and Applied Analysis 3 Theorem C. Let Aj z /≡ 0 j 0, 1 be entire functions with σ Aj < 1, and let a, b be complex constants such that ab / 0 and arga/ arg b or a cb 0 < c < 1 . If ψ z /≡ 0 is an entire function with finite order, then every solution f /≡ 0 of 1.2 satisfies λ f − ψ λ f ′ − ψ λ f ′′ − ψ ∞. Furthermore, let d0 z , d1 z ,and d2 z be polynomials that are not all equal to zero, and l...

• ei, fi. e1 = 1⊗ E1, e2 = 1⊗ E2; f1 = 1⊗ F1, f2 = 1⊗ F2. • e0, f0, h0. e0 = t⊗ [F1, F2] = t⊗ E31. f0 = t −1 ⊗ [E1, E2] = t−1 ⊗ E13. h0 = [e0, f0] = −1⊗ (H1 +H2) + c = −1⊗ (E11 + E33) + c • H. H = 1⊗H ⊕ Cc⊕ Cd. Note that c is just the central element c = h0 + h1 + h2. • Π. α1 = 1 = 2, α1(c) = α1(d) = 0 α2 = 1 = 2, α1(c) = α1(d) = 0 θ = α1 + α2 = 1 − 3, θ(c) = θ(d) = 0 δ : δ(1⊗H) = δ(c) = 0, δ(d...

let x be a random variable from a normal distribution with unknown mean θ and known variance σ2. in many practical situations, θ is known in advance to lie in an interval, say [−m,m], for some m > 0. as the usual estimator of θ, i.e., x under the linex loss function is inadmissible, finding some competitors for x becomes worthwhile. the only study in the literature considered the problem of min...

Let Θ(G) denote the Shannon capacity of a graph G. We give an elementary proof equivalence, for any graphs G and H, inequalities Θ(G⊔H)>Θ(G)+Θ(H) Θ(G⊠H)>Θ(G)Θ(H). This was shown independently by Wigderson Zuiddam (2022) using Kadison–Dubois duality Axiom choice.

Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras Θ. For every algebra H in Θ one can consider algebraic geometry in Θ over H. Correspondingly, algebras in Θ are considered with the emphasis on equations and geometry. We give examples of geometric properties of algebras in Θ and of geometric relations between them. The main problem considered i...

Let X be a random variable from a normal distribution with unknown mean θ and known variance σ2. In many practical situations, θ is known in advance to lie in an interval, say [−m,m], for some m > 0. As the usual estimator of θ, i.e., X under the LINEX loss function is inadmissible, finding some competitors for X becomes worthwhile. The only study in the literature considered the problem of min...

4. Spatial Point Processes with Dependent Marks for Fingerprint Minutiae. Let xn ≡ { xi, i = 1, 2, · · · , n } denote the collection of n minutiae locations, and for each x ∈ xn, the minutiae orientation wx denotes the corresponding mark, which takes values in (0, π]. The distribution of minutiae in a fingerprint image is best described in terms of a hierarchical model involving all random enti...

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form sup ∫ D θ dx=m inf u∈H1 0 (D) ∫ D (1 + θ 2 |∇u| − fu ) dx. We prove the existence of an optimal reinforcement θ and that it has some higher integrability properties. We also provide some numerical computations for θ and u.

Lemma 3. Theorem 1 is true if and only if Theorem 2 is true. Proof. Assume that Theorem 2 is true and let θ′ 1, . . . , θ ′ k, 1 be real numbers that are linearly independent over Z, let α1, . . . , αk be real numbers, let N > 0 and let 0 < < 1. Let θm = θ ′ m − qm with 0 < θm ≤ 1. Because θ′ 1, . . . , θ′ k, 1 are linearly independent over Z, so are θ1, . . . , θk, 1. Using Theorem 2 with k + ...

1. Introduction. Let M be a compact smooth manifold of dimension n, and let T * M → τ M be its cotangent bundle. T * M carries a canonical 1-form θ, which in canonical coordinates (q i , p i) is given by θ = p i dq i. Then ω := dθ is a symplectic form on T * M. To a smooth Hamiltonian H ∈ C ∞ (S 1 ×T * M, R), 1-periodic in time, we associate the Hamiltonian system ˙ x = X H (t, x), (HS) where t...