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## Windows into Elementary Mathematics:Alternate public images of mathematics and mathematicians

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Artigo de Revista Científica
Formato: 8-23

Português

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Research on students’ (and teachers’) images of mathematics and mathematicians reveals a number of stereotypical images, most of which are negative. In this paper we present an overview of some these images and stereotypes and consider the questions: (1) how might the image of mathematics and mathematicians be a problem in mathematics education, and (2) what can be done to remedy the situation? Also, we consider an outreach project called Windows into Elementary Mathematics. In this project mathematicians are interviewed about their perspectives on elementary mathematics topics and their interviews are videotaped and are posted online, along with supporting images and interactive content. In this context we consider the questions: (3) what is the Windows project about, and (4) how might it offer an alternate (and perhaps better) image of mathematics and mathematicians? Lastly, we share an example where activities from the project were used in a math-for-teachers course.; Pesquisas sobre as imagens de estudantes e professores sobre matemática e sobre matemáticosrevelam uma série de imagens estereotipadas, das quais muitas são negativas. Neste artigo, apresentamosuma visão geral de algumas dessas imagens e estereótipos e consideramos as seguintes questões: (1)Como a imagem da matemática e dos matemáticos pode ser um problema em educação matemática...

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## A New Elementary Mathematics Curriculum: Practice, Learning and Assessment - Some Classroom Episodes

Fonte: The Mathematics Education into the 21st Century Project – MEC 21
Publicador: The Mathematics Education into the 21st Century Project – MEC 21

Tipo: Artigo de Revista Científica

Português

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The aim of this paper is to present the new and innovative mathematics curriculum for elementary levels that is being implemented in the Portuguese basic education system (students from 1st to 9th grade) through an overview of an ongoing study of implementation/experimentation of this curriculum. A speciﬁc mechanism was implemented in the ﬁeld to provide scientiﬁc and pedagogical support to the development of the new elementary mathematics curriculum (NPMEB) implementation at all grade levels and all over the country. In particular, the NPMEB is being experimented by a set of teachers that teach in their own classes and that have been trained and accompanied along the experience by the diﬀerent authors of the program. We will focus on some classroom practices, sharing innovative and creative ideas of teachers and students, grounded on some of the tasks used by the teachers. The preliminary results suggest that some improvements are already visible, namely regarding students’ attitudes and mathematical competences and teachers’ practice

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## Evaluation of the implementation process of a new mathematics curriculum for Grades 1-‐9: A discussion of the main results

Fonte: Universidade de Évora
Publicador: Universidade de Évora

Tipo: Aula

Português

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Recently, a new Mathematics curriculum for basic education (Grades 1-9) has been introduced in the Portuguese educational system. This curriculum was designed to bring together some disperse curricular documents and to substitute the former one that was being used since the early 90s. But, above all, the new curriculum was aimed at providing a more sustained development of students’ mathematical learning based upon recent international recommendations for mathematics teaching and learning.
The new curriculum implementation was strongly supported by a special designed programme aimed at providing scientific and pedagogical support to all teachers of Grades 1-9 throughout all the schools of the country. Before its generalization to all basic education schools of the country, the new curriculum had been put into practice by 40 teachers of the different grade levels for a three-year period. These teachers had the opportunity to attend a number of workshops and to share their experiences in a variety of contexts. In fact, they functioned as a true network, working in close collaboration. Besides they were also in touch with the authors of the new curriculum to insure that it was put into practice according to the authors’ proposal.
At the same time the Ministry of Education appointed a team of three mathematics educators to conduct a three-year evaluation study of the whole process of implementation of the new curriculum. An overview of this evaluation study will be presented: Besides...

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## Groups, Wavelets, and Wavelet Sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/07/2003
Português

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Wavelet and frames have become a widely used tool in mathematics, physics,
and applied science during the last decade. This article gives an overview over
some well known results about the continuous and discrete wavelet transforms
and groups acting on $\mathbb{R}^n$. We also show how this action can give rise
to wavelets, and in particular, MSF wavelets)in $L^2(\mathbb{R}^n)$.

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## Blaschke addition and convex polyhedra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/02/2005
Português

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#Mathematics - Metric Geometry#Mathematics - Optimization and Control#52A20, 52B11 (Primary), 52A38, 52B55, 49Q20, 65D17, 68U07 (Secondary)

This is an extended version of a talk on October 4, 2004 at the research
seminar ``Differential geometry and applications'' (headed by Academician A. T.
Fomenko) at Moscow State University. The paper contains an overview of
available (but far from well-known) results about the Blaschke addition of
convex bodies, some new theorems on the monotonicity of the volume of convex
bodies (in particular, convex polyhedra with parallel faces) as well as
description of a software for visualization of polyhedra with prescribed
outward normals and face areas.; Comment: 17 pages, 7 figures

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## Finite-dimensional algebras and quivers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/05/2005
Português

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This is an overview article on finite-dimensional algebras and quivers,
written for the Encyclopedia of Mathematical Physics. We cover path algebras,
Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.; Comment: 9 pages, 7 figures

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## The use of geometric and quantum group techniques for wild quivers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/04/2003
Português

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#Mathematics - Representation Theory#Mathematics - Algebraic Geometry#Mathematics - Rings and Algebras

This overview paper reviews several results relating the representation
theory of quivers to algebraic geometry and quantum group theory. (Potential)
applications to the study of the representation theory of wild quivers are
discussed. To appear in the Proceedings of the International Conference on
Representations of Algebras and Related Topics ICRA X, The Fields Institute,
July/August 2002.; Comment: 32 pages

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## Differential Isomorphism and Equivalence of Algebraic Varieties

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/04/2003
Português

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#Mathematics - Algebraic Geometry#Mathematical Physics#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras

In this survey article we discuss the question: to what extent is an
algebraic variety determined by its ring of differential operators? In the case
of affine curves, this question leads to a variety of mathematical notions such
as the Weyl algebra, Calogero-Moser spaces and the adelic Grassmannian. We give
a fairly detailed overview of this material.; Comment: 22 pages, to appear in a volume dedicated to Graeme Segal's 60th
birthday

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## Simultaneous Linear Inequalities: Yesterday and Today

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This is a short overview of some recent tendencies in the theory of linear
inequalities that are evoked by Boolean valued analysis.; Comment: A talk at the International Conference "Order Analysis and Related
Problems of Mathematical Modeling," Vladikavkaz, July 19--24, 2010, dedicated
to the 10th anniversary of the Vladikavkaz Scientific Center of the Russian
Academy of Sciences

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## Theory and Applications of N-Fold Integer Programming

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Optimization and Control#Computer Science - Discrete Mathematics#Computer Science - Data Structures and Algorithms#Mathematics - Combinatorics#05A, 15A, 51M, 52A, 52B, 52C, 62H, 68Q, 68R, 68U, 68W, 90B, 90C

We overview our recently introduced theory of n-fold integer programming
which enables the polynomial time solution of fundamental linear and nonlinear
integer programming problems in variable dimension. We demonstrate its power by
obtaining the first polynomial time algorithms in several application areas
including multicommodity flows and privacy in statistical databases.; Comment: IMA Volume on Mixed Integer Nonlinear Programming, Frontier Series,
Springer, to appear

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## Barrier methods for critical exponent problems in geometric analysis and mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/07/2011
Português

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#Mathematical Physics#General Relativity and Quantum Cosmology#Mathematics - Analysis of PDEs#Mathematics - Numerical Analysis#Mathematics - Optimization and Control#Physics - Computational Physics

We consider the design and analysis of numerical methods for approximating
positive solutions to nonlinear geometric elliptic partial differential
equations containing critical exponents. This class of problems includes the
Yamabe problem and the Einstein constraint equations, which simultaneously
contain several challenging features: high spatial dimension n >= 3, varying
(potentially non-smooth) coefficients, critical (even super-critical)
nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and
spatial domains that are typically Riemannian manifolds rather than simply open
sets in Rn. These problems may exhibit multiple solutions, although only
positive solutions typically have meaning. This creates additional complexities
in both the theory and numerical treatment of such problems, as this feature
introduces both non-uniqueness as well as the need to incorporate an inequality
constraint into the formulation. In this work, we consider numerical methods
based on Galerkin-type discretization, covering any standard bases construction
(finite element, spectral, or wavelet), and the combination of a barrier method
for nonconvex optimization and global inexact Newton-type methods for dealing
with nonconvexity and the presence of inequality constraints. We first give an
overview of barrier methods in non-convex optimization...

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## Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/07/2003
Português

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#Mathematics - Rings and Algebras#Mathematics - Combinatorics#Primary 05E35#Secondary 33C45,05E30,33D45

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite
positive dimension. We consider an ordered pair of linear transformations
$A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below.
(i) There exists a basis for $V$ with respect to which the matrix
representing $A$ is irreducible tridiagonal and the matrix representing $A^*$
is diagonal.
(ii) There exists a basis for $V$ with respect to which the matrix
representing $A$ is diagonal and the matrix representing $A^*$ is irreducible
tridiagonal.
We call such a pair a Leonard pair on $V$. We give an overview of the theory
of Leonard pairs.; Comment: 14 pages, 1 figure

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## Metric spaces: The definition, and some examples

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2010
Português

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These informal notes were prepared in connection with a lecture at a high
school mathematics tournament, and provide an overview of some examples of
metric spaces and a few of their basic properties.; Comment: 10 pages

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## Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Numerical Analysis#Computer Science - Computer Vision and Pattern Recognition#Computer Science - Learning#Mathematics - Optimization and Control#G.1.6#I.4#I.5

Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness.

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## Transport and Concentration Problems with Interaction Effects

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/08/2005
Português

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After presenting an overview about variational problems on probability
measures for functionals involving transport costs and extra terms encouraging
or discouraging concentration, we look for optimality conditions, regularity
properties and explicit computations in the case where Wasserstein distances
and interaction energies are considered.; Comment: 12 pages

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## Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

45.36%

#Physics - Data Analysis, Statistics and Probability#Mathematics - Optimization and Control#Statistics - Applications#Statistics - Machine Learning

Imaging spectrometers measure electromagnetic energy scattered in their
instantaneous field view in hundreds or thousands of spectral channels with
higher spectral resolution than multispectral cameras. Imaging spectrometers
are therefore often referred to as hyperspectral cameras (HSCs). Higher
spectral resolution enables material identification via spectroscopic analysis,
which facilitates countless applications that require identifying materials in
scenarios unsuitable for classical spectroscopic analysis. Due to low spatial
resolution of HSCs, microscopic material mixing, and multiple scattering,
spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus,
accurate estimation requires unmixing. Pixels are assumed to be mixtures of a
few materials, called endmembers. Unmixing involves estimating all or some of:
the number of endmembers, their spectral signatures, and their abundances at
each pixel. Unmixing is a challenging, ill-posed inverse problem because of
model inaccuracies, observation noise, environmental conditions, endmember
variability, and data set size. Researchers have devised and investigated many
models searching for robust, stable, tractable, and accurate unmixing
algorithms. This paper presents an overview of unmixing methods from the time
of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models
are first discussed. Signal-subspace...

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## Root systems and generalized associahedra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Combinatorics#Mathematics - Rings and Algebras#Mathematics - Representation Theory#20F55

These lecture notes for the IAS/Park City Graduate Summer School in Geometric
Combinatorics (July 2004) provide an overview of root systems, generalized
associahedra, and the combinatorics of clusters.
Lectures 1-2 cover classical material: root systems, finite reflection
groups, and the Cartan-Killing classification. Lectures 3-4 provide an
introduction to cluster algebras from a combinatorial perspective. Lecture 5 is
devoted to related topics in enumerative combinatorics.; Comment: 69 pages, many figures. Version 3 is the final version appearing in
the PCMI Lecture Series volume.

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## Jordan structures in mathematics and physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/06/2011
Português

Relevância na Pesquisa

45.39%

#Mathematics - Differential Geometry#Mathematical Physics#Mathematics - Rings and Algebras#16T25, 17B60, 17C40, 17C50, 17C65, 17C90, 17D92, 35Q51, 35Q53,
44A12, 51A35, 51C05, 53C35, 81T05, 81T30, 92D10

The aim of this paper is to offer an overview of the most important
applications of Jordan structures inside mathematics and also to physics,
up-dated references being included. For a more detailed treatment of this topic
see - especially - the recent book Iordanescu [364w], where sugestions for
further developments are given through many open problems, comments and remarks
pointed out throughout the text.
Nowadays, mathematics becomes more and more nonassociative and my prediction
is that in few years nonassociativity will govern mathematics and applied
sciences.
Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, JB*-, JBW-,
JBW*-, JH*-algebra, Ricatti equation, Riemann space, symmetric space, R-space,
octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum
group, B\"acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation,
KP equation, Sato Grassmann manifold, genetic algebra, random quadratic form.

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## Nonstandard Models and Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/05/2008
Português

Relevância na Pesquisa

45.38%

This is an overview of a few possibilities that are open by model theory in
applied mathematics. Most attention is paid to the present state and frontiers
of the Cauchy method of majorants, approximation of operator equations with
finite-dimensional analogs, and the Lagrange multiplier principle in
multiobjective decision making.; Comment: An extended version of a talk prepared for the International
Conference "Methods of Logic in Mathematics V," June 1--6, 2008, St.
Petersburg

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## Homogenization problems in the Calculus of Variations: an overview

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/03/2015
Português

Relevância na Pesquisa

45.42%

#Mathematics - Optimization and Control#Mathematics - Analysis of PDEs#Primary 35B27, Secondary 49J40, 35E99, 49-02

In this note we present a brief overview of variational methods to solve
homogenization problems. The purpose is to give a first insight on the subject
by presenting some fundamental theoretical tools, both classical and modern. We
conclude by mentioning some open problems.; Comment: 11 pages

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