Results 21  30
of
104
Gröbner Bases and Invariant Theory
"... This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion
On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations
, 2005
"... Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defect ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the socalled Convection Theorem. Both approaches are thought to be necessary for crossverification of the standard differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general 4dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary crossverification for a reconsidered differential form of continuity equation. Although the approach is developed for onecomponent (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored.
Gauss, Statistics, and Gaussian Elimination
, 1994
"... This report gives a historical survey of Gauss's work on the solution of linear systems. This report is available by anonymous ftp from thales.cs.umd.edu in the directory pub/reports. It will appear in the proceedings of Interface 94. y Department of Computer Science and Institute for Advanced ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This report gives a historical survey of Gauss's work on the solution of linear systems. This report is available by anonymous ftp from thales.cs.umd.edu in the directory pub/reports. It will appear in the proceedings of Interface 94. y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Gauss, Statistics, and Gaussian Elimination G. W. Stewart Department of Comuter Science and Institute for Advanced Computer Studies University of Maryland at College Park 1. Introduction Everyone knows that Gauss invented Gaussian elimination, and, excepting a quibble, everyone is right. 1 What is less well known is that Gauss introduced the procedure as a mathematical tool to get at the precision of least squares estimates. In fact the computational component in the original description is so little visible, that it takes some doing to see an algorithm in it. Gaussian elimination, therefore, was not conceived as a gene...
The Changing Relationships Between Science and Mathematics: From Being Queen of Sciences to Servant of Sciences
, 2003
"... this paper, we will give a brief historical overview of the relationship between science and mathematics then we will move on to the impact of computers in this relationship and finally we will talk about the future of this relationship. The origin of the sciences is rooted in tool making and agricu ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
this paper, we will give a brief historical overview of the relationship between science and mathematics then we will move on to the impact of computers in this relationship and finally we will talk about the future of this relationship. The origin of the sciences is rooted in tool making and agriculture. It is fair to say that making and using tools and the cultural transmission of scientific knowledge became essential to the existence of the human species and was practiced in all human societies. The history of science and mathematics starts with the Neolithic era. In the Neolithic Revolution, although mathematical knowledge was limited to counting and arithmetical operations, scientific knowledge was more advanced than mathematical knowledge; for example, potters possessed practical knowledge of the behavior of clay and fire, and, although they may not have had explanations for the phenomena of their crafts, they toiled without any 2 systematic science of materials or selfconscious
CauchyBinet for pseudo determinants
, 2013
"... Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants. More specifically, after proving some properties for pseudodeterminants, we show that for any two n × m matrices F, G, the formula Det(F T G) = P det(FP)det(GP) holds, where det(FP) runs over all k×k minors of A with k = min(rank(F T G), rank(GF T)). A consequence is the following Pythagoras theorem: for any selfadjoint matrix A of rank k one has Det 2 (A) = ∑ P det2 (AP), where det(AP) runs over all k × k minors of A. 1.
The Bridge Between The Continuous And The Discrete Via Original Sources
"... this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula in rel ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula in relation to sums of infinite series
How applied mathematics became pure
 Review of Symbolic Logic
"... Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics in contemporary science. My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one
Some Notes on Applied Mathematics for Machine Learning
 Advanced Lectures on Machine Learning
, 2004
"... This chapter 1 describes Lagrange multipliers and some selected subtopics from matrix analysis from a machine learning perspective. The goal is to give a detailed description of a number of mathematical constructions that are widely used in applied machine learning. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This chapter 1 describes Lagrange multipliers and some selected subtopics from matrix analysis from a machine learning perspective. The goal is to give a detailed description of a number of mathematical constructions that are widely used in applied machine learning.
NEGATIVE NUMBERS AS AN EPISTEMIC DIFFICULT CONCEPT: SOME LESSONS FROM HISTORY
"... Historical studies on the development of mathematical concepts will serve mathematics teachers to relate their students ’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Historical studies on the development of mathematical concepts will serve mathematics teachers to relate their students ’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only the division by negative numbers poses problems for the number line, but also the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics we argue for the introduction of negative numbers in education within the context of symbolic operations. Mon enthousiasme pour les mathématiques avaient peutêtre eu pour base principale mon horreur pour l’hypocrisie; l’hypocrisie à mes yeux, c’était ma tante Séraphie, Mme Vignon, et leurs prêtres. Suivant moi, l’hypocrisie était impossible en mathématiques, et, dans ma simplicité juvénile, je pensais qu’il en était ainsi dans toutes les sciences où j’avais ouï dire qu’elles s’appliquaient. Que devinsje quand je m’aperçus que personne ne pouvait m’expliquer comment il se faisait que: moins par moins donne plus?