exceptional vision). Additionally a “defect status” variable can be attached at each site in the visual field. This variable would indicate whether the site is normal or defective. Using Bayesian methodologies, the “defect status” process can be regarded as a parameter of the probability distribution of the thresholds and can be estimated as the maximum of its posterior distribution. The stochastic model assumed for the observed “threshold”, given the “defect status”, is a first order autoregressive integrated model (VARI(1,1)) in time, with first order homogeneous spatial correlation. The defect status is modeled by using a Spatiotemporal Autologistic Model with non-homogeneous spatial dependence. This dependence assumes that the propagation of the lesions follows the directions taken by the nerve fibers. MCMC methods are used to jointly estimate the defect status, and parameters and hyperparameters of the model.     

On Galilean connections and the first jet bundle

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations — sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse — the “fundamental theorem” — that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

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.     

Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

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.     

Why vagueness is a mystery

This paper considers two “mysteries” having to do with vagueness. The first pertains to existence. An argument is presented for the following conclusion: there are possible cases in which ‘There exists something that is F’ is of indeterminate truth-value and with respect to which it is not assertable that there are borderline-cases of “being F.” It is contended that we have no conception of vagueness that makes this result intelligible. The second mystery has to do with “ordinary” vague predicates, such as ‘tall’. An argument is presented for the conclusion that although there are people who are “tall to degree 1”—definitely tall, tall without qualification—, no greatest lower bound can be assigned to the set of numbers n such that a man who is n centimeters tall is tall to degree 1. But, since this set is bounded from below, this result seems to contradict a well-known property of the real numbers.     

Study on “double dawn”

The ancient record, “During the first year of King Yi, the day dawned twice at Zheng”, has provided important clues to early Chinese chronicles. The astronomical conditions and visible area distributions related to such a “double dawn” event are discussed, and the precision and current problems in the calculations of ancient astronomical phenomena are shown. On such a basis, all the solar eclipses from 1000 BC to 840 BC are calculated and their associated “double dawn” features investigated. The conclusion that the “double dawn” was a solar eclipse occurring on April 21st, 899 BC is corfirmed to be the most reasonable. Project supported by the Xia-Shang-Zhou Chronology Project and Director Foundation of the Chinese Academy of Sciences.     

Explorative and dynamic visualization of data in virtual reality

Summary  A software system has been developed for the study of dynamic glyph visualizations in the context of Visual Data Mining in Virtual Reality. The system uses parallel processing to calculate data visualizations in real-time, with real-time interaction and dynamic changes to the view. The system allows morphing between different visualizations, the use of dynamic features like “vibrations” and “rotations” of thousands of objects individually, and dynamic visualization, where the influence of any variable of a dataset with a “reasonable” distribution, can be shown as a dynamic development. It appears that these facilities for dynamic data visualization have a very promising potential, but their optimal use will depend on further developments in the context of their individual practical application.     

Human or Cylon? Group testing on ‘Battlestar Galactica’

As in the real world, statistics could have played a significant role in solving the sci-fi problem in “Battlestar Galactica.” However, the consequences for implementing these procedures might have prematurely stifled fans’ enthusiasm for the show because the humanoid Cylons would have been identified earlier. While the series ended in 2009, a prequel, “Caprica,” premieres in 2010. “Caprica” will investigate topics such as how Cylons were first developed by humans. Of course, this makes one wonder if the use of statistical science could have played a role in preventing the Cylon attack on the humans in the first place. One can only hope the producers will ask a statistician to serve as a consultant in the writing of the new show.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of Cⁿ and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

Transplantation harmonique,transplantation par modules,et théorèmes isopérimétriques

Summary The “harmonic transplantation” allows to extend some isoperimetric theorems, so far proved by conformal mapping, to higher connectivity and to higher dimensions; for the first eigenvalue λ₁ of a membrane, it again can give only upper bounds.—The “transplantation by moduli” is much more flexible; for example, it leads to a simple one-dimensional interpretation of the Rayleigh-Faber-Krahn theorem.