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Truncated Power Function
In mathematics, the truncated power function with exponent n is defined as :x_+^n = \begin x^n &:\ x > 0 \\ 0 &:\ x \le 0. \end In particular, :x_+ = \begin x &:\ x > 0 \\ 0 &:\ x \le 0. \end and interpret the exponent as conventional power. Relations * Truncated power functions can be used for construction of B-splines. * x \mapsto x_+^0 is the Heaviside 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 argume .... * \chi_{ ,b)}(x) = (b-x)_+^0 - (a-x)_+^0 where \chi is the indicator function. * Truncated power functions are refinable function">refinable. See also * Macaulay brackets External linksTruncated Power Function on MathWorld References Numerical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Function
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\ ex& = \underbrace_ \times \underbrace_ \\ ex& = b^n \times b^m \end In other words, when multiplying a base raised to one exp ... [...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] |
Heaviside 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] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refinable Function
In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function \varphi is called refinable with respect to the mask h if :\varphi(x)=2\cdot\sum_^ h_k\cdot\varphi(2\cdot x-k) This condition is called refinement equation, dilation equation or two-scale equation. Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator D one can write more concisely: :\varphi=2\cdot D_ (h * \varphi) It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems and de Rham curves. The operator \varphi\mapsto 2\cdot D_ (h * \varphi) is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not uniquely defined. That is, if \varphi is a refinable function, then for every c the function c\cdot\varphi is refinable, too. These functions play ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macaulay Brackets
Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.'' . Introduction to Aerospace Structures. Department of Aerospace Engineering Sciences, University of Colorado at Boulder Another commonly used notation is + or + for the part of , which avoids conflicts with for |
