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The Spectral Topology in Rings.

紀錄類型:	書目-語言資料,手稿 : Monograph/item
正題名/作者:	The Spectral Topology in Rings./
作者:	Wessels, Francois.
面頁冊數:	1 online resource (93 pages)
附註:	Source: Masters Abstracts International, Volume: 84-04.
Contained By:	Masters Abstracts International84-04.
標題:	Eigenvalues. -
電子資源:	click for full text (PQDT)
ISBN:	9798352610275

The Spectral Topology in Rings.
Wessels, Francois.

The Spectral Topology in Rings. - 1 online resource (93 pages)

Source: Masters Abstracts International, Volume: 84-04.

Thesis (M.Sc.)--University of Johannesburg (South Africa), 2022.

Includes bibliographical references

A careful study of Hilbert [22] and Cauchy's [6] original work reveals that the word characteristique in French translates to "eigenschaften" in German ("eienskap" in Afrikaans) which may be interpreted as intrinsic or property in the sense that it associates a(n) (set of) eigenvalue(s) λ [thought of as the seed] belonging to an operator T [thought of as the stem]. It is due to this operator-eigenvalue tug'o war that most of the modern theory has become so well developed to model physical problems leading to insightful results. The word "integralgleichung" is used frequently throughout the text which translates to English asbintegral equation. In his article Hilbert discusses the symmetric kernel on page 52, characteristic determinant equations on page 53, linear combinations and algebraic solutions to the integral equations on page 57, convergent power series solution of Fredholm determinant expression on page 58, orthogonality and eigenfunction solutions of integral equations on page 67, essentially the foundation for classical operator theory.Long before the power of vector space algebra was postulated into existence and became common practice, the eigenvalue method was one of the tools used to study systems of linear equations as roots of determinant equations, but made some of its first appearance in the studies of differential and integral equations originating from variational calculus during the celestial era of mathematics, giving birth to operator theory. Operator theory can be chronologically tracked through Leonard Euler (1707 - 1783), Jean le Rond d'Alembert (1717 - 1783), Joseph Louis Lagrange (1736 - 1813), Daniel (1700 - 1782) & the three Bernoulli brothers (1710 - 1790, 1744 - 1807, 1759 - 1789), Pierre Simon Laplace (1749 - 1827), Joseph Fourier (1768 - 1830), Johann Peter Gustav Lejeune Dirichlet (1805 - 1859), who all played a big part in pursuit of an even bigger picture - Modelling, analysing, solving and generalising functional equations.Sometimes eigenvalues are referred to as the roots of characteristic equations, those equations deeply associated with its homogeneous auxiliary equations and in modern mathematics related to the spectrum, the set of values in a scalar field associated with algebraic invertibility of an operator expression of the form T − λI. In fact, eigenvalues arise naturally in the generalization of operator equations as the following examples illustrate:In each example above, the equations were reducible to an ordinary linear eigenvalue problem. The foundations of the solvability theory for such problems had already mostly been explored by researchers such as the three prodigies, Johann Carl Friedrich Gauss (1777 - 1855) (in his Disquisitiones Arithmeticae), Evariste ´ Galois (1811 - 1832), and Niels Hendrik Abel (1802 - 1829). This is the insight which motivated mathematicians to study operator equations collectively as a subject. However, a great deal of the behaviour of operator equations is due to the underlying structure space on which the equation is defined. For example continuity, convergence, invertibility, integrability and differentiability are all in one way or another, dependent, not only on the form of the equation, but also largely on the underlying algebraic and topological aspects of the space.The demand for clear insight about the underlying structure space for operator equations further stimulated research in the fields of topology and algebra. The story diverts to set theory and the axiomatization of mathematics. Georg Ferdinand Ludwig Philipp Cantor's (1845 - 1918) set theory had been digested by the mathematics community and axiomatized by Ernst Zermelo (1871 - 1953) in his 1908 paper Untersuchengen ¨uber die Grundlagen der Mengenlehre ["mengenlehre" in German translates to set theory in English] while Giuseppe Peano fully axiomatized linear spaces in his 1888 book, titled Calcolo. Cantor had already loosely defined the notions of open, closed and derived sets, influencing Ren´e-Louis Baire (1874 - 1932), Emile Borel (1871 - 1956) and Henri Lebesgue (1875 - 1941) to write his phenom- ´ enal piece, titled Sur l'approximation des fonctions, thereby extending the ideas of George Friedrich Bernhard Riemann (1826 - 1866) with strong reliance on set theory, consequently giving rise to the subjects of measure and integration theory as it is taught in current times.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798352610275Subjects--Topical Terms:

527710
Eigenvalues.
Index Terms--Genre/Form:

554714
Electronic books.

The Spectral Topology in Rings.
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504 $a Includes bibliographical references
520 $a A careful study of Hilbert [22] and Cauchy's [6] original work reveals that the word characteristique in French translates to "eigenschaften" in German ("eienskap" in Afrikaans) which may be interpreted as intrinsic or property in the sense that it associates a(n) (set of) eigenvalue(s) λ [thought of as the seed] belonging to an operator T [thought of as the stem]. It is due to this operator-eigenvalue tug'o war that most of the modern theory has become so well developed to model physical problems leading to insightful results. The word "integralgleichung" is used frequently throughout the text which translates to English asbintegral equation. In his article Hilbert discusses the symmetric kernel on page 52, characteristic determinant equations on page 53, linear combinations and algebraic solutions to the integral equations on page 57, convergent power series solution of Fredholm determinant expression on page 58, orthogonality and eigenfunction solutions of integral equations on page 67, essentially the foundation for classical operator theory.Long before the power of vector space algebra was postulated into existence and became common practice, the eigenvalue method was one of the tools used to study systems of linear equations as roots of determinant equations, but made some of its first appearance in the studies of differential and integral equations originating from variational calculus during the celestial era of mathematics, giving birth to operator theory. Operator theory can be chronologically tracked through Leonard Euler (1707 - 1783), Jean le Rond d'Alembert (1717 - 1783), Joseph Louis Lagrange (1736 - 1813), Daniel (1700 - 1782) & the three Bernoulli brothers (1710 - 1790, 1744 - 1807, 1759 - 1789), Pierre Simon Laplace (1749 - 1827), Joseph Fourier (1768 - 1830), Johann Peter Gustav Lejeune Dirichlet (1805 - 1859), who all played a big part in pursuit of an even bigger picture - Modelling, analysing, solving and generalising functional equations.Sometimes eigenvalues are referred to as the roots of characteristic equations, those equations deeply associated with its homogeneous auxiliary equations and in modern mathematics related to the spectrum, the set of values in a scalar field associated with algebraic invertibility of an operator expression of the form T − λI. In fact, eigenvalues arise naturally in the generalization of operator equations as the following examples illustrate:In each example above, the equations were reducible to an ordinary linear eigenvalue problem. The foundations of the solvability theory for such problems had already mostly been explored by researchers such as the three prodigies, Johann Carl Friedrich Gauss (1777 - 1855) (in his Disquisitiones Arithmeticae), Evariste ´ Galois (1811 - 1832), and Niels Hendrik Abel (1802 - 1829). This is the insight which motivated mathematicians to study operator equations collectively as a subject. However, a great deal of the behaviour of operator equations is due to the underlying structure space on which the equation is defined. For example continuity, convergence, invertibility, integrability and differentiability are all in one way or another, dependent, not only on the form of the equation, but also largely on the underlying algebraic and topological aspects of the space.The demand for clear insight about the underlying structure space for operator equations further stimulated research in the fields of topology and algebra. The story diverts to set theory and the axiomatization of mathematics. Georg Ferdinand Ludwig Philipp Cantor's (1845 - 1918) set theory had been digested by the mathematics community and axiomatized by Ernst Zermelo (1871 - 1953) in his 1908 paper Untersuchengen ¨uber die Grundlagen der Mengenlehre ["mengenlehre" in German translates to set theory in English] while Giuseppe Peano fully axiomatized linear spaces in his 1888 book, titled Calcolo. Cantor had already loosely defined the notions of open, closed and derived sets, influencing Ren´e-Louis Baire (1874 - 1932), Emile Borel (1871 - 1956) and Henri Lebesgue (1875 - 1941) to write his phenom- ´ enal piece, titled Sur l'approximation des fonctions, thereby extending the ideas of George Friedrich Bernhard Riemann (1826 - 1866) with strong reliance on set theory, consequently giving rise to the subjects of measure and integration theory as it is taught in current times.
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