{"id":1608,"date":"2022-10-03T13:24:30","date_gmt":"2022-10-03T13:24:30","guid":{"rendered":"http:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1608"},"modified":"2023-02-01T23:56:48","modified_gmt":"2023-02-01T23:56:48","slug":"undergraduate-research-2","status":"publish","type":"page","link":"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1608","title":{"rendered":"Student Research"},"content":{"rendered":"<h2>Master Students<\/h2>\n<h3>\n<div id=\"jaziel1\" class=\"wp-block-image\"><\/div>\n<p><strong>Jaziel Torres Fuentes<\/strong>&nbsp;(2020-2022)<\/p>\n<\/h3>\n<ul>\n<li><a name=\"ACMLCPA\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/05\/PaperMultidimArrays-Revised-WCC2022.pdf\" target=\"_blank\"><em>Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/em><\/a>. (2022). Rafael Arce, Carlos Hern\u00e1ndez, Jos\u00e9 Ortiz, Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Linear complexity is an important parameter for arrays that are used in applications related to information security. \u00a0In this work we present new results on the multidimensional linear complexity of periodic arrays obtained using the definition and method proposed in [2,6,11]. The results include a generalization of a bound for the linear complexity, a comparison with the measure of complexity for multisequences, and computations of the complexity of arrays with periods that are not relatively prime for which the &#8220;unfolding method&#8221; does not work. We also present conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions. <\/em><\/li>\n<\/ul>\n<ul>\n<li><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Torrres_Presentation_Multi_LinearComplexity.pdf\" target=\"_blank\"><em>Multidimensional Costas Arrays and Periodic Properties<\/em><\/a>. Thesis. (2022). Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Costas arrays were first introduced for SONAR detection applications, but later became an interesting object of mathematical research. Several generalizations of Costas arrays to multiple dimensions have been proposed. In this thesis, we lay the ground for the study of multidimensional Costas arrays by proposing concepts and showing results that extend into the higher-dimensional realm most of what is known about the periodicity of two-dimensional Costas arrays. Among the most important results is, for large classes of arrays, the non-existence of multidimensional Costas arrays that preserve the Costas property in every commensurate window when extended periodically in all directions. We also extend to higher-dimensions the Golomb-Moreno Conjecture that asserts all circular Costas maps are those from the Welch construction, which had been proved in the two-dimensional case. We prove a weaker version of this conjecture in the higher-dimensional context. <\/em><\/li>\n<\/ul>\n<ul>\n<li><a name=\"MCATP\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/10\/2022-MDCostas-ArXiv.pdf\" target=\"_blank\"><em>Multidimensional Costas Arrays and Their Periodicity <\/em><\/a>. (2022). Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A novel higher-dimensional definition for Costas arrays is introduced. This defini- tion works for arbitrary dimensions and avoids some limitations of previous definitions. Some non-existence results are presented for multidimensional Costas arrays preserv- ing the Costas condition when the array is extended periodically throughout the whole space. In particular, it is shown that three-dimensional arrays with this property must have the least possible order; extending an analogous two-dimensional result by H. Taylor. Said result is conjectured to extend for Costas arrays of arbitrary dimensions.<\/em>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"lillian1\" class=\"wp-block-image\"><\/div>\n<p><strong>Lillian Gonz\u00e1lez Albino<\/strong>&nbsp;(2019-2022)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Final_Thesis_LillianGA.pdf\" target=\"_blank\">Involutions of Finite Fields Obtained From Binomials of the Form $x^m(x^{\\frac{q\u22121}{2}} + a)$<\/a><\/em>. Thesis. (2022). Lillian Gonz\u00e1lez.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Permutations of finite fields Fq have many applications ranging from cryptography and combinatorics to the theory of computation. In many of these applications, a permutation and its inverse are stored in memory. A good option to reduce the memory footprint is to generate the permutation with a polynomial at the time of implementation. A better option is to use a permutation polynomial that is its own inverse; in this case, the permutation is called an involution. In applications to cryptography, the number of fixed points is correlated with its cryptographic properties.<br \/>\nIn 2018, Zheng et al. characterized involutions of the form $x^mh(x^s)$ over $F_q$, and, in 2017, Castro et al. gave explicit formulas for monomial involutions of $F_q$ and their fixed points. The next simplest polynomials to study would be the binomials.<br \/>\n&#038;nbsp In this work we characterize involutions of $F_q$ of the form $x^m(x^{\\frac{q\u22121}{2}} + a)$ in terms of the number of fixed points. We present explicit formulas for obtaining these involutions and formulas for their fixed points. Additionally, we give an improvement of Zheng et al.\u2019s result for polynomial involutions of $F_q$ of the form $x^mh(x^{\\frac{q\u22121}{2}})$.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"seda\" class=\"wp-block-image\"><\/div>\n<p><strong>Carlos E. Seda Damiani<\/strong>&nbsp;(2018-2020)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Tesis-UPRRP_Mate_SedaDamiani_2020.pdf\" target=\"_blank\">Solvability of systems of polynomial equations with multivariate polynomials as coefficients<\/a><\/em>. Thesis. (2020). Carlos Seda.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> In [3] Castro, Moreno and Rubio generalize the results of Moreno-Moreno\u2019s theorem that gives a bound for the power of a prime p to divide the number of common zeros of the multivariate polynomials $F_1, &#8230;, F_t$ over a finite field $F_q$. This generalization regarded the coefficients of the polynomials to be uni-variate polynomials over a finite field instead of plain elements of the finite field. The result led to improve a theorem of Carlitz, for the estimation of the number of variables needed so that a system of polynomial equations with coefficients in $F_q[X]$ can have non-trivial zeros. We generalize the results of Castro, Moreno and Rubio to polynomials whose coefficients are multivariate polynomials over finite fields.<\/em><\/li>\n<\/ul>\n<hr\/\/\/>\n<h2>Undergraduate Students<\/h2>\n<h3>\n<div id=\"carlos\" class=\"wp-block-image\"><\/div>\n<p><strong>Carlos Hern\u00e1ndez Franco<\/strong>&nbsp;(2020-2022)<\/p>\n<\/h3>\n<ul>\n<li><a name=\"ACMLCPA\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/10\/2022-AnalysisComputationMDLinearComplexity-ArXiv.pdf\" target=\"_blank\"><em>Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/em><\/a>. (2022). Rafael Arce, Carlos Hern\u00e1ndez, Jos\u00e9 Ortiz, Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Linear complexity is an important parameter for arrays that are used in applications related to information security. \u00a0In this work we present new results on the multidimensional linear complexity of periodic arrays obtained using the definition and method proposed in [2,6,11]. The results include a generalization of a bound for the linear complexity, a comparison with the measure of complexity for multisequences, and computations of the complexity of arrays with periods that are not relatively prime for which the &#8220;unfolding method&#8221; does not work. We also present conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions. <\/em>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"javier\" class=\"wp-block-image\"><\/div>\n<p><strong>Javier I. Santiago V\u00e9lez<\/strong>&nbsp;(2019-2021)<\/p>\n<\/h3>\n<ul>\n<li><a name=\"PB\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/06\/FFA-21-63-Masuda-Rubio-Santiago-Revised-Formated.pdf\" target=\"_blank\" rel=\"noreferrer noopener\"><em>Permutation binomials of the form x<sup>r<\/sup>(x<sup>q &#8211; 1<\/sup>&nbsp;+&nbsp;a) over F<sub>q<sup>e<\/sup><\/sub><\/em>.<\/a> (2022). Ariane Masuda, Ivelisse Rubio, &amp; Javier Santiago.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> We present several existence and nonexistence results for permutation binomials of&nbsp;the form&nbsp;x<sup>r<\/sup>(x<sup>q &#8211; 1<\/sup>&nbsp;+&nbsp;a), where&nbsp;e&nbsp;\u2265&nbsp;2 and&nbsp;a&nbsp;\u2208&nbsp;F<sup>*<\/sup><sub>q<sup>e<\/sup><\/sub>. As a consequence, we obtain a complete characterization of such permutation binomials over&nbsp;F<sub>q<sup>2<\/sup><\/sub>&nbsp;,&nbsp;F<sub>q<sup>3<\/sup><\/sub>&nbsp;,&nbsp;F<sub>q<sup>4<\/sup><\/sub>&nbsp;,&nbsp;F<sub>p<sup>5<\/sup><\/sub>&nbsp;, and&nbsp;F<sub>p<sup>6<\/sup><\/sub>&nbsp;, where&nbsp;p&nbsp;is an odd prime.<\/em>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Technical_Report-Karla-Javier.pdf\" target=\"_blank\">Sums of Permutation Monomials<\/a><\/em>. Technical Report. (2019). Karla Borges &#038; Javier Santiago.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Las permutaciones juegan un papel importante en las comunicaciones, ya que se utilizan en una variedad de aplicaciones, desde la teor\u00eda de c\u00f3digos hasta la criptograf\u00eda. Los polinomios que generan permutaciones se llaman polinomios de permutaci\u00f3n. Nosotros estudiamos binomios de la forma $f(x) = x^m +Ax^n = x^n(x^{m\u2212n} +A)$ sobre cuerpos finitos $F_p$, donde $x^m$ y $Ax^n$ son tambi\u00e9n monomios de permutaci\u00f3n. El objetivo es encontrar las condiciones para que $f(x)$ permute $F_p$. Aqu\u00ed demostramos que $f(x)$ nunca es polinomio de permutaci\u00f3n para $F_p$, donde $q = 2p+1$, $p$ primo, y conjeturamos que si $x^n(x^{m\u2212n} +A)$ es un polinomio de permutaci\u00f3n, donde  $m \u2212 n = \\frac{p\u22121}{d}$ , entonces $A^d + (\u22121)^{d+1}$ es un residuo ed\u00e9simo.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"andres\" class=\"wp-block-image\"><\/div>\n<p><strong>Andr\u00e9s Ramos Rodr\u00edguez<\/strong>&nbsp;(2017-2021)>\/p><\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/PresentacionBoca2021-Difference_of_functions-1.pdf\" target=\"_blank\">Differences of Functions with the Same Value Multiset<\/a><\/em>. Presentation. (2021). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  In a recent article, Ullman and Velleman studied functions a from an abelian group $G$ to itself that can be expressed as a difference of two bijections $b$, $c$ from $G$ to itself. In this work we relax the condition that $b$ and $c$ are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions $b$, $c$ with same value multiset, such that $a = b \u2212 c$. As a consequence, we obtain a stronger version of Hall\u2019s theorem, which gives a description of $b$ and $c$ in terms of $a$. We conclude by presenting further directions and questions that relate this new approach to applications of Hall\u2019s theorem.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/PosterAndresDylan2019.pdf\" target=\"_blank\">Binomials that do not produce permutations<\/a><\/em>. Poster. (2019). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Permutations have important applications in fields such as coding theory and cryptography. We study binomials of the form $x^m(x^{\\frac{q-3}{2} }+B)$ over finite fields $F_q$. In this research we prove that this class of binomials never permutes a finite fields when: (a) $A = \\alpha^k$, where $k$ is odd and $alpha$ is a primitive root of the field, (b) $A = \\alpha^k$, where $k$ is even, $\\alpha$ is a primitive root of the field and $q = 4h+1$, $h \u2208 N$. We are left with the case $A = \\alpha^k$ where $k$ is even and $q = 4h + 3$, $h \u2208 N$. We conjecture that the remaining class of polynomials do not permute $F_q$.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/AndresDylan-TechReportDic2018.pdf\" target=\"_blank\">Involutions of Finite Fields Obtained from Binomials<\/a><\/em>. Technical Report. (2019). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Las permutaciones de cuerpos finitos tienen aplicaciones en varios campos de las Matem\u00e1ticas y Ciencias de C\u00f3mputos, como lo es la Criptograf\u00eda. En espec\u00edfico, las permutaciones que son su propia inversa (involuciones) son de inter\u00e9s porque ofrecen una ventaja al implementarlas ya que requieren menos memoria. Estudiamos binomios de la forma $x^m(x \\frac{q\u22123}{2} +A)$ sobre$F_q$ con el prop\u00f3sito de encontrar las condiciones en $A$ y $q$ tal que el binomio sea una involuci\u00f3n. Aqu\u00ed probaremos que, si $m = 1$, esta familia de binomios nunca produce permutaciones cuando de $F_q$ donde $q$ es potencia de un primo impar. Conjeturamos que esto es cierto para toda $m$.<\/em><\/li>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"dylan\" class=\"wp-block-image\"><\/div>\n<p><strong>Dylan Cruz Fonseca<\/strong>&nbsp;(2017 &#8211; 2020)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/PresentacionBoca2021-Difference_of_functions-1.pdf\" target=\"_blank\">Differences of Functions with the Same Value Multiset<\/a><\/em>. Presentation. (2021). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  In a recent article, Ullman and Velleman studied functions a from an abelian group $G$ to itself that can be expressed as a difference of two bijections $b$, $c$ from $G$ to itself. In this work we relax the condition that $b$ and $c$ are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions $b$, $c$ with same value multiset, such that $a = b \u2212 c$. As a consequence, we obtain a stronger version of Hall\u2019s theorem, which gives a description of $b$ and $c$ in terms of $a$. We conclude by presenting further directions and questions that relate this new approach to applications of Hall\u2019s theorem.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/PosterAndresDylan2019.pdf\" target=\"_blank\">Binomials that do not produce permutations<\/a><\/em>. Poster. (2019). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Permutations have important applications in fields such as coding theory and cryptography. We study binomials of the form $x^m(x^{\\frac{q-3}{2} }+B)$ over finite fields $F_q$. In this research we prove that this class of binomials never permutes a finite fields when: (a) $A = \\alpha^k$, where $k$ is odd and $alpha$ is a primitive root of the field, (b) $A = \\alpha^k$, where $k$ is even, $\\alpha$ is a primitive root of the field and $q = 4h+1$, $h \u2208 N$. We are left with the case $A = \\alpha^k$ where $k$ is even and $q = 4h + 3$, $h \u2208 N$. We conjecture that the remaining class of polynomials do not permute $F_q$.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/AndresDylan-TechReportDic2018.pdf\" target=\"_blank\">Involutions of Finite Fields Obtained from Binomials<\/a><\/em>. Technical Report. (2019). Dylan Cruz  &#038; Andr\u00e9s Ramos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Las permutaciones de cuerpos finitos tienen aplicaciones en varios campos de las Matem\u00e1ticas y Ciencias de C\u00f3mputos, como lo es la Criptograf\u00eda. En espec\u00edfico, las permutaciones que son su propia inversa (involuciones) son de inter\u00e9s porque ofrecen una ventaja al implementarlas ya que requieren menos memoria. Estudiamos binomios de la forma $x^m(x \\frac{q\u22123}{2} +A)$ sobre $F_q$ con el prop\u00f3sito de encontrar las condiciones en $A$ y $q$ tal que el binomio sea una involuci\u00f3n. Aqu\u00ed probaremos que, si $m = 1$, esta familia de binomios nunca produce permutaciones cuando de $F_q$ donde $q$ es potencia de un primo impar. Conjeturamos que esto es cierto para toda $m$.<\/em><\/li>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"quinones\" class=\"wp-block-image\"><\/div>\n<p><strong>Luis Qui\u00f1ones<\/strong>&nbsp;(2019)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Torrres_Presentation_Multi_LinearComplexity.pdf\" target=\"_blank\">Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/a><\/em>. Presentation. (2019). Luis Qui\u00f1ones &#038; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Sequences and multidimensional periodic arrays with entries in finite fields have important applications in coding theory and cryptography. The correlations and the linear complexity of the sequences and multidimensional arrays are important parameters for many applications, especially those related to information security, and hardware implementation. The general goal of this research is to study different constructions of sequences and multidimensional periodic arrays and their correlation and complexity parameters. We give a proof for the exact value of the complexity of an array constructed using the composition method that was previously conjectured. <\/em><\/li>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"jaziel\" class=\"wp-block-image\"><\/div>\n<p><strong>Jaziel Torres Fuentes<\/strong>&nbsp;(2019-2020)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Torrres_Presentation_Multi_LinearComplexity.pdf\" target=\"_blank\">Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/a><\/em>. Presentation. (2019). Luis Qui\u00f1ones &#038; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  Sequences and multidimensional periodic arrays with entries in finite fields have important applications in coding theory and cryptography. The correlations and the linear complexity of the sequences and multidimensional arrays are important parameters for many applications, especially those related to information security, and hardware implementation. The general goal of this research is to study different constructions of sequences and multidimensional periodic arrays and their correlation and complexity parameters. We give a proof for the exact value of the complexity of an array constructed using the composition method that was previously conjectured.<\/em><\/li>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"borges\" class=\"wp-block-image\"><\/div>\n<p><strong>Karla Borges<\/strong>&nbsp;(2019)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Technical_Report-Karla-Javier.pdf\" target=\"_blank\">Sums of Permutation Monomials<\/a><\/em>. Technical Report. (2019). Karla Borges &#038; Javier Santiago.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Las permutaciones juegan un papel importante en las comunicaciones, ya que se utilizan en una variedad de aplicaciones, desde la teor\u00eda de c\u00f3digos hasta la criptograf\u00eda. Los polinomios que generan permutaciones se llaman polinomios de permutaci\u00f3n. Nosotros estudiamos binomios de la forma $f(x) = x^m +Ax^n = x^n(x^{m\u2212n} +A)$ sobre cuerpos finitos $F_p$, donde $x^m$ y $Ax^n$ son tambi\u00e9n monomios de permutaci\u00f3n. El objetivo es encontrar las condiciones para que $f(x)$ permute $F_p$. Aqu\u00ed demostramos que $f(x)$ nunca es polinomio de permutaci\u00f3n para $F_p$, donde $q = 2p+1$, $p$ primo, y conjeturamos que si $x^n(x^{m\u2212n} +A)$ es un polinomio de permutaci\u00f3n, donde  $m \u2212 n = \\frac{p\u22121}{d}$ , entonces $A^d + (\u22121)^{d+1}$ es un residuo ed\u00e9simo.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"lillian\" class=\"wp-block-image\"><\/div>\n<p><strong>Lillian Gonz\u00e1lez Albino<\/strong>&nbsp;(2015-2019)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/Lillian-Tech-Report-con-Apendices2018.pdf\" target=\"_blank\">Involutions of finite fields obtained from binomials <\/a><\/em>. Technical Report. (2018). Lillian Gonz\u00e1lez.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Las permutaciones sobre cuerpos finitos son importantes ya que tienen aplicaciones desde cifrados de voz hasta teor\u00eda de computabilidad y criptograf\u00eda. Polinomios que generan permutaciones son llamados polinomios de permutaci\u00f3n; si estos polinomios son su propia inversa, son llamados involuciones. En esta investigaci\u00f3n, queremos caracterizar binomios de involuci\u00f3n la forma $x^m(x^{\\frac{q\u22121}{2}} + a)$ sobre cuerpos finitos de caracter\u00edstica $p$.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/11\/LillianGonzalez-ImplemAlgoriCompLineal-Mayo2016.pdf\" target=\"_blank\">Implementation of an algorithm to compute linear complexity of periodic arrays<\/a><\/em>. Technical Report. (2016). Lillian Gonz\u00e1lez.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  Moreno y Terkel presentaron una construcci\u00f3n de arreglos peri\u00f3dicos multidimensionales con propiedades de buena correlaci\u00f3n y complejidad. Para analizar la complejidad de estos arreglos se \u201cdesenvolv\u00edan&#8221; utilizando el Teorema del Residuo Chino y luego se analizaban con el algoritmo Berlekamp-Massey. Pero este m\u00e9todo presentaba una restricci\u00f3n en las dimensiones del arreglo pues ten\u00edan que ser coprimos. En [4, 5] fue propuesto una teor\u00eda nueva para analizar la complejidad linear de arreglos multidimensionales que no tiene esta restricci\u00f3n; provee una definici\u00f3n de complejidad linear de arreglos multidimensionales que es consistente con la definici\u00f3n de complejidad linear en una dimensi\u00f3n. En esta investigaci\u00f3n se implementa el algoritmo desarrollado en [3] sobre la teor\u00eda presentada en [4, 5].<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"jeffry\" class=\"wp-block-image\"><\/div>\n<p><strong>Jeffry Matos<\/strong>&nbsp;(2015-2016)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/TesinaJeff.pdf\" target=\"_blank\">Complexity of Multidimensional Periodic Arrays<\/a><\/em>. Bachelor Thesis. (2016). Jeffry Matos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Para obtener arreglos que puedan tener aplicaciones en sistemas que utilizan marcas de agua digitales, criptograf\u00eda o se\u00f1ales de radar multi-blanco, los mismos deben poseer buenas propiedades de correlaci\u00f3n y complejidad [5]. Por esto es imperativo el poder tener una medida de la complejidad de un arreglo multidimensional que no presente limitaciones. Las bases de Gr\u00f6bner [3] son conjuntos de polinomios que poseen propiedades algor\u00edtmicas muy ricas. Al utilizarlas, es posible generalizar la definici\u00f3nn de la medida de la complejidad lineal de una sucesi\u00f3n a un arreglo. Si calculamos alguna base de Gr\u00f6ebner que genere los polinomios asociados al arreglo peri\u00f3dico multidimensional, entonces podremos examinar la complejidad del arreglo porque esto permite obtener un conjunto $\\Delta_{Val(A)}$ cuyo tama\u00f1o define la complejidad lineal del arreglo. En este trabajo se estudian algunas propiedades de arreglos construidos utilizando un m\u00e9todo propuesto por Moreno y Tirkel [7]. Se utiliz\u00f3 el algoritmo de Rubio-Sweedler [9] para computar las bases de Gr\u00f6bner del arreglo y con las mismas examinar la complejidad de arreglos peri\u00f3dicos multidimensionales obtenidos con construcciones en [7], comparar con la complejidad de los arreglos vistos como multisucesiones [8], formular conjeturas y obtener resultados.  <\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"pacheco\" class=\"wp-block-image\"><\/div>\n<p><strong>Natalia Pacheco-Tallaj<\/strong>&nbsp;(2015-2016) <\/p>\n<\/h3>\n<ul>\n<li><a name=\"EFMIFF\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/2017-Involuciones-PublicadoAdvMathComm.pdf\" target=\"_blank\" rel=\"noreferrer noopener\"><em>Explicit Formulas for Monomial Involutions over Finite Fields<\/em>.<\/a>&nbsp;(2017). Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj &amp; Ivelisse Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of Fq that are given by monomials and for their fixed points.<\/em><\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/DicksonPolynomials.pdf\" target=\"_blank\">Dickson Polynomials that Generate Involutions with More Than 3 Fixed Points<\/a><\/em>. Technical Report. (2016). Natalia Pacheco.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Let $F_q$ be a finite field of prime order $q$. This research studies the Dickson Polynomials $D_i(x, a)$ that generate a permutation of $F_q$ such that $D_i$ decomposes into cycles of uniform length $2$, i.e. $D_i$ is an involution. Our goal is to provide a necessary and sufficient formula to construct $I$ such that the Dickson Polynomial $D_i(x, a)$ has a particular amount d of fixed points. We wish to find a formula for $I$ in terms of $d$ and $q$, similarly to what we were able to achieve with Permutation Monomials in [1].<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"oscar\" class=\"wp-block-image\"><\/div>\n<p><strong>Oscar E. Gonz\u00e1lez<\/strong>&nbsp;(2014-2016)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/Raul-Oscar-paper.pdf\" target=\"_blank\">Exact divisibility of exponential sums associated to elementary symmetric Boolean functions<\/a><\/em>. (2017). Oscar E. Gonz\u00e1lez, Ra\u00fal E. Negr\u00f3n, Francis N. Castro, Luis A. Medina &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/AficheOscarRaul.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>In this paper, we present an elementary method to compute or estimate the exact 2-divisibility of exponential sums associated to symmetric Boolean functions. As a direct consequence of these results, we prove some of the open cases of Cusick-Li-St\u0103nic\u0103\u2019s conjecture about balanced symmetric Boolean functions.<\/em><\/li>\n<\/ul>\n<ul>\n<li><a name=\"NFBSFGCLSC\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2017-GeneralizationCusick-DesignsCodesCrypto.pdf\" target=\"_blank\"><em>New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica&#8217;s conjecture<\/em>.<\/a> (2015). Rafael A. Arce-Nazario, Francis N. Castro, Oscar E. Gonz\u00e1lez, Luis A. Medina &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> In general, the methods to estimate the p-divisibility of exponen- tial sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact p-divisibility of exponential sums over a prime field. The results here allow us to compute the exact p-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.<\/em>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"negron\" class=\"wp-block-image\"><\/div>\n<p><strong>Ra\u00fal Negr\u00f3n Otero<\/strong>&nbsp;(2014-2016)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/Raul-Oscar-paper.pdf\" target=\"_blank\">Exact divisibility of exponential sums associated to elementary symmetric Boolean functions<\/a><\/em>. (2017). Oscar E. Gonz\u00e1lez, Ra\u00fal E. Negr\u00f3n, Francis N. Castro, Luis A. Medina &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/AficheOscarRaul.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>In this paper, we present an elementary method to compute or estimate the exact 2-divisibility of exponential sums associated to symmetric Boolean functions. As a direct consequence of these results, we prove some of the open cases of Cusick-Li-St\u0103nic\u0103\u2019s conjecture about balanced symmetric Boolean functions.<\/em><\/li>\n<\/ul>\n<hr\/>\n<h3>\n<div id=\"collazo\" class=\"wp-block-image\"><\/div>\n<p><strong>Ram\u00f3n L. Collazo<\/strong>&nbsp;(2013-2014)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/AficheOscarRaul.pdf\" target=\"_blank\">Exact $p$-divisibility  of exponential sums of polynomials over finite fields<\/a><\/em>. Poster. (2015). Ram\u00f3n L. Collazo, Julio J. del Cruz, Daniel E. Ram\u00edrez, Francis N. Castro &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\\{x \\in F|f(x) = 1\\}| = 2^{n\u22121}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-St\u0103nic\u0103\u2019s conjecture about the non-existence of certain balanced Boolean funtions.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/Articulo_Ingenios_Version_2-12.pdf\" target=\"_blank\">Solvability of Systems of Polynomial Equations over Finite Fields<\/a><\/em>.  Technical Report. (2014). Ram\u00f3n L. Collazo, Julio J.  de la Cruz &amp; Daniel E. Ram\u00edrez.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro &#038; Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"delacruz\" class=\"wp-block-image\"><\/div>\n<p><strong>Julio De La Cruz<\/strong>&nbsp;(2013-2014)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/AficheOscarRaul.pdf\" target=\"_blank\">Exact $p$-divisibility  of exponential sums of polynomials over finite fields<\/a><\/em>. Poster. (2015). Ram\u00f3n L. Collazo, Julio J. del Cruz, Daniel E. Ram\u00edrez, Francis N. Castro &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\\{x \\in F|f(x) = 1\\}| = 2^{n\u22121}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-St\u0103nic\u0103\u2019s conjecture about the non-existence of certain balanced Boolean funtions.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/Articulo_Ingenios_Version_2-12.pdf\" target=\"_blank\">Solvability of Systems of Polynomial Equations over Finite Fields<\/a><\/em>.  Technical Report. (2014). Ram\u00f3n L. Collazo, Julio J.  de la Cruz &amp; Daniel E. Ram\u00edrez.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro &#038; Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"calderon\" class=\"wp-block-image\"><\/div>\n<p><strong>Daniel Ramirez Calderon<\/strong>&nbsp;(2013-2014)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/AficheOscarRaul.pdf\" target=\"_blank\">Exact $p$-divisibility  of exponential sums of polynomials over finite fields<\/a><\/em>. Poster. (2015). Ram\u00f3n L. Collazo, Julio J. del Cruz, Daniel E. Ram\u00edrez, Francis N. Castro &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\\{x \\in F|f(x) = 1\\}| = 2^{n\u22121}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-St\u0103nic\u0103\u2019s conjecture about the non-existence of certain balanced Boolean funtions.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/Articulo_Ingenios_Version_2-12.pdf\" target=\"_blank\">Solvability of Systems of Polynomial Equations over Finite Fields<\/a><\/em>.  Technical Report. (2014). Ram\u00f3n L. Collazo, Julio J.  de la Cruz &amp; Daniel E. Ram\u00edrez.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro &#038; Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"christian\" class=\"wp-block-image\"><\/div>\n<p><strong>Christian A. Rodr\u00edguez Encarnaci\u00f3n<\/strong>&nbsp;(2011-2014)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/PermutationTrinomials.pdf\" target=\"_blank\">Generating Permutation Trinomials over Finite Fields<\/a><\/em>. Technical Report. (2014). Christian A. Rodr\u00edguez &#038; Alex D. Santos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>Permutation polynomials over finite fields have many applications in areas such as coding theory and cryptography. We consider polynomials of the form $F_{a,b} (X) = X(X^{\\frac{q\u22121}{d_1}} + aX^{\\frac{q\u22121}{d_2}} + b)$, where $a, b \\in {F_q}^*$ and $d_1 < d_2$. We construct partitions of these polynomials where polynomials in the same partition have value sets of equal cardinality. As a consequence we provide families of permutation polynomials.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/Poster-JMM-Version-Final.pdf\" target=\"_blank\">Number of Permutation Polynomials<\/a><\/em>. Poster. (2014). Christian A. Rodr\u00edguez &#038; Alex D. Santos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>Given $q = p^r$, $d_1$ and $d_2$, we construct partitions of polynomials of the form $F_{a,b} (X) = X(X^{\\frac{q\u22121}{d_1}} + aX^{\\frac{q\u22121}{d_2}} + b)$, where $a, b \\in {F_q}^*$, that have value sets of the same cardinality. As a consequence we provide families of permutation polynomials and of polynomials with small value sets.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"alex\" class=\"wp-block-image\"><\/div>\n<p><strong>Alex D. Santos Sosa<\/strong>&nbsp;(2011-2014)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/PermutationTrinomials.pdf\" target=\"_blank\">Generating Permutation Trinomials over Finite Fields<\/a><\/em>. Technical Report. (2014). Christian A. Rodr\u00edguez &#038; Alex D. Santos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>Permutation polynomials over finite fields have many applications in areas such as coding theory and cryptography. We consider polynomials of the form $F_{a,b} (X) = X(X^{\\frac{q\u22121}{d_1}} + aX^{\\frac{q\u22121}{d_2}} + b)$, where $a, b \\in {F_q}^*$ and $d_1 < d_2$. We construct partitions of these polynomials where polynomials in the same partition have value sets of equal cardinality. As a consequence we provide families of permutation polynomials.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/Poster-JMM-Version-Final.pdf\" target=\"_blank\">Number of Permutation Polynomials<\/a><\/em>. Poster. (2014). Christian A. Rodr\u00edguez &#038; Alex D. Santos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>Given $q = p^r$, $d_1$ and $d_2$, we construct partitions of polynomials of the form $F_{a,b} (X) = X(X^{\\frac{q\u22121}{d_1}} + aX^{\\frac{q\u22121}{d_2}} + b)$, where $a, b \\in {F_q}^*$, that have value sets of the same cardinality. As a consequence we provide families of permutation polynomials and of polynomials with small value sets.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"accetta\" class=\"wp-block-image\"><\/div>\n<p><strong>Jean-Karlo Accetta<\/strong>&nbsp;(2010-2011)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/LastVersionWNeditedIve.pdf\" target=\"_blank\">N\u00famero de Waring en Cuerpos Finitos<\/a><\/em>.  Technical Report. (2011). Zahir Mejias &amp; Jean-Karlo Accetta.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/Afiche-2.0.pdf\" target=\"_blank\">[Poster]<\/a>&nbsp;<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/PRISM_last.pdf\" target=\"_blank\">[Slideshow]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> El n\u00famero de Waring es el n\u00famero m\u00ednimo de variables que necesita una ecuaci\u00f3n de la forma &nbsp;x<sub>1<\/sub><sup>d<\/sup>&nbsp;+ \u00b7\u00b7\u00b7 + x<sub>n<\/sub><sup>d<\/sup>&nbsp;= \u03b2&nbsp; para que tenga soluci\u00f3n en los n\u00fameros naturales para cualquier valor de la constante en los n\u00fameros naturales. Trabajamos en el estudio de generalizaciones de este problema cuando los valores de la constante y las soluciones se definen en cuerpos finitos. Hemos desarrollado un programa para calcular el n\u00famero de Waring, y con el mismo hemos mejorado en resultados anteriormente publicados.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"zahir\" class=\"wp-block-image\"><\/div>\n<p><strong>Zahir Mejias<\/strong>&nbsp;(2010-2011)<\/p>\n<\/h3>\n<ul>\n<li><em><a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/LastVersionWNeditedIve.pdf\" target=\"_blank\">N\u00famero de Waring en Cuerpos Finitos<\/a><\/em>.  Technical Report. (2011). Zahir Mejias &amp; Jean-Karlo Accetta.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/Afiche-2.0.pdf\" target=\"_blank\">[Poster]<\/a>&nbsp;<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2011\/04\/PRISM_last.pdf\" target=\"_blank\">[Slideshow]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> El n\u00famero de Waring es el n\u00famero m\u00ednimo de variables que necesita una ecuaci\u00f3n de la forma &nbsp;x<sub>1<\/sub><sup>d<\/sup>&nbsp;+ \u00b7\u00b7\u00b7 + x<sub>n<\/sub><sup>d<\/sup>&nbsp;= \u03b2&nbsp; para que tenga soluci\u00f3n en los n\u00fameros naturales para cualquier valor de la constante en los n\u00fameros naturales. Trabajamos en el estudio de generalizaciones de este problema cuando los valores de la constante y las soluciones se definen en cuerpos finitos. Hemos desarrollado un programa para calcular el n\u00famero de Waring, y con el mismo hemos mejorado en resultados anteriormente publicados.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"jeranfer\" class=\"wp-block-image\"><\/div>\n<p><strong>Jeranfer Berm\u00fadez<\/strong>&nbsp;(2007-2010)<\/p>\n<\/h3>\n<ul>\n<li> <em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/1_Poster_Study-of-r-Orthogonality-of-latin-squares_March2010.pdf\" target=\"_blank\">Study of r-Orthogonality of Latin Squares<\/a><\/em>. Poster. (2010). Jeranfer Berm\u00fadez &amp; Lourdes Morales.<br \/>\n<br \/>\n<a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/2_Presentation_Study-of-r-Orthogonality-of-latin-squares_March2010_C.pdf\" target=\"_blank\">[Presentation]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A Latin square (LS) of order <em>n<\/em>, is an <em>n \u00d7 n<\/em> array of <em>n<\/em> different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory, Projective Geometry and others. Two Latin squares of order <em>n<\/em> are said to be <em>r<\/em>-orthogonal if when the squares are superimposed we get <em>r<\/em> distinct ordered pairs of symbols. We study generalizations of the <em>r<\/em>-orthogonality to sets of LS\u02bcs. In this work we present preliminary results on some properties of these generalizations. <\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/tech_report_final_draft.pdf\" target=\"_blank\">Some Properties of Latin Squares \u2013 Study of Mutually Orthogonal Latin Squares<\/a><\/em>. Technical Report. (2009). Jeranfer Berm\u00fadez &amp; Lourdes Morales.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A latin square of order <em>n<\/em> is an <em>n x n<\/em> matrix containing <em>n<\/em> distinct symbols (usually denoted by the non-negative integers from 0 to <em>n-1<\/em>) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be orthogonal if, when superimposed, all the pairs that are formed are different. In our research we look for new constructions of mutually orthogonal latin squares (MOLS). We present some partial results and conjectures related to this.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/5_Technical-Report-_Study-of-r-Orthogonality-for-Latin-Squares_JernaRichardReynaldo_May2009.pdf\" target=\"_blank\">Study of r-Orthogonality for Latin Squares<\/a><\/em>. Technical Report. (2009). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A Latin square (LS) of order n, is an n x n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory and Cryptography. The famous Sudoku squares are examples of Latin squares. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. In this work we study generalizations of r-orthogonality to sets of LSs. Also, we will present some preliminary results on some of the properties of these generalizations.<\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/3_Technical-Report_Study-of-Latin-Square-Generating-Polynomials_Jeranfer_July2009.pdf\" target=\"_blank\">Study of Latin Square Generating Polynomials<\/a><\/em>. Technical Report. (2009). Jeranfer Berm\u00fadez.<br \/>\n<br \/>\n<em><strong>Abstract<\/strong>: A Latin Square of order <em>n<\/em> is an <em>n x n<\/em> matrix of <em>n<\/em> distinct elements (usually represented with the numbers from 0 to <em>n<\/em>-1), where each element appears in each row and in each column exactly once. Their various applications in Coding Theory, Cryptography and Processor Scheduling, just to mention a few, make Latin Squares a very interesting field to study. We our interested in patterns or tendencies that could relate Latin Squares, or sets of Latin Squares, or could give Maximum Orthogonality. For that reason we look for another way of constructing Latin Squares which we had not studied previously, using polynomials over finite fields.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/LDPC-Paper.pdf\" target=\"_blank\">Low-Density Parity-Check Codes<\/a><\/em>. Poster. (2007). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Poster-LDPC2007final.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Los c\u00f3digos correctores de errores se utilizan en la comunicaci\u00f3n digital para detectar y corregir errores en la transmisi\u00f3n o almacenamiento de la informaci\u00f3n. En esta investigaci\u00f3n estudiamos c\u00f3digos Low-Density Parity-Check (LDPC). Estos c\u00f3digos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro prop\u00f3sito es encontrar construcciones que resulten en c\u00f3digos LDPC eficientes. Para esto estudiamos si existe relaci\u00f3n entre la descomposici\u00f3n c\u00edclica de la permutaci\u00f3n y el girth del grafo.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"lourdes\" class=\"wp-block-image\"><\/div>\n<p><strong>Lourdes Morales<\/strong>&nbsp;(2007-2010)<\/p>\n<\/h3>\n<ul>\n<li> <em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/1_Poster_Study-of-r-Orthogonality-of-latin-squares_March2010.pdf\" target=\"_blank\">Study of r-Orthogonality of Latin Squares<\/a><\/em>. Poster. (2010). Jeranfer Berm\u00fadez &amp; Lourdes Morales.<br \/>\n<br \/>\n<a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/2_Presentation_Study-of-r-Orthogonality-of-latin-squares_March2010_C.pdf\" target=\"_blank\">[Presentation]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A Latin square (LS) of order <em>n<\/em>, is an <em>n \u00d7 n<\/em> array of <em>n<\/em> different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory, Projective Geometry and others. Two Latin squares of order <em>n<\/em> are said to be <em>r<\/em>-orthogonal if when the squares are superimposed we get <em>r<\/em> distinct ordered pairs of symbols. We study generalizations of the <em>r<\/em>-orthogonality to sets of LS\u02bcs. In this work we present preliminary results on some properties of these generalizations. <\/em><\/li>\n<\/ul>\n<ul>\n<li> <a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/tech_report_2_final_draft.pdf\" target=\"_blank\"><em>Some Properties of Latin Squares &#8211; Study of Maximal Sets of Latin Squares<\/em><\/a>. Technical Report. (2010). Lourdes Morales.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A latin square of order <em>n<\/em> is an <em>n x n<\/em> matrix containing <em>n<\/em> distinct symbols (usually denoted by the non-negative integers from 0 to <em>n-1<\/em>) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be <em>r<\/em>-orthogonal if you get <em>r<\/em> distinct ordered pairs when you superimpose them. In our research we look for new constructions of maximal sets of latin squares. We present some partial results and conjectures related to this.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/tech_report_final_draft.pdf\" target=\"_blank\">Some Properties of Latin Squares \u2013 Study of Mutually Orthogonal Latin Squares<\/a><\/em>. Technical Report. (2009). Jeranfer Berm\u00fadez &amp; Lourdes Morales.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A latin square of order <em>n<\/em> is an <em>n x n<\/em> matrix containing <em>n<\/em> distinct symbols (usually denoted by the non-negative integers from 0 to <em>n-1<\/em>) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be orthogonal if, when superimposed, all the pairs that are formed are different. In our research we look for new constructions of mutually orthogonal latin squares (MOLS). We present some partial results and conjectures related to this.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"claudio\" class=\"wp-block-image\"><\/div>\n<p><strong>Armando Claudio Jim\u00e9nez<\/strong>&nbsp;(2008-2009)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/FSLS_armando.pdf\" target=\"_blank\">De Cuadrados de Frecuencia a Cuadrados Latinos<\/a><\/em>. Slideshow. (2009). Armando Claudio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A Latin Square of order $n$ is an $n \\times n$ matrix of $n$ distinct elements in which each element appears exactly once in each row and once in each column. These Latin Squares have various applications such as Coding Theory, Cryptography, Processor Scheduling, among others. By an $F(n; \\lambda_1, \\cdots , \\lambda_m)$ frequency square is meant an $n \u00d7 n$ array in which each of the numbers $i$ with $1 \\leq i \\leq m$ appears exactly $\\lambda_i$ times in each row and each column. Let $(F; \\lambda_1, \\cdots , \\lambda_m)$ be a frequency square of order $n$. For $i = 1, \u00b7 \u00b7 \u00b7 , m$, by an $i \u2212 transversal$ is meant a set of $n$ cells, one in each row and one in each column, each containing the symbol $i$.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"richard\" class=\"wp-block-image\"><\/div>\n<p><strong>Richard Garc\u00eda Lebr\u00f3n<\/strong>&nbsp;(2008-2009)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/12\/5_Technical-Report-_Study-of-r-Orthogonality-for-Latin-Squares_JernaRichardReynaldo_May2009.pdf\" target=\"_blank\">Study of r-Orthogonality for Latin Squares<\/a><\/em>. Technical Report. (2009). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A Latin square (LS) of order n, is an n x n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory and Cryptography. The famous Sudoku squares are examples of Latin squares. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. In this work we study generalizations of r-orthogonality to sets of LSs. Also, we will present some preliminary results on some of the properties of these generalizations.<\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/LDPC-Paper.pdf\" target=\"_blank\">Low-Density Parity-Check Codes<\/a><\/em>. Poster. (2007). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Poster-LDPC2007final.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Los c\u00f3digos correctores de errores se utilizan en la comunicaci\u00f3n digital para detectar y corregir errores en la transmisi\u00f3n o almacenamiento de la informaci\u00f3n. En esta investigaci\u00f3n estudiamos c\u00f3digos Low-Density Parity-Check (LDPC). Estos c\u00f3digos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro prop\u00f3sito es encontrar construcciones que resulten en c\u00f3digos LDPC eficientes. Para esto estudiamos si existe relaci\u00f3n entre la descomposici\u00f3n c\u00edclica de la permutaci\u00f3n y el girth del grafo.<\/em><\/li>\n<\/ul>\n<hr \/>\n<h3>\n<div id=\"reynaldo\" class=\"wp-block-image\"><\/div>\n<p><strong>Reynaldo L\u00f3pez<\/strong>&nbsp;(2007-2009)<\/p>\n<\/h3>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/LDPC-Paper.pdf\" target=\"_blank\">Low-Density Parity-Check Codes<\/a><\/em>. Poster. (2007). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Poster-LDPC2007final.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Los c\u00f3digos correctores de errores se utilizan en la comunicaci\u00f3n digital para detectar y corregir errores en la transmisi\u00f3n o almacenamiento de la informaci\u00f3n. En esta investigaci\u00f3n estudiamos c\u00f3digos Low-Density Parity-Check (LDPC). Estos c\u00f3digos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro prop\u00f3sito es encontrar construcciones que resulten en c\u00f3digos LDPC eficientes. Para esto estudiamos si existe relaci\u00f3n entre la descomposici\u00f3n c\u00edclica de la permutaci\u00f3n y el girth del grafo.<\/em><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Master Students Jaziel Torres Fuentes&nbsp;(2020-2022) Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays. (2022). Rafael Arce, Carlos Hern\u00e1ndez, Jos\u00e9 Ortiz, Ivelisse Rubio, &amp; Jaziel Torres. Abstract: Linear complexity is an important parameter for arrays that are used in &hellip; <a href=\"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1608\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"onecolumn-page.php","meta":{"footnotes":""},"class_list":["post-1608","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1608"}],"version-history":[{"count":90,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1608\/revisions"}],"predecessor-version":[{"id":1611,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1608\/revisions\/1611"}],"wp:attachment":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}