Handwritten character recognition using some (anti)-diagonal structural features

In this paper, we present a methodology for off-line handwritten character recognition. The proposed methodology relies on a new feature extraction technique based on structural characteristics, histograms and profiles. As novelty, we propose the extraction of new eight histograms and four profiles from the $32\times 32$ matrices that represent the characters, creating 256-dimension feature vectors. These feature vectors are then employed in a classification step that uses a $k$-means algorithm. We performed experiments using the NIST database to evaluate our proposal. Namely, the recognition system was trained using 1000 samples and 64 classes for each symbol and was tested on 500 samples for each symbol. We obtain promising accuracy results that vary from 81.74\% to 93.75\%, depending on the difficulty of the character category, showing better accuracy results than other methods from the state of the art also based on structural characteristics.Comment: Revised version with a number of improvements and update references, 9 page

More on five commutator identities

It is proved that the five well-known identities universally satisfied by commutators in a group generate all universal commutator identities for commutators of weight 4.Comment: to be published in "Journal of Homotopy and Related Structures

Universal central extensions of superdialgebras of matrices

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $\mathfrak{sl}(m, n, D)$ when $m+n \geq 3$ and $D$ is a superdialgebra, solving in particular the problem when $D$ is an associative algebra, superalgebra or dialgebra. To accomplish this task we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension.Comment: 13 page

Flow of finite-dimensional algebras

Each finite-dimensional algebra can be identified to the cubic matrix given by structural constants defining the multiplication between the basis elements of the algebra. In this paper we introduce the notion of flow (depending on time) of finite-dimensional algebras. This flow can be considered as a particular case of (continuous-time) dynamical system whose states are finite-dimensional algebras with matrices of structural constants satisfying an analogue of Kolmogorov-Chapman equation. These flows of algebras (FA) can also be considered as deformations of algebras with the rule (the evolution equation) given by Kolmogorov-Chapman equation. We mainly use the multiplications of cubic matrices which were introduced by Maksimov and consider Kolmogorov-Chapman equation with respect to these multiplications. If all cubic matrices of structural constants are stochastic (there are several kinds of stochasticity) then the corresponding FA is called stochastic FA (SFA). We define SFA generated by known quadratic stochastic processes. For some multiplications of cubic matrices we reduce Kolmogorov-Chapman equation given for cubic matrices to the equation given for square matrices. Using this result many FAs are given (time homogenous, time non-homogenous, periodic, etc.). For a periodic FA we construct a continuum set of finite-dimensional algebras and show that the corresponding discrete time FA is dense in the set. Moreover, we give a construction of an FA which contains algebras of arbitrary (finite) dimension. For several FAs we describe the time depending behavior (dynamics) of the properties to be baric, limit algebras, commutative, evolution algebras or associative algebras.Comment: 19 page

Peiffer Elements in Simplicial Groups and Algebras

The main objectives of this paper are to give general proofs of the following two facts: A. For an operad \oo in \ab, let $A$ be a simplicial \oo-algebra such that $A_m$ is the \oo-subalgebra generated by $(\sum_{i = 0}^{m} s_i(A_{m-1}))$, for every $n$, and let $\N A$ be the Moore complex of $A$. Then d (\N_m A) = \sum_{I} \gamma(\oo_{p} \otimes \bigcap_{i \in I_1}\ker d_i \otimes ... \otimes \bigcap_{i \in I_{p}}\ker d_i) where the sum runs over those partitions of $[m-1]$, $I = (I_1,...,I_p)$, $p \geq 1$, and $\gamma$ is the action of \oo on $A$. B. Let $G$ be a simplicial group with Moore complex $\N G$ in which the normal subgroup of $G_n$ generated by the degenerate elements in dimension $n$ is the proper $G_n$. Then $d(\N_nG) = \prod_{I,J}[\bigcap_{i \in I}\ker d_i, \bigcap_{j \in J}\ker d_j]$, for $I,J \subseteq [n-1]$ with $I \cup J = [n-1]$. In both cases, $d_i$ is the $i-th$ face of the corresponding simplicial object. The former result completes and generalizes results from Ak\c{c}a and Arvasi, and Arvasi and Porter; the latter, results from Mutlu and Porter. Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the adjoint inverse of the normalization functor \N: \sab \to \ch. For the case of simplicial groups, we have then adapted the construction for the adjoint inverse used for algebras to get a simplicial group G \boxtimes \lb from the Moore complex of a simplicial group $G$. This construction could be of interest in itself.Comment: 18 page

Evolution algebra of a "chicken" population

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.Comment: 15 page

Evolution algebra of a bisexual population

We introduce an (evolution) algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. The basic properties of the algebra are studied. We prove that this algebra is commutative (and hence flexible), not associative and not necessarily power associative. We show that the evolution algebra of the bisexual population is not a baric algebra, but a dibaric algebra and hence its square is baric. Moreover, we show that the algebra is a Banach algebra. The set of all derivations of the evolution algebra is described. We find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra. We also establish upper estimate of the limit points set for trajectories of the evolution operator. Using the necessary conditions we give a detailed analysis of a special case of the evolution algebra (bisexual population of which has a preference on type "1" of females and males). For such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements.Comment: 21 page

Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras

In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.Comment: 24 page

Construction of flows of finite-dimensional algebras

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the KCE with respect to the fixed multiplication. The existence of a solution for the KCE provides the existence of an FA. In this paper our aim is to find sufficient conditions on the multiplications under which the corresponding KCE has a solution. Mainly our conditions are given on the algebra of cubic matrices (ACM) considered with respect to a fixed multiplication of cubic matrices. Under some assumptions on the ACM (e.g. power associative, unital, associative, commutative) we describe a wide class of FAs, which contain algebras of arbitrary finite dimension. In particular, adapting the theory of continuous-time Markov processes, we construct a class of FAs given by the matrix exponent of cubic matrices. Moreover, we remarkably extend the set of FAs given with respect to the Maksimov's multiplications of our previous paper (J. Algebra 470 (2017) 263--288). For several FAs we study the time-dependent behavior (dynamics) of the algebras. We derive a system of differential equations for FAs.Comment: 11 page

Algebras of cubic matrices

We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff $m=1$. We introduce a notion of accompanying algebra (which is $m^2$-dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.Comment: 11 page