on an arbitrary time scale T with supT =∞, subject to the following conditions: (C) t ∈ T and [t,∞)T := {t ∈ T : t ≥ t} is a time scale interval in T; (C) r ∈ Crd( [t,∞)T , (,∞)) and ∫ ∞ t  r(t) t =∞; (C) p,q ∈ Crd( [t,∞)T , [,∞)); (C) ξ , δ ∈ Crd(T,T), limt→∞ ξ (t) = limt→∞ δ(t) = ∞, δ has the inverse function δ– ∈ Crd(T,T), v := δ– ◦ ξ ∈ Crd(T,T), ξ , v ∈ Crd( [t,∞)T , (,∞)),...

The approximation of a general d-variate function f by the shifts φ(· − ξ), ξ ∈ Ξ ⊂ R d , of a fixed function φ occurs in many applications such as data fitting , neural networks, and learning theory. When Ξ = hZ d is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem [6, 18] using Fourier techniques. However, in most applications the center s...

We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor’s derivative operator on sets of ordinals, starting form the usual interval topology. We characterize the non-isolated points in the ξ-th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call ξ-simultaneous-reflection. W...

The approximation of a general d-variate function f by the shifts φ(· − ξ), ξ ∈ Ξ ⊂ Rd, of a fixed function φ occurs in many applications such as data fitting, neural networks, and learning theory. When Ξ = hZd is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem [6, 18] using Fourier techniques. However, in most applications the center set Ξ...

The basic notions of the theory of term rewriting are defined for terms that may involve function letters of infinite arity. A sufficient condition for completeness is derived, and its use demonstrated by the example of abstract clones over infinitary signatures. Introduction In general, to see that two terms of some equational theory A are not equivalent requires a model of A in which their in...

1), find the explicit fundamental solution to the heat equation ∆u + b · ∇u − u t = 0 in R n × (0, ∞). (1) Letting G be what you find, show u 0 (x) = lim t→0 + R n G(x, t; y, 0)u 0 (y) dy, (2) when u 0 is bounded and continuous on R n. Taking the Fourier transform (with respect to x) of the equation gives −|ξ| 2 ˆ u + ib · ξ ˆ u − ˆ u t = 0 − |ξ| 2 + ib · ξ ˆ u = ˆ u t =⇒û = ce −(|ξ| 2 +ib·ξ)t ...

Given a finite subset Ξ ⊂ R and data |Ξ , the surface spline interpolant to the data |Ξ is a function s which minimizes a certain seminorm subject to the interpolation conditions |Ξ = |Ξ . It is known that surface spline interpolation is stable on the Sobolev space W in the sense that ‖s‖L∞(Ω) ≤ const ‖f‖Wm , where m is an integer parameter which specifies the surface spline. In this note we sh...

and Applied Analysis 3 Theneuron activation functions,g p (y p (⋅)) (p = 1, . . . , n), are assumed to be nondecreasing, bounded, and globally Lipschitz; that is, l − p ≤ g p (ξ p ) − g p (ξ q ) ξ p − ξ q ≤ l + p , ∀ξ p , ξ q ∈ R, ξ p ̸ = ξ q , (5) where l− p and l+ p are constant values. For simplicity, in stability analysis of the network (1), the equilibrium point y∗ = [y∗ 1 , . . . , y ∗ n ]...

Let ξ ∈ R, let ι : [1,∞)→ (0, 1] be a positive decreasing function, and let E ξ (ι, Q) be the set of integer pairs (p, q) satisfying |p + qξ| < ι(q), 0 < q < Q. In a series of papers starting in 1959 Erdős [14], Schmidt [22, 23], Lang [18, 19, 9], Adams [1, 2, 3, 4, 5, 7, 6, 8], Sweet [26], and others, considered the problem of finding the asymptotics for the cardinality |E ξ (ι, Q)| as Q gets ...

The weak decays Ξ b → Λ b π and Ξ c → Λ c π, in which the heavy quark is not destroyed, are discussed. The branching fractions for these decays, corresponding to an absolute rate of order 0.01 ps −1 , should be at a one percent level for the b hyperons and at a (few) per mill level for Ξ c , possibly making feasible their experimental study in future. It is shown, through an application of the ...