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Power series. In mathematics, a power series (in one variable) is an infinite series of the form where an 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.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
A formal power series version of Zariski's main theorem says that if x is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain (Mumford 1999, III.9).
A normal distribution or Gaussian distribution (also known as the "bell-shaped curve") is a concept used in probability theory and statistics. [ 2 ] The normal distribution concept is applied in numerous disciplines, including education, psychology, economics, business, the sciences and nursing.
In mathematics, the convolution power is the n -fold iteration of the convolution with itself. Thus if is a function on Euclidean space Rd and is a natural number, then the convolution power is defined by. ⋯ ⏟ {\displaystyle x^ {*n}=\underbrace {x*x*x*\cdots *x*x} _ {n},\quad x^ {*0}=\delta _ {0}} where ∗ denotes the convolution operation ...
There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere. Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit. The Weierstrass M-test is a useful ...
The terms "distribution" and "family" are often used loosely: Specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; [a] however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families ...
Specifically, if k is a field and X is an indeterminate, then the ring of formal power series k[[X]] is a regular local ring having (Krull) dimension 1. If p is an ordinary prime number, the ring of p-adic integers is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field.