We have showed previously that the Fourier transform is an isometric isomorphism on L2 = CcR ∩ L2Rn. In particular, we have a ∀u ∈ L2 ||u||0 = ||û||0 Parseval relation b ∀u,v ∈ L2, u,v0 = û, v̂0 Plancherel relation The results (a) and (b) hold for every test function, and since the test functions are dense in L2Rn, they extend to L2Rn by continuity. Then the Fourier transform can...

Let p be a prime number; for x, y ∈ Rn, we will write x ≡ y (mod p) to mean that there exists a z ∈ Rn such that x− y = pz. Otherwise put, this is congruence modulo the principal ideal pRn of Rn. Since Z ⊂ Rn, if x and y are ordinary integers, the notation x ≡ y (mod p) is ambiguous: interpreting it as a usual congruence in the integers, it means that there exists an integer n such that x− y = ...

The proof is based on the optimality conditions on β̂ for the original problem, and those on β̌ for the restricted problem. We first observe that β̌n exists, since the function x 7→ ‖x‖1 is coercive. Recall that β̌n is assumed to be uniquely defined. To achieve sparsistency, it suffices that β̂n = β̌n and supp β̌n = suppβ ∗. We derive sufficient conditions for β̂n = β̌n in Lemma 2.1, and make this suffi...

Proof Let f : R → R be any smooth function that is identically 1 for t ≤ 0 and identically 0 for t ≥ 1. Since H k(R) is the completion of C∞(Rn), it is enough to show that every function φ ∈ C∞(Rn) ∩ H k(R) can be approximated in H k(R n) by functions in C∞ c (Rn). Consider the sequence φj(x) := φ(x)f(|x| − j). We have that φj ∈ C∞ c (Rn): in fact |x| is not differentiable at x = 0, but f(t) is...

A subsetA of a Boolean algebra B is said to be (n,m)reaped if there is a partition of unity P ⊂ B of size n such that |{b ∈ P : b∧a 6= ∅}| ≥ m for all a ∈ A. The reaping number rn,m(B) of a Boolean algebra B is the minimum cardinality of a set A ⊂ B\ {0} such which cannot be (n,m)-reaped. It is shown that, for each n ∈ ω, there is a Boolean algebra B such that rn+1,2(B) 6= rn,2(B). Also, {rn,m(...

We consider a number of decision problems, that appear in the dynamical systems and database literature, concerning the termination of iterates of real functions. These decision problems take a function f :Rn → Rn as input and ask, for example, whether this function is mortal, nilpotent, terminating, or reaches a fixed point on a given point in Rn. We associate topologies to functions f :Rn →Rn...

Faculty Hsiao-Lan Wang, PhD, RN, CMSRN, HFS, College of Nursing University of South Florida Ming Ji, PhD, College of Nursing University of South Florida Connie Visovsky, PhD, RN, ACNP-BC, College of Nursing University of South Florida Carmen S. Rodriguez, PhD, RN, ACNP-BC, AOCN®, College of Nursing University of South Florida Amanda F. Elliott, PhD, RN, ARNP, College of Nursing University of So...

A procedure which accelerates the convergence of iterative methods for the numerical solution of systems of nonlinear algebraic and/or transcendental equations in Rn is introduced. This procedure uses a rotating hyperplane in Rnil, whose rotation axis depends on the current approximation of n 1 components of the solution. The proposed procedure is applied here on the traditional Newton's method...

0883-2927/$ see front matter 2008 Elsevier Ltd doi:10.1016/j.apgeochem.2008.05.021 * Corresponding author. Tel.: +1 919 681 6577; fa E-mail address: [email protected] (D.S. Vinson). Estimation of Rn transfer from water to indoor air based on multi-day measurements may underestimate alpha exposure that occurs at short time scales in confined spaces, such as from showering, in houses with high Rn act...

Monte Carlo simulation was used to determine how violation of the independence assumption affects the empirical probability distribution and Type I error rates of Revusky's Rn statistical test. Simulation results show that the probability distribution of Rn was distorted when the data were autocorrelated. A corrected Rn statistic was proposed to reach a reasonable fit between theoretical (exact...