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Encyclopedia > Dirac delta function
Parameters Probability density function Schematic representation of the Dirac delta function for x0 = 0. A line surmounted by an arrow is usually used to schematically represent the Dirac delta function. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead. Cumulative distribution function Using the half-maximum convention, with x0 = 0 $x_0,$ location (real) $x in [x_0; x_0]$ $delta(x-x_0),$ $H(x-x_0),$   (Heaviside) $x_0,$ $x_0,$ $x_0,$ $0,$ $0,$ (undefined) $-infty$ $e^{tx_0}$ $e^{itx_0}$

The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. The integral of the Dirac delta from any negative limit to any positive limit is 1. The discrete analog of the Dirac delta "function" is the Kronecker delta which is sometimes called a delta function even though it is a discrete sequence. It is also often referred to as the discrete unit impulse function. Note that the Dirac delta is not strictly a function, but a distribution that is also a measure. Image File history File links Download high resolution version (1300x975, 59 KB) File links The following pages link to this file: Dirac delta function ... Image File history File links Dirac_distribution_CDF.svgâ€Ž Please see the file description page for further information. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... The infinity symbol âˆž in several typefaces. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... Discrete time is non-continuous time. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a measure is a function that assigns a number, e. ...

Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph of the delta function is usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.) The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...

Despite its name, the delta function is not truly a function. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... It has been suggested that this article or section be merged into Logical biconditional. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

The Dirac delta function was named after the Kronecker delta [citation needed], since it can be used as a continuous analogue of the discrete Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

## Definitions

The Dirac delta function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $delta(x) = begin{cases} infty, & x = 0 0, & x ne 0 end{cases}$

and which is also constrained to satisfy the identity $int_{-infty}^infty delta(x) , dx = 1.$

This heuristic definition should not be taken too seriously though. Firstly, the Dirac delta is not a function, as no function has the above properties. Moreover there exist descriptions which differ from the above conceptualization. For example, sinc(x / a) / a (where sinc is the sinc function) behaves as a delta function in the limit of $arightarrow 0$, yet this function does not approach zero for values of x outside the origin. The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...

The defining characteristic $int_{-infty}^infty f(x) , delta(x) , dx = f(0)$

where f is a suitable test function, cannot be achieved by any function, but the Dirac delta function can be rigorously defined either as a distribution or as a measure. This page deals with mathematical distributions. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a measure is a function that assigns a number, e. ...

In terms of dimensional analysis, this definition of δ(x) implies that δ(x) has dimensions reciprocal to those of dx. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...

## The delta function as a measure

As a measure, δ(A) = 1 if $0in A$, and δ(A) = 0 otherwise. Then, $int_{-infty}^infty f(x) , delta(x) , dx = f(0)$

for all continuous f. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

## The delta function as a probability density function

As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... $delta[phi] = phi(0),$

for every test function $phi$. It is a distribution with compact support (the support being {0}). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral. In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Thus, the Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

Equivalently, one may define $delta : mathbb{R} ni &# 0;longrightarrow delta ( &# 0;)in delta(mathbb{R})$ as a distribution δ(ξ) whose indefinite integral is the function In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... $int^{x}_{-infin} delta (t) dt = h(x) equiv frac{1+{rm sgn}(x)}{2}$

for all real numbers x.

## Delta function of more complicated arguments

A helpful identity is the scaling property: $int_{-infty}^infty delta(alpha x),dx =int_{-infty}^infty delta(u),frac{du}{|alpha|} =frac{1}{|alpha|}$

and so $delta(alpha x) = frac{delta(x)}{|alpha|}$

The scaling property may be generalized to: $delta(g(x)) = sum_{i}frac{delta(x-x_i)}{|g'(x_i)|}$

where xi are the real roots of g(x) (assumed simple roots). Thus, for example $delta(x^2-alpha^2) = frac{1}{2|alpha|}[delta(x+alpha)+delta(x-alpha)]$

In the integral form the generalized scaling property may be written as $int_{-infty}^infty f(x) , delta(g(x)) , dx = sum_{i}frac{f(x_i)}{|g'(x_i)|}$

In an n-dimensional space with position vector $mathbf{r}$, this is generalized to: $int_V f(mathbf{r}) , delta(g(mathbf{r})) , d^nr = int_{partial V}frac{f(mathbf{r})}{|mathbf{nabla}g|},d^{n-1}r$

where the integral on the right is over $partial V$, the n-1  dimensional surface defined by $g(mathbf{r})=0$.

The integral of the time-shifted Dirac delta is given by: $intlimits_{-infty}^infty f(t) delta(t-T),dt = f(T)$

Thus, the delta function is said to "shift out" the function $f(t),$ at the value $t=T,$, when integrated over all time.
Similarly, the convolution: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... $f(t) * delta(t-T) = intlimits_{-infty}^infty f(tau) cdot delta(t-T-tau) dtau = f(t-T)$

means that the effect of convolving with the time-shifted Dirac delta is to time-shift $f(t),$ by the same amount.

## Fourier transform

Using Fourier transforms, one has In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... $int_{-infty}^infty 1 cdot e^{-i 2pi f t},dt = delta(f)$

and therefore: $int_{-infty}^infty e^{i 2pi f_1 t} left[e^{i 2pi f_2 t}right]^*,dt = int_{-infty}^infty e^{-i 2pi (f_2 - f_1) t} ,dt = delta(f_2 - f_1)$

which is a statement of the orthogonality property for the Fourier kernel.

## Laplace transform

The direct Laplace transform of the delta function is: In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... $int_{0}^{infty}delta (t-a)e^{-st} , dt=e^{-as}$

a curious identity using Euler's formula 2cos(as) = e ias + eias allows us to find the Laplace inverse transform for the cosine $2frac{1}{2pi {i}}int_{c-iinfty}^{c+iinfty} cos(as)e^{st} , ds=2[delta (t+ia) +delta (t-ia)]$ and a similar identity holds for sin(as).

## Informal introduction to derivatives of the Dirac Delta Function

One of the most commonly used representations of the dirac delta function is $f(x)=lim_{a to 0}frac{1}{a sqrt{pi}} mathrm{e}^{-x^2/a^2}$

One way to think of the derivatives is to form the commutative diagram (which is just a box that one can move around in either direction from the upper left hand corner to the lower right hand corner) and where the starting point function = f(x): In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... $function ,,, rightarrow,,,,,, distribution$ $downarrow ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,downarrow$ $derivative rightarrow derivative,of,distribution$

The right-arrows represent computing the limit as a goes to 0, and the down-arrows represent differentiation. So one simply differentiates the function, takes the limit as a goes to 0, and this results in the derivatives of the function. This process can be repeated any number of times.

It is interesting that the derivatives of the normal distribution look very similar to the wavefunctions of the quantum harmonic oscillator (see the instructive illustrations on that page). One can (poetically) imagine the wavefunctions as becoming "more and more squished", and finally ending up as the derivatives of the dirac delta function. The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function &#968; defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

## Distributional derivatives

As a tempered distribution, the Dirac delta distribution is infinitely differentiable. Let U be an open subset of Euclidean space Rn and let S(U) denote the Schwartz space of smooth, rapidly decaying real-valued functions on U. Let a be a point of U and let δa be the Dirac delta distribution centred at a. If α = (α1, ..., αn) is any multi-index and ∂α denotes the associated mixed partial derivative operator, then the αth derivative ∂αδa of δa is given by In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, Schwartz space is the function space of rapidly decreasing functions. ... The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... $leftlangle partial^{alpha} delta_{a}, varphi rightrangle = (-1)^{| alpha |} leftlangle delta_{a}, partial^{alpha} varphi rightrangle = left. (-1)^{| alpha |} partial^{alpha} varphi (x) right|_{x = a} mbox{ for all } varphi in S(U).$

That is, the αth derivative of δa is the distribution whose value on any test function φ is the αth derivative of φ at a (with the appropriate positive or negative sign). This is rather convenient, since the Dirac delta distribution δa applied to φ is just φ(a).

## Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions $delta (x) = lim_{ato 0} delta_a(x),$

where δa(x) is sometimes called a nascent delta function. This limit is in the sense that $lim_{ato 0} int_{-infty}^{infty}delta_a(x)f(x)dx = f(0)$

for all continuous f. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (also on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions. In functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence) such that for every element of , the net (or sequence) has limit . ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Illustration of a unit circle. ...

Some nascent delta functions are: $delta_a(x) = frac{1}{a sqrt{pi}} mathrm{e}^{-x^2/a^2}$ Limit of a Normal distribution $delta_a(x) = frac{1}{pi} frac{a}{a^2 + x^2} =frac{1}{2pi}int_{-infty}^{infty}mathrm{e}^{mathrm{i} k x-|ak|};dk$ Limit of a Cauchy distribution $delta_a(x)=frac{e^{-|x/a|}}{2a} =frac{1}{2pi}int_{-infty}^{infty}frac{e^{ikx}}{1+a^2k^2},dk$ Cauchy $varphi$(see note below) $delta_a(x)= frac{textrm{rect}(x/a)}{a} =frac{1}{2pi}int_{-infty}^infty textrm{sinc} left( frac{a k}{2 pi} right) e^{ikx},dk$ Limit of a rectangular function $delta_a(x)=frac{1}{pi x}sinleft(frac{x}{a}right) =frac{1}{2pi}int_{-1/a}^{1/a} cos (k x);dk$ rectangular function $varphi$(see note below) $delta_a(x)=partial_x frac{1}{1+mathrm{e}^{-x/a}} =-partial_x frac{1}{1+mathrm{e}^{x/a}}$ Derivative of the sigmoid (or Fermi-Dirac) function $delta_a(x)=frac{a}{pi x^2}sin^2left(frac{x}{a}right)$ $delta_a(x) = frac{1}{a}A_ileft(frac{x}{a}right)$ Limit of the Airy function $delta_a(x) = frac{1}{a}J_{1/a} left(frac{x+1}{a}right)$ Limit of a Bessel function

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ... Sigmoid generally means resembling the letter S or the lower-case Greek letter sigma (&#962;). Specific uses include: In mathematics, either a specific function &#8212; the logistic curve &#8212; or any real function whose graph has a sigmoid shape: see sigmoid function. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... In mathematics, the Airy function Ai(x) is a special function, i. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred to...

Note: If δ(ax) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δφ(ax) can be built from its characteristic function as follows: In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... $delta_varphi(a,x)=frac{1}{2pi}~frac{varphi(1/a,x)}{delta(1/a,0)}$

where $varphi(a,k)=int_{-infty}^infty delta(a,x)e^{-ikx},dx$

is the characteristic function of the nascent delta function δ(ax). This result is related to the localization property of the continuous Fourier transform. In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

## The Dirac comb

Main article: Dirac comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ð¨). From the orthogonality of the Fourier series... In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ð¨). From the orthogonality of the Fourier series... In signal processing, sampling is the reduction of a continuous signal to a discrete signal. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...

In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ð¨). From the orthogonality of the Fourier series... Like the standard Dirac comb, the logarithmically-spaced Dirac comb consists of an infinite sequence of Dirac delta functions. ... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... In mathematics, a Dirac measure is a measure &#948;x on a set X that gives a given element x measure 1, so that &#948;x({x}) = 1 and in general &#948;x(Y) = 0 for any subset Y of X not containing x, &#948;x(Z) = 1 for any... Results from FactBites:

 PlanetMath: Dirac delta function (82 words) is not a true function since it cannot be defined completely by giving the function value for all values of the argument can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width. This is version 3 of Dirac delta function, born on 2002-01-19, modified 2007-07-02.
 help - DiracDelta Science & Engineering Encyclopedia (0 words) Dirac Delta Consultants Ltd. can tailor the Encyclopaedia to your corporate needs (and colour schemes, logos etc.) and because all the information is hard-coded it can be uploaded to anywhere on your intranet server without worry. Dirac Delta Consultants Ltd. have endeavoured to ensure that the information presented here is accurate and that the calculations are correct but can accept no responsibility for any consequential damages or faults that may arise from the use of this software. In no event will Dirac Delta Consultants Ltd. be liable for direct, indirect, special, incidental or consequential damages arising out of the use or inability to use the software or information presented within even if advised of the possibility of such damages.
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