|
Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example. Notation and interpretation Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Function Gnuplot
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example. Notation and interpretation Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDIFF
In geometric topology, PDIFF, for ''p''iecewise ''diff''erentiable, is the category of piecewise- smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between them) and PL (the category of piecewise linear manifolds and piecewise linear maps between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them. Motivation PDIFF is mostly a technical point: smooth maps are not piecewise linear (unless linear), and piecewise linear maps are not smooth (unless globally linear) – the intersection is linear maps, or more precisely affine maps (because not based) – so they cannot directly be related: they are separate generalizations of the notion of an affine map. However, while a smooth manifold is not a PL manifold, it carrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spline (mathematics)
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term ''spline'' more frequently refers to a piecewise polynomial ( parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. Introduction The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-spline
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data. In computer-aided design and computer graphics, spline functions are constructed as linear combinations of B-splines with a set of control points. Introduction The term "B-spline" was coined by Isaac Jacob Schoenberg and is short for basis spline. A spline function of order n is a piecewise polynomial function of degree n - 1 in a variable x. The places where the pieces meet are known as knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Step Function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. Definition and first consequences A function f\colon \mathbb \rightarrow \mathbb is called a step function if it can be written as :f(x) = \sum\limits_^n \alpha_i \chi_(x), for all real numbers x where n\ge 0, \alpha_i are real numbers, A_i are intervals, and \chi_A is the indicator function of A: :\chi_A(x) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \\ \end In this definition, the intervals A_i can be assumed to have the following two properties: # The intervals are pairwise disjoint: A_i \cap A_j = \emptyset for i \neq j # The union of the intervals is the entire real line: \bigcup_^n A_i = \mathbb R. Indeed, if that is not the case to start with, a different set of intervals can be picked for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as ''the'' triangular function. Triangular functions are useful in signal processing and ''communication systems engineering'' as representations of idealized signals, and the triangular function specifically as an integral transform kernel function from which more realistic signals can be derived, for example in kernel density estimation. It also has applications in pulse-code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also used to define the triangular window sometimes called the Bartlett window. Definitions The most common definition is as a piecewise function: : \begin \operatorname(x) = \Lambda(x) \ &\overset \ \max\big(1 - , x, , 0\big) \\ &= \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Continuation
Simplicial continuation, or piecewise linear continuation (Allgower and Georg),Eugene L. Allgower, K. Georg, "Introduction to Numerical Continuation Methods", ''SIAM Classics in Applied Mathematics'' 45, 2003.E. L. Allgower, K. Georg, "Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations", ''SIAM Review'', Volume 22, 28-85, 1980. is a one-parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher-dimensional manifolds by (Allgower and Gnutzman)Eugene L. Allgower, Stefan Gnutzmann, "An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces", ''SIAM Journal on Numerical Analysis'', Volume 24, Number 2, 452-469, 1987. and (Allgower and Schmidt).Eugene L. Allgower, Phillip H. Schmidt, "An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold", ''SIAM Journal on Numerical Analysis'', Volume 22, N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, June 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, June 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of \R, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the codomain, and the set of values attained by the function (which is a subset of ) is called its range or image. Any function can be restricted to a subset of its domain. The restriction of f \colon X \to Y to A, where A\subseteq X, is written as \left. f \_A \colon A \to Y. Natural domain If a real function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broken Power Law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Empirical examples The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, volcanic eruptions, human judgments of stimulus intensity and many other qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
