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Level Set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is called a level curve, also known as ''contour line'' or ''isoline''; so a level curve is the set of all real-valued solutions of an equation in two variables and . When , a level set is called a level surface (or '' isosurface''); so a level surface is the set of all real-valued roots of an equation in three variables , and . For higher values of , the level set is a level hypersurface, the set of all real-valued roots of an equation in variables. A level set is a special case of a fiber. Alternative names Level sets show up in many applications, often under different names. For example, an implicit curve is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an implici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isochrone Map
An isochrone map in geography and urban planning is a map that depicts the area accessible from a point within a certain time threshold. An isochrone (iso = equal, chrone = time) is defined as "a line drawn on a map connecting points at which something occurs or arrives at the same time". In hydrology and transportation planning isochrone maps are commonly used to depict areas of equal travel time. The term is also used in cardiology as a tool to visually detect abnormalities using body surface distribution. History Early examples of Isochrone maps include the Galton's ''Isochronic Postal Charts'' and ''Isochronic Passage Charts'' of 1881 and 1882, Bartholomew's ''Isochronic Distance Map and Chart'' first published 1889 and Albrecht Penck's ''Isochronenkarte'' first published 1887. Where as Galton and the Bartholomews published maps depicting the days or weeks it took to travel long distances, Albrecht further developed the idea to not only depict long distances and world tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level Grad
Level or levels may refer to: Engineering *Level (instrument), a device used to measure true horizontal or relative heights *Spirit level, an instrument designed to indicate whether a surface is horizontal or vertical *Canal pound or level *Regrading or levelling, the process of raising and/or lowering the levels of land *Storey or level, a vertical unit of a building or a mine * Level (coordinate), vertical position Gaming *Level (video games), a stage of the game * Level (role-playing games), a measurement of character development Music *Level (music), similar to but more general and basic than a chord * ''Levels'' (album), an album by AKA * "Levels" (Avicii song) * "Levels" (Bilal song) * "Levels" (Nick Jonas song) * "Levels" (Meek Mill song) * "Level" (The Raconteurs song) * "Levels" (NorthSideBenji song), featuring Houdini Places *Level Mountain, a volcano in northern British Columbia, Canada *Levél, Győr-Moson-Sopron, Hungary * Levels, New Zealand * Level, Ohio, United ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Himmelblau Contour
Himmelblau is a German language surname, which means " skyblue".''Behind the Name''"Surname Blau" Retrieved on 26 January 2016. The name may refer to: * David Himmelblau (1924–2011), American scientist * Fabian Himmelblau (1860–1931), Polish bookseller and publisher See also *Coop Himmelb(l)au, an architectural firm *Himmelblau's function In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by: : f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\quad It has one local maximum at x = ..., in mathematical optimization * PerfektBreitHimmelblau, a set of songs References {{surname German-language surnames German words and phrases Jewish surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Himmelblau's Function
In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by: : f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\quad It has one local maximum at x = -0.270845 and y = -0.923039 where f(x,y) = 181.617 , and four identical local minima: * f(3.0, 2.0) = 0.0, \quad * f(-2.805118, 3.131312) = 0.0, \quad * f(-3.779310, -3.283186) = 0.0, \quad * f(3.584428, -1.848126) = 0.0. \quad The locations of all the minima can be found analytically. However, because they are roots of cubic polynomials, when written in terms of radicals, the expressions are somewhat complicated. The function is named after David Mautner Himmelblau (1924–2011), who introduced it. See also *Test functions for optimization In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as: * Convergence rate. * Precision. * Robustness. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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