|
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation :\begin x &= f(t)\\ y &= g(t), \end a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope \lim (g'(t)/f'(t))). Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation :F(x,y) = 0, which is smooth, cusps are points where the terms of lowest degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp At (0,0
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurcation theory in the study of dynamical systems * Cusp form, in modular form theory * Cusp neighborhood, a set of points near a cusp * Cuspidal representation, a generalization of cusp forms in the theory of automorphic representations Science and medicine * Beach cusps, a pointed and regular arc pattern of the shoreline at the beach * Behavioral cusp, a change in behavior with far-reaching consequences * Caltech-USGS Seismic Processing, software for analyzing earthquake data * Center for Urban Science and Progress, a graduate school of New York University focusing on urban informatics * CubeSat for Solar Particles, a satellite launched in 2022 * Cusp (anatomy), a pointed structure on a tooth * Cusps of heart valves, leaflets of a hea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (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 (topology), interior. Intuitively speaking, a neighbourhood of a point is a Set (mathematics), 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 Interior (topology)#Interior point, 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable (mathematics)
In mathematics, a variable (from Latin language, Latin ) is a Mathematical symbol, symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable ''represents'' or ''denotes'' the object, and that any valid candidate for the object is the value (mathematics), value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a Set (mathematics), set, such as the set of real numbers. The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables and and require that the value of the square of is twice the square of , which in algebraic notation can be written . A definitive proof that this relationship is impossible to satisfy when and are restricted to integer numbers isn't obvious, but it has been known since ancient times and has had a big ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real-valued Function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real functions'') and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Algebraic structure Let (X,) be the set of all functions from a set to real numbers \mathbb R. Because \mathbb R is a field, (X,) may be turned into a vector space and a commutative algebra over the reals with the following operations: *f+g: x \mapsto f(x) + g(x) – vector addition *\mathbf: x \mapsto 0 – additive identity *c f: x \mapsto c f(x),\quad c \in \mathbb R – scalar multiplication *f g: x \mapsto f(x)g(x) – pointwise multiplication These operations extend to partial functions from to \mathbb R, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (topology)
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 neighbourhoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definition Given two differentiable manifolds M and N, a Differentiable manifold#Differentiability of mappings between manifolds, continuously differentiable map f \colon M \rightarrow N is a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. Two C^r-differentiable manifolds are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a ... [...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] |
Vladimir Arnold
Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem. His first main result was the solution of Hilbert's thirteenth problem in 1957 when he was 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), symplectic topology and KAM theory. Arnold was also a populariser of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as '' Mathematical Methods of Clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as the founder of catastrophe theory (later developed by Christopher Zeeman). Life and career René Thom grew up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After the German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
