For a set A of nonnegative integers, let R2(A,n) and R3(A,n) denote the number solutions to n=a+a′ with a,a′∈A, a<a′ a≤a′, respectively. In this paper, we prove that, if A⊆N N is positive integer such that R2(A,n)=R2(N∖A,n) for all n≥2N−1, then any θ 0<θ<2log⁡2−log⁡342log⁡2−9log⁡3, cardinality n≤x R2(A,n)=n8+O(n1−θ) equals x+O(x1−δ) some constant δ. The similar result holds R3(A,n). These impro...

It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found. 1. Let g be a semisimple complex Lie algebra, let G be the adjoint group of g, and let θ ∈ Aut g be an element of order 2. We set k := {x ∈ g | θ(x) = x}, p := {x ∈ g | θ(x) = −x...

0.1. Let K be a nonarchimedean local field with a residue field of cardinal q. Let G(K) be the group of K-rational points of a connected, adjoint simple algebraic group G defined over K which becomes split over an unramified extension of K. Let U(G(K)) be the set of isomorphism classes of unipotent representations of G(K) (see [L4, 1.21]). Let G be a simply connected almost simple algebraic gro...

In this appendix we show that 1 2 ∆ F (θ)∆ is a second order Taylor approximation of D KL (p(θ)p(θ + ∆)). First, let g q (θ) :=D KL (qp(θ)) = ω∈Ω q(ω) ln q(ω) p(ω|θ). We begin by deriving equations for the Jacobian and Hessian of g q at θ: ∂g q (θ) ∂θ = ω∈Ω q(ω) p(ω|θ) q(ω) ∂ ∂θ q(ω) p(ω|θ) = ω∈Ω q(ω) p(ω|θ) q(ω) −q(ω) ∂p(ω|θ) ∂θ p(ω|θ) 2 = ω∈Ω − q(ω) p(ω|θ) ∂p(ω|θ) ∂θ , (4) and so: ∂ 2 g q (θ)...

Let A be a unital simple C∗-algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2 : C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ C(X) and every trace τ of A.Adopting a theorem of Tomiyama, we introduce a notion of approximate con...

Let us assume that x ∈ R,θ ∈ R and fθ(·) : R → R is a vector valued function parameterized by θ. Moreover, due to the assumption of separability fθ(xt−1) = l(xt−1) h(θ), where we assume that l(·) : R → Rm×d and h(·) : R → R and m is an arbitrary constant. The stochastic perturbance will have the logpolynomial density p(vt) ∝ exp(Λ1vt + v t Λ2vt + . . . ) Let us analyze the case of p(vt) ∝ exp(Λ...

Let G be a graph of order n and size m. A labeling of G is a bijective mapping θ : V(G) → {1, 2, . . . , n}, and we call Θ(G) the set of all labelings of G. For any graph G and any labeling θ ∈ Θ(G), let SL(G, θ) = ∑ e∈E(G)min{θ(u) : u ∈ e}. In this paper, we consider the S-Labeling problem, defined as follows: Given a graph G, find a labeling θ ∈ Θ(G) that minimizes SL(G, θ). The S-Labeling pr...

Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A denote a tridiagonal pair on V . Let θ0, θ1, . . . , θd (resp. θ ∗ 0, θ ∗ 1, . . . , θ ∗ d) denote a standard ordering of the eigenvalues of A (resp. A). We assume there exist nonzero scalars a, a in K such t...

where ǫ, the noise, is a random variable over some probability space (Ω,F ,P). The gradient of L, i.e. Lθ(θ) is not available, not even via physical measurements. To estimate it let θ be fixed, and take a random simultaneous perturbation (SP) of θ by a multiple of a vector ∆ = (∆1, ...,∆p) . Here ∆ is defined over an independent probability space (Ω,F ,P ). We assume that the components ∆i are ...

Let X,Y be locally compact Hausdorff spaces and M,N be Banach algebras. Let θ : C0(X,M) → C0(Y,N ) be a zero-product preserving bounded linear map with dense range. We show that θ is given by a continuous field of algebra homomorphisms from M into N if N is irreducible. As corollaries, such a surjective θ arises from an algebra homomorphism, provided thatM is a W*-algebra and N is a semi-simple...