A Survey of Modern Algebra

Birkhoff, Garrett; Lane, Saunders Mac

doi:10.1201/9781315275499

Cited by 260 publications

(110 citation statements)

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“…(Birkhoff 1948, p. 4) Then he reworded the original theorem and for the construction used in the proof cites Birkhoff and MacLane (1941), not van der Waerden (the English edition of van der Waerden's book was published in 1949, a year after the publication of the second edition of Birkhoff's Lattice Theory).…”

Section: A Portray Of the Evolution Of The Terminologiesmentioning

confidence: 99%

Equivalence: an attempt at a history of the idea

Asghari

2018

Synthese

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This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the 'first' modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class.

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Section: A Portray Of the Evolution Of The Terminologiesmentioning

confidence: 99%

Equivalence: an attempt at a history of the idea

Asghari

2018

Synthese

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“…If all roots of a polynomial are known, then the polynomial can be written in another basic form x P n (w) = (w -w 1 )(w -w 2 ) (w -w n ) (2) where w 1 , w 2 , w n are complex numbers and the polynomial in form (2) obviously reduces to zero if w is equal with one of the roots. If the mathematical operations are performed, then (2) reduces to (1) because all operations are unique.…”

Section: G the Second Basic Form Of A Polynomialmentioning

confidence: 99%

“…If the mathematical operations are performed, then (2) reduces to (1) because all operations are unique.…”

Section: G the Second Basic Form Of A Polynomialmentioning

confidence: 99%

Theory Of The Root Lines Of The General Polynomial

Schmidt¹

2016

ARA 40th Congress Proceedings

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This paper describes the main characteristics of the root lines of polynomials with complex coefficients. Root lines are defined as the geometric loci of the roots of the real, respective imaginary part of the polynomial. A graphical representation of these lines offers a clear picture of the positions of the polynomial's roots. The coordinates of various points of the root lines can be obtained by use of a computer program. Each root line leads to a given root, this way the polynomial can be solved and graphically represented. A short description of the rules of variation and properties of these root lines follows depending on the position of the roots of the polynomial. This study contains many numerical examples which show not only how the root lines, but also the polynomials with complex roots and coefficients work, how they can be studied, transformed and solved. A. IntroductionThis study is a continuation and completion of a study about the root lines presented at the 27-th ARA Congress in Oradea, Romania, May 29 -June 2, 2002. The length of the study was then restricted to 5 pages, therefore some observations in the study could not be mathematically proved or demonstrated. This paper presents all those observations and new explanations regarding the properties of root lines. B. Importance of the root lines.Root lines are characteristic for polynomials with complex roots. A polynomial of degree n has n roots and all roots can be complex. According to some scientific works the concept of complex roots has a particular importance to the physical and engineering sciences. See Reference [4] Volume II, for the list of such cases. The same [4], on page 413 and ff. also presents some curves which are root lines, but are called by the author μ(x,y) = c1 and ν(x,y) = c2. (μ and ν are obviously the real and imaginary part of the polynomial). The name "root lines"of this theory is given by me, and probably is not found in any other book or article. In the same chapter of that book [4] is also mentioned that the angles between these curves at a multiple root are equal. It explains this with the vector dot product of the gradients. This study gives a more simple explanation for this. The studies mentioned in [4] are also restricted to polynomials with real coefficients. My studies have no such restriction, because I found that they can easily be extended to cases with complex coefficients, so they are more general, without restriction. C. The basic form of a polynomial isP n (w) = C n w n +C n-1 w n-1 + + C 1 x+ C 0( 1) w is the independent variable of the polynomial which can be a real or a complex number; regarding w see also Par. I. The coefficients C n to C 0 are complex or real numbers. D. Roots of a polynomial.Roots (or zeros) of a polynomial are those values of the variable w for which the polynomial's value (both the real and the imaginary part) reduces to zero. E. Real roots of a polynomial.If the variable w in relation (1) is a real number, and P(w) is represented along a straight reference line, then the roo...

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“…In particular, the tensor product ⊗ universal law [1,Theorem 22] states that every bilinear map β : V × W → X is factorizable as the composition β = σ · ρ of a linear map σ : V ⊗ W → X after the tensor embedding ρ : V × W → V ⊗ W which combines a pair of inputs V × W , two spaces V and W , into a unique input, the tensor space V ⊗ W .…”

Section: Introductionmentioning

confidence: 99%

Gaussian elimination is not optimal, revisited

Macedo

2016

Journal of Logical and Algebraic Methods in Programming

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We refactor the universal law for the tensor product to express matrix multiplication as the product M · N of two matrices M and N thus making possible to use such matrix product to encode and transform algorithms performing matrix multiplication using techniques from linear algebra. We explore such possibility and show two stepwise refinements transforming the composition M · N into the Naïve and Strassen's matrix multiplication algorithms.The inspection of the stepwise transformation of the composition of matrices M ·N into the Naïve matrix multiplication algorithm evidences that the steps of the transformation correspond to apply Gaussian elimination to the columns of M and to the lines of N therefore providing explicit evidence on why "Gaussian elimination is not optimal", the aphorism serving as the title to the succinct paper introducing Strassen's matrix multiplication algorithm.Although the end results are equations involving matrix products, our exposition builds upon previous works on the category of matrices (and the related category of finite vector spaces) which we extend by showing: why the direct sum (⊕, 0) monoid is not closed, a biproduct encoding of Gaussian elimination, and how to further apply it in the derivation of linear algebra algorithms.

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A Survey of Modern Algebra

Cited by 260 publications

References 0 publications

Equivalence: an attempt at a history of the idea

Equivalence: an attempt at a history of the idea

Theory Of The Root Lines Of The General Polynomial

Gaussian elimination is not optimal, revisited

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