Blockchain technology started as the backbone for cryptocurriencies and it has emerged as one of the most interesting technologies of the last decade. It is a new paradigm able to modify the way how industries transact. Today, the industries’ concern is about their ability to handle a high volume of data transactions per second while preserving both decentralization and security. Both decentralization and security are guaranteed by the mathematical strength of cryptographic primitives. There are two main approaches to achieve consensus: the Proof-of-Work based blockchains—PoW—and the Proof-of-Stake—PoS. Both of them come with some pros and drawbacks, but both rely on cryptography. In this survey, we present a review of the main consensus procedures, including the new consensus proposed by Algorand: Pure Proof-of-Stake—Pure PoS. In this article, we provide a framework to compare the performances of PoW, PoS and the Pure PoS, based on throughput and scalability.

In this paper, we consider a monomial ideal J ⊳ P := A[x 1 , . . . , x n ], over a commutative ring A, and we face the problem of the characterization for the family Mf (J) of all homogeneous ideals I ⊳ P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Janet in [19,20,21] and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied in [1,6] for the special case of a strongly stable monomial ideal J.Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F (J), called stably complete, that allows an explicit description of the family Mf (J). We obtain stronger results if J is quasi stable, proving that F (J) is a Pommaret basis and Mf (J) has a natural structure of affine scheme.The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.

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For classical matroid, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of q-polymatroids and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

Aim of this paper is to count 0-dimensional stable and strongly stable ideals in 2 and 3 variables, given their (constant) affine Hilbert polynomial.To do so, we define the Bar Code, a bidimensional structure representing any finite set of terms M and allowing to desume many properties of the corresponding monomial ideal I, if M is an order ideal. Then, we use it to give a connection between (strongly) stable monomial ideals and integer partitions, thus allowing to count them via known determinantal formulas. P(n, k) = 0 for k > n P(n, n) = 1 P(n, 0) = 0We define now the notion of plane partition.