On rank, underdetermined systems and consistency

The concepts of rank, underdetermined systems and consistency in linear algebra are discussed in the context of a puzzle. The article begins with a specific example, moving on to a generalization of the example and then to the general n x n case. As well as providing a solution of the puzzle, the article aims to provide students with a greater understanding of these abstract ideas in linear algebra through the study of the puzzle.

Rank histograms of stratified Monte-Carlo ensembles

The application of forecast ensembles to probabilistic weather prediction has spurred considerable interest in their evaluation. Such ensembles are commonly interpreted as Monte Carlo ensembles meaning that the ensemble members are perceived as random draws from a distribution. Under this interpretation, a reasonable property to ask for is statistical consistency, which demands that the ensemble members and the verification behave like draws from the same distribution. A widely used technique to assess statistical consistency of a historical dataset is the rank histogram, which uses as a criterion the number of times that the verification falls between pairs of members of the ordered ensemble. Ensemble evaluation is rendered more specific by stratification, which means that ensembles that satisfy a certain condition (e.g., a certain meteorological regime) are evaluated separately. Fundamental relationships between Monte Carlo ensembles, their rank histograms, and random sampling from the probability simplex according to the Dirichlet distribution are pointed out. Furthermore, the possible benefits and complications of ensemble stratification are discussed. The main conclusion is that a stratified Monte Carlo ensemble might appear inconsistent with the verification even though the original (unstratified) ensemble is consistent. The apparent inconsistency is merely a result of stratification. Stratified rank histograms are thus not necessarily flat. This result is demonstrated by perfect ensemble simulations and supplemented by mathematical arguments. Possible methods to avoid or remove artifacts that stratification induces in the rank histogram are suggested.

Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

Global analytic regularity for structures of co-rank one

We consider real analytic involutive structures V, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of V. We prove that analytic regularity propagates along them. With a further assumption on the level sets of V we characterize the global analytic hypoellipticity of a differential operator naturally associated to V. As an application we study a case of tube structures.

Expressão imuno-hitoquímica das proteínas RANK, RANKL e OPG em cistos radiculares e cistos dentígeros

Receptor ativador nuclear κappa B (RANK), ligante do receptor ativador nuclear κappa B (RANKL) e osteoprotegerina (OPG) são membros da família do fator de necrose tumoral relacionados com o metabolismo ósseo. A formação, diferenciação e atividade dos osteoclastos são reguladas por estas três proteínas. RANK é um receptor transmembrana presente em diversos tipos celulares, principalmente em células de linhagem macrofágica, linfócitos, células dendríticas e fibroblastos e quando ativado pelo seu ligante, RANKL, promove a diferenciação e ativação de células osteoclásticas responsáveis pelo processo de reabsorção óssea. A OPG impede a ligação RANK/RANKL atuando como um receptor inibitório para a atividade osteolítica. O objetivo deste estudo foi comparar a expressão imuno-histoquímica destes biomarcadores em cistos radiculares (n=20) e cistos dentígeros (n=20). A expressão imuno-histoquímica destes marcadores foi avaliada no epitélio e na cápsula dos cistos por escores e percentuais médios de imunomarcação. Para o epitélio, a análise semi-quantitativa revelou um padrão similar dos escores de imunomarcação de RANK, RANKL e OPG nas lesões, não havendo diferença estatística significante (p=0.589, p=0.688, p=0.709, respectivamente). Para a cápsula cística a análise quantitativa, mostrou diferença estatística significante entre os percentuais médios de imunomarcação do RANK e RANKL (p=0,001 e p=0,005, respectivamente) nos cistos. A correlação dos escores de imunomarcação de RANKL e OPG no epitélio do CR e do CD revelou diferença estatística significante (p=0,029, p=0,003, respectivamente). No epitélio dos CRs e dos CDs observou-se uma maior imunoexpressão da OPG comparada a do RANKL. Os resultados apontam a presença de RANK, RANKL e OPG nos cistos radiculares e cistos dentígeros, sugerindo a atuação destas proteínas no desenvolvimento e expansão das lesões no osso adjacente

Eye color as an indicator of social rank in the fish Nile tilapia

We investigated the association of eye color with the dominant-subordinate relationship in the fish Nile tilapia, Oreochromis niloticus. Eye color pattern was also examined in relation to the intensity of attacks. We paired 20 size-matched fish (intruder: 73.69 ± 11.49 g; resident: 75.42 ± 8.83 g) and evaluated eye color and fights. These fish were isolated in individual aquaria for 10 days and then their eye color was measured 5 min before pairing (basal values). Twenty minutes after pairing, eye color and fights were quantified for 10 min. Clear establishment of social hierarchy was observed in 7 of 10 pairs of fish. Number of attacks ranged from 1 to 168 among pairs. The quartile was calculated for these data and the pairs were then divided into two classes: low-attack (1 to 111 attacks - 2 lower quartiles) or high-attack (112 to 168 attacks - 2 higher quartiles). Dominance decreased the eye-darkening patterns of the fish after pairing, while subordinance increased darkening compared to dominance. Subordinate fish in low-attack confrontations presented a darker eye compared to dominant fish and to the basal condition. We also observed a paler eye pattern in dominants that shared low-attack interactions after pairing compared to the subordinates and within the group. However, we found no differences in the darkening pattern between dominants and subordinates from the high-attack groups. We conclude that eye color is associated with social rank in this species. Moreover, the association between eye color and social rank in the low-attack pairs may function to reduce aggression.

NDIM achievements: massive, arbitrary tensor rank and N-loop insertions in Feynman integrals

One of the main difficulties in studying quantum field theory, in the perturbative regime, is the calculation of D-dimensional Feynman integrals. In general, one introduces the so-called Feynman parameters and, associated with them, the cumbersome parametric integrals. Solving these integrals beyond the one-loop level can be a difficult task. The negative-dimensional integration method (NDIM) is a technique whereby such a problem is dramatically reduced. We present the calculation of two-loop integrals in three different cases: scalar ones with three different masses, massless with arbitrary tensor rank, with and N insertions of a two-loop diagram.

Expression of opg, rank, and rankl proteins in tooth repair processes after immediate and delayed tooth

This study evaluated the immunohistochemical expression of OPG, RANK, and RANKL proteins in the repair after immediate and delayed replantation of rat teeth. Fifty-six Wistar rats (Rattus norvegicus albinus) had their maxillary right lateral incisor extracted and then replanted, according to the following conditions: group I (control; n = 8), teeth were not extracted; group II (n = 16), immediate replantation; group III (n = 16), delayed replantation without treatment; and group IV (n = 16), delayed replantation after root surface treatment (periodontal ligament removal and immersion in 2% acidulated-phosphate sodium fluoride) and calcium hydroxide intracanal dressing. Rats in group I were euthanized on the first day of the experiment, while the animals in the other groups were euthanized 10 and 60 days after replantation (n = 8/period). Hematoxylin and eosin-stained sections were obtained for histological analysis. The immunohistochemical analysis revealed expression of OPG and RANKL proteins in all groups and both postreplantation times, except for group II at 60 days. In the experimental groups, RANK expression was observed only at 10 days. In conclusion, there was strong immunostaining for the OPG-RANK-RANKL system at the earlier postreplantation time, suggesting a more effective participation of these proteins at the start of the healing process, as their expression decreased at 60 days. Copyright © 2013 by Mutaz B. Habal, MD.

Asentamientos humanos intermedios en las estrategias de organización territorial en América Latina y el Caribe = Middle-rank human settlements in territorial organization strategies in Latin America and the Caribbean

Incluye Bibliografía

Stable singularities of co-rank one quasi homogeneous map germs from (ℂn+1, 0) to (ℂn, 0), n = 2, 3

In this article, we investigate the geometry of quasi homogeneous corank one finitely determined map germs from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. We give a complete description, in terms of the weights and degrees, of the invariants that are associated to all stable singularities which appear in the discriminant of such map germs. The first class of invariants which we study are the isolated singularities, called 0-stable singularities because they are the 0-dimensional singularities. First, we give a formula to compute the number of An points which appear in any stable deformation of a quasi homogeneous co-rank one map germ from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. To get such a formula, we apply the Hilbert's syzygy theorem to determine the graded free resolution given by the syzygy modules of the associated iterated Jacobian ideal. Then we show how to obtain the other 0-stable singularities, these isolated singularities are formed by multiple points and here we use the relation among them and the Fitting ideals of the discriminant. For n = 2, there exists only the germ of double points set and for n = 3 there are the triple points, named points A1,1,1 and the normal crossing between a germ of a cuspidal edge and a germ of a plane, named A2,1. For n = 3, there appear also the one-dimensional singularities, which are of two types: germs of cuspidal edges or germs of double points curves. For these singularities, we show how to compute the polar multiplicities and also the local Euler obstruction at the origin in terms of the weights and degrees. © 2013 Pushpa Publishing House.

Massive spin-2 particle from a rank-2 tensor

Here we obtain all possible second-order theories for a rank-2 tensor which describe a massive spin-2 particle. We start with a general second-order Lagrangian with ten real parameters. The absence of lower-spin modes and the existence of two local field redefinitions leads us to only one free parameter. The solutions are split into three one-parameter classes according to the local symmetries of the massless limit. In the class which contains the usual massive Fierz-Pauli theory, the subset of spin-1 massless symmetries is maximal. In another class where the subset of spin-0 symmetries is maximal, the massless theory is invariant under Weyl transformations and the mass term does not need to fit into the form of the Fierz-Pauli mass term. In the remaining third class neither the spin-1 nor the spin-0 symmetry is maximal and we have a new family of spin-2 massive theories. © 2013 American Physical Society.