An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b^? of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y+V of a vector subspace of b^? such that the restriction map induces an isomorphism of Y onto the algebra R[y+V] of regular functions on y+V. This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C(2) no algebraic slice exists.Very surprisingly the exception of type C(2) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B(2m), C(n) and F(4).Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

On the number of groups in clustering

Clustering is the problem of partitioning data into a finite number k of homogeneous and separate groups, called clusters. A good choice of k is essential for building meaningful clusters. In this paper, this task is addressed from the point of view of model selection via penalization. We design an appropriate penalty shape and derive an associated oracle-type inequality. The method is illustrated on both simulated and real-life data sets.     

The Influence of Positivism on the Teaching of Mathematics in Brazil: 1870–1930

This article describes the strong influence of positivism on the teaching of mathematics in Brazil. The dissemination of positivism occurred in a very intensive way from 1870 to 1930, due mainly to the strong leadership of teachers at the military and engineering academies. From its firmly entrenched position in these institutions, the positivistic ideology affected the social, political, pedagogical, and ideological life in Brazil. Here, I identify the main representatives of positivism, who focused their research on Auguste Comte's concept of mathematics. They oriented curricula and programs according to Comte's principles as well as produced mathematics with a distinct positivist bent. Although a marked decline occurred after 1930, the positivistic phenomenon was not exhausted as a research topic, and, indeed, it still has not been entirely extinguished in Brazilian life. Copyright 1999 Academic Press.Este trabalho descreve a forte influência do positivismo no ensino da Matemática no Brasil. A difusão do positivismo aconteceu de forma muita intensa entre 1870 e 1930, devido principalmente a atuação dos docentes-militares, que mantinham uma liderança forte nas academias militares e de engenharia. Nestas instituições a ideologia positivista encontrou uma forte sustentação e pode, então, ter efeitos na vida social, polı́tica, pedagógica e ideológica brasileira. Identificamos os principais representantes do positivismo no cı́rculo acadêmico. Detectamos as primeiras manifestações da concepção de Matemática de Auguste Comte em livros-texto. Identificamos a orientação de currı́culos e programas segundo os preceitos de Comte e analisamos principalmente as obras de Matemática de autores positivistas. O declı́nio do positivismos depois de 1930 também é registrado. O fenômeno positivismo não foi esgotado como tema de pesquisa e tudo indica que ainda não se extinguiu completamente da vida brasileira. Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A55, 01A70.     

A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows

We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.     

Sufficiently general framework for simple models of the net immune response

A general framework for modelling the immune response is proposed and analyzed. The models have simple formulation in terms of only few measurable quantities and the complexity of the immune response is accounted for by using nonlinear functions with time-delayed arguments. It is illustrated that such models are capable of describing various typical cases of the immune response. Standard analytical tools are used to show how such models can assist in the problem of parameter estimation.     

A Contextualized Historical Analysis of the Kuhn–Tucker Theorem in Nonlinear Programming: The Impact of World War II

When Kuhn and Tucker proved the Kuhn–Tucker theorem in 1950 they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, which was unpublished; in 1948 by F. John in a paper that was at first rejected by the Duke Mathematical Journal; and possibly earlier by Ostrogradsky and Farkas. The questions of whether the Kuhn-Tucker theorem can be seen as a multiple discovery and why the different occurences of the theorem were so differently received by the mathematical communities are discussed on the basis of a contextualized historical analysis of these works. The significance of the contexts both mathematically and socially for these questions is discussed, including the role played by the military in the shape of Office of Naval Research (ONR) and operations research (OR). Copyright 2000 Academic Press.En démontrant, en 1950, le théorème qui porte aujourd'hui leur nom, Kuhn et Tucker ont donné naissance à la théorie de la programmation non-linéaire. Cependant, en un sens, ce théorème avait été démontré auparavant, d'abord par W. Karush en 1939 dans un mémoire de maîtrise inédit, par la suite par F. John en 1948 dans un article qui avait d'abord été rejeté par le Duke Mathematical Journal, et peut-être même plus tôt par Ostrogradsky et aussi par Farkas. Le présent article cherche à élucider deux questions: Peut-on considérer le théorème Kuhn–Tucker comme un exemple de découverte multiple? Et pourquoi le théorème a-t-il été reçu si différemment dans les diverses communautés mathématiques? Notre discussion se base sur une analyse historique contextuelle des différents ouvrages. Nous examinons ici l'importance du contexte, tant du point de vue des mathématiques que du point de vue social, y compris le rôle joué par le secteur militaire dans le cadre de l'Office of Naval Research et de la recherche opérationnelle. Copyright 2000 Academic Press.MSC 1991 subject classification: 01A60; 49-03; 52-03; 90-03; 90C30.     

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

A. N. Whitehead (1861–1947) contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910–1913): he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of “process,” especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. © 2002 Elsevier Science (USA).A. N. Whitehead (1861–1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversas álgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesse âmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910–1913): estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de “processo” especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05.     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.     

Mathematics for Independence: From Spanish Liberal Exile to the Young American Republics

This paper studies one period in the history of mathematical relationships between Spain and Latin America, when right after the first liberating movements at the beginning of the 19th century, the Spanish-exiled liberals in London produced a peculiar kind of mathematical textbook for the new republics, the catechisms. Copyright 1999 Academic Press.Este trabajo estudia un periodo de la historia de los intercambios matemáticos entre España y Latinoamérica inmediatamente posterior a los movimientos liberadores de principios del siglo XIX, cuando los liberales españoles exiliados en Londres produjeron un tipo muy peculiar de textos matemáticos para las nuevas repúblicas americanas, los catecismos. Copyright 1999 Academic Press.AMS 1991 subject classifications: 00A05, 01A55, 01A80.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints

We study a spline-based likelihood method for the partly linear model with monotonicity constraints. We use monotone B-splines to approximate the monotone nonparametric function and apply the generalized Rosen algorithm to compute the estimators jointly. We show that the spline estimator of the nonparametric component achieves the possible optimal rate of convergence under the smooth assumption and that the estimator of the regression parameter is asymptotically normal and efficient. Moreover, a spline-based semiparametric likelihood ratio test is established to make inference of the regression parameter. Also an observed profile information method to consistently estimate the standard error of the spline estimator of the regression parameter is proposed. A simulation study is conducted to evaluate the finite sample performance of the proposed method. The method is illustrated by an air pollution study.     

On the minor-minimal 3-connected matroids having a fixed minor

Let N be a minor of a 3-connected matroid M such that no proper 3-connected minor of M has N as a minor. This paper proves a bound on |E(M)−E(N)| that is sharp when N is connected.     

Gevrey regularity of subelliptic Monge–Ampère equations in the plane

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge–Ampère equations in the plane. Under the assumptions that one principal entry of the Hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy, we prove that the solution of the degenerate Monge–Ampère equation will be smooth in Gevrey classes.     

On the minimizers of the Ginzburg-Landau energy for high kappa: the axially symmetric case

The Ginzburg-Landau theory of superconductivity is examined in the case of a special geometry of the sample, the infinite cylinder. We restrict to axially symmetric solutions and consider models with and without vortices. First putting the Ginzburg-Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg-Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg-Landau energy is analyzed and different convergence results are obtained. Our main result states that, when κ is large, the minimum of the energy is reached when there are about κ vortices at the center of the cylinder. Numerical computations illustrate the various behaviours.     

Two classes of Quadratic Assignment Problems that are solvable as Linear Assignment Problems

The Quadratic Assignment Problem is one of the hardest combinatorial optimization problems known. We present two new classes of instances of the Quadratic Assignment Problem that can be reduced to the Linear Assignment Problem and give polynomial time procedures to check whether or not an instance is an element of these classes.     

The topology of moduli spaces of group representations: The case of compact surface

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.