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Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle X\rangle$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\ldots, a_n) = 0$ for every $a_1, \ldots, a_n \in A$. Consider $\mathcal{A}$ the subalgebra of $UT_3(K)$ given by: \[ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , \] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $\mathbb{Z}_2$-graded identities of $\mathcal{A}$, and also for the $\mathbb{Z}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra.

Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be two simple algebras with involution over $K$. In this note we study the embedding problem for algebras with involution. More specifically, if the algebra $A$ satisfies the polynomial identities with involution of the algebra $B$, we investigate whether there exists an embedding of $A$ into $B$ that preserves the involutions.

Let $K$ be a field of characteristic 0, and let $E$ be the infinite-dimensional Grassmann algebra over $K$. We consider $E$ as a $\mathbb{Z}_2$-graded algebra, where the grading is given by the vector subspaces $E_0$ and $E_1$, consisting of monomials of even and odd lengths, respectively. Thus, if $A = A_0 \oplus A_1$ is an associative $\mathbb{Z}_2$-graded algebra, we can consider the $\mathbb{Z}_2$-graded algebra $A \hat{\otimes} E = (A_0 \otimes E_0) \oplus (A_1 \otimes E_1)$. In case both $E$ and $A$ are endowed with superinvolutions, we can define a $\mathbb{Z}_2$-graded involution on $A \hat{\otimes} E$ induced by the respective superinvolutions. In this paper, we consider the $\mathbb{Z}_2$-graded matrix algebras $M_{1,1}(K)$, $UT_{1,1}(K)$, and $UT_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes} E$, $UT_{1,1}(K)\hat{\otimes} E$, and $UT_{(0,1,0)}(K)\hat{\otimes} E$, considering these resulting algebras as $\mathbb{Z}_2$-graded algebras with graded involution.

Blockchain security is becoming increasingly relevant in today's cyberspace as it extends its influence in many industries. This paper focuses on protecting the lowest level layer in the blockchain, particularly the P2P network that allows the nodes to communicate and share information. The P2P network layer may be vulnerable to several families of attacks, such as Distributed Denial of Service (DDoS), eclipse attacks, or Sybil attacks. This layer is prone to threats inherited from traditional P2P networks, and it must be analyzed and understood by collecting data and extracting insights from the network behavior to reduce those risks. We introduce Tikuna, an open-source tool for monitoring and detecting potential attacks on the Ethereum blockchain P2P network, at an early stage. Tikuna employs an unsupervised Long Short-Term Memory (LSTM) method based on Recurrent Neural Network (RNN) to detect attacks and alert users. Empirical results indicate that the proposed approach significantly improves detection performance, with the ability to detect and classify attacks, including eclipse attacks, Covert Flash attacks, and others that target the Ethereum blockchain P2P network layer, with high accuracy. Our research findings demonstrate that Tikuna is a valuable security tool for assisting operators to efficiently monitor and safeguard the status of Ethereum validators and the wider P2P network, Comment: 15 pages, 2 figures, submitted to ISPEC 2023 Conference

The multi-point Taylor polynomial, which is the general, unique and of minimum degree ($mk+m-1$) polynomial $P_{k,m}(x)$ which interpolates a function's derivatives in multiple points is presented in its explicit form. A proof that this expression satisfies the multi-point Taylor polynomial's defining property is given. Namely, it is proven that for a k-differentiable function $f$ and a set of different m-points $\{a_1,...,a_m\}$, this polynomial satisfies $P^{(n)}_{k,m}(a_i) = f^{(n)}(a_i)\quad \forall \, i = 1,...,m\quad \&\quad \forall \, n = 0,...,k$. A discussion regarding previous expressions presented in the literature, which mostly consisted in recursion formulas and not explicit formulas, is made., Comment: 9 pages, no figures. Keywords: multi-point Taylor polynomial, multi-point polynomial interpolation, Hermite interpolation, Osculatory interpolation

In this paper, we introduce the class of finitely semi-graded algebras which extends the connected graded algebras finitely generated in degree one. The Koszul behavior of finitely semi-graded algebras is investigated by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincar\'e series and the Yoneda algebra are defined for this class of algebras. Finitely semi-graded algebras include many important examples of non $\mathbb{N}$-graded algebras finitely generated in degree one coming from mathematical physics, and for these concrete examples the Koszulity will be established, as well as, the explicit computation of its Hilbert and Poincar\'e series.

Machine Learning (ML) and Deep Learning (DL) are two technologies used to extract representations of the data for a specific purpose. ML algorithms take a set of data as input to generate one or several predictions. To define the final version of one model, usually there is an initial step devoted to train the algorithm (get the right final values of the parameters of the model). There are several techniques, from supervised learning to reinforcement learning, which have different requirements. On the market, there are some frameworks or APIs that reduce the effort for designing a new ML model. In this report, using the benchmark DLBENCH, we will analyse the performance and the execution modes of some well-known ML frameworks on the Finis Terrae II supercomputer when supervised learning is used. The report will show that placement of data and allocated hardware can have a large influence on the final timeto-solution.