W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang–Baxter equation and cycle sets. J. S. Carter, M. Elhamdadi, Saito, meanwhile, introduced homology theory set-theoretic in order to define cocycle invariants classical knots. In this paper, we introduce normalized an solution compute cyclic racks. Moreover, explicitly calcul...

The purpose of this exercise is to show that, for an arbitrary renewal process, N(t), the number of renewals in (0, t], has finite expectation. a) Let the inter-renewal intervals have the distribution FX (x), with, as usual, FX (0) = 0. Using whatever combination of mathematics and common sense is comfortable for you, show that numbers � > 0 and δ > 0 must exist such that FX (δ) ≤ 1 − �. In oth...

The paper deals with the properties of a monotone operator defined on a subset of an ordered Banach space. The structure of the set of fixed points between the minimal and maximal ones is described.

|x′ = v, y′ = u 〉 = |x = u, y = v 〉 . (1) We assume a similar relationship exists in momentum space (to be completely general, we should also include a possible phase factor, but for this problem it won’t be necessary): |px = a, py = b 〉 = |px = c, py = d 〉 . (2) Because the inner product in conserved under a change of coordinates, we have: 〈x′ = u, y′ = v |px = a, py = b 〉 = 〈x = u, y = v |px ...

The purpose of this problem is to illustrate that for an arrival process with independent but not identically distributed interarrival intervals, X1, X2, . . . ,, the number of arrivals N(t) in the interval (0, t] can be a defective rv. In other words, the ‘counting process’ is not a stochastic process according to our definitions. This illustrates that it is necessary to prove that the countin...