(a) [Problem 2.2 in DFT] For u 1 : For u 2 : For u 3 : Note that u 1 = u 2 2 = ∞ 0 1 · dt = ∞, while u ∞ = 1 and P ow(u) = lim T →∞ 1 2T T 0 1 · dt = 1 2. T −T u 2 (t)dt = T 0 u 2 (t)dt = T 0 |u(t)|dt = k−1 i=0 2 2i for 2 2k−1 ≤ T ≤ 2 2k , k ≥ 1. Hence, u 1 = u 2 = ∞ while u ∞ = 1. Finally: P ow(u) = lim T →∞ 1 2T T −T u 2 (t)dt =    lim k→∞ 1 2 2k k−1 i=0 2 2i = 1 4 when T = 2 2k−1 , k ≥ 1 ...

Problem 1 (a) Recall that the null space and the range of a matrix A ∈ C m×n are defined as: N (A) = {x ∈ C n |Ax = 0} R(A) = {y ∈ C m |y = Ax, for some x ∈ C n } Note that N (A) is a subspace of C n since it contains 0 ∈ C n (A0 = 0) and is closed under addition and scalar multiplication: x 1 , x 2 ∈ N (A) ⇒ Ax 1 = 0 and Ax 2 = 0 ⇒ Ax 1 + Ax 2 = 0 ⇒ A(x 1 + x 2) = 0 ⇒ x 1 + x 2 ∈ N (A) x ∈ N (...

Let A be a non empty set, let B be a set, let B1 be a set, and let y1 be a function from B1 into A. Let us assume that B1 ⊆ B. The functor cylinder0(A,B,B1, y1) yields a non empty subset of A B and is defined by: (Def. 1) cylinder0(A,B,B1, y1) = {y : B → A: y B1 = y1}. Let A be a non empty set and let B be a set. A non empty subset of AB is said to be a thin cylinder of A and B if: (Def. 2) The...

The Pb region is characterised by the existence of collective octupole states. Here we populated such states in Pb + Pb deep-inelastic reactions. γ-ray angular distribution measurements were used to infer the octupole character of several E3 transitions. The octupole character of the 2318 keV 17− → 14 in Pb, 2485 keV 19/2− → 13/2 in Pb, 2419 keV 15/2− → 9/2 in Pb and 2465 keV 17/2 → 11/2− in Tl...

background: recently, some new nanobiosensors using different nanoparticles or microarray systems for detection of mycotoxins have been designed . however, rapid, sensitive and early detection of aflatoxicosis would be very helpful to distinguish high-risk persons.objectives: we report a highly sensitive competitive immunoassay using magnetic/silica core shell as a signal intensifier for the de...

Problem 1 (a) We will compute theH2 norm ofG using state-space methods (see Section 2.6 in DFT). We begin by finding a minimal state-space representation of the system with transfer function G(s). Assuming u(0) = y(0) = 0 as usual, we have: sy(s) + 2sy(s) + sy(s) + y(s) = u(s) ⇔ ... y (t) + 2ÿ(t) + ẏ(t) + y(t) = u(t) Defining x1 = y, x2 = ẏ, x3 = ÿ, the corresponding state-space representation ...

A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a 00 > (a 0 0) 0 (i.e. a 2 GL 2) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.). Many results for GL 2 deg...

Biochemical measurements were carried out on 20 workers entering employment in a lead industry. Blood lead (Pb-B) urinary delta-aminolevulinic acid (ALA), and coproporphyrin (CP) levels were determined before entry and then at approximately weekly intervals for the first 12 weeks of employment. Levels of blood lead began to increase in all men within one week rising from an average pre-employme...