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== '''[[Hausdorff dimension]]''' ==
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''by  [[User:Melchior Grutzmann|Melchior Grutzmann]] (and [[User:Brandon Piercy|Brandon Piercy]] and [[User:Hendra I. Nurdin|Hendra I. Nurdin]])
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==Footnotes==
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In [[mathematics]], the '''Hausdorff dimension''' is a way of defining a possibly fractional exponent for all  figures in a [[metric space]] such that the dimension describes partially the amount to that the set fills the space around it.  For example, a [[plane (geometry)|plane]] would have a Hausdorff dimension of 2, because it fills a 2-parameter subset.  However, it would not make sense to give the [[Sierpiński triangle]] [[fractal]] a dimension of 2, since it does not fully occupy the 2-dimensional realm.  The Hausdorff dimension describes this mathematically by measuring the size of the set.  For self-similar sets there is a relationship to the number of self-similar subsets and their scale.
 
=== Informal definition ===
Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point.  This is made rigorously with the notion of ''d''-dimensional (topological) [[manifold]] which are particularly regular sets.  The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or ''d'') real numbers.  The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way.
 
The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number.
 
Benoît Mandelbrot discovered<ref>B.B. Mandelbrot: ''The fractal geometry of nature'', Freemann '''(1983)''', ISBN 978-0-716-711-865</ref> that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of '''R'''<sup>''N''</sup> whose Hausdorff dimension is strictly greater than its topological dimension.
 
 
=== Hausdorff measure and dimension ===
Let ''d'' be a non-negative real number and ''S'' ⊂ ''X'' a subset of a metric space (''X'',''ρ'').  The ''d''-dimesional Hausdorff measure of scale ''δ''>0 is
:<math> H^{d*}_\delta(S) := \inf \{\sum_{i=1}^\infty r_i^d : S\subset\bigcup_{i=1}^\infty B_{r_i}(x_i), r_i\le\delta \}</math>
where B<sub>''r''<sub>''i''</sub>(''x''<sub>''i''</sub>) is the open ball around ''x''<sub>''i''</sub> ∈ ''X'' of radius ''r''<sub>''i''</sub>.  The ''d''-dimensional Hausdorff measure is now the limit
:<math> H^{d*}(S) := \lim_{\delta\to0+} H^{d*}_\delta(S)</math>.
As in the Carathéodory construction a set  ''S'' ⊂ ''X'' is called ''d''-measurable iff
:<math> H^{d*}(T) = H^{d*}(S\cap T)+ H^{d*}(T\cap X\setminus S)</math> for all  ''T'' ⊂ ''X''.
A set ''S'' ⊂ ''X'' is called Hausdorff measurable if it is H<sup>''d''</sup>-measurable for all ''d''≥0.
 
''[[Hausdorff dimension|.... (read more)]]''
 
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Latest revision as of 10:19, 11 September 2020

Nuclear weapons proliferation is one of the four big issues that have held back worldwide deployment of peaceful nuclear power. This article will address the proliferation questions raised in Nuclear power reconsidered.

As of 2022, countries with nuclear weapons have followed one or both of two paths in producing fissile materials for nuclear weapons: enrichment of uranium to very high fractions of U-235, or extraction of fissile plutonium (Pu-239) from irradiated uranium nuclear reactor fuel. The US forged the way on both paths during its World War II Manhattan Project. The fundamental aspects of both paths are well understood, but both are technically challenging. Even relatively poor countries can be successful if they have sufficient motivation, financial investment, and, in some cases, direct or illicit assistance from more technologically advanced countries.

The International Non-proliferation Regime

The International Atomic Energy Agency (IAEA) has a vigorous program to prevent additional countries from acquiring nuclear weapons. The Treaty on the Non-Proliferation of Nuclear Weapons (NPT) is the cornerstone arrangement under which strategic rivals can trust, by independent international verification, that their rivals are not developing a nuclear weapons threat. The large expense of weapons programs makes it very unlikely that a country would start its own nuclear weapons program, if it knows that its rivals are not so engaged. With some notable and worrying exceptions, this program has been largely successful.

Paths to the Bomb

It is frequently claimed that building a civil nuclear power program adds to the weapons proliferation risk. There is an overlap in the two distinct technologies, after all. To build a bomb, one needs Highly Enriched Uranium (HEU) or weapons-grade plutonium (Pu-239). Existing reactors running on Low Enriched Uranium (LEU, under 5% U-235) or advanced reactors running on High Assay LEU (HALEU,up to 20% U-235) use the same technology that can enrich uranium to very high levels, but configured differently. Enrichment levels and centrifuge configurations can be monitored using remote cameras, on-site inspections, and installed instrumentation -- hence the value of international inspections by the IAEA. Using commercial power reactors as a weapons plutonium source is an extremely ineffective, slow, expensive, and easily detectable way to produce Pu. Besides the nuclear physics issues, refueling pressurized water reactors is both time-consuming and obvious to outside observers. That is why the US and other countries developed specialized Pu production reactors and/or uranium enrichment to produce fissile cores for nuclear weapons.

Future Threats and Barriers

Minimizing the risk of future proliferation in states that want to buy nuclear reactors or fuel might require one or more barriers:
1) Insisting on full transparency for all nuclear activities in buyer states, including monitoring and inspections by the International Atomic Energy Agency (IAEA).
2) Limiting fuel processing to just a few supplier states that already have weapons or are approved by the IAEA.
3) Ensuring that fuel at any stage after initial fabrication has an isotopic composition unsuitable for weapons. "Spiking" the initial fuel with non-fissile isotopes, if necessary.
4) Limiting the types of reactors deployed to buyer states. In general, breeders are less secure than burners. Sealed reactor modules are more secure than reactors with on-site fuel processing.
5) Providing incentives and assurances for buyer states to go along with all of the above.
6) Application of diplomatic pressure, sanctions, and other economic measures to non-compliant states.
7) Agreement that any reactor declared rogue by the IAEA will be "fair game" for any state feeling threatened.

Footnotes