a priori bound (original) (raw)
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2011
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Galilean Classification of Curves
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Galilean transformation group has an important place in classic and modern physics for instance: in quantum theory, gauge transformations in electromag-netism, in mechanics [1], and conductivity tensors in fluid dynamics [2], also, in mathematical fields such as ...
WHAT ARE DIFFERENTIAL EQUATIONS: A Review of Curve Families
This paper is a review of curve families in light of their importance in a course on differential equations. Many texts depict curve families but do not treat them as an important mathematical concept that can greatly add to a student's comprehension of differential equations and continuous mathematics in general. It is not an accident that the solution of an nth order differential equation is an n parameter curve family. It is not an accident that the solutions of linear differential equations are linear curve families. The forms, features and properties of curve families are discussed. Also to be discussed are the relations of one parameter curve families to surfaces in three dimensional space.
Eckes and Giardino (2018), The Classificatory Function of Diagrams. Two Examples from Mathematics.
2018
In a recent paper, DeToffoli and Giardino analyzed the practice of knot theory, by focusing in particular on the use of diagrams to represent and study knots. To this aim, they distinguished between illustrations and diagrams. An illustration is static; by contrast, a diagram is dynamic, that is, it is closely related to some specific inferential procedures. In the case of knot diagrams, a diagram is also a well-defined mathematical object in itself, which can be used to classify knots. The objective of the present paper is to reply to the following questions: Can the classificatory function characterizing knot diagrams be generalized to other fields of mathematics? Our hypothesis is that dynamic diagrams that are mathematical objects in themselves are used to solve classification problems. To argue in favor of our hypothesis, we will present and compare two examples of classifications involving them: (i) the classification of compact connected surfaces (orientable or not, with or without boundary) in combinatorial topology; (ii) the classification of complex semisimple Lie algebras.
The Alphabet of Knowledge -ABC Philosophy and Theory of Science General Principles & Items in Terms
Universe of Interactions , 2012
Energy is everything and vice - versa. Power is potential (quantum) state of energy (charge). Force is real (physical) condition of energy (field and elements). The scalar properties of energy are, dynamic relativity (interchange between physical and quantum states) and derivative limitation, variation, dimension, mass, motion and information. Energy originates from gravitational singularity, which was the starting point of the universal evolution and the composition of dimensional mass, via the Big Bang. Gravitational singularity is the original quantum (power) state of energy. Dimensional expansion is the transformation of energy from gravitational singularity to the material plurality of the interchanging, state of quantum power and the condition of physical force, of energy. In dimensional and material expansion, energy powers variations, which configure into particles and interactions (interactive particles). Interactions are relations therefore they are forces.
In La Géométrie, Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one — geometry became a mean of visualization, no longer of construction. Descartes’s foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has, however, been extended beyond algebraic limits, albeit in two different periods. In the late 17th century, the synthetic aspect was extended by “tractional motion” (construction of transcendental curves with idealized machines). In the first half of the 20th century, the analytical part was extended by “differential algebra,” now a branch of computer algebra. This thesis seeks to prove that it is possible to obtain a new balance between these synthetic and analytical extensions of Cartesian tools for a class of transcendental problems. In other words, there is a possibility of a new convergence of machines, algebra, and geometry that gives scope for a foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. The peculiarity of this work lies in the attention to the constructive role of geometry as idealization of machines for foundational purposes. This approach, after the “de-geometrization” of mathematics, is far removed from the mainstream discussions of mathematics, especially regarding foundations. However, though forgotten these days, the problem of defining appropriate canons of construction was very important in the early modern era, and had a lot of influence on the definition of mathematical objects and methods. According to the definition of Bos [2001], these are “exactness problems” for geometry. Such problems about exactness involve philosophical and psychological interpretations, which is why they are usually considered external to mathematics. However, even though lacking any final answer, I propose in conclusion a very primitive algorithmic approach to such problems, which I hope to explore further in future research. From a cognitive perspective, this approach to calculus does not require infinity and, thanks to idealized machines, can be set with suitable “grounding metaphors” (according to the terminology of Lakoff and Nuñez [2000]). This concreteness can have useful fallouts for math education, thanks to the use of both physical and digital artifacts (this part will be treated only marginally).