Agustin Moreno

National University of Colombia | UNAL · Mathematics

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras...

UDC 512.5 The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras. In addition, it is shown how the number of these homological ideals can be connected with the process of categorification of Fibonacci numbers defined by Ringel and Fahr...

Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group H. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some config- urations called Brauer configurations. These configurations define so...

Dyck path categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin-type [Formula: see text]. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of...

Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra A. In this setting, we prove that the indecomposable uniserial A-modules with a mast of multiplicity one...

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) T1,T2,⋯,Tn. In this case, for 1≤j≤n−1, two consecutive tiles Tj and Tj+1 share exactly one edge, either the edge at the east (west) of Tj (Tj+1) or the edge at the north (south) of Tj (Tj+1). Finding the number of perfect matchings associated with a...

In this paper, it is proved that the algorithms of differentiation VIII-X (introduced by A.G. Zavadskij to classify equipped posets of tame representation type) induce categorical equivalences between some quotient categories, in particular, an algorithm A z is introduced to build equipped posets with a pair of points (a, b) suitable for differenti...

Recently, Çanakçi and Schroll proved that associated with a string module $ M(w) $ there is an appropriated snake graph $ \mathscr{G} $. They established a bijection between the corresponding perfect matching lattice $ \mathscr{L}(\mathscr{G}) $ of $ \mathscr{G} $ and the canonical submodule lattice $ \mathscr{L}(M(w)) $ of $ M(w) $. We introduce B...

The enumeration of Dyck paths is one of the most remarkable problems in Catalan combinatorics. Recently introduced categories of Dyck paths have allowed interactions between the theory of representation of algebras and cluster algebras theory. As another application of Dyck paths theory, we present Brauer configurations, whose polygons are defined...

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categor...

Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargam...

Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers i...

In this paper, suitable Brauer configuration algebras are used to give an explicit formula for the number of perfect matchings of a snake graph. Some relationships between Brauer configuration algebras with path problems as the Lindstr\"om problem are described as well.

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determi...

We introduce an algorithm based on posets and tiled orders to generate emerging images. Experimental results allow concluding that images obtained with these kinds of tools are easy to detect by human beings. It is worth pointing out that the emergence phenomenon is a Gestalt grouping law associated with AI open problems. For this reason, emerging...

Mutations on Brauer configurations are introduced and associated with some suitable automata to solve generalizations of the Chicken McNugget problem. Additionally, based on marked order polytopes, the new Diophantine equations called Gelfand–Tsetlin equations are also solved. The approach allows algebraic descriptions of some properties of the AES...

The notion of visual cryptography was introduced without formalisms by Naor and Shamir in 1994. It provides a very powerful technique by which one secret can be distributed into two or more shares, when the shares on transparencies are superimposed exactly together, the original secret can be discovered without computer participation. In this paper...

Mutations on Brauer configurations are introduced and associated with some suitable automata in order to solve generalizations of the Chicken McNugget problem. Besides, based on marked order polytopes a new class of diophantine equations called Gelfand-Tsetlin equations are also solved. The approach allows giving an algebraic description of the sch...

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type \({\mathbb {A}}_{n}\). Such diamonds constitute a tool to build integral frieze patterns.

In this paper homological ideals associated to some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras. Besides, it is shown how the number of such homological ideals can be connected with the categorification process of Fibonacci numbers defined by Ringel and Fahr.

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type An. Such diamonds constitute a tool to build integral frieze patterns.

Bijections between invariants associated to indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated to solutions of the Kronecker problem and the four subspace problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and the...

Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An i...

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type.

In this paper, Krawtchouk-Zavadskij matrices are used to solve systems of differential equations (linear of second order and non-linear), to do that, it is given a matrix interpretation to some identities arising from the classical calculus. We recall that, Krawtchouk-Zavadskij matrices were introduced in the late 1920s by Krawtchouk, however we a...

The apparatus of differentiation DI–DV was introduced by A.G. Zavadskij to classify different kind of posets, in particular, Zavadskij et al. described categorical properties of algorithms of differentiation DI and DII as Gabriel did for the algorithm of differentiation with respect to a maximal point introduced by Nazarova and Roiter. In this pape...

In this paper, Delannoy numbers are interpreted as dimensions of suitable representations of some equipped posets induced by compositions of integer numbers.

In this paper, k-linear representations of posets are used to define lattice-based schemes of visual cryptography for color images.

A categorification in the sense of Ringel and Fahr is given to the sequences A016116 and A000034 in the OEIS by using τ-orbits in the Auslander-Reiten quiver of some Dynkin algebras.

Categorification of integer sequences A083329, A000295 and A049611 in the OEIS is given by using the number of sections in the Auslander-Reiten quiver of path algebras of type kΔ, where Δ is an oriented Dynkin diagram of type An, Dn, E6, E7 and E8. In particular, the sequence A000295 counts the number of Dyck paths of semilength n having exactly on...

Indecomposable representations of the modular lattice L(An0) where An0 is an equipped Dynkin diagram, are used to give a categorification for Catalan numbers.

A categorification of the sequence A052558 in the OEIS is given by defining new invariants for indecomposable Kronecker modules.

Tiled orders are used to establish properties of the sequence whose elements count the number of two-point antichains in the powerset of an n-element set ordered by inclusion. In particular, we find a partition formula for numbers in this sequence.

The structure of some path algebras of infinite representation type is used to obtain a formula for the number of two-points antichains in the powerset 2ⁿ of an n-element set ordered by inclusion.

We prove two triangle equivalences. One is the triangle equivalence between the homotopy category of the bounded below complexes of Extinjectives objects of a closed by subobjects and co-resolving subcategory B of an abelian category and the derived category of the bounded below complexes over B. The other triangle equivalence is between the homoto...

Matrix problems and in particular matrix representations of partially ordered sets (posets) are used to formally define and generate emerging and multistable images. Images induced by such representations are mosaics which can be used to design different types of Human Interaction Proofs (HIPs).

In this paper, a Zavadskij’s differentiation algorithm is used to reduce some tiled orders to (0, 1) -tiled orders. © 2015 Pushpa Publishing House, Allahabad, India Published Online: April 2015

A complete description of the indecomposable representations and irreducible morphisms of some equipped posets of finite growth representation type is provided.

In this paper, we prove that the number of some 3-dimensional partitions is given by the number of indecomposable objects of some posets of finite representation type.

k-linear representations of posets are used to define some partitions of multiples of polygonal numbers. Furthermore, the number of indecomposable representations of some posets of finite representation type is used to obtain a partition formula for elements of the sequence A002662 in the On-Line Encyclopedia of Integer Sequences (OEIS).

We describe numbers of a given form which cannot be written as a sum of three n-gonal numbers, 5 ≤ n ≤ 30. Furthermore, we conjecture via computation upper bounds for numbers which are not a sum of such types.

Properties of a \(k\) -vector space \(\mathrm {Hom}_{k}(U_{0}, V_{0})\) of linear maps between fixed \(k\) -vector spaces \(U_{0}\) and \(V_{0}\) are used to define perfect black visual cryptography schemes for sharing secret images, such images can be revealed by stacking qualified sets of transparencies induced by a linear combination of some bas...

In this paper, we define entropy of a partially ordered set (poset) and use it to establish the properties of endomorphism rings of posets of tame representation type.

We use techniques of the theory of algorithms of differentiation of posets and P-partitions to describe identities of some one-dimensional compositions involving polygonal and cubic numbers. We also describe with these techniques numbers which can be written as a sum of three square of numbers of a given shape or sequences of numbers which can be w...

Recently, lattice theory and poset representation theory have been used to obtain mathematical models for the war on terrorism and applications in visual authentication for mobile and non-mobile devices. In this paper, we use the algorithm of differentiation with respect to a maximal point and the algorithm with respect to a suitable pair of points...

Doodles are used in several systems of visual authentication for mobile and non-mobile devices, in this paper, we create doodles which can be used in systems of visual authentication. The proposed algorithm is based on the structure of the space of k-linear maps Homk (U0, V0) between fixed k-linear vector spaces U0 and V0.

In this paper, we give a brief biography of Alexander Georgievich Zavadskij describing his work in the eld of the theory of representations of algebras and the impact of such work in the investigation of the theory of representation in Colombia. In particular, we describe the role of professor Zavadskij in the introduction of the theory of matrix p...

As fourth part of a series of papers concerning morphisms of equipped posets [for earlier work, see the authors ibid. 26, No. 2, 173-196 (2012; Zbl 1281.16024)], in this paper, the authors discuss some categorical properties of the short generalized version of the algorithm of differentiation VII for equipped posets introduced by C. Rodríguez Beltr...

Via linear extensions of posets, P-partitions, and some higher dimensional compositions defined previously by the first author et al., we obtain solutions to a generalization of a problem on partitions proposed by G. E. Andrews in 1987. Furthermore, it is given a bijective correspondence between higher dimensional compositions whose parts are polyg...

We prove that the algorithm of differentiation VIII for equipped posets induces a categorical equivalence between quotient categories.

We establish a categorical equivalence induced by the algorithm of differentiation D-IX for equipped posets.

In 2009, Mitra et al. proposed a synthesis technique to generate emerging images of 3D objects. Such images are gestalts with the property of to be detectable by humans but difficult to process by computer vision algorithms [1]. Therefore, emerging images can be used to design a blind test to tell humans and machines apart (CAPTCHA). In this paper,...

Recently, Vanegas and the first author introduced an algorithm to generate a large amount of emerging images. Such an algorithm uses linear representations of posets and admissible transformations of matrix representations to obtain different kind of gestalts. In this paper, we present an algorithm to extract gestalts of different types from these...

In 2009, N. J. Mitra, H. Kuo Chu, T. Yee Lee, L. Wolf, H. Yeshurun and D. Cohen-Or proposed a synthesis technique to generate emerging images of 3D objects. Such images are gestalts with the property of to be detectable by humans but difficult to process by computer vision algorithms [1]. Therefore, emerging images can be used to design a blind tes...

A. G. Zavadskij called completion to one of the algorithms of differentiation introduced by him to classify equipped posets of finite growth. In this paper, we describe the categorical properties of such an algorithm.

We establish that the algorithm VII for equipped posets induces a categorical equivalence between some quotient categories. Such an algorithm belongs to the apparatus of differentiation introduced by A. G. Zavadskij to classify equipped posets of finite growth.

We describe some properties of morphisms in categories of representations of equipped posets which can be derived for some of the algorithms of differentiation VII-XVII introduced by A. G. Zavadskij to describe tame and wild equipped posets.

It is presented a complete categorical description of one of the main differentiation algorithms for representations of posets with involution — Differentiation II — constructed originally by the second author at the end of the 80's on the base of the matrix approach. A similar categorical description of some simpler additional differentiations is...

We use P -partitions and the number of linear extensions of a given poset P, in order to obtain formulas for the number of some restricted higher dimensional compositions of a positive integer.

Visual cryptography schemes have been introduced in 1994 by Naor and Shamir [9]. These kind of schemes have been also well described by C. Blundo, A. De Santis and D.R. Stinson in [3]. In this case, a secret image I may be encoded into n shadowimages called the shares, and to give exactly one such shadowimage to each member of a group P of n person...

We use some recent results concerning universal mixed sums of squares and triangular numbers in order to prove that all positive integers n of the form n = 8m + k, m ∈ N, k ∈ {1, 2, 5, 6}, can be written as a sum of three squares of numbers of a given shape. We also use some P-partitions in order to obtain formulas for the number of some restricted...

We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral numbers, three polygonal numbers of positive rank or four cubes with two of them equal. Some identities of the Roge...

Usamos conjuntos parcialmente ordenados (posets) y grafos para obtener una fórmula para el número de particiones de un entero positivo n en cuatro cubos con dos de ellos iguales. Palabras Clave: composiciones, números cúbicos, grafos, particiones, trayectorias, posets. We use partially ordered sets (posets) and graphs in order to obtain a formula f...