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A Mathematical Theory of Origami Constructions and Numbers
 J. Math
, 2000
"... Abstract. In this article we giveasimpli ed set of axioms for mathematical origami and numbers. The axioms are hierarchically structured so that the addition of each axiom, allowing new geometrical complications, is mirrored in the eld theory of the possible constructible numbers. The elds of Thalia ..."
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Abstract. In this article we giveasimpli ed set of axioms for mathematical origami and numbers. The axioms are hierarchically structured so that the addition of each axiom, allowing new geometrical complications, is mirrored in the eld theory of the possible constructible numbers. The elds of Thalian, Pythagorean, Euclidean and Origami numbers are thus obtained using this set of axioms. The other new ingredient here relates the last axiom to the algebraic geometry of pencils of conics. It is hoped that the elementary nature of this article will also be useful for advanced algebra students in understanding more
Test Functions for Elliptic Problems Satisfying Linear Essential Edge Conditions on Both Convex and Concave Polygons
, 2003
"... Interpolations which are smooth and bounded can be constructed over any two dimensional polygonal domain, including those with concavities and inclusions. Like boundary element test functions, they depend only on the boundary values. Unlike the boundary element method formulations, they satisfy line ..."
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Interpolations which are smooth and bounded can be constructed over any two dimensional polygonal domain, including those with concavities and inclusions. Like boundary element test functions, they depend only on the boundary values. Unlike the boundary element method formulations, they satisfy linear essential boundary conditions exactly and do not depend on a Greens' function solution to the governing field equation. In other words, they are Ritz type coordinate functions which apply to any polygonal domain. The interpolations satisfy element level constancy and linear patch tests and perform well in approximation of potential field solutions. Similar functions, applicable only to convex polygons, have been applied successfully to biomedical problems including skull growth and heart function analyses. No smooth kinematic concave polygonal element description of any type is presented in the available finite element, boundary element or computational geometry literature.
AXIOMATIC AND COORDINATE GEOMETRY
"... At some point between high school and college we ¯rst make the transition between Euclidean (or synthetic) geometry and coordinate (or analytic) geometry. Later, during graduate studies we are introduced to di®erential geometry of many dimensions. The justi¯cation given in the ¯rst instance is that ..."
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At some point between high school and college we ¯rst make the transition between Euclidean (or synthetic) geometry and coordinate (or analytic) geometry. Later, during graduate studies we are introduced to di®erential geometry of many dimensions. The justi¯cation given in the ¯rst instance is that coordinates are a natural outcome of the axioms of Euclidean geometry; and in the second case
unknown title
, 1992
"... Abstract In the "little theories " version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach ..."
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Abstract In the &quot;little theories &quot; version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
Modeling of unbounded media with concave finite elements using the cloning algorithm
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ON CONSTRUCTIVE AXIOMATIC METHOD
"... Abstract. The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory ..."
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Abstract. The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory in the standard sense of the word. However these theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert constructive. I show that the formal axiomatic method always requires a support of some more basic constructive method. 1.