Series solution of the differential equation determining the nth-shell one-electron density of a bare Coulomb problem in quantum physics

The differential equation determining the nth-shell one-electron density of a bare Coulomb problem is analytically investigated by using the differential function method and extended homogeneous balance idea. New series solution of the differential equation has been obtained in quantum physics for the first time. The explicit series analytical expression for the mathematical characteristic of the bare Coulomb problem in the density-functional theory can further describe the dynamical behaviors of the nth-shell one-electron density.     

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Positivity of the K-entropy on non-abelian K-flows

A non-commutative extension of certain aspects of classical probability theory is presented in such a manner that the notion of Kolmogorov entropy can be extended to a large class of non-classical dynamical systems. In particular, the generalized K-entropy so defined is shown to be strictly positive on the class of non-abelian K-flows.     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.     

On the topology of spherically symmetric space-times

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below.     

A Survey of Total Positivity

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmüller theory. Leonardo da Vinci lecture held on May 28, 2007 Received: June 2007     

Positivity of the principal minors of a nonsingularM-matrix

One gives an elementary proof of the known fact that if the principal minors of an M -matrix with a positive determinant are nonnegative, then they are positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, p. 88, 1981.In conclusion, we note that this problem was suggested to the author by V. P. If'in.     

Positivity of the schur-weyl invariants of projective varieties

We prove the positivity of Schur-Weyl invariants of projective varieties under certain conditions.     

Positivity of quasi-local mass II

We prove the following stronger version of the positivity of quasi-local energy (mass) stated by Liu and Yau: the quasi-local energy of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface and the boundary of the spacelike hypersurface is connected.

     

Positivity of the Density for Rough Differential Equations

Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive. In this paper we study this problem from the viewpoint of Aida–Kusuoka–Stroock’s general theory.

     

Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤ^d. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ T^d (where T^d is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ T^d. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 381–387, December, 2007.     

Positivity and boundedness of trigonometric sums

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.