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Tacnode
In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point. The canonical example is :y^2-x^4= 0. A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency locally diffeomorphic to the point at the origin of this curve. Another example of a tacnode is given by the links curve shown in the figure, with equation :(x^2+y^2-3x)^2 - 4x^2(2-x) = 0. More general background Consider a smooth real-valued function of two variables, say where and are real numbers. So is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point. For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is equivalent to :y^2 = x^2 (x-1) and x^2(x-1) is non-negative only when x ≥ 1 or x = 0. Thus, over the ''real'' numbers the equation has no solutions for x < 1 except for (0, 0). In contrast, over the complex numbers the origin is not isolated since square roots of negative real numbers exist. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point. An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives and vanish. Furth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crunode
In mathematics, a crunode (archaic; from Latin ''crux'' "cross" + ''node'') or node of an algebraic curve is a type of singular point at which the curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ''ordinary double point''. In the case of a smooth real plane curve , a point is a crunode provided that both first partial derivatives vanish \frac = \frac = 0 and the Hessian determinant is negative: \frac \frac - \left(\frac\right)^2 < 0. See also * * Acnode * * ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Point Of A Curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curves in the plane may be defined as the set of points satisfying an equation of the form f(x,y) = 0, where is a polynomial function If is expanded as f = a_0 + b_0 x + b_1 y + c_0 x^2 + 2c_1 xy + c_2 y^2 + \cdots If the origin is on the curve then . If then the implicit function theorem guarantees there is a smooth function so that the curve has the form near the origin. Similarly, if then there is a smooth function so that the curve has the form near the origin. In either case, there is a smooth map from to the plane which defines the curve in the neighborhood of the origin. Note that at the origin b_0 = \frac, \; b_1 = \frac, so the curve is non-singular or ''regular'' at the origin if at least one of the partial derivatives o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ..., college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 11 Dupont in the Dupont Circle, Washington, D.C., Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the ''American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Node (algebraic Geometry)
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 that is not singular is said to be regular. An algebraic variety that has no singular point is said to be non-singular or smooth. The concept is generalized to smooth schemes in the modern language of scheme theory. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ak Singularity
In mathematics, and in particular singularity theory, an singularity, where is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let f: \R^n \to \R be a smooth function. We denote by \Omega (\R^n,\R) the infinite-dimensional space of all such functions. Let \operatorname(\R^n) denote the infinite-dimensional Lie group of diffeomorphisms \R^n \to \R^n, and \operatorname(\R) the infinite-dimensional Lie group of diffeomorphisms \R \to \R. The product group \operatorname(\R^n) \times \operatorname(\R) acts on \Omega (\R^n,\R) in the following way: let \varphi : \R^n \to \R^n and \psi : \R \to \R be diffeomorphisms and f: \R^n \to \R any smooth function. We define the group action as follows: : (\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^ The orbit of , denoted , of this group action is given by : \mbox(f) = \ \ . The members of a given orbit of this action have the following fact in common: we can find a diffeomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
