https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=134.148.85.110formulasearchengine - User contributions [en]2026-07-04T04:06:01ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Conjugate_residual_method&diff=26087Conjugate residual method2013-10-13T22:13:28Z<p>134.148.85.110: Modified x_0, r_0, and p_0 syntax in math section to fix the parse error.</p>
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<div>The term '''''semi-infinite''''' has several related meanings in various branches of pure and applied [[mathematics]]. It typically describes objects which are [[Infinity|infinite]] or [[unbounded]] in some but not all possible ways.<br />
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==In ordered structures and Euclidean spaces==<br />
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Generally, a semi-infinite set is [[bounded]] in one direction, and [[unbounded]] in another. For instance, the [[natural numbers]] are semi-infinite considered as a subset of the integers; similarly, the [[interval (mathematics)|intervals]] <math>(c,\infty)</math> and <math>(-\infty,c)</math> and their closed counterparts are semi-infinite subsets of <math>\R</math>. [[Half-space (geometry)|Half-space]]s are sometimes described as semi-infinite regions.<br />
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Semi-infinite regions occur frequently in the study of [[differential equations]].<ref>Bateman, [http://projecteuclid.org/euclid.bams/1183492736 Transverse seismic waves on the surface of a semi-infinite solid composed of heterogeneous material], Bull. Amer. Math. Soc. Volume 34, Number 3 (1928), 343–348.</ref><ref>Wolfram Demonstrations Project, [http://demonstrations.wolfram.com/HeatDiffusionInASemiInfiniteRegion/ Heat Diffusion in a Semi-Infinite Region] (accessed November 2010).</ref> For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar.<br />
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A semi-infinite [[integral]] is an [[improper integral]] over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.<ref>Cator, Pimentel, [http://arxiv.org/abs/1001.4706v3 A shape theorem and semi-infinite geodesics for the Hammersley model with random weights], 2010.</ref><br />
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Most forms of semi-infiniteness are [[bounded]]ness properties, not [[cardinality]] or [[measure (mathematics)|measure]] properties: semi-infinite sets are typically infinite in cardinality and measure.<br />
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==In optimisation==<br />
{{Main|Semi-infinite programming}}<br />
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Many [[optimisation (mathematics)|optimisation]] problems involve some set of variables and some set of constraints. A problem is called semi-infinite if one (but not both) of these sets is finite. The study of such problems is known as [[semi-infinite programming]].<ref>Reemsten, Rückmann, [http://books.google.ca/books?id=sJgX5jQZnQcC&lpg=PP1&ots=gvz6-MY_t1&dq=semi-infinite%20programming&pg=PP1#v=onepage&q&f=false Semi-infinite Programming], Kluwer Academic, 1998. ISBN 0-7923-5054-5</ref><br />
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==References==<br />
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[[Category:Infinity]]</div>134.148.85.110