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Saddle Point
In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and a relative maxima and minima, maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface (mathematics), surface that ''curves up'' in one direction, and ''curves down'' in a different dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface (mathematics), surface of the graph where all its partial derivatives are zero (equivalently, the gradient has zero vector norm, norm). The notion of stationary points of a real-valued function is generalized as ''Critical point (mathematics), critical points'' for complex-valued functions. Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., Parallel (geometry), parallel to the Abscissa, -axis). For a function of two variables, they correspond to the points on the gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monkey Saddle Surface
Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes. Thus monkeys, in that sense, constitute an incomplete paraphyletic grouping; alternatively, if apes (Hominoidea) are included, ''monkeys'' and ''simians'' are synonyms. In 1812, Étienne Geoffroy grouped the apes and the Cercopithecidae group of monkeys together and established the name Catarrhini, "Old World monkeys" ("''singes de l'Ancien Monde''" in French). The extant sister of the Catarrhini in the monkey ("singes") group is the Platyrrhini (New World monkeys). Some nine million years before the divergence between the Cercopithecidae and the apes, the Platyrrhini emerged within "monkeys" by migration to South America likely by ocean. Apes are thus deep in the tree of extant and extinct monkeys, and any of the apes is distinctly closer related to the Cercopithecida ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Hyperboloid
''Ruled'' is the fifth full-length LP album, LP by The Giraffes (Brooklyn band), The Giraffes. Drums, bass and principal guitar tracks recorded at The Bunker in Brooklyn, NY. Vocals and additional guitars recorded at Strangeweather in Brooklyn, NY. Mixed at Studio G in Brooklyn, NY by Joel Hamilton. Mastered by Julian Silva at On Air Mastering. Produced by The Giraffes and Joel Hamilton. Track listing All songs written by The Giraffes. #"The Border" - 3:10 #"The Bed" - 5:58 #"The City" - 6:47 #"The Kids" - 3:35 #"The Invasion" - 2:23 #"The Store" - 6:32 #"The War of Hormones" - 7:59 #"The Occupation" - 13:24 Personnel * Aaron Lazar - vocals * Damien Paris - guitar * Jens Carstensen - bass * Andrew Totolos - drums * James SK Wān – bamboo flute * Joel Hamilton - engineer, mixer * Marc Alan Goodman - engineer * Titfinger - art direction References External links The Giraffes // Ruled at Crustacean Records {{Authority control 2011 albums The Giraffes (Brookl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflection Point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (its first derivative , and its second derivative , exist and are continuous), the condition can also be used to find an inflection point since a point of must be passed to change from a positive value (concave upward) to a negative value (concave downward) or vice versa as is continuous; an inflection point of the curve is where and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X Cubed Plot
X, or x, is the twenty-fourth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ex'' (pronounced ), plural ''exes''."X", ''Oxford English Dictionary'', 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "ex", ''op. cit''. History The letter , representing , was inherited from the Etruscan alphabet. It perhaps originated in the of the Euboean alphabet or another Western Greek alphabet, which also represented . Its relationship with the of the Eastern Greek alphabets, which represented , is uncertain. The pronunciation of in the Romance languages underwent sound changes, with various outcomes: * French: (e.g. ''laisser'' from ''laxare'') * Italian: (e.g. ''asse'' from ''axem'') and, in some cases, (e.g. ''lasciare'' from ''laxare'') * Portuguese: (e.g. ''eixo'' from ''axem'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the tangent space at a point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a neighbourhood of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need not be an open subset of X. When V is open (resp. closed, compact, etc.) in X, it is called an (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open, so it is important to note their conventions. A set that is a neighbourhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
