We provide an ultrafilter proof of the 2-dimensional Halpern–Läuchli Theorem in the following sense. If T0 and T1 are trees and T0 ⊗ T1 denotes their level product, we exhibit an ultrafilter U ∈ β(T0 ⊗ T1) so that every A ∈ U contains a subset of the form S0 ⊗ S1 for suitable strong subtrees of T0 and T1. We then discuss obstacles to extending our method of proof to higher dimensions.

and Applied Analysis 3 input signal v t v1 t , v2 t , . . . , vq t T of the pulse-width sampler satisfy the following dynamic relation: ui t ⎧ ⎨ ⎩ signαni , nT0 ≤ t < n |αni | T0, i 1, 2, . . . , q; 0, n |αni | T0 ≤ t < n 1 T0, n 0, 1, . . . , 1.4 αni ⎧ ⎨ ⎩ vi nT0 , |vi nT0 | ≤ 1, i 1, 2, . . . , q; signvi nT0 , |vi nT0 | ≥ 1, n 0, 1, . . . , 1.5 where T0 is called the sampling period of the pu...

we call a the symbol of the sequence {Tn} and denote Tn by Tn(a). The case where a is in L∞(T) is easy, since then ‖Tn(a)‖ → ‖a‖∞ as n → ∞. Things are more complicated for a ∈ L(T) \L∞(T). We here focus our attention on so-called FisherHartwig symbols with a single singularity, that is, we consider functions a of the form a(t) = |t− t0|φβ,t0(t)b(t) (t ∈ T) where t0 ∈ T, α is a complex number su...

Conditions are found upon satisfaction of which the differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 has solutions which are asymptotically equivalent to the solutions of the equation x(n)(t)− λx(n)(t− σ) = 0. § 0. Introduction We consider the neutral functional differential equation x(n)(t)− λx(n)(t− σ) + f(t, x(g(t))) = 0 (A) under the assumptions that (i) n ≥ 1 is an integer; λ...

The nonlinear neutral differential equation dn dtn [x(t)+ h(t)x(τ (t))] + f (t, x(g(t))) = q(t), (1) is considered under the following conditions: n ∈ N; h ∈ C[t0,∞); τ ∈ C[t0,∞) is strictly increasing, limt→∞ τ(t) = ∞ and τ(t) < t for t ≥ t0; g ∈ C[t0,∞) and limt→∞ g(t) = ∞; f ∈ C([t0,∞) × R); q ∈ C[t0,∞). It is shown that if f is small enough in some sense, Eq. (1) has a solution x(t) which b...

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...

Open AccessEngineering NotesState Transition Tensors for Continuous-Thrust Control of Three-Body Relative MotionJackson Kulik, William Clark and Dmitry SavranskyJackson KulikCornell University, Ithaca, New York 14850*Ph.D. Candidate, Center Applied Mathematics; .Search more papers by this author, ClarkCornell 14850†Visiting Assistant Professor, Mathematics.Search author SavranskyCornell 14850‡A...

(2) x∆(t) = G(t, x(s); 0 ≤ s ≤ t) := G(t, x(·)) on a time scale T that is unbounded above with 0 ∈ T, where x ∈ R andG : [0,∞)T× R 7→ R is a is rd-continuous function in t and x with G(t, 0) = 0. Throughout this paper, for each continuous function φ : [0, t0]T 7→ R there exists at least one continuous function x(t) = x(t, t0, φ) on an interval [t0, a]T such that it satisfies (2) for t ∈ [t0, a]...

The terminal set M is a closed subset of Rn+1. The admissible control set U is assumed to be the set of piecewise continuous function on [t0, t1]. The performance function J is assumed to be C1. The function V (·, ·) is called the value function and we shall use the convention V (t0, x0) = ∞ if the control problem above admits no feasible solution. We will denote by U(x0, t0), the set of feasib...