We have developed an artificial neural network, whose purpose is to automatically find in a datab... more We have developed an artificial neural network, whose purpose is to automatically find in a database of synthetic stellar spectra the one which best reproduces an observed spectrum. Using the equivalent widths of selected spectral lines, the network fits a set of lines related to the physical parameters in the stellar atmosphere (i.e., temperature, gravity and mass loss rate). The main advantage of this approach is its scalability.
An intercalate matrix $M$ of type $[r,s,n]$ is an $r\times s$ matrix with entries in $\{1,2,\dots... more An intercalate matrix $M$ of type $[r,s,n]$ is an $r\times s$ matrix with entries in $\{1,2,\dotsc,n\}$ such that all entries in each row are distinct, all entries in each column are distinct, and all $2 \times 2$ submatrices of $M$ have either $2$ or $4$ distinct entries. Yuzvinsky's Conjecture on intercalate matrices claims that the smallest $n$ for which there is an intercalate matrix of type $[r,s,n]$ is the Hopf-Stiefel function $r \circ s$. In this paper we prove that Yuzvinsky's Conjecture is asimptotically true for $\frac{5}{6}$ of integer pairs $(r,s)$. We prove the Conjecture for $r\le 8$, and we study it in the range $r,s\le 32$.
Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requiremen... more Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-f...
Let G be a graph without isolated vertices, α(G) and τ (G) be the stability number and the coveri... more Let G be a graph without isolated vertices, α(G) and τ (G) be the stability number and the covering number of G, respectively. The paper is divided in two parts: In the first part we study the minimum number of edges that a k-connected graph can have as a function of α(G) and τ (G). In particular, we obtain the following lower bound: q(G) ≥ α(G) − c(G) + Γ(α(G), τ (G)), where q(G) is the number of edges of G, c(G) is the number of connected components of G and Γ(α(G), τ (G)) = (α(G) − s) r 2 + s r + 1 2 , where α(G) + τ (G) = rα(G) + s with 0 ≤ s < α(G). This is a solution to an open question posed by Ore in his book [11, pag. 216], which indeed is a variant for connected graphs of a celebrated theorem of Turán [12]. In the second part of this paper, we study the relations between α(G), τ (G) and δ(G) = α(G) − σ v (G), where the σ v-cover number of a graph, denoted by σ v (G), is the maximum natural number m, such that every vertex of G belongs to a maximal independent set with at least m vertices. The main theorem of this part states that α(G) ≤ τ (G)[1 + δ(G)]. In the last section, we discuss some conjectures related to this theorem.
Communications in Computer and Information Science, 2016
The results described in this chapter are part of a larger project. The long term goal of this pr... more The results described in this chapter are part of a larger project. The long term goal of this project is to help physicians predict the hemodynamic changes, and associated risks, caused by different treatment options for brain arteriovenous malformations. First, we need to build a model of the vascular architecture of each specific patient, our approach to build these models is described in this chapter. Later we will use the model of the vascular architecture to simulate the velocity and pressure gradients of the blood flowing within the vessels, and the stresses on the blood vessel walls, before and after treatment. We are developing a program to describe each blood vessel as a parametric curve, where each point within this curve includes a normal vector that points in the opposite direction of the pressure gradient. The shape of the cross section of the vessel in each point is described as an ellipse. Our program is able to describe the geometry of a blood vessel using as an input a cloud of dots. The program allows us to model any blood vessel, and other tubular structures.
We prove terminal-Y reducibility of planar graphs with at most three terminals. The most importan... more We prove terminal-Y reducibility of planar graphs with at most three terminals. The most important consequence of our proof is that this implicitly gives an efficient algorithm with time complexity O(|E (G)| 4) for reducibility of planar graphs G with at most three terminals. It also can be used for restricted reducibility problems with more terminals. Our proof uses a very well-known translation from these operations to transformations on the medial graph.
ABSTRACT Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number... more ABSTRACT Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\alpha(G)\leq \tau(G)[1+\alpha(G)-\alpha_{v}(G)]$. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability number $\alpha(G)$, the covering number $\tau(G)$ and the number of connected components $c(G)$ of $G$. Namely, let $\alpha$ and $\tau$ be two natural numbers and let $$ \Gamma(\alpha,\tau)= \min{\sum_{i=1}^{\alpha}\bin{z_i}{2} | z_1+...+z_{\alpha}= \alpha+\tau {and} z_i \geq 0 \forall i=1,..., \alpha}. $$ Then if $G$ is any graph, we have: $$ |E(G)| \geq \alpha(G)-c(G)+ \Gamma(\alpha(G), \tau(G)). $$
... Page 306. 278 ISIDORO GITLER. ENRIQUE REYES, AND RAFAEL H. VILLARREAL belongs to R] It]. Sinc... more ... Page 306. 278 ISIDORO GITLER. ENRIQUE REYES, AND RAFAEL H. VILLARREAL belongs to R] It]. Since xatbmxi... ... [7] M. Hochster. Rings of invariants of tori. Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972), 318-337. ...
We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clut... more We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornu\'ejols \cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \cite{Seymour}) and dyadic clutters (Cornu\'ejols, Guenin and Margot \cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with th...
Let I = (x v1 ,. .. , x vq) be a square-free monomial ideal of a polynomial ring K[x 1 ,. .. , x ... more Let I = (x v1 ,. .. , x vq) be a square-free monomial ideal of a polynomial ring K[x 1 ,. .. , x n ] over an arbitrary field K and let A be the incidence matrix with column vectors v 1 ,. .. , v q. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedra and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornuéjols using an algebraic approach.
Let R = K[x 1 ,. .. , x n ] be a polynomial ring over a field K and let I be an ideal of R genera... more Let R = K[x 1 ,. .. , x n ] be a polynomial ring over a field K and let I be an ideal of R generated by a set x 1 ,. .. , x q of square-free monomials of degree two such that the graph G defined by those monomials is bipartite. We study the Rees algebra R(I) of I, by studying both the Rees cone R + A generated by the set A = {e 1 ,. .. , e n , (1 , 1),. .. , (q , 1)} and the matrix C whose columns are the vectors in A. It is shown that C is totally unimodular. We determine the irreducible representation of the Rees cone in terms of the minimal vertex covers of G. Then we compute the a-invariant of R(I).
We have developed an artificial neural network, whose purpose is to automatically find in a datab... more We have developed an artificial neural network, whose purpose is to automatically find in a database of synthetic stellar spectra the one which best reproduces an observed spectrum. Using the equivalent widths of selected spectral lines, the network fits a set of lines related to the physical parameters in the stellar atmosphere (i.e., temperature, gravity and mass loss rate). The main advantage of this approach is its scalability.
An intercalate matrix $M$ of type $[r,s,n]$ is an $r\times s$ matrix with entries in $\{1,2,\dots... more An intercalate matrix $M$ of type $[r,s,n]$ is an $r\times s$ matrix with entries in $\{1,2,\dotsc,n\}$ such that all entries in each row are distinct, all entries in each column are distinct, and all $2 \times 2$ submatrices of $M$ have either $2$ or $4$ distinct entries. Yuzvinsky's Conjecture on intercalate matrices claims that the smallest $n$ for which there is an intercalate matrix of type $[r,s,n]$ is the Hopf-Stiefel function $r \circ s$. In this paper we prove that Yuzvinsky's Conjecture is asimptotically true for $\frac{5}{6}$ of integer pairs $(r,s)$. We prove the Conjecture for $r\le 8$, and we study it in the range $r,s\le 32$.
Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requiremen... more Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-f...
Let G be a graph without isolated vertices, α(G) and τ (G) be the stability number and the coveri... more Let G be a graph without isolated vertices, α(G) and τ (G) be the stability number and the covering number of G, respectively. The paper is divided in two parts: In the first part we study the minimum number of edges that a k-connected graph can have as a function of α(G) and τ (G). In particular, we obtain the following lower bound: q(G) ≥ α(G) − c(G) + Γ(α(G), τ (G)), where q(G) is the number of edges of G, c(G) is the number of connected components of G and Γ(α(G), τ (G)) = (α(G) − s) r 2 + s r + 1 2 , where α(G) + τ (G) = rα(G) + s with 0 ≤ s < α(G). This is a solution to an open question posed by Ore in his book [11, pag. 216], which indeed is a variant for connected graphs of a celebrated theorem of Turán [12]. In the second part of this paper, we study the relations between α(G), τ (G) and δ(G) = α(G) − σ v (G), where the σ v-cover number of a graph, denoted by σ v (G), is the maximum natural number m, such that every vertex of G belongs to a maximal independent set with at least m vertices. The main theorem of this part states that α(G) ≤ τ (G)[1 + δ(G)]. In the last section, we discuss some conjectures related to this theorem.
Communications in Computer and Information Science, 2016
The results described in this chapter are part of a larger project. The long term goal of this pr... more The results described in this chapter are part of a larger project. The long term goal of this project is to help physicians predict the hemodynamic changes, and associated risks, caused by different treatment options for brain arteriovenous malformations. First, we need to build a model of the vascular architecture of each specific patient, our approach to build these models is described in this chapter. Later we will use the model of the vascular architecture to simulate the velocity and pressure gradients of the blood flowing within the vessels, and the stresses on the blood vessel walls, before and after treatment. We are developing a program to describe each blood vessel as a parametric curve, where each point within this curve includes a normal vector that points in the opposite direction of the pressure gradient. The shape of the cross section of the vessel in each point is described as an ellipse. Our program is able to describe the geometry of a blood vessel using as an input a cloud of dots. The program allows us to model any blood vessel, and other tubular structures.
We prove terminal-Y reducibility of planar graphs with at most three terminals. The most importan... more We prove terminal-Y reducibility of planar graphs with at most three terminals. The most important consequence of our proof is that this implicitly gives an efficient algorithm with time complexity O(|E (G)| 4) for reducibility of planar graphs G with at most three terminals. It also can be used for restricted reducibility problems with more terminals. Our proof uses a very well-known translation from these operations to transformations on the medial graph.
ABSTRACT Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number... more ABSTRACT Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\alpha(G)\leq \tau(G)[1+\alpha(G)-\alpha_{v}(G)]$. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability number $\alpha(G)$, the covering number $\tau(G)$ and the number of connected components $c(G)$ of $G$. Namely, let $\alpha$ and $\tau$ be two natural numbers and let $$ \Gamma(\alpha,\tau)= \min{\sum_{i=1}^{\alpha}\bin{z_i}{2} | z_1+...+z_{\alpha}= \alpha+\tau {and} z_i \geq 0 \forall i=1,..., \alpha}. $$ Then if $G$ is any graph, we have: $$ |E(G)| \geq \alpha(G)-c(G)+ \Gamma(\alpha(G), \tau(G)). $$
... Page 306. 278 ISIDORO GITLER. ENRIQUE REYES, AND RAFAEL H. VILLARREAL belongs to R] It]. Sinc... more ... Page 306. 278 ISIDORO GITLER. ENRIQUE REYES, AND RAFAEL H. VILLARREAL belongs to R] It]. Since xatbmxi... ... [7] M. Hochster. Rings of invariants of tori. Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972), 318-337. ...
We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clut... more We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornu\'ejols \cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \cite{Seymour}) and dyadic clutters (Cornu\'ejols, Guenin and Margot \cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with th...
Let I = (x v1 ,. .. , x vq) be a square-free monomial ideal of a polynomial ring K[x 1 ,. .. , x ... more Let I = (x v1 ,. .. , x vq) be a square-free monomial ideal of a polynomial ring K[x 1 ,. .. , x n ] over an arbitrary field K and let A be the incidence matrix with column vectors v 1 ,. .. , v q. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedra and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornuéjols using an algebraic approach.
Let R = K[x 1 ,. .. , x n ] be a polynomial ring over a field K and let I be an ideal of R genera... more Let R = K[x 1 ,. .. , x n ] be a polynomial ring over a field K and let I be an ideal of R generated by a set x 1 ,. .. , x q of square-free monomials of degree two such that the graph G defined by those monomials is bipartite. We study the Rees algebra R(I) of I, by studying both the Rees cone R + A generated by the set A = {e 1 ,. .. , e n , (1 , 1),. .. , (q , 1)} and the matrix C whose columns are the vectors in A. It is shown that C is totally unimodular. We determine the irreducible representation of the Rees cone in terms of the minimal vertex covers of G. Then we compute the a-invariant of R(I).
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