The ultimate random number generators are those certified to be unpredictable -- including an adversary. use of simple quantum processes promises provide numbers that no physical observer could predict but, in practice, unwanted noise and imperfect devices can compromise fundamental randomness protocol security. Certified protocols have been developed which remove the need for trust by taking a...

Abstract This paper proposes a numerical model of the network of two-Chinese-character compound words (two-character network, for short). In this network, a Chinese character is a node and a twoChinese-character compound word links two nodes. The basic framework of the model is that an important character gets many edges. As the importance of a character, we use the frequency of each character ...

Problem 2. (a) For any ring R, define a ring structure on the abelian group R̃ = R×Z such that (0, 1) is the identity of R̃ and the inclusion map R ↪→ R̃, r 7→ (r, 0) is a rng homomorphism. (b) Let C be the category of rings without identity. Show that the functor F : C → Rings such that F (R) = R̃ is a left adjoint to the forgetful functor G : Rings→ C taking a ring with identity R to R considered...

In this paper X denotes a set, R, R1, R2 denote binary relations, x, y, z denote sets, and n, m, k denote natural numbers. Let us consider a binary relation R on X. Now we state the propositions: (1) fieldR ⊆ X. (2) If x, y ∈ R, then x, y ∈ X. Now we state the propositions: (3) Let us consider sets X, Y . Then (idX)Y = X ∩ Y . (4) 〈x, y〉 ∈ R |2 X if and only if x, y ∈ X and 〈x, y〉 ∈ R. (5) dom(...