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2. Medians are below joins in semimodular lattices of breadth 2

Author

Czédli, Gábor, Powers, Robert C., and White, Jeremy M.

Subjects
Mathematics - Rings and Algebras, 06C10
Abstract

Let $L$ be a lattice of finite length and let $d$ denote the minimum path length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in L^k$, an element $y$ belonging to $L$ is called a median of $\xi$ if the sum $d(y,x_1)+\cdots+d(y,x_k)$ is minimum. The lattice $L$ satisfies the $c_1$-median property if, for any $\xi=(x_1,\dots,x_k)\in L^k$ and for any median $y$ of $\xi$, $y\leq x_1\vee\dots\vee x_k$. Our main theorem asserts that if $L$ is an upper semimodular lattice of finite length and the breadth of $L$ is less than or equal to $2$, then $L$ satisfies the $c_1$-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that $2$ cannot be replaced by $3$., Comment: 12 pages, 1 figure

Published
2019

7. A strong triangle inequality in hyperbolic geometry

Author

Biró, Csaba and Powers, Robert C.

Subjects
Mathematics - Metric Geometry, Primary 51M16, Secondary 51M09
Abstract

For a triangle in the hyperbolic plane, let $\alpha,\beta,\gamma$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $\alpha,\beta, \gamma$ can be chosen uniformly in the interval $(0,\pi)$ and it is given that $\alpha+\beta+\gamma<\pi$, we show that the strong triangle inequality $a + b > c + h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded.

Published
2015

13. Consensus of Star Tree Hypergraphs

Author

McMorris, Fred R., Powers, Robert C., Bock, H. -H., editor, Gaul, W., editor, Vichi, M., editor, Arabie, Ph., editor, Baier, D., editor, Critchley, F., editor, Decker, R., editor, Diday, E., editor, Greenacre, M., editor, Lauro, C., editor, Meulman, J., editor, Monari, P., editor, Nishisato, S., editor, Ohsumi, N., editor, Opitz, O., editor, Ritter, G., editor, Schader, M., editor, Weihs, C., editor, Brito, Paula, editor, Cucumel, Guy, editor, Bertrand, Patrice, editor, and de Carvalho, Francisco, editor

Published
2007
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19. An Ordered Set Approach to Neutral Consensus Functions

Author

Crown, Gary D., Janowitz, Melvin F., Powers, Robert C., Bock, H. H., editor, Opitz, O., editor, Schader, M., editor, Diday, Edwin, editor, Lechevallier, Yves, editor, Schader, Martin, editor, Bertrand, Patrice, editor, and Burtschy, Bernard, editor

Published
1994
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40. A strong triangle inequality in hyperbolic geometry

Author

Bir��, Csaba and Powers, Robert C.

Subjects
Primary 51M16, Secondary 51M09, Mathematics - Metric Geometry, FOS: Mathematics, Metric Geometry (math.MG), Computer Science::Databases
Abstract

For a triangle in the hyperbolic plane, let $\alpha,\beta,\gamma$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $\alpha,\beta, \gamma$ can be chosen uniformly in the interval $(0,\pi)$ and it is given that $\alpha+\beta+\gamma c + h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded.

Published
2015

45. BRAC 2005

Author

Powers, Robert C., Doyle, Richard B., Thomas, Gail F., Naval Postgraduate School (U.S.), and Leadership and Human Resource Development

Subjects
Military base closures, United States
Abstract

On December 13, 2001, the 107th Congress authorized a round of base realignment and closure (BRAC) for 2005. This policy decision was preceded by years of political dispute and dialogue between Congress and the executive branch. Much of this debate centered on the Clinton Administration's privatization-in-place of two bases scheduled for closure by the 1995 BRAC Commission and the dispute over estimated BRAC costs and savings. After painstaking compromise and a national security crisis, reformed BRAC legislation was passed, balancing political leadership and national strategy with job loss and disruption to local communities. A comprehensive analysis of journal articles, books, relevant congressional records, government reports, and legislation identifies the variables that explain Congress's decision to amend the Defense Base Closure and Realignment Act of 1990 and authorize a BRAC round in 2005. Congress eventually approved BRAC 2005 because: (1) a new presidential administration concurred with DoD's argument regarding excessive infrastructure and anticipated savings from BRAC; (2) national economic conditions could not support both spending for excess infrastructure and the war on terrorism; (3) studies confirmed that most communities can rebound economically after a base closure; and (4) the improved BRAC law purportedly reduced the parochial politics of base closings. http://archive.org/details/baserealignmentn10945926 Captain, United States Marine Corps Approved for public release; distribution is unlimited.

Published
2003

47. Planar Graded Lattices and the c1-Median Property.

Author

Czédli, Gábor, Powers, Robert C., and White, Jeremy M.

Abstract

Let L be a lattice of finite length, ξ = (x1, . . ., xk) ∈ Lk, and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x1)+· · ·+d(y, xk), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x1∨· · ·∨xk. In other words, L satisfies the so-called c1-median property. [ABSTRACT FROM AUTHOR]

Published
2016
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