The information divergence of a probability measure P from an exponential family E over a nite set is de ned as in mum of the divergences of P from Q subject to Q ∈ E . All directional derivatives of the divergence from E are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The rst order conditions for P to be a maxi...

In her thesis [RB], Ranee Brylinski (then Gupta) studied the orbit structure of the projective variety of abelian subalgebras of a …xed dimension, k, in a simple Lie algebra, g, over C under its adjoint group, G. Fix a Borel subalgebra, b, of g and let B be the closed subgroup of G corresponding to b. Then the Borel …xed point theorem implies that the closed G-orbits are precisely the orbits of...

This paper investigates the solutions of two-point fuzzy boundary value problems as the form x′′ = f(t, x(t)), x(0) = A and x(l) = B, where A and B are fuzzy numbers. There are four different solutions for the problems when the lateral type of H-derivative is employed to solve the problems. As f(t, x) is a monotone function of x, these four solutions are reduced to two different solutions. As f...

The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two diffe...

With industrial modelling tools, topological structures and free form surfaces are often managed separately, and patches used for embedding are limited to dimension 2. A new approach is to combine topology structures of any dimension with embedding of same dimension, and use topological operations to modify the shape and the properties of surfaces and volumes. The use of Chains of map as topolo...

1. Non-Banach limits C(R), C∞(R) of Banach spaces C[a, b] 2. Banach completion C o (R) of C c (R) 3. Rapid-decay functions, Schwartz functions 4. Non-Fréchet colimit C∞ c (R) of Fréchet spaces 5. LF-spaces of moderate-growth functions 6. Strong operator topology 7. Generalized functions (distributions) on R 8. Tempered distributions and Fourier transforms on R 9. Test functions and Paley-Wiener...

Procedures designed to select alternatives on the basis of the results of pairwise contests between them have received much attention in literature. The particular case of tournaments has been studied in depth. More recently weak tournaments and valued generalizations thereof have been investigated. The purpose of this paper is to investigate to what extent these choice procedures may be meanin...

Recently, Huang and Ng presented second-order sufficient conditions for error bounds of continuous and Gâteaux differentiable functions in Banach spaces. Wu and Ye dropped the assumption of Huang and Ng on Gâteaux differentiability but required the space to be a Hilbert space. We carry on this research in two directions. First we extend Wu and Ye’s result to some non-Hilbert spaces; second, sam...

where [g]j,l=1 defines a C ∞-smooth Riemannian metric and b = (b1, ..., bm) and q are, correspondingly, C∞-smooth complex-valued 1-form and function on M . σ is a C∞-smooth complex-valued function on ∂M and ∂ν stands for the normal derivative. Let Rλ be the resolvent of (1), (2) which is meromorphic for λ ∈ C (see Sect. 3 and [1]) and let Rλ(x, y) be its Schwartz kernel. A natural analog of the...

The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing literature on the Kac–Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N = 2 supergravity the simple al...