Prawo wiernych do wyrażania wobec pasterzy Kościoła swoich potrzeb i życzeń (kan. 212 § 2 KPK)

Quasi-kan Extensions for 2-categories

Polar Lattices: Where Ar{\i}kan Meets Forney

In this paper, we propose the explicit construction of a new class of lattices based on polar codes, which are provably good for the additive white Gaussian noise (AWGN) channel. We follow the multilevel construction of Forney et al. (i.e., Construction D), where the code on each level is a capacityachieving polar code for that level. The proposed polar lattices are efficiently decodable by usi...

CDS 212 - Solutions to Homework # 2

(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 ...

Kan Extensions of Institutions

Institutions were introduced by Goguen and Burstall 8, 9, 10, 11] to formally capture the notion of logical system. Interpreting institutions as functors, and morphisms and representations of institutions as natural transformations, we give elegant proofs for the completeness of the categories of institutions with morphisms and representations, respectively, show that the duality between morphi...

2-absorbing $I$-prime and 2-absorbing $I$-second submodules

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we will introduce the notions of 2-absorbing $I$-prime and 2-absorbing $I$-second submodules of an $R$-module $M$ as a generalization of 2-absorbing and strongly 2-absorbing second submodules of $M$ and explore some basic properties of these classes of modules.