Instruct: Given a question, retrieve documents that can help answer the question.
Query: What is exponentiation?
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
As with the other results in this family, the strong three exponentials conjecture implies the sharp three exponentials conjecture which implies the three exponentials conjecture. However, the strong and sharp three exponentials conjectures are implied by their four exponentials counterparts, bucking the usual trend. And the three exponentials conjecture is neither implied by nor implies the four exponentials conjecture.
Double exponential function Exponential decay Exponential growth List of exponential topics Modular exponentiation Scientific notation Unicode subscripts and superscripts xy=yx Zero to the power of zero
The matrix "P" is a projection onto the stable subspace and "I" − "P" is a projection onto the unstable subspace. What the exponential dichotomy says is that the norm of the projection onto the stable subspace of any orbit in the system decays exponentially as "t" → ∞ and the norm of the projection onto the unstable subspace of any orbit decays exponentially as "t" → −∞, and furthermore that the stable and unstable subspaces are conjugate (because formula_4).
The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents.
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. This corresponds to the sequence A003313 on the Online Encyclopedia of Integer Sequences. It works by creating the shortest addition chain that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. More generally, "addition-chain exponentiation" may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very difficult to find).
In other words, by what amounts to the same, by the series expansion of the exponential:
The complete exponential Bell polynomial is defined by formula_15, or in other words:
The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras.
It is critical, when considering the examples in this section, to remember the discussion above about what it means to say that a "distribution" is an exponential family, and in particular to keep in mind that the set of parameters that are allowed to vary is critical in determining whether a "distribution" is or is not an exponential family.
In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes, which is an exponential time analogue of the polynomial hierarchy. As elsewhere in complexity theory, “exponential” is used in two different meanings (linear exponential bounds formula_1 for a constant "c", and full exponential bounds formula_2), leading to two versions of the exponential hierarchy:
The exponential family arises naturally as the answer to the following question: what is the maximum-entropy distribution consistent with given constraints on expected values?
The raw data sequence is often represented by formula_1 beginning at time formula_2, and the output of the exponential smoothing algorithm is commonly written as formula_3, which may be regarded as a best estimate of what the next value of formula_4 will be. When the sequence of observations begins at time formula_2, the simplest form of exponential smoothing is given by the formulas:
Kurzweil explains that evolutionary progress is exponential because of positive feedback; the results of one stage are used to create the next stage. Exponential growth is deceptive, nearly flat at first until it hits what Kurzweil calls "the knee in the curve" then rises almost vertically. In fact Kurzweil believes evolutionary progress is super-exponential because more resources are deployed to the winning process. As an example of super-exponential growth Kurzweil cites the computer chip business. The overall budget for the whole industry increases over time, since the fruits of exponential growth make it an attractive investment; meanwhile the additional budget fuels more innovation which makes the industry grow even faster, effectively an example of "double" exponential growth.
There is nothing particularly special about C here, exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of "e" above. Similarly, there is no reason to have one variable, and an exponential polynomial in "n" variables would be of the form "P"("x"...,"x","e"...,"e"), where "P" is a polynomial in 2"n" variables.
In combinatorial mathematics, a "q"-exponential is a "q"-analog of the exponential function,
namely the eigenfunction of a "q"-derivative. There are many "q"-derivatives, for example, the classical "q"-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, "q"-exponentials are not unique. For example, formula_1 is the "q"-exponential corresponding to the classical "q"-derivative while formula_2 are eigenfunctions of the Askey-Wilson operators.
More generally, for a generic -dependent exponent, ,
Taking the above expression outside the integral sign and expanding the integrand with the help of the Hadamard lemma one can obtain the following useful expression for the derivative of the matrix exponent,
Note that the coefficients in the expression above are different from what appears in the exponential. For a closed form, see derivative of the exponential map.
In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures.
The exponential formula is a power-series version of a special case of Faà di Bruno's formula.
Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by
So what will be the next operation after exponentiation? We defined multiplication so that formula_8, and defined exponentiation so that formula_9 so it seems logical to define the next operation, tetration, so that formula_10 with a tower of three 'a'. Analogously, the pentation of (a, 3) will be tetration(a, tetration(a, a)), with three "a" in it.
The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change "proportional" to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to exponential growth or exponential decay.
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.
Simple exponential smoothing does not do well when there is a trend in the data, which is inconvenient. In such situations, several methods were devised under the name "double exponential smoothing" or "second-order exponential smoothing.", which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing". This nomenclature is similar to quadruple exponential smoothing, which also references its recursion depth. 
The basic idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series
exhibiting some form of trend. This slope component is itself updated via exponential smoothing.
What is important to note, and what characterizes all exponential family variants, is that the parameter(s) and the observation variable(s) must factorize (can be separated into products each of which involves only one type of variable), either directly or within either part (the base or exponent) of an exponentiation operation. Generally, this means that all of the factors constituting the density or mass function must be of one of the following forms:
where ħ is the reduced Planck constant, treated as a formal parameter here. This is a special case of what is known as the Berezin formula on the algebra of symbols and can be given a closed form (which follows from the Baker–Campbell–Hausdorff formula). The closed form can be obtained by using the exponential:
where formula_12 is the multiplication map, formula_13, and the exponential is treated as a power series:
"Today, of course, there is exponentially more news coverage than there was in 1964 — not to mention exponentially more attempts to slot attention-getting events into simple, truthy stories about human nature. The gap between the famous version of the Genovese story and the true one, then, is worth remembering, if only as a check on our natural tendency to hear a story and then nod along and say, 'Of course that’s what happened!'”