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Mathematical Creativity and School Mathematics

In Rising Above The Gathering Storm: Energizing and Employing America for a Brighter Economic Future (Committee on Science, Engineering, and Public Policy, 2005) members of the National Academy of Science developed a list of recommended actions needed to ensure that the United States can continue to compete globally. The top recommendation was to increase America's talent pool by vastly improving K-12 mathematics and science education (pp. 91-110).

One of the strengths of the United States economic growth has been the creativity of its citizens. Inherent in the recommendations above is the need for growth and innovation, both of which are fueled by creativity. This study investigates several means of identifying mathematical creativity as a first step in identifying and nurturing this talent.One of the strengths of the United States economic growth has been the creativity of its citizens. Inherent in the recommendations above is the need for growth and innovation, both of which are fueled by creativity. This study investigates several means of identifying mathematical creativity as a first step in identifying and nurturing this talent.

Data mining and Neural Networks from a Commercial Perspective Ebook

Companies have been collecting data for decades, building massive data warehouses in which to store it. Even though this data is available, very few companies have been able to realize the actual value stored in it. The question these companies are asking is how
to extract this value. The answer is Data mining.

There are many technologies available to data mining practitioners, including Artificial Neural Networks, Regression, and Decision Trees. Many practitioners are wary of Neural Networks due to their black box nature, even though they have proven themselves in many situations.

In our current research we are attempting to compare the aforementioned technologies and determine if Neural Networks outperform more traditional statistical techniques. This paper is an overview of artificial neural networks and questions their position as a preferred tool by data mining practitioners.

Decoding by Linear Programming

This paper considers the model problem of recovering an input vector f ? R n from corrupted measurements y = Af +e. Here, A is an m by n matrix (we will assume throughout the paper that m > n), and e is an arbitrary and unknown vector of errors. The problem we consider is whether it is possible to recover f exactly from the data y. And if so, how?

Random Loewner chains in Riemann surfaces

The thesis describes an extension of O. Schramm's SLE processes to complicated plane domains and Riemann surfaces. First, three kinds of new SLEs are defined for simple conformal types. They have properties similar to traditional SLEs. Then harmonic random Loewner chains (HRLC) are defined in finite Riemann surfaces. They are measures on the space of Loewner chains, which are increasing families of closed subsets satisfying certain properties. An HRLC is first defined on local charts using Loewner's equation. Since the definitions in different charts agree with each other, these local HRLCs can be put together to construct a global HRLC. An HRLC in a plane domain can be described by differential equations involving canonical plane domains. Those old and new SLEs are special cases of HRLCs. An HRLC is determined by a parameter [kappa] >= 0, a starting point and a target set. When [kappa] = 6, the HRLC satisfies the locality property.

Mathematical Intuition vs. Mathematical Monsters

Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which?according to some mathematicians?seriously challenge the reliability of intuition. We examine several famous geometrical, topological and set-theoretical examples of such monsters in order to see to what extent, if at all, intuition is undermined in its everyday roles.

A Prospective On Mathematics And Artificial Intelligence: Problem Solving = Modeling + Theorem Proving

Mathematics and artificial intelligence (AI) have had a symbiotic relationship since Allen Turing dreamed of taking Hilbert’s tenth problem into the realm of computation that would blur the distinction between human and machine reasoning. Every aspect of AI has mathematical roots, and here have been some bilateral developments. For example, efforts to improve computational logic led to new results in mathematical logic, itself. Intelligent tutors have improved, particularly with modern programming paradigms, and some contribute to mathematics education. I see this as a part of mathematics because I view mathematics as not just a collection of facts, but as a process mathematical reasoning. Once this is accepted, mathematics and artificial intelligence interleave throughout each of their branches.

Measures in topological spaces

Let X be a completely-regular topological space and let C*(X) denote the space of all bounded, real-valued continuous functions on X. For a positive linear functional [...] on C*(X), consider the following two continuity conditions. [...] is said to be a B-integral if whenever [...] and [...] for all [...], then [...]. [...] is said to be B-normal if whenever [...] is a directed system with [...] for all [...], then [...]. It is obvious that a B-normal functional is always a B-integral. The main concern of this paper is what can be said in the converse direction.

Structure theorems for local Noether lattices

A local Noether lattice of dimension n is regular if and only if its maximal element is a join of n principal elements. A set of n principal elements whose join is the maximal element is called a regular system of parameters. An element of a regular system of parameters is called a regular parameter.

The foundations of general arithmetic

We assume as known the whole machinery of formal logic, such as the notion of set class, relation, propositional function, correspondence, formal equivalence, counting, and the like. We shall use this machinery to investigate a kind of class called a "collection" whose elements consist of "entities" which in a given collection are either all "objects" or all "marks," and certain special types of propositional functions associated with the collection.

Conformal laminations

A lamination on a circle is an equivalence relation on the points of the circle. Laminations can be induced on a circle by a map that is continuous on the closed disc and injective in the interior. Such laminations are characterized topologically, as being flat and closed. In this paper we investigate the conditions under which a closed, flat lamination is induced by a conformal mapping.

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