• For any x, y ∈ X, we have ρ(x, y) = 0 if and only if x = y. • For any x, y ∈ X, we have ρ(x, y) = ρ(y, x). • For any x, y, z ∈ X, we have ρ(x, y) ≤ ρ(x, z) + ρ(z, y). For any r ≥ 0, a r-partition of M is a partition of X, such that every cluster has diameter at most r. That is, for any cluster C in the partition, and for any x, y ∈ C, we have ρ(x, y) ≤ r. Let β > 0. A (β, r)-Lipschitz partiti...

The system Q undergoes a completely positive trace-preserving transformation or quantum operation E and R is assumed to be isolated and its state remains the same. This quantum operation is also represented by IR ⊗ E , where IR is the identity superoperator on R. We add subscript ‘1’ to denote the state of the system (joint or otherwise) after this quantum operation. So the state of the joint s...

A b s t r a c t We predict a strikingly different A and Q 2 dependence of quasielastic leptoproduction of the ρ 0 (1S)-meson and its radial excitation ρ ′ (2S) on nuclei. Whereas for the ρ 0 production nuclear transparency T A decreases monotonically with A, for the ρ ′ nuclear transparency T A can have the counterintuitive nonmonotonic A-dependence, having the minimum for light nuclei, and inc...

Let W [n] be an independent random sequence with constant mean µ W = 0 and variance σ 2 W. Define a new random sequence X[n] as follows: X[0] = 0 X[n] = ρX[n − 1] + W [n] for n ≥ 1. (a) Find the mean value of X[n] for n ≥ 0. (b) Find the autocovariance of X[n], denoted as K XX [m, n]. (c) For what values of ρ does K XX [m, n] tend to G[m − n], for some finite-valued function G, as m and n becom...