102,808 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
Letter, [Author unclear] to Paulina T. Merritt
No full textHandwritten letter to Paulina Merritt from an unknown author, October 1, 1876.
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
Nolan-Pollak Type CN Counters in the Vienna Aerosol Workshop
No full textThree standard Nolan-Pollak (N-P) and modified N-P design condensation nucleus (CN) counters were included in the Vienna Workshop on Intercomparison of Condensation Nuclei and Aerosol Particle counters. These counters came from diverse backgrounds, namely programs in USA, Europe and Australia. In this work, principles of the operation and previous history of calibration of the N-P expansion counter are briefly reviewed and comparisons between the particular counters used in the workshop are presented and discussed. Counting agreement was found to be very good between the N-P counters, typically better than ± 12% for a range of aerosol sizes and compositions from a minimum diameter of 4 nm. The independently calibrated GIV CNC-440 (modified N-P type counter) also agreed well with the N-P counters. The minimum size sensitivity of the N-P counter was examined showing a lower detection limit for insoluble (Ag) particles of around 2.6 ± 0.3 nm diameter. © 2002 Elsevier Science B.V. All rights reserved
Handwritten biographical information on Paulina T. McClung Merritt
No full textA handwritten biography of Paulina T. McClung Merritt by an unknown author, 1892.
Heterogeneous and tissue-specific regulation of effector T cell responses by IFN-gamma during Plasmodium berghei ANKA infection.
No full textIFN-γ and T cells are both required for the development of experimental cerebral malaria during Plasmodium berghei ANKA infection. Surprisingly, however, the role of IFN-γ in shaping the effector CD4(+) and CD8(+) T cell response during this infection has not been examined in detail. To address this, we have compared the effector T cell responses in wild-type and IFN-γ(-/-) mice during P. berghei ANKA infection. The expansion of splenic CD4(+) and CD8(+) T cells during P. berghei ANKA infection was unaffected by the absence of IFN-γ, but the contraction phase of the T cell response was significantly attenuated. Splenic T cell activation and effector function were essentially normal in IFN-γ(-/-) mice; however, the migration to, and accumulation of, effector CD4(+) and CD8(+) T cells in the lung, liver, and brain was altered in IFN-γ(-/-) mice. Interestingly, activation and accumulation of T cells in various nonlymphoid organs was differently affected by lack of IFN-γ, suggesting that IFN-γ influences T cell effector function to varying levels in different anatomical locations. Importantly, control of splenic T cell numbers during P. berghei ANKA infection depended on active IFN-γ-dependent environmental signals--leading to T cell apoptosis--rather than upon intrinsic alterations in T cell programming. To our knowledge, this is the first study to fully investigate the role of IFN-γ in modulating T cell function during P. berghei ANKA infection and reveals that IFN-γ is required for efficient contraction of the pool of activated T cells
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