On Some Expectation and Derivative Operators Related to Integral Representations of Random Variables with Respect to a PII Process

Given a process with independent increments X (not necessarily a martingale) and a large class of square integrable r.v. H = f(X _T ), f being the Fourier transform of a finite measure μ, we provide a direct expression for Kunita-Watanabe and Föllmer-Schweizer decompositions of H. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of X. We also evaluate the expression for the variance optimal error when hedging the claim H with underlying process X. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.     

A geometrical take on invariants of low-dimensional manifolds found by integration

An elementary geometrical proof of the fact that the Euler characteristic is the only topological invariant of a surface that can be found by integration (using Gauss–Bonnet) is given. A similar method is also applied to three-dimensional manifolds.     

Involutions Fixing Products of Projective Spaces

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Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.     

Isothermal production from a supercritical gas container: A new characteristic curve

When emptying isothermally a supercritical gas container, as already well known, the cumulative production varies linearly with the pressure to compressibility factor ratio. However, the instantaneous production (that is to say, the instantaneous standard volume of the released gas) varies in a totally different manner when looked as a function of pressure; in particular, at any temperature above the critical, this production goes through a maximum which is more and more pronounced when approaching the critical point. The locus of these maxima defines in pressure–temperature coordinates a curve which starts at the critical point, goes through a maximum and ends at the Boyle temperature. This new characteristic curve is analyzed and some rather surprising features are presented in this paper. For pure component systems, this curve coincides with the locus of maximum density fluctuations.     

On a characteristic equation of well-poised Bailey chains

In this paper, we find by inverse technique two solutions of a system of linear equations which together serve as a sufficient and necessary condition for well-poised Bailey chains. Using these two solutions, we establish a new well-poised Bailey chain, two usual Bailey chains, and a well-poised extension of Bailey’s lemma. Their applications to q-series are also investigated. X. Ma was supported by Natural Science Foundation of China (No. 10771156).     

Multistep characteristic method for incompressible flow in porous media

In this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method and a five-point finite difference scheme for space discretization. We prove that the convergence is of order O(h²+(Δt)²), which is in contrast to the convergence of order O(h+Δt) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.     

Full 3D finite element Cosserat formulation with application in layered structures

In this paper, a full three-dimensional (3D) finite element Cosserat formulation is developed within the principles of continuum mechanics in the small deformation framework. The developed finite element formulation is general; however, the proposed constitutive laws incorporate the effect of the internal length parameter of 3D layered continua. The extension of the existing two-dimensional (2D) Cosserat formulation to the 3D framework is novel and is consistent with plate theory which can be considered as the 3D version of beam theory. The results demonstrate a high level of consistency with the analytical solutions predicted by plate theory as well as predictions by alternative numerical techniques such as the discrete element method.