In this paper, we cite two weighted rank distances (Wilcoxon rank and Log rank) to analyze how many times must a deck of 52 cards be shuffled to become sufficiently randomized. Bayer and Diaconis (1992) used the variation distance as a measure of randomness to analyze the card-shuffling. Lin (2006) used the deviation distance to analyze card-shuffling without complicated mathematics formulas. We provide two new ideas to measure the distance for card-shuffling analysis.

Let F be a field, G = (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G)be the minimum rank over all matrices in S(F,G). We give a field independent proof of a well-known result that for a tree the sum of its path cover number and minimal rank is equal to the number of vertices in the tree. The rank-spread of a vertex v of G is the difference between the minimal ranks of G and G - v, the graph obtained by deleting v and all its incident edges from G. The rank-spread of an edge is defined similarly. We derive a formula that expresses the minimal rank of a tree as the difference of sums of rank-spreads, the first being the sum of the rank-spreads of all the vertices and the second the sum of the rank-spreads of all the edges. We show that this is a special case of a more general inequality for all graphs. In proving the above results we explore how rank-spreads change as graphs are vertex-summed.

Trabajo de Investigación Requisito para optar al Título de Cirujano Dentista / Introducción: El blanqueamiento intracoronario es una opción de tratamiento estético mínimamente invasivo para aquellos pacientes con cambio de coloración de una pieza dentaria no vital. Los agentes blanqueadores que se utilizan en este procedimiento poseen como agente activo el peróxido de hidrógeno. Esta sustancia se caracteriza por su fuerte poder oxidante y alta difusión entre las estructuras dentarias. Diversos autores sugieren que podría generar algún tipo de respuesta inflamatoria al difundir y alcanzar los tejidos periodontales. Es por esto que resulta de gran interés evaluar los niveles de marcadores de destrucción tisular involucrados en procesos inflamatorios como RANKL en el espacio extraradicular, en pacientes sometidos a blanqueamiento intracoronario, ya que es un tratamiento cada vez más frecuente y no hay estudios clínicos ni registro de esto en la literatura. Material y Métodos: Se incluyeron 50 dientes tratados endodónticamente con cambio de coloración. Se conformaron de forma aleatorizada dos grupos de estudio según el agente blanqueador utilizado, G1: peróxido de hidrógeno al 35% (n=25), G2: peróxido de carbamida al 37% (n=25). El blanqueamiento intracoronario se realizó mediante la técnica walking bleach con un protocolo de 4 sesiones de blanqueamiento. Las muestras de fluido crevicular gingival para determinar los niveles de RANKL se tomaron con papel absorbente periopaper ® en 6 sitios por diente en los siguientes tiempos: antes del inicio del blanqueamiento (baseline), al finalizar cada sesión de blanqueamiento, a la semana y al primer mes de finalizado el tratamiento. Se cuantificaron los niveles de proteínas totales usando el sistema Bradford® y a partir de 100 μl de muestra eluída se midieron los niveles de RANKL mediante ELISA (Quantikine®; R&D Systems Inc.) Resultados: Los niveles de RANKL aumentaron significativamente respecto al baseline en todos los tiempos evaluados, en ambos grupos (p<0,05). No hay diferencia estadísticamente significativa entre los niveles de RANKL al comparar los grupos entre sí (p>0,05). Conclusiones: Los niveles de RANKL aumentan gradualmente luego de cada sesión de blanqueamiento y hasta el mes post blanqueamiento. / Adscrito a Proyecto PRI-ODO No. 03/016.

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A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The minimum rank of a sign pattern matrix A is the minimum of the rank of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n - 1,(where A is mxn), the conjecture is shown to hold.Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either -1 or 1 are explored. Sign patterns that almost require unique rank are also investigated.

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The purpose of this study was to determine what criteria were being used or for use in determining rank assignment of non-degree technical-vocational junior college instructors.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

We introduce the notion of diagonal ranks of semigroups,which are numerical characteristics of semigroups. Some base propertiesof diagonal ranks are obtained. A new criterion for a monoidbeing a group is obtained using diagonal ranks.For some semigroup classes we investigate whether their diagonal acts are finitely generatedor not. For the semigroups of full transformations, partial transformations andbinary relations we find the general form of the generating pairs.