The information divergence of a probability measure P from an exponential family E over a nite set is deened as innmum of the divergences of P from Q subject to Q in E. For convex exponential families the local maximizers of this function of P are found. General exponential family E of dimension d is enlarged to an exponential family E of the dimension at most 3d + 2 such that the local maximiz...

We define a ternary product and more generally a (2k +1)-ary product on the vector space T p q (E) of tensors of type (p, q) that is contravariant of order p, covariant of order q and total order (p + q). This product is totally associative up to a permutation sk of order k (we call this property a sk-totally associativity). When p = 2 and q = 1, we obtain a (2k + 1)-ary product on the space of...

Consider in a real Hilbert space H the differential equation (E) : p(t)u′′(t) + q(t)u′(t) ∈ Au(t) + f(t), for a.a. t ∈ R+ = [0,∞), with the condition u(0) = x ∈ D(A), where A : D(A) ⊂ H → H is a maximal monotone operator, with [0, 0] ∈ A (or, more generally, 0 ∈ R(A)); p, q ∈ L∞(R+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0; and f : R+ → H is a given function. Recall that eq...

Proof of Lemma 1: This retailer would see (y/L) of the total demand, hence the cumulative distribution function of its demand is H(s) = F(sL/y) where F() gives the cumulative distribution function for total market demand, x. Differentiating H(s) gives the required result. () parts) by on (integrati) () () () () () () () () () () () 0 , max(0 0 0 0 0 0 ds s F L y ds s f s y Lq L y ds s f s y Lq ...

Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic Zp-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the ...

An extension of tree automata framework, called equational tree automata, is presented. This theory is useful to deal with uni cation modulo equational rewriting. In the manuscript, we demonstrate how equational tree automata can be applied to uni cation problems. 1 Equational Tree Languages Uni cation modulo equational theory is a central topic in automated reasoning. Tree automata are the pow...

We follow the notation of [7,10]. In particular, if E[τ ] is a locally convex space (in short l.c.s.), σ(E, E∗), μ(E,E∗) and β(E,E∗) will denote, respectively, the weak, Mackey and strong topology corresponding to the dual pair 〈E, E∗〉. If U is a neighbourhood of 0 in E[τ ], EU denotes the Banach space associated to U and φU : E → EU denotes the corresponding quotient map. Definition 1. Let E[τ...

This article deals with some special cases which are extension of the strong summability of Fourier series with constant factor. We obtain a new equivalent form of inequalities A 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) φ (z) 2 dρ r/2 dθ ≤ B 2π 0 φ(e iθ) r dθ, (1) 2π 0 1 0 (1 − ρ) q−1 φ (z) q dρ r/q dθ ≤ C 2π 0 φ(e iθ) r dθ, (2) D 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) p−1 φ (z) p dρ r/p dθ.