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imported>John Stephenson |
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| The '''[[Dirac delta function]]''' is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass ''M''—the integral over the mass distribution. When the distribution becomes smaller and smaller, while ''M'' is constant, the mass distribution shrinks to a ''point mass'', which by definition has zero extent and yet has a finite-valued integral equal to total mass ''M''. In the limit of a point mass the distribution becomes a Dirac delta function.
| | {{:{{FeaturedArticleTitle}}}} |
| | | <small> |
| Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of <font style="vertical-align: 13%"> <math>\mathbb{Z}</math></font>) to real indices (elements of <font style="vertical-align: 13%"><math>\mathbb{R}</math></font>). Note that the Kronecker delta acts as a "filter" in a summation:
| | ==Footnotes== |
| :<math> | | {{reflist|2}} |
| \sum_{i=m}^n \; f_i\; \delta_{ia} =
| | </small> |
| \begin{cases}
| |
| f_a & \quad\hbox{if}\quad a\in[m,n] \sub\mathbb{Z} \\
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| 0 & \quad \hbox{if}\quad a \notin [m,n].
| |
| \end{cases}
| |
| </math> | |
| | |
| In analogy, the Dirac delta function δ(''x''−''a'') is defined by (replace ''i'' by ''x'' and the summation over ''i'' by an integration over ''x''),
| |
| :<math>
| |
| \int_{a_0}^{a_1} f(x) \delta(x-a) \mathrm{d}x =
| |
| \begin{cases}
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| f(a) & \quad\hbox{if}\quad a\in[a_0,a_1] \sub\mathbb{R}, \\
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| 0 & \quad \hbox{if}\quad a \notin [a_0,a_1].
| |
| \end{cases}
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| </math>
| |
| | |
| The Dirac delta function is ''not'' an ordinary well-behaved map <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a distribution, also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.
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| | |
| ==Properties==
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| Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this will be done.
| |
| :<math>
| |
| \begin{align}
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| \int_{-\infty}^{\infty} \delta(x)\mathrm{d}x &= 1, \\
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| \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \mathrm{d}k &= \delta(x) \\
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| \delta(x-a) &= \delta(a-x), \\
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| (x-a)\delta(x-a) &= 0, \\
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| \delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\
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| f(x) \delta(x-a) &= f(a) \delta(x-a), \\
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| \int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a)
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| \end{align}
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| </math> | |
| The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function ''f''(''x'') = 1 (the second property) will be proved below.
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| The last property is the analogy of the multiplication of two identity matrices,
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| :<math>
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| \sum_{j=1}^n \;\delta_{ij}\;\delta_{jk} = \delta_{ik}, \quad i,k=1,\ldots, n.
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| </math>
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| ''[[Dirac delta function|.... (read more)]]''
| |
The Mathare Valley slum near Nairobi, Kenya, in 2009.
Poverty is deprivation based on lack of material resources. The concept is value-based and political. Hence its definition, causes and remedies (and the possibility of remedies) are highly contentious.[1] The word poverty may also be used figuratively to indicate a lack, instead of material goods or money, of any kind of quality, as in a poverty of imagination.
Definitions
Primary and secondary poverty
The use of the terms primary and secondary poverty dates back to Seebohm Rowntree, who conducted the second British survey to calculate the extent of poverty. This was carried out in York and was published in 1899. He defined primary poverty as having insufficient income to “obtain the minimum necessaries for the maintenance of merely physical efficiency”. In secondary poverty, the income “would be sufficient for the maintenance of merely physical efficiency were it not that some portion of it is absorbed by some other expenditure.” Even with these rigorous criteria he found that 9.9% of the population was in primary poverty and a further 17.9% in secondary.[2]
Absolute and comparative poverty
More recent definitions tend to use the terms absolute and comparative poverty. Absolute is in line with Rowntree's primary poverty, but comparative poverty is usually expressed in terms of ability to play a part in the society in which a person lives. Comparative poverty will thus vary from one country to another.[3] The difficulty of definition is illustrated by the fact that a recession can actually reduce "poverty".
Causes of poverty
The causes of poverty most often considered are:
- Character defects
- An established “culture of poverty”, with low expectations handed down from one generation to another
- Unemployment
- Irregular employment, and/or low pay
- Position in the life cycle (see below) and household size
- Disability
- Structural inequality, both within countries and between countries. (R H Tawney: “What thoughtful rich people call the problem of poverty, thoughtful poor people call with equal justice a problem of riches”)[4]
As noted above, most of these, or the extent to which they can be, or should be changed, are matters of heated controversy.
- ↑ Alcock, P. Understanding poverty. Macmillan. 1997. ch 1.
- ↑ Harris, B. The origins of the British welfare state. Palgrave Macmillan. 2004. Also, Oxford Dictionary of National Biography.
- ↑ Alcock, Pt II
- ↑ Alcock, Preface to 1st edition and pt III.
