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List Of Mathematical Properties Of Points
In mathematics, the following appear: * Algebraic point * Associated point * Base point * Closed point * Divisor point * Embedded point * Extreme point * Fermat point * Fixed point * Focal point * Geometric point * Hyperbolic equilibrium point * Ideal point * Inflection point * Integral point * Isolated point * Generic point * Heegner point * Lattice hole, Lattice point * Lebesgue point * Midpoint * Napoleon points * Non-singular point * Normal point * Parshin point * Periodic point * Pinch point * Point (geometry) * Point source * Rational point * Recurrent point * Regular point, Regular singular point * Saddle point * Semistable point * Separable point * Simple point * Singular point of a curve * Singular point of an algebraic variety * Smooth point * Special point * Stable point * Torsion point * Vertex (curve) * Weierstrass point Calculus * Critical point (aka stationary point), any value ''v'' in the domain of a differentiable function of any real or com ... [...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 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 abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' is a point ''P'' whose closure is all of ''X'', that is, a point that is dense in ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Source
A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources can usually be approximated as a mathematical point to simplify analysis. The actual source need not be physically small, if its size is negligible relative to other length scales in the problem. For example, in astronomy, stars are routinely treated as point sources, even though they are in actuality much larger than the Earth. In three dimensions, the density of something leaving a point source decreases in proportion to the inverse square of the distance from the source, if the distribution is isotropic, and there is no absorption or other loss. Mathematics In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point (geometry)
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy; for example, ''"there is exactly one line that passes through two different points"''. Points in Euclidean geometry Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (, ) of numbers, where the first number conventionally represents the horizontal and is often denoted by , and the second number conventionally represents the vertical and is often denoted by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinch Point (mathematics)
frame, Section of the Whitney umbrella, an example of pinch point singularity. In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface. The equation for the surface near a pinch point may be put in the form : f(u,v,w) = u^2 - vw^2 + \, where denotes terms of degree 4 or more and v is not a square in the ring of functions. For example the surface 1-2x+x^2-yz^2=0 near the point (1,0,0), meaning in coordinates vanishing at that point, has the form above. In fact, if u=1-x, v=y and w=z then is a system of coordinates vanishing at (1,0,0) then 1-2x+x^2-yz^2=(1-x)^2-yz^2=u^2-vw^2 is written in the canonical form. The simplest example of a pinch point is the hypersurface defined by the equation u^2-vw^2=0 called Whitney umbrella. The pinch point (in this case the origin) is a limit of normal crossings singular points (the v-axis in this case). These singular points are intimately related in the sense that in order to resolve the pinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Point
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a mapping ''f'' from a set ''X'' into itself, :f: X \to X, a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that :\ f_n(x) = x where f_n is the ''n''th iterate of ''f''. The smallest positive integer ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a periodic function). If there exist distinct ''n'' and ''m'' such that :f_n(x) = f_m(x) then ''x'' is called a preperiodic point. All periodic points are preperiodic. If ''f'' is a diffeomorphism of a differentiable manifold, so that the derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parshin Point
Parshin (Russian: Паршин) (feminine: Parshina) is a Russian-language surname. It may refer to: *Aleksandr Parshin, Russian footballer *Aleksei Parshin, Russian mathematician, the namesake of the terms "Parshin chain" and "Parshin's conjecture" * Denis Parshin, Russian ice hockey player *Georgy Parshin, Soviet World War II pilot, twice Hero of the Soviet Union *Nikolai Parshin, Soviet footballer *Lana Parshina, Russian-American journalist and filmmaker *Daria Parshina, Russian swimmer *Valentina Parshina Valentina Romanovna Parshina (russian: Валентина Романовна Паршина; 16 March 1937 – 21 December 2020) was a Soviet and Russian agronomist and politician. She was a foreman of vegetable growers at the Detskoselsky Order ..., Russian and Soviet agronomist and politician {{surname Russian-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Point
Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * '' Normal!'', a 2011 Algerian film * ''The Normals'' (film), a 2012 American comedy film * "Normal" (''New Girl''), an episode of the TV series Mathematics * Normal (geometry), an object such as a line or vector that is perpendicular to a given object * Normal basis (of a Galois extension), used heavily in cryptography * Normal bundle * Normal cone, of a subscheme in algebraic geometry * Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) * Normal distribution, the Gaussian continuous probability distribution * Normal equations, describing the solution of the linear least s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-singular Point
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Napoleon Points
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers. The name "Napoleon points" has also been applied to a different pair of triangle centers, better known as the isodynamic points. Definition of the points First Napoleon point Let ''ABC'' be any given plane triangle. On the sides ''BC'', ''CA'', ''AB'' of the triangle, construct outwardly drawn equilateral triangles ''DBC'', ''ECA'' and ''FAB'' respectively. Let the centroids of these triangles be ''X'', ''Y'' and ''Z'' respectively. Then the lines ''AX'', ''BY'' and ''CZ'' are concurrent. The point of concurrence ''N1'' is the first Napoleon point, or the outer Nap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimensional space whose endpoints are A = (a_1, a_2, \dots , a_n) and B = (b_1, b_2, \dots , b_n) is given by :\frac. That is, the ''i''th coordinate of the midpoint (''i'' = 1, 2, ..., ''n'') is :\frac 2. Construction Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction. The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Point
In mathematics, given a locally Lebesgue integrable function f on \mathbb^k, a point x in the domain of f is a Lebesgue point if :\lim_\frac\int_ \!, f(y)-f(x), \,\mathrmy=0. Here, B(x,r) is a ball centered at x with radius r > 0, and \lambda (B(x,r)) is its Lebesgue measure. The Lebesgue points of f are thus points where f does not oscillate too much, in an average sense. The Lebesgue differentiation theorem In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named fo ... states that, given any f\in L^1(\mathbb^k), almost every x is a Lebesgue point of f.. References {{DEFAULTSORT:Lebesgue Point Mathematical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
