2,721 research outputs found
Generalizaciones de la fórmula de Graham-Pollak. Una prueba combinatoria.
No full textA famous formula of Graham and Pollak (1971) describes the determinant of the distance matrix of a
tree T of order n:
detM(T) = (−1)n−1(n − 1)2n−2.
Remarkably, this formula shows that the value of this determinant only depends on n, the number of
vertices of T, and not on its tree structure.
In this senior thesis, we present a long sought after combinatorial proof for the elegant formula of Graham
and Pollak. Moreover, we show how our framework can be used to derive combinatorially many of its
existing generalizations, and even to obtain suggestions for new ones.Una famosa fórmula de Graham y Pollak (1971) describe el determinante de la distancia matriz de un
árbol T de orden n.
detM(T) = (−1)n−1(n − 1)2n−2.
Sorprendentemente, esta fórmula revela que el valor de este determinante solo depende de n, su número
de vértices, pero no de la estructura del árbol.
En este trabajo de fin de grado presentamos la tan buscada prueba combinatoria de la elegante fórmula
de Graham y Pollak. Además, mostramos como nuestro marco de trabajo puede usarse para derivar
combinatoriamente varias de sus generalizaciones e, incluso, sugerir otras nuevas.Universidad de Sevilla. Grado en Matemática
From Eggs to the Stars
No full textJane Pollak is a Westport, Connecticut, artist who started her career as a high school art teacher. She has now branched out into public speaking, is the author of two books, and embraces the life of entrepreneur as a sole proprietor of her rapidly expanding business of decorating eggs. For Jane, her life path has been one of hope and unexpected personal and business achievements.</jats:p
From Eggs to the Stars
No full textInterview of artist Jane Pollak by Shawn Blau and Laurence Weinstein.
Jane Pollak is a Westport, Connecticut, artist who started her career as a high school art teacher. She has now branched out into public speaking, is the author of two books, and embraces the life of entrepreneur as a sole proprietor of her rapidly expanding business of decorating eggs. For Jane, her life path has been one of hope and unexpected personal and business achievements
A Generalization of the Graham-Pollak Tree Theorem to Steiner Distance
No full textGraham and Pollak showed that the determinant of the distance matrix of a
tree depends only on the number of vertices of . Graphical distance, a
function of pairs of vertices, can be generalized to ``Steiner distance'' of
sets of vertices of arbitrary size, by defining it to be the fewest edges
in any connected subgraph containing all of . Here, we show that the same is
true for trees' {\em Steiner distance hypermatrix} of all odd orders, whereas
the theorem of Graham-Pollak concerns order . We conjecture that the
statement holds for all even orders as well.Comment: 7 page
Gary Becker's Contributions to Family and Household Economics
No full textGary Becker's influence on the economics of the family has been pervasive. His ideas have dominated research in the economics of the family, shaping the tools we use, the questions we ask, and the answers we give. The foundational assumptions of Becker's economic approach to the family -- maximizing behavior and equilibrium -- as well as such primary auxiliary assumptions as household production and interdependent preferences, are now widely accepted not only by economists but also by family sociologists, demographers, and others who study the family. Yet the interesting and provocative implications of Becker's economic approach to the family do not follow from the foundational assumptions or from the primary auxiliary assumptions. Instead they depend on contested auxiliary assumptions to which neoclassical economics has no commitment and which lack empirical support. This paper discusses the crucial role of auxiliary assumptions in Becker's analysis of the family, first in the context of preferences, then in the context of household production, and finally in the context of family or household collective choice.
Lower Bounds for Nonrelativistic Atomic Energies
No full textA recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H-) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory
A new result similar to the Graham-Pollak theorem
No full textLet be an integer, and let be a tree with vertices
, where and are two leaves of . For each
edge of , assign a complex number as its weight. We obtain that
where
is the weighted distance between and in the
tree . This is similar to the celebrated Graham-Pollak theorem on
determinants of distance matrices for trees. Actually, a more general result is
deduced in this paper.Comment: 9 pages. For new additions, see the current (1.7) and (1.9
Talking Trade: China's Super Consumers: Changing China, Changing the World
No full textFIT's International Trade and Marketing Department, in partnership with the New York District Export Council at World Trade Week, presents Talking Trade: China's Super Consumers. Distinguished speakers discuss how marketing to China has been transformed, how China's consumers themselves are influencing marketing around the world, and how companies can enhance their sales and exports to the burgeoning Chinese market.Master of Ceremonies: Thomas Pollak, President, Tally-Ho Creations, Pollak Import/Export Corp. and ITM Advisory Board member. Speakers: Mr. Michael A. Zakkour, Vice President, China/Asia Pacific Practice Leader, Tompkins International and co-author of book, China's Super Consumers; Mr. Jason Merritt, Assistant Vice President, Corporate Foreign Exchange Sales, HSBC Bank USA, N
1995 Sub-Librarians Meeting: Let a Woman in Your Life: the Women in Conan Doyle\u27s Life and Fiction
No full textAt the 23rd (Irregular) meeting, the Sub-Librarians greeted members of multiple Chicago area scion societies at the Harold Washington Library Center. The meeting began with a champagne and dessert reception in the lower lobby and then moved into the video theater for the program. Toasts were given by Katherine Rankin, Deborah Schlesinger and others.
Marsha Pollak, ASH, welcomed everyone and introduced Ely. M. Liebow, professor of English at Northeastern Illinois University and author of Dr. Joe Bell: Model for Sherlock Holmes. Liebow spoke on the topic Let a Woman in Your Life: The Women in Arthur Conan Doyle\u27s Life and Fiction
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