Let K be a number field, and let F be a symmetric bilinear form in 2N variables over K. Let Z be a subspace of K . A classical theorem of Witt states that the bilinear space (Z,F ) can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of...

and Applied Analysis 3 Remark 2.2. For each t0 ∈ T \ {maxT}, the single-point set {t0} is Δ-measurable, and its Δmeasure is given by μΔ {t0} σ t0 − t0 μ t0 . 2.2 Obviously, E1 ⊂ A does not have any right-scattered points. For a set E ⊂ T, define the Lebesgue Δ-integral of f over E by ∫ Ef t Δt and let f ∈ LT E,R see 8 . Lemma 2.3 see 8 . Let f : a, b T → R. f̃ : a, b → R is the extension of f to...

and Applied Analysis 3 the autoregressive coefficient estimator. For the nonstationary of continuous-valued time series, we often need to examine whether the characteristic polynomial of AR(1) process has a unit root. Thus, we want to see if we can find the limiting distribution of the autoregressive coefficient estimator. Let us present a result that is needed later on. Lemma 3. Suppose that Z...

and Applied Analysis 3 Definition 4. Let X be a Banach space and ΩX the bounded set ofX. The Kuratowski’s measure of noncompactness is the map α : ΩX → [0,∞) defined by α (D) = inf {d > 0 : D ⊆ n ⋃ i=1 Di, diam (Di) ≤ d} , (8) hereD ∈ ΩX. One will use the following basic properties of the α measure and Sadovskii’s fixed point theorem here (see [37– 39]). Lemma 5. Let D1 and D2 be two bounded se...

We prove a result on certain finite index subgroups of rank one lattices which is motivated by cusp closing constructions. Let X always denote a rank one symmetric space of non-compact type, i.e., X is the hyperbolic space KH, n ≥ 2, where K is either R,C,H or O and n = 2 in the latter case. By Σ we always denote a lattice in the isometry group Iso(X), that is, a discrete subgroup of the isomet...

and Applied Analysis 3 The paper is organized as follows. In Section 2, we recall preliminary results. In Section 3, we present uniqueness of the solution to the atmospheric circulation equations. In Section 4, we obtain global attractor of the equations. ‖ · ‖X denote norm of the space X; C and Ci are variable constants. Let H {φ u, T, q ∈ L2 Ω, R4 | φ satisfy 1.4 , 1.6 }, and H1 {φ u, T, q ∈ ...

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R generalized periodic-like if for all x ∈ R \ (N ∪ J ∪ Z) there exist positive integers m, n of opposite parity for which xm − xn ∈ N ∩ Z. We identify some basic properties of such rings and prove some results on commutativity. Let R be a ring; and let N = N(R), Z = Z(R) and J = J(R) denote respective...

. (1.2) In the manipulations above we have used the linearity of the superoperator M in the first line. In the second line we have used Lemma 0.4 with γ = I ⊗ σ , noting that |X|= s. The third line follows from the concavity of the square root. We will now use a helpful identity for the trace of an operator squared: TrZ =Tr(Z ⊗ Z)F , where F is defined as follows. Definition 1.1. The swap opera...