mathematical analysis

Main article: limit of a function Suppose f(x) is a real function and c is a real number. The expression: means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f(x), as x approaches c". Note that this statement can be true even if f(c) L. Indeed, the function f(x) need not even b ...

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Limit mathematics, Limit mathematics - Limit of a function, Limit mathematics - Formal definition, Limit mathematics - Limit of a function at infinity, Limit mathematics - Limit of a sequence, Limit mathematics - Topological net, Limit mathematics - Limit in category theory

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To every Lie group, we can associate a Lie algebra which completely captures the local structure of the group, at least if the Lie group is connected. This is done as follows. Conventionally, one can regard any field X of tangent vectors on a Lie group as a partial differential operator, denoting by Xf the Lie derivative (the directional derivative) of the scalar field f in the direction of X. Then a vector field on a Lie group G is said to be left-invariant if it commutes with left translatio ...

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Lie group, Lie group - Types of Lie groups, Lie group - Homomorphisms and isomorphisms, Lie group - The Lie algebra associated to a Lie group, Lie group - Alternative definitions

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Tonelli's theorem (named after Leonida Tonelli) is a predecessor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumptions are different. Tonelli's theorem states that a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral. In fact, the existence of the first integral above (the integral of th ...

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Fubini's theorem, Fubini's theorem - Tonelli's theorem, Fubini's theorem - Applications

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The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integration has two distinct parts: A theory of measurable sets and measures on these sets. A theory of measurable functions and integrals on these functions. Lebesgue integration - Measure theory. Measure theory initially was created to provide a detailed analysis of the notion of length of subsets of the real line and more generally area and volum ...

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Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

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Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35000 BC and 20000 BC, indicate early attempts to quantify time Evidence exists that early counting involved women who kept records of their monthly biological ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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For , every function can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below. One says that h(z) is an inner (interior) function if and only if on the unit disc and the limit exists for almost all θ. One says that G(z) is an outer (exterior) function if it takes the form for some real value and some real-valued function g( ...

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Hardy space, Hardy space - Applications, Hardy space - Factorization

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Let G be a locally compact topological group. In this article, the σ-algebra generated by all compact subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If a is an element of G and S is a subset of G, then we define the left and right translates of S as follows: Left translate: Right translate: Left and ri ...

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Haar measure, Haar measure - Preliminaries, Haar measure - Existence of the left Haar measure, Haar measure - The right Haar measure, Haar measure - The Haar integral, Haar measure - Uses, Haar measure - Examples, Haar measure - The modular function

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The Legendre polynomials are solutions to the Sturm-Liouville problem and because of the theory, these polynomials are eigenfunctions of the problem and are solutions orthogonal with respect to the inner product above with unit weight. So we can form a generalized Fourier series (known as a Fourier-Legendre series) involving the Legendre polynomials, and As an example, let us calculate the Fourier-Legendre series for f(x)=cos x over [−1,1]. Now, See also:

Generalized Fourier series, Generalized Fourier series - Example Fourier-Legendre series, Generalized Fourier series - Coefficient theorems, Generalized Fourier series - Bessel's inequality, Generalized Fourier series - Parseval's theorem

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After his schooling at Winchester, Hardy entered Trinity College, Cambridge in 1896 after standing fourth in the Tripos examination. Years later, Hardy sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than being a means to an end. While at university, Hardy joined the Cambridge Apostles, an elite, intellectual secret society. Hardy was Sadleirian Professor at Cambridge from 1931 to 1942; he had left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertran ...

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G. H. Hardy, G. H. Hardy - Life, G. H. Hardy - Work, G. H. Hardy - Attitudes, G. H. Hardy - Books

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List of publications in mathematics - Euclid's Elements. Euclid Publication data: c. 300 BC Online version: Interactive Java version Description: This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The Elements is still a valuable resource and a good introduction to algorithm. More tha ...

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List of publications in mathematics, List of publications in mathematics - Geometry, List of publications in mathematics - Euclid's Elements, List of publications in mathematics - La Géométrie, List of publications in mathematics - Logic, List of publications in mathematics - Begriffsschrift, List of publications in mathematics - Formulario mathematico, List of publications in mathematics - Principia Mathematica, List of publications in mathematics - Gödel's incompleteness theorem, List of publications in mathematics - Information theory, List of publications in mathematics - Number theory, List of publications in mathematics - Disquisitiones Arithmeticae, List of publications in mathematics - On the Number of Primes Less Than a Given Magnitude, List of publications in mathematics - Vorlesungen über Zahlentheorie, List of publications in mathematics - Number Theory An approach through history from Hammurapi to Legendre, List of publications in mathematics - Calculus, List of publications in mathematics - Philosophiae Naturalis Principia Mathematica, List of publications in mathematics - Newton's Principia for the Common Reader, List of publications in mathematics - Calculus and Calculus on Manifolds, List of publications in mathematics - Numerical analysis, List of publications in mathematics - Method of Fluxions, List of publications in mathematics - Game theory, List of publications in mathematics - Evolution and the Theory of Games, List of publications in mathematics - Theory of Games and Economic Behavior, List of publications in mathematics - On Numbers and Games, List of publications in mathematics - Winning Ways for your Mathematical Plays, List of publications in mathematics - Fractals, List of publications in mathematics - How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, List of publications in mathematics - Early manuscripts, List of publications in mathematics - Rhind Mathematical Papyrus, List of publications in mathematics - The Nine Chapters on the Mathematical Art, List of publications in mathematics - Archimedes Palimpsest, List of publications in mathematics - The Sand Reckoner, List of publications in mathematics - Textbooks, List of publications in mathematics - Course of Pure Mathematics, List of publications in mathematics - Art of Problem Solving, List of publications in mathematics - Metalogic: an Introduction to the Metatheory of Standard First Order Logic, List of publications in mathematics - Popular writing, List of publications in mathematics - Gödel Escher Bach, List of publications in mathematics - The World of Mathematics, List of publications in mathematics - Arithmetic, List of publications in mathematics - Arithmetick: or The Grounde of Arts, List of publications in mathematics - The Schoolmaster's Assistant Being a Compendium of Arithmetic both Practical and Theoretical, List of publications in mathematics - Abstract algebra, List of publications in mathematics - Moderne Algebra, List of publications in mathematics - Linear algebra, List of publications in mathematics - Algebraic geometry, List of publications in mathematics - Faisceaux Algébriques Cohérents, List of publications in mathematics - Géométrie Algébrique et Géométrie Analytique, List of publications in mathematics - Éléments de géométrie algébrique, List of publications in mathematics - Séminaire de géométrie algébrique, List of publications in mathematics - Algebraic Geometry, List of publications in mathematics - Universal algebra, List of publications in mathematics - Universal algebra, List of publications in mathematics - Group theory, List of publications in mathematics - Monoid, List of publications in mathematics - Topology, List of publications in mathematics - Topologie, List of publications in mathematics - Topology, List of publications in mathematics - General Topology, List of publications in mathematics - Graph theory, List of publications in mathematics - Category theory, List of publications in mathematics - Categories for the Working Mathematician, List of publications in mathematics - Category Theory for Computing Science, List of publications in mathematics - Order theory, List of publications in mathematics - Trigonometry, List of publications in mathematics - Differential geometry, List of publications in mathematics - Differential topology, List of publications in mathematics - Topology from the Differentiable Viewpoint, List of publications in mathematics - Algebraic topology, List of publications in mathematics - Algebraic Topology, List of publications in mathematics - Fractal geometry, List of publications in mathematics - Discrete mathematics, List of publications in mathematics - Combinatorics, List of publications in mathematics - Set theory, List of publications in mathematics - Grundzüge der Mengenlehre, List of publications in mathematics - Naive Set Theory, List of publications in mathematics - Cardinal and Ordinal Numbers, List of publications in mathematics - The Consistency of the Continuum Hypothesis, List of publications in mathematics - Set Theory and the Continuum Hypothesis, List of publications in mathematics - Set Theory: An Introduction to Independence Proofs, List of publications in mathematics - Optimization, List of publications in mathematics - The New Variational Method, List of publications in mathematics - Decomposition Principle for Linear Programs., List of publications in mathematics - Network Flows and General Matchings, List of publications in mathematics - Paths trees and Flowers, List of publications in mathematics - The complexity of theorem proving procedures, List of publications in mathematics - Reducibility among combinatorial problems, List of publications in mathematics - How good is the simplex algorithm?, List of publications in mathematics - Linear Programming and Polynomial time algorithms, List of publications in mathematics - New polynomial-time algorithm for linear programming, List of publications in mathematics - Interior Point Polynomial Algorithms in Convex Programming

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Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, characterized by both its magnitude represented by length, and its direction. Vectors can be used to represent physical entities such as forces, and they can be added and multiplied with scalars, thus forming the first example of a real vector space. Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n ...

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Linear algebra, Linear algebra - History, Linear algebra - Elementary introduction, Linear algebra - Some useful theorems, Linear algebra - Generalization and related topics

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Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first conceived of logarithms. The method of natural logarithms was first propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston in Scotland, four years after the publication of his memorable invention. This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constant ...

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Logarithm, Logarithm - Logarithms and exponentials: inverses, Logarithm - Using logarithms, Logarithm - Unspecified bases, Logarithm - Change of base, Logarithm - Uses of logarithms, Logarithm - Exponential functions, Logarithm - Easier computations, Logarithm - Calculus, Logarithm - History, Logarithm - Tables of logarithms, Logarithm - Algorithm, Logarithm - Trivia, Logarithm - Unicode glyph, Logarithm - Alternate notation, Logarithm - Relationships between binary natural and common logarithms

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Kerala School - Narayana Pandit c. 1340-1400. Narayana Pandit, the earliest of the notable Keralese mathematicians, is known to have definitely written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. He was strongly influenced by the work of Bhaskara II, which proves work from the classic period was known to Keralese mathematicians and was thus influential in the continued progress of the subject. Due to this influence Narayana is also th ...

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Kerala School, Kerala School - Keralese Mathematicians, Kerala School - Narayana Pandit c. 1340-1400, Kerala School - Madhava of Sangamagramma 1340-1425, Kerala School - Parameshvara c. 1370-1460, Kerala School - Nilakantha Somayaji 1444-1544, Kerala School - Jyesthadeva c. 1500-1575, Kerala School - Sankara Varman Early 1800s, Kerala School - Possible transmission of Keralese mathematics to Europe

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To say that "0/0" is an indeterminate form does not just mean that "0/0" by itself can represent any number, or may represent no number. Those points are true, in a certain sense, but of limited practical significance when stated in those terms. It means also that the ratio of two functions f and g that approach zero can approach any member of a range of well-defined values, depending on which functions f and g are. Whether such a value exists, and what it might be, d ...

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Indeterminate form, Indeterminate form - Discussion, Indeterminate form - Examples on 0/0, Indeterminate form - List of indeterminate forms

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In order to calculate: Let: u = x, so that du = dx, dv = cos(x) dx, so that v = sin(x). Then: where C is an arbitrary constant of integration. By repeatedly using integration by parts, integrals such as can be computed in the same fashion: each application of the rule lowers the power of x by one. An interesting examp ...

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Integration by parts, Integration by parts - The rule, Integration by parts - Examples, Integration by parts - The LIATE rule, Integration by parts - Recursive formulation, Integration by parts - Higher dimensions

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Formal power series - Two definitions of the formal power series ring. We start with a commutative ring R. We want to define the ring of formal power series over R in the variable X, denoted by R[[X]]; elements of this ring should be thought of as power series whose coefficients are elements of R. Perhaps the most efficient definition of R[[X]] is as the completion of the polynomial ring R[X] with respect to the I-adic topolog ...

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Formal power series, Formal power series - Informal introduction, Formal power series - Formal development, Formal power series - Two definitions of the formal power series ring, Formal power series - Universal property, Formal power series - Operations on formal power series, Formal power series - Algebraic properties of the formal power series ring, Formal power series - Topological properties of the formal power series ring, Formal power series - Applications, Formal power series - Interpreting formal power series as functions, Formal power series - Generalizations, Formal power series - Formal Laurent series, Formal power series - Power series in several variables, Formal power series - Replacing the index set by an ordered abelian group, Formal power series - Examples and related topics

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If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an equivariant fashion. Jets do not transform as tensors. In fact, jets of functions between two manifolds belong to a Jet bundle. This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a vector bundle, analog ...

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Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

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In biological systems such as organisms, ecosystems, or the biosphere, most parameters must stay under control within a narrow range around a certain optimal level under certain environmental conditions. The deviation of the optimal value of the controlled parameter can result from the changes in internal and external environments. A change of some of the environmental conditions may also require change of that range to change for the system to function. The value of the parameter to maintain is recorded by a reception system and conveyed to ...

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Feedback, Feedback - Types of feedback, Feedback - Feedback in electronic engineering, Feedback - Feedback in mechanical engineering, Feedback - Feedback in economics and finance, Feedback - Feedback in nature, Feedback - Feedback in organizations, Feedback - Feedback in gaming, Feedback - Sources

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He discovered the infinite series for arctan and sin and many methods for calculating the circumference of the circle. One of Madhava's series is known from the text Yuktibhasa which describes - The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd num ...

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Madhava of Sangamagrama, Madhava of Sangamagrama - Contributions, Madhava of Sangamagrama - Kerala School of Astronomy and Mathematics

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Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sutra Bhashya by Jaina philosopher Omaswati (135 BC) and the book titled Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time. Indian mathematicians during this period used notations for squares, cube and other exponents of numbers. They gave shape to Beezganit Samikaran (Algebraic Equations). ...

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Indian mathematics, Indian mathematics - Indian contributions to mathematics, Indian mathematics - Harappan Mathematics 3300 BC - 1700 BC, Indian mathematics - Vedic Mathematics 1500 BC - 500 BC, Indian mathematics - Vedas 1500 BC - 500 BC, Indian mathematics - Samhitas 1500 BC - 500 BC, Indian mathematics - Lagadha 1350 BC - 800 BC, Indian mathematics - Yajnavalkya 1000 BC - 600 BC, Indian mathematics - Sulba Sutras 800 BC - 500 BC, Indian mathematics - Ancient Period 500 BC - 400 CE, Indian mathematics - Panini 500 BC - 400 BC, Indian mathematics - Pingala 400 BC - 200 BC, Indian mathematics - Vaychali Ganit 300 BC - 200 BC, Indian mathematics - Katyayana 200 BC, Indian mathematics - Jaina Mathematics 400 BC - 400 CE, Indian mathematics - Surya Siddhanta 300 CE - 400 CE, Indian mathematics - Classical Period 400 CE - 1200 CE, Indian mathematics - Aryabhata I 476-550, Indian mathematics - Bhaskara I 600-680, Indian mathematics - Brahmagupta 598-668, Indian mathematics - Shridhara Acharya 650-850, Indian mathematics - Mahavira Acharya 850, Indian mathematics - Aryabhata II 920-1000, Indian mathematics - Shripati Mishra 1019-1066, Indian mathematics - Nemichandra Siddhanta Chakravati 1100, Indian mathematics - Bhaskara Acharya Bhaskara II 1114-1185, Indian mathematics - Keralese Mathematics 1300 CE -1600 CE, Indian mathematics - Narayana Pandit 1340-1400, Indian mathematics - Madhava of Sangamagramma 1340-1425, Indian mathematics - Parameshvara 1370-1460, Indian mathematics - Nilakantha Somayaji 1444-1544, Indian mathematics - Jyesthadeva 1500-1575, Indian mathematics - Charges of Eurocentrism

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A fairly natural question to ask is, does there exist an operator H, or half-derivative, such that ? It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that , or to put it another way, is well-defined for all real values of n > 0. A similar result applies to integration. To delve into a little detail, start with the Gamma function , which is defined su ...

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Fractional calculus, Fractional calculus - Fractional derivative, Fractional calculus - Heuristics, Fractional calculus - Half derivative of a simple function, Fractional calculus - Laplace transform, Fractional calculus - Riemann-Liouville differintegral, Fractional calculus - Functional calculus

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William Thomson 1st Baron Kelvin - Family. William's father was Dr. James Thomson, the son of a Belfast farmer. James received little youthful instruction in Ireland but, when 24 years old, started to study for half the year at the University of Glasgow, Scotland, while working as a teacher back in Belfast for the other half. On graduating, he became a mathematics teacher at the Royal Belfast Academical Institution. He married Margaret Gardner ...

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William Thomson 1st Baron Kelvin, William Thomson 1st Baron Kelvin - Early life and work, William Thomson 1st Baron Kelvin - Family, William Thomson 1st Baron Kelvin - Youth, William Thomson 1st Baron Kelvin - Cambridge, William Thomson 1st Baron Kelvin - Thermodynamics, William Thomson 1st Baron Kelvin - Transatlantic cable, William Thomson 1st Baron Kelvin - Calculations on data-rate, William Thomson 1st Baron Kelvin - Scientist to engineer, William Thomson 1st Baron Kelvin - Disaster and triumph, William Thomson 1st Baron Kelvin - Later expeditions, William Thomson 1st Baron Kelvin - Other activities and contributions, William Thomson 1st Baron Kelvin - Geology and theology, William Thomson 1st Baron Kelvin - Honours, William Thomson 1st Baron Kelvin - Notes, William Thomson 1st Baron Kelvin - Bibliography, William Thomson 1st Baron Kelvin - Kelvin's works, William Thomson 1st Baron Kelvin - Biography

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