Let z be the available data which are assumed to have been generated as one random observation from model Mz = {p(z |ω),z ∈ Z,ω ∈ Ω}. Often, but not always, data will consist of a random sample z = {x1, . . . ,xn} from some distribution q(x |ω), with x ∈ X ; in this case p(z |ω) = Qn i=1 q(xi |ω) and Z = X . Let θ(ω) be the vector of interest. Without loss of generality, the model may explicitl...

and Applied Analysis 3 The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases see, e.g., 4, 5 . Numerous problems in physics, optimization, and economics reduce to find a solution of problem 1.2 . S...

Let k ≥ 4, and let H1, H2 be connected graphs with |V (Hi)| ≥ 3 for i = 1, 2. A graph G is said to be {H1, . . . , Hl}-free if none of H1, . . . , Hl is an induced subgraph of G. We prove that if there exists a positive integer n0 such that every {H1, H2}-free graph G with |V (G)| ≥ n0 and δ(G) ≥ 3 contains k vertex-disjoint claws, then {H1, H2} ∩ {K1,t | t ≥ 2} = ∅. Also, we prove that every K...

We show that the Fourier transform on the Jacobian of a curve interchanges " δ functions " on the curve and the theta divisor. The Torelli theorem is an immediate consequence. 1. Statement of the theorem. 1.1. We live over an algebraically closed base field k. Let J be an abelian variety equipped with a principal polarization θ : J ∼ → J • = Pic 0 (J), so we have the corresponding Fourier trans...

Let A be an algebra, and let θ, φ be ring automorphisms of A. An additive mapping H : A → A is called a θ, φ -derivation if H xy H x θ y φ x H y for all x, y ∈ A. Moreover, an additive mapping F : A → A is said to be a generalized θ, φ -derivation if there exists a θ, φ derivation H : A → A such that F xy F x θ y φ x H y for all x, y ∈ A. In this paper, we investigate the superstability of gene...

Let M be a separable compact Hausdorff space with dim M ≤ 2 and θ : M → M be a homeomorphism with prime period p (p ≥ 2). Set Mθ = {x ∈ M | θ(x) = x} 6= ∅ and M0 = M\Mθ. Suppose that M0 is dense in M and H(M0/θ,Z) ∼= 0, H(χ(M0/θ),Z) ∼= 0. Let M ′ be another separable compact Hausdorff space with dim M ′ ≤ 2 and θ′ be the self–homeomorphism of M ′ with prime period p. Suppose that M ′ 0 = M ′\M ...

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

1. Note that Y1 is independent of {Xn}n=1. Now, observe that H0 is a composite hypothesis, and we are in a Bayesian situation with the rv θ (= the rv P0) taking values in Θ0 = {1/4, 3/4} with P0[θ = 1 4 ] = 14 = 1 − P0[θ = 34 ], where P0 denotes the conditional pmf of θ given H = H0. H1 is a simple hypothesis. Under H0: For each t ∈ Θ0 = { 1 4 , 4}, {Yn}n=1 is a 1st order Markov process with tr...

type ref t = id:N{witnessed (has_a_t id t)} let has (id:N) (H h _) = match h id with Used _ _ (Untyped _)→⊤ | _→⊥ abstract type uref = id:N{witnessed (has id)}type uref = id:N{witnessed (has id)} To enforce these invariants on state-manipulating operations, we define a preorder rel on heap, that constrains the heap evolution. It states that every Used identifier remains Used; every Typed refere...

• Let Ak denote the set of methods that observe a sequence of data pairs Y t = (F (θ, X ), F (τ , X )), 1 ≤ t ≤ k, and return an estimate θ̂(k) ∈ Θ. • Let FG denote the class of functions we want to optimize, where for each (F, P ) ∈ FG the subgradient g(θ;X) satisfies EP [‖g(θ;X)‖2∗] ≤ G. • For each A ∈ Ak and (F, P ) ∈ FG, consider the optimization gap: k(A, F, P,Θ) := f (θ̂(k))− inf θ∈Θ f (θ) ...