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Piecewise Smooth
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 pair ... [...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 pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bump Function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain \R^n forms a vector space, denoted \mathrm^\infty_0(\R^n) or \mathrm^\infty_\mathrm(\R^n). The dual space of this space endowed with a suitable topology is the space of distributions. Examples The function \Psi:\R \to \R given by \Psi(x) = \begin \exp\left( -\frac\right), & x \in (-1,1) \\ 0, & \text \end is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded closed support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian fu ... [...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 map In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessaril ... [...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] |
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 data ... [...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 quantit ... [...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] |
Sign Function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoid confusion with the sine function, this function is usually called the signum function. Definition The signum function of a real number is a piecewise function which is defined as follows: \sgn x :=\begin -1 & \text x 0. \end Properties Any real number can be expressed as the product of its absolute value and its sign function: x = , x, \sgn x. It follows that whenever is not equal to 0 we have \sgn x = \frac = \frac\,. Similarly, for ''any'' real number , , x, = x\sgn x. We can also ascertain that: \sgn x^n=(\sgn x)^n. The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero. More formally, in integration theory it is a weak derivative, and in convex function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := 0.html" ;"title=">0">>0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boxcar Function
In mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, ''A''. The boxcar function can be expressed in terms of the uniform distribution as \operatorname(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)), where is the uniform distribution of ''x'' for the interval and H(x) is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points. These values are probably best chosen for each individual application. When a boxcar function is selected as the impulse response of a filter, the result is a moving average filter. The function is named after its graph's resemblance to a boxcar, a type of railroad car. See also * Boxcar averager * Rectangular function * Step function * Top-hat filter The name Top-hat filter refers to several real-space or Fourier space filtering techniques (not to be confused with the top-hat transf ... [...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 whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
