Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

We consider some subsets of the Euclidean plane R ², having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on R ².     

Approximate Factorization in Shallow Water Applications

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the right-hand side of the resulting system of ordinary differential equations can be split into terms f ₁, f ₂, f ₃ and f ₄, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term f ₄ is nonstiff and that the function f ₃ corresponding with the vertical spatial direction is much more stiff than the functions f ₁ and f ₂ corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported.     

Zero-sum problems with subgroup weights

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider ℤ _p ^d for a general d and subgroups of Z* _p as weights.     

An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n of P ²=P _k ² ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position. This paper was written while the author was member of C.N.R., Sez. 3 of G.N.S.A.G.A. and was supported by M.P.I. funds     

Minimum Convex Partition of Polygonal Domains by Guillotine Cuts

Let P be a multiply connected polygonal domain with holes (possibly degenerate) and let F be a certain family of directions in the plane. We give a formula for the minimum number of convex pieces in which P is partitioned with the help of guillotine cuts along the directions of F. Received January 5, 1996, and in revised form December 4, 1996.     

Conformally Flat Hypersurfaces of E4

We consider 3-dimensional conformally flat hypersurfaces of E ⁴ with 2 different principal curvatures such that the coordinate directions are principal directions. We describe explicitly those which allow an immersion with constant mean curvature. They are shown to be in close correspondence with solutions of the nonlinear integrable sine-Gordon and sinh-Gordon equations. Conversely, this provides a geometrical characterization for this particular class of conformally flat hypersurfaces of E ⁴.     

FORMULE DE TRACE SEMI-CLASSIQUE SUR UNE VARIETE DE DIMENSION 3 AVEC UN POTENTIEL DE DIRAC

ABSTRACT

On a riemannian manifold of dimension 3, it is possible to change the laplacian Δ at a single point p. This procedure gives the so-called Dirac potential, and we study the associated wave equation. We first show that the propagator can be written as a sum in which each operator takes into account n diffractions at p. We then show that the curve obtained by following, one after another, n geodesic segments emanating from, and returning to p, gives a singularity in the trace formula. The principal part of this singularity is also computed.     

Some calculations on the conditional densities of well-admissible measures on linear spaces

When a probability measurem on a topological vector spaceE is well-admissible in a directionk E, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.     

Influence Analysis for Linear Measurement Error Models

In this paper, we present a unified diagnostic method for linear measurement error models based upon the corrected likelihood of Nakamura (1990, Biometrika, 77, 127–137). Both global influence and local influence are discussed. The case-deletion model and mean-shift outlier model are considered, and they are shown to be approximately equivalent. Several diagnostic measures are derived and discussed. It is found that they can be written in terms of the residual and leverage measure. Some existing results are improved. Numerical example illustrates that our method is useful for diagnosing influential observations.     

Applied algebraic dynamics

A review of the book Applied Algebraic Dynamics (De Gruyter Expositions in Mathematics 49, (2009)), written by Vladimir Anashin and Andrei Khrennikov, is presented.     

ASYMPTOTIC LIMITS FOR QUANTUM TRAJECTORY MODELS

ABSTRACT

The semi-classical and the inviscid limit in quantum trajectory models given by a one-dimensional steady-state hydrodynamic system for quantum fluids are rigorously performed. The model consists of the momentum equation for the particle density in a bounded domain, with prescribed current density, and the Poisson equation for the electrostatic potential. The momentum equation can be written as a dispersive third-order differential equation which may include viscous terms. It is shown that the semi-classical and inviscid limit commute for sufficiently small data (i.e. current density) corresponding to subsonic states, where the inviscid non-dispersive solution is regular. In addition, we show that these limits do not commute in general. The proofs are based on a reformulation of the problem as a singular second-order elliptic system and on elliptic and W ^1,1 estimates.     

Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.     

Geometric Tomography of Convex Cones

The parallel X-ray of a convex set K⊂ℝ ⁿ in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex polyhedral cones in ℝ³ are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal.     

Utility Indifference Pricing: A Time Consistent Approach

Abstract

This article considers the optimal portfolio selection problem in a dynamic multi-period stochastic framework with regime switching. The risk preferences are of exponential (CARA) type with an absolute coefficient of risk aversion that changes with the regime. The market model is incomplete and there are two risky assets: tradable and non-tradable. In this context, the optimal investment strategies are time inconsistent. Consequently, the subgame perfect equilibrium strategies are considered. The utility indifference ask price of a contingent claim written on the risky assets is computed through an indifference valuation algorithm. By running numerical experiments, we examine how this price varies in response to changes in model parameters.     

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

On Power Series having Sections with Multiply Positive Coefficients

Pólya's theorem of 1913 says that if all sections of power series have real negative zeros only, then the series converges in the whole complex plane and its sum satisfies a certain growth condition. Here we show that the assertion of Pólya's theorem remains valid for a much larger class of formal power series and, moreover, a better growth estimate holds.     

The Number of Directions Determined by Points in the Three-Dimensional Euclidean Space

Abstract. Let X be a set of n points in the three-dimensional Euclidean space such that no three points in X are on the same line and there is no plane containing all points in X . An old conjecture states that pairs of points in X determine at least 2n-3 directions. We prove the weaker result that X determines at least 1.75n-2 directions.     

Symmetries,Lagrangian and Conservation Laws for the Maxwell Equations

The evolution equations of Maxwell’s equations has a Lagrangian written in terms of the electric E and magnetic H fields, but admit neither Lorentz nor conformal transformations. The additional equations ∇⋅E=0, ∇⋅H=0 guarantee the Lorentz and conformal invariance, but the resulting system is overdetermined, and hence does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties and provide a correspondence between symmetries and conservation laws using the Lagrangian for the evolutionary part of Maxwell’s equations.