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.     

COMPOSITION OPERATORS FROM Bα TO F(p, q, s)

Necessary and sufficient conditions are given for a composition operator Cφ f ＝f o φ to be bounded and compact from Bα to F(p, q, s).     

Arc-transitive cubic Cayley graphs on PSL(2,p)

For a prime p at least 5,let T=PSL(2,p).This paper gives a classification of the connected arc-transitive cubic Cayley graphs on T and a determination of the gener- ated pairs ((?),(?)) of T such that o((?))=2 and o((?))=3.     

Polynomiality of Hurwitz numbers,Bouchard–Mariño conjecture,and a new proof of the ELSV formula

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of the Bouchard–Mariño conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard–Mariño conjecture and the ELSV formula (it is a refinement of an argument by Eynard).     

带边紧流形上热核的Harnack不等式

１９８６年,Ｐ．Ｌｉ与丘成桐给出了带凸边界的紧黎曼流形上关于热核的一个Ｈａｒｎａｃｋ不等式（可参看［６］）,而该文的目的正是将他们的工作推广到可能带非凸边界的紧黎曼流形上．     

Mathematical videos,social semiotics and the changing classroom

With the advancement of digital technology, the roles of teachers and students are slowly changing. The classroom is on the verge of becoming a new, more o     

Bi-solitons, breather solution family and rogue waves for the (2+1)-dimensional nonlinear Schr\"{o}dinger equation

In this paper, bi-solitons, breather solution family and rogue waves for the (2+1)-Dimensional nonlinear Schr\"{o}dinger equations are obtained by using Exp-function method. These solutions derived from one unified formula which is solution of the standard (1+1) dimension nonlinear Schr\"{o}dinger equation. Further, based on the solution obtained by other authors, higher-order rational rogue wave solution are obtained by using the similarity transformation. These results greatly enriched the diversity of wave structures for the (2+1)-dimensional nonlinear Schr\"{o}dinger equations     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

AMlet,RAMlet, and GAMlet: Automatic Nonlinear Fitting of Additive Models,Robust and Generalized,With Wavelets

A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.     

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.     

Simple groups,maximal subgroups,and probabilistic aspects of profinite groups

We show that a finite simple group has at mostn ^1.875+o(1) maximal subgroups of indexn. This enables us to characterise profinite groups which are generated with positive probability by boundedly many random elements. It turns out that these groups are exactly those having polynomial maximal subgroup growth. Related results are also established.     

Metropolis, Simulated Annealing, and Iterated Energy Transformation Algorithms: Theory and Experiments

In this paper, we compare from the theoretical and experimental points of view three stochastic optimization algorithms: the Metropolis, simulated annealing, and iterated energy transformation algorithms. We give the optimal exponents for the concentration of the marginal distribution of the final state of these algorithms around the global minima of the virtual energy function. Experiments are performed on an N.P. complete benchmark which tries to retain the main aspects of scheduling problems. They lead to the same qualitative ranking of algorithms as the theory does.     

Thermally coupled,stationary, incompressible MHD flow; existence,uniqueness, and finite element approximation

This article addresses the questions of existence, uniqueness, and finite element approximation (including some computational aspects) of solutions to the equations of steady-state magnetohy-drodynamic (MHD) when buoyancy effects due to temperature differences in the flow cannot be neglected. We couple the MHD equations to the heat equation and employ the well-known Boussinesq approximation. We consider the equations posed on a bounded three-dimensional domain. The boundary conditions for the velocity are of Dirichlet type; the boundary conditions for the temperature are mixed (of Dirichlet type and of Neumann type); we also specify the normal component of the magnetic field and tangential component of the electric field on the boundary. We point out that these problems are relevant to many physical phenomena such as the cooling of nuclear reactors by electrically conducting fluids, continuous metal casting, crystal growth, and semi-conductor manufacture. © 1995 John Wiley & Sons, Inc.     

Real-World Contexts,Multiple Representations,Student-Invented Terminology,and Y-Intercept

One classroom using two units from a Standards-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher's interpretation of different activities and his assumptions about the transparency of different representations, as well as students' past experiences left the student-generated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself.     

Observing,reporting, and deciding in networks of sentences

In prior work [7] we considered networks of agents who have knowledge bases in first order logic, and report facts to their neighbors that are in their common languages and are provable from their knowledge bases, in order to help a decider verify a single sentence. In report complete networks, the signatures of the agents and the links between agents are rich enough to verify any decider?s sentence that can be proved from the combined knowledge base. This paper introduces a more general setting where new observations may be added to knowledge bases and the decider must choose a sentence from a set of alternatives. We consider the question of when it is possible to prepare in advance a finite plan to generate reports within the network. We obtain conditions under which such a plan exists and is guaranteed to produce the right choice under any new observations.     

Faster,but weaker,relaxations for quadratically constrained quadratic programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.     

on O-simple semirings,semigroup semirings,and two kinds of division semirings

In this paper we consider O-simple semirings S, where O denotes the multiplicative zero of S, which may be in particular the additive neutral o of S at the same time. In this context we give some statements on matrix semirings and introduce contracted semigroup semirings in §3, a matter of interest of its own. We further use our results to compare the usual concept of division semirings with a new one introduced in [18], and we show that a corresponding theorem in [18] is in general only valid for division semirings in the usual meaning. Dedicated to E.S. Lyapin on his 70th birthday     

有理数乘群的两类子群及其相互关系

本文对任意的素数p 及自然数n,构造出有理数乘群G= (Q,·)的两类子群Gp,n和Gp,0,G关于它们的商群分别为Zn 和Z,它们之间有许多很好的关系,特别是其中的同构关系.这些对我们进一步认识有理数域及近世代数的入门教学有一定的参考价值.     

Development of Theme-based,Interdisciplinary, Integrated Curriculum: A Theoretical Model

Interest in interdisciplinary, integrated curriculum development continues to increase. However, teachers who have been given primary responsibility for developing these materials are often working with little guidance. A model for developing theme-based, interdisciplinary, integrated curriculum is presented in this article. The model includes two phases, a theme creation phase and an activity refinement phase. Validity criteria from the interdisciplinary curriculum literature are used to assess the power of potential themes. An integration continuum model is used to facilitate refinement of activities included in the theme-based instruction. The curriculum development model is used to clarify the meaning of the terms theme-based, interdisciplinary, and integrated. The model is intended to be used by curriculum developers to facilitate the creation of powerful interdisciplinary lessons.