Let g be the Lie algebra of a compact Lie group and let θ be any automorphism of g. Let k denote the fixed point subalgebra g . In this paper we present LiE programs that, for any finite dimensional complex representation π of g, give the explicit branching π|k of π on k. Cases of special interest include the cases where θ has order 2 (corresponding to compact Riemannian symmetric spaces G/K), ...

Given two graphs H1 and H2, a graph is (H1,H2)-free if it contains no induced subgraph isomorphic to or H2. Let Pt be the path on t vertices. A bull obtained from triangle with disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical (P5, bull)-free graphs.

Example 1B. Here we extend the analysis of Example 1B to censored regression models with random coefficients. Let us assume for simplicity that B (x, θ) has full-column rank q, for all θ, almost surely in x. Let V be a T × q matrix such that Q = V V ′ and V ′V = Iq. Let also U be a T × (T − q) matrix such that W = UU ′, and U ′U = IT−q. Lastly, let (μ, ν) = ( V ′Σ− 1 2 y, U ′Σ− 1 2 y ) . Then, ...

Let θ 7→ P (θ) be a set-valued mapping from R into the family of closed compact polyhedra in R. Let ξ be a R valued random variable. Many stochastic optimization problems in computer networking, system reliability, transportation, telecommunication, finance etc. can be formulated as a problem to minimize (or maximize) the probability P{ξ ∈ P (θ)} under some constraints on the decision variable ...

Let A be a principally polarized abelian variety of dimension four and let Θ ⊂ A be a symmetric theta-divisor, which we assume to be smooth. Using the Hodge structure on H(Θ) we associate to A two abelian subvarieties J(K) ⊂ J(H) of the intermediate jacobian J(Θ) of Θ of dimensions five and nine respectively. We show that J(H) is generated by the image under the Abel-Jacobi map of the family F ...

Strong convexity can be characterized in a number of ways. The following facts provide some conditions that are equivalent to Definition 1. For the 2-norm, Fact 2 means that the minimum eigenvalue of the Hessian is lower-bounded by κ. Recall that˜µ(θ) and˜µ(θ) are the gradients of˜Φ(θ) and˜Φ(θ), respectively. Since the conjugate function, ˜ Φ * , is assumed to be κ-strongly convex, we have via ...

Let A be a finite or countable alphabet and let θ be literal (anti)morphism onto A∗ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ (θ-invariant for short). We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin θ-invariant codes, maximality and complet...

Abstract. Let Θ be a smooth compact oriented manifold without boundary, imbedded in a euclidean space E, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + ǫξ where ǫ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic e...

Let G be a semisimple algebraic group defined over an algebraically closed field k whose characteristic is very good for G and not equal to 2. Suppose θ is an involution on G. We also denote the induced involution on g by θ. Let K = {g ∈ G : θ(g) = g} and let p be the −1-eigenspace of θ in g. The adjoint action of G on g induces an action of K on p and on the variety N (p), which consists of th...

We have recently been studying the case where have a training set T generated from an underlying distribution and our goal is to find some good hypothesis, with respect to the true underlying distribution, using the training set T . We now examine how to use online learning algorithms (which work on individual, arbitrary sequences) in a stochastic setting. Let us consider the training set T as ...