Summing Squares and Cubes of Integers

Recreational mathematics can provide students with opportunities to explore mathematics in meaningful ways. Elementary number theory is one area of mathematics that lends itself readily to recreational mathematics. In this article, the author provides two examples from elementary number theory with results that students might find surprising, and which may be used to motivate them to study additional topics from number theory

Local Availability of mathematics and number scaling: Effects on quantum physics

Local availability of mathematics and number scaling provide an approach to a coherent theory of physics and mathematics. Local availability of mathematics assigns separate mathematical universes, U_{x}, to each space time point, x. The mathematics available to an observer, O_{x}, at x is contained in U_{x}. Number scaling is based on extending the choice freedom of vector space bases in gauge theories to choice freedom of underlying number systems. Scaling arises in the description, in U_{x}, of mathematical systems in U_{y}. If a_{y} or \psi_{y} is a number or a quantum state in U_{y}, then the corresponding number or state in U_{x} is r_{y,x}a_{x} or r_{y,x}\psi_{x}. Here a_{x} and \psi_{x} are the same number and state in U_{x} as a_{y} and \psi_{y} are in U_{y}. If y=x+\hat{\mu}dx is a neighbor point of x, then the scaling factor is r_{y,x}=\exp(\vec{A}(x)\cdot\hat{\mu}dx) where \vec{A} is a vector field, assumed here to be the gradient of a scalar field. The effects of scaling and local availability of mathematics on quantum theory show that scaling has two components, external and internal. External scaling is shown above for a_{y} and \psi_{y}. Internal scaling occurs in expressions with integrals or derivatives over space or space time. An example is the replacement of the position expectation value, \int\psi^{*}(y)y\psi(y)dy, by \int_{x}r_{y,x}\psi^{*}_{x}(y_{x})y_{x}\psi_{x}(y_{x})dy_{x}. This is an integral in U_{x}. The good agreement between quantum theory and experiment shows that scaling is negligible in a space region, L, in which experiments and calculations can be done, and results compared. L includes the solar system, but the speed of light limits the size of L to a few light years. Outside of $L$, at cosmological distances, the limits on scaling are not present.Comment: 21 pages, 2 figures, To appear in SPIE conference proceedings, Quantum information and computation X, April 26,27, 201

Set-Theoretic Completeness for Epistemic and Conditional Logic

The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of formulas in the language (the semantics), a collection of axioms and rules of inference characterizing reasoning (the proof theory), and then relate the proof theory to the semantics via soundness and completeness results. Here we consider an approach that is more common in the economics literature, which works purely at the semantic, set-theoretic level. We provide set-theoretic completeness results for a number of epistemic and conditional logics, and contrast the expressive power of the syntactic and set-theoretic approachesComment: This is an expanded version of a paper that appeared in AI and Mathematics, 199

Mathematics in Hands-On Science for Liberal Arts Students

We describe a number of experiments from the courses called, General Science 9, part of the science program for elementary education majors at Brooklyn College. These courses provide hands-on learning experiences for students who are insecure and weak in science and mathematics. Quantitative thinking is a central element in most of the students’ work. Mathematics is taught in a concrete and intuitive way, as a direct outgrowth of their needs; first, in analysis of data, and second, in discovering underlying theory. The science program has been developed through cooperation among faculty from the School of Education and the science departments