Types of subgroups of E(n): Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category. Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically ...

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Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes

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Hernández was first spotted by Luis Fuenmayor, a part-time Mariners scout who saw him pitching at age 14 in a tournament near Maracaibo, Venezuela. Fuenmayor recommended Hernández to fellow scouts Pedro Avila and Emilio Carrasquel, who were impressed with the youngster who could already throw 94 mph. The Mariners continued to follow Hernández for over a year, but baseball rules prohibit teams from ...

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Félix Hernández, Félix Hernández - Discovery as a prospect, Félix Hernández - Minor league career, Félix Hernández - Major league debut

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The definition of the Lebesgue covering dimension can be used to build some unusual topological sets, such as the Sierpinski carpet. A construction can proceed as follows. Consider, for example, a finite open covering for the two-dimensional unit disk. This covering can always be refined so that no point in the disk belongs to more than three sets. Fixing this covering, remove all of the points in the disk that belong to three sets. Depending on the refinement, this will leave possibly one or more holes in the disk. The remaining obje ...

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Lebesgue covering dimension, Lebesgue covering dimension - Some unusual topological constructions, Lebesgue covering dimension - History, Lebesgue covering dimension - Historical references, Lebesgue covering dimension - Modern references

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The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is akin to a derivative of a multivariate function. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component funct ...

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Jacobian, Jacobian - Jacobian matrix, Jacobian - Example, Jacobian - Jacobian determinant, Jacobian - Example

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The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. Ordinary differential equation - Homogeneous linear ODEs with constant coefficients. The first method of integrating linear ordinary differential equations with constant coefficients is due to E ...

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Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

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An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The first mathematical formulation of gravity was Isaac Newton's law of universal gravitation, published in his 1687 work Principia Mathematica. Professor William Whewell of Cambridge University, author of History of the Inductive Sciences (1837) stated: "The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of the truth disclosed, or the fundamental and satisfactory nature of this truth." [In A Treasury o ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

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A two-dimensional spiral may be described using polar coordinates by saying that the radius r is a continuous monotonic function of θ. The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). Some of the more important sorts of two-dimensional spirals include: The Archimedean spiral: r = a + bθ The Cornu spiral or clothoid Fermat's spiral: r = θ1/2 The hyperbolic spiral: r = ...

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Spiral, Spiral - Two-dimensional spirals, Spiral - Three-dimensional spirals, Spiral - Spherical spiral

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Émile Clapeyron - Thermodynamics. In 1834, he made his first contribution to the creation of modern thermodynamics by publishing a report entitled the Driving force of the heat, in which it developed the work of the physicist Nicolas Léonard Sadi Carnot, deceased two years before. Though Carnot had developed a compelling anaysis of a generalised heat engine, he had employed the clumsy and already unfashionable caloric theory. Clapeyron, in his memoire, presented Carnot's work in a more accessible and analytic graphical form, showing the Carnot cycle as a closed curve on an indicator dia ...

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Émile Clapeyron, Émile Clapeyron - Life, Émile Clapeyron - Work, Émile Clapeyron - Thermodynamics, Émile Clapeyron - Other work, Émile Clapeyron - Honors

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Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties. If X is simply connected, then this holds globally ...

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Covering map, Covering map - Examples, Covering map - Elementary properties, Covering map - Universal covers, Covering map - Deck transformation group regular covers, Covering map - Monodromy action, Covering map - Group structure redux

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A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearin ...

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Angle, Angle - Units of measure for angles, Angle - Conventions on measurement, Angle - Types of angles, Angle - Some facts, Angle - A formal definition, Angle - Angles in different contexts, Angle - Angles in Riemannian geometry, Angle - Angles in astronomy, Angle - Angles in maritime navigation

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Roof tiles are designed mainly to keep out rain, and are traditionally made from locally available materials such as clay, slate, or wood (wooden tiles are called shingles). Modern materials such as concrete and plastic are also used. Some clay tiles have a waterproof glaze. Because of their long history, a large number of shapes (or "profiles") of roof tiles have evolved. These include: Flat tiles - the simplest type, which are laid in regular overlapping rows. This profile is suitable for stone and wooden tiles, ...

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Tile, Tile - Roof tiles, Tile - Floor tiles, Tile - Wall tiles, Tile - Decorative tilework, Tile - Islamic tilework, Tile - The mathematics of tiling, Tile - History of tiles

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Conventional economic paradigms acknowledge the basic notion of the Laffer curve, but argue that government was operating on the left-hand side of the curve, so a tax cut would thus lower revenue. The central question is the elasticity of work with respect to tax rates. For example, Pecorino (1995) argued that the peak occurred at tax rates around 65%, and summarized the controversy as: Just about everyone can agree that if an increase in tax rates leads to a decrease in tax revenues, then taxes are too high. It is also g ...

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Laffer curve, Laffer curve - Context in US History, Laffer curve - Critiques of the Laffer Curve, Laffer curve - Supporting Examples, Laffer curve - Difficulties of measurement, Laffer curve - Keynesian critique, Laffer curve - The wrong incentives?, Laffer curve - Estimates of the effectiveness of the Laffer Curve, Laffer curve - Precedents to the Laffer Curve

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Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the three functions h, T and M from above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper. To be more precise, we say that the function f is continuous at some point c when the following two re ...

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Continuous function, Continuous function - Real-valued continuous functions, Continuous function - Epsilon-delta definition, Continuous function - Heine definition of continuity, Continuous function - Examples, Continuous function - Facts about continuous functions, Continuous function - Continuous functions between metric spaces, Continuous function - Continuous functions between topological spaces, Continuous function - Continuous functions between partially ordered sets

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Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electrical work", an electric field does work on charged particles as they move through a medium. One mechanism of heat conduction is collisions between fast-moving atoms in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, it averages to nearly zero in bulk, so conduction is not considered to be mechanical work.

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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The 1909 edition of Webster's unabridged dictionary (citation: Webster's New International Dictionary, G. & C. Merriam Co. copyright 1909, pp1752) defines a quaternion as 5. Math. The quotient of two vectors ... Such is the view of the inventor, Sir Wm. Rowan Hamilton, and his disciple, Prof. P. G. Tait; but authorities are not yet quite agreed as to what a quaternion is or ought to be. This definition together with the excised "..." is technically correct if the the definition of "vector" is restricted to its ...

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Quaternions and spatial rotation, Quaternions and spatial rotation - Introducion, Quaternions and spatial rotation - Non-commutativity, Quaternions and spatial rotation - Double covering, Quaternions and spatial rotation - Chirality, Quaternions and spatial rotation - Definitions, Quaternions and spatial rotation - Concepts, Quaternions and spatial rotation - Terminology, Quaternions and spatial rotation - Notation, Quaternions and spatial rotation - Reflections and Rotations, Quaternions and spatial rotation - Analytic form of a reflection, Quaternions and spatial rotation - Rotation: the composition of two reflections, Quaternions and spatial rotation - Quaternion representation of a rotation, Quaternions and spatial rotation - General rotations in four dimensional space, Quaternions and spatial rotation - Algebraic rules, Quaternions and spatial rotation - Other properties, Quaternions and spatial rotation - Quaternion rotation, Quaternions and spatial rotation - An example, Quaternions and spatial rotation - Quaternions versus other representations of rotations, Quaternions and spatial rotation - Pairs of unit quaternions as rotations in 4D space

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There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below. Tangent space - Definition as directions of curves. Suppose M is a Ck manifold (k ≥ 1) and p is a point in M. Pick a chart φ : U → Rn where See also:

Tangent space, Tangent space - Informal description, Tangent space - Formal definitions, Tangent space - Definition as directions of curves, Tangent space - Definition via derivations, Tangent space - Definition via the cotangent space, Tangent space - Properties, Tangent space - Tangent vectors as directional derivatives, Tangent space - The derivative of a map

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The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula: for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation. Differential calculus can be used to determine the instantaneous speed at any given instant, while the f ...

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Calculus, Calculus - Differential calculus, Calculus - Integral calculus, Calculus - Foundations, Calculus - Fundamental theorem of calculus, Calculus - Applications, Calculus - History, Calculus - Footnotes

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Gordon started his career as a Royal, and was signed away by Boston where he was converted from a starter to a closer. In 1998, Gordon set the club's single-season saves record (46) and was named to his first All-Star Team. His success continued in 1999 setting a major league record with his 54th consecutive save in June, but a nagging elbow injury limited him to just 21 appearances, which required ulnar collateral ligament reconstruction (o ...

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Tom Gordon, Tom Gordon - High School Career, Tom Gordon - Major League Baseball Career, Tom Gordon - Highlights, Tom Gordon - Trivia

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Chloracne normally results from direct skin contact with chloracnegens, although ingestion and inhalation are also possible causative routes. Chloracnegens are fat-soluble, meaning they persist in the body fat for a very long period following exposure. Chloracne is a chronic inflammatory condition that results from this persistence, in combination with the toxin's chemical properties. It is believed, at least from rodent models, that the toxin activates a series of receptors promoting macrophage proliferation, inducing neutrophilia an ...

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Chloracne, Chloracne - Etiology and progression, Chloracne - Treatment, Chloracne - Related conditions, Chloracne - Notable cases

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Fixed curve is a catenary, rolling curve is a line: f(t) = t + icosh(t)   f'(t) = 1 + isinh(t) r(t) = sinh(t)   r'(t) = cosh(t) if p=-i the expression is real and the roulette is a horizontal line. In other words, a square wheel could roll without bouncing in a road t ...

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Roulette curve, Roulette curve - Example, Roulette curve - External link

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In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. The midpoint of a line segment is its 'middle' point: the unique point at an equal distance from the two end points. ...

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Line mathematics, Line mathematics - Line segment, Line mathematics - Ray

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