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Exponentials
Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Exponential discounting, a specific form of the discount function, used in the analysis of choice over time *Exponential growth, where the growth rate of a mathematical function is proportional to the function's current value * Exponential map (Riemannian geometry), in Riemannian geometry *Exponential map (Lie theory), in Lie theory * Exponential notation, also known as scientific notation, or standard form * Exponential object, in category theory *Exponential time, in complexity theory *in probability and statistics: **Exponential distribution, a family of continuous probability distributions **Exponentially modified Gaussian distribution, describes the sum of independent normal and exponential random variables **Exponential family, a parametr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Exponential Topics
{{Short description, none This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. * Accelerating change * Approximating natural exponents (log base e) * Artin–Hasse exponential * Bacterial growth * Baker–Campbell–Hausdorff formula * Cell growth * Barometric formula * Beer–Lambert law * Characterizations of the exponential function * Catenary * Compound interest * De Moivre's formula * Derivative of the exponential map * Doléans-Dade exponential * Doubling time * ''e''-folding * Elimination half-life * Error exponent * Euler's formula * Euler's identity * e (mathematical constant) * Exponent * Exponent bias * Exponential (other) * Exponential backoff * Exponential decay * Exponential dichotomy * Exponential discounting * Exponential diophantine equation * Exponential dispersion model * Exponential distribution * Exponential error * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Family
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. 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; 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 is sometimes lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac = -\lambda N. The solution to this equation (see derivation below) is: :N(t) = N_0 e^, where is the quantity at time , is the initial quantity, that is, the quantity at time . Measuring rates of decay Mean lifetime If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime), where the exponential time constant, \tau, relates to the decay rate constant, λ, in the following way: :\tau = \frac. The mean lifetime can be looked at as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law. Definition Let \mathbf be a category, let Z and Y be objects of \mathbf, and let \mathbf have all binary products with Y. An object Z^Y together with a morphism \mathrm\colon (Z^Y \times Y) \to Z is an ''exponential object'' if for any object X and morphism g \colon X\times Y \to Z there is a unique morphism \lambda g\colon X\to Z^Y (called the ''transpose'' of g) such that the following diagram commutes: This assignment of a unique \lambda g to each g establishes an isomorphism (bijection) of hom-sets, \mathrm(X\times Y,Z) \cong \mathrm(X,Z^Y). If Z^Yexists for all objects Z, Y in \mathbf, then the functor (-)^Y \colon \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. Definitions Probability density function The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin \lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Technology
Exponential Technology was a vendor of PowerPC microprocessors. The company was founded by George Taylor and Jim Blomgren in 1993. The company's plan was to use BiCMOS technology to produce very fast processors for the Apple Computer market. Logic used 3-level ECL circuits (single-ended for control logic, and differential for datapaths) while RAM structures used CMOS. The company was originally named Renaissance Microsystems. Rick Shriner was the CEO. Their chips were manufactured by Hitachi. Their product, the Exponential X704, was advertised to run at 533 MHz, but the first version of the device only ran at about 400 MHz, still significantly faster than the 233 MHz PowerPC 604e used in Macintosh computers at the time. This lower frequency along with small level-one caches, produced systems which had good but not stellar performance. This allowed Motorola (Apple's traditional processor vendor), to convince the computer maker that Motorola's future roadmap would pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Type
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function ''e''''C'', ''z'', for some real-valued constant ''C'' as , ''z'', → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(''z'') as opposed to ''e''''z''. Basic idea A function ''f''(''z'') defined on the complex plane is said to be of exponential type if there exist real-valued constants ''M'' and ''τ'' such that :\left, f\left(re^\right)\ \le Me^ in the limit of r\to\infty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Type
In computer science and mathematical logic, a function type (or arrow type or exponential) is the type of a variable or parameter to which a function has or can be assigned, or an argument or result type of a higher-order function taking or returning a function. A function type depends on the type of the parameters and the result type of the function (it, or more accurately the unapplied type constructor , is a higher-kinded type). In theoretical settings and programming languages where functions are defined in curried form, such as the simply typed lambda calculus, a function type depends on exactly two types, the domain ''A'' and the range ''B''. Here a function type is often denoted , following mathematical convention, or , based on there existing exactly (exponentially many) set-theoretic functions mappings ''A'' to ''B'' in the category of sets. The class of such maps or functions is called the exponential object. The act of currying makes the function type adj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Smoothing
Exponential smoothing is a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data. Exponential smoothing is one of many window functions commonly applied to smooth data in signal processing, acting as low-pass filters to remove high-frequency noise. This method is preceded by Poisson's use of recursive exponential window functions in convolutions from the 19th century, as well as Kolmogorov and Zurbenko's use of recursive moving averages from their studies of turbulence in the 1940s. The raw data sequence is often represented by \ beginning at time t = 0, and the outp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
