We analyse the data for the proton structure function F 2 over the entire Q 2 domain, including especially low Q 2 , in terms of perturbative and non-perturbative QCD contributions. The small distance configurations are given by perturbative QCD, while the large distance contributions are given by the vector dominance model and, for the higher mass qq states, by the additive quark approach. The...

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points—and the distan...

1.1 Hypercube and Hamming distance Definition 1.1 The set of points Hd = {0, 1} is the d-dimensional hypercube. A point p = (p1, . . . , pd) ∈ Hd can be interpreted, naturally, as a binary string p1p2 . . . pd. The Hamming distance dH(p, q) between p, q ∈ Hd, is the number of coordinates where p and q disagree. It is easy to verify that the Hamming distance comply with the triangle inequality, ...

In this paper we determine an upper bound for the covering radius of a q-ary MacDonald codeCk;u(q). Values of nq(4; d), the minimal length of a 4-dimensional q-ary code with minimum distance d is obtained for d = q 1 and q 2. These are used to determine the covering radius of C3;1(q),C3;2(q) and C4;2(q).

Let V be a finite set with q distinct elements. For a subset C of V n, denote var(C) the variance of the average Hamming distance of C. Let T (n,M; q) andR(n,M; q) denote the minimum and maximum variance of the average Hamming distance of subsets of V n with cardinalityM, respectively. In this paper, we study T (n,M; q) and R(n,M; q) for general q. Using methods from coding theory, we derive up...

We consider the problem of estimating the L1 distance between two discrete probability measures P and Q from empirical data in a nonasymptotic and large alphabet setting. When Q is known and one obtains n samples from P , we show that for every Q, the minimax rate-optimal estimator with n samples achieves performance comparable to that of the maximum likelihood estimator (MLE) with n lnn sample...

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely partition-distance D(·, ·). The comparison achieves by checking their restriction to modular elements of the partition lattice, as well as in terms of suitable classifiers. Two of the new measures obtain through the size, a function mapping every ...