Talk:Support (mathematics)

In case I bugger it up, the union of the support and the kernel of a numerical function make up the domain of that function, correct? Dysprosia 10:12, 17 Jul 2004 (UTC)

Hello I am confused by the statement in "Support": " Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails - essentially because the singular supports of two distributions to multiply should be disjoint)." Seems to me if you multiply two distributions whose support is disjoint, you will get zero. Is that not true? Seems to me that the problem in squaring a Dirac Delta has more to do with orders of infinity and limiting processes. Thanks Peter Pdn Feb 20, 2005 4:22 PM EST or 9:22 PM UTC I suppose

But the singular support is not the support. Charles Matthews 22:11, 20 Feb 2005 (UTC)

The "sing supp" is the support w.r.t. the subsheaf C&infin; while "supp" is the support w.r.t. the subsheaf {0}. At least some links to sheaf seem necessary to me. MFH 02:14, 12 Mar 2005 (UTC)

Can the concept of support be applied to geometry? I think that would make a good graphical example for newcomers to the idea, if so. The intro is a little hard to parse — Omegatron 20:17, 19 October 2005 (UTC)


 * I'm doing some groundwork for expanding the GJK page. One of the core features of the algorithm is that it only relies on the support function for a specific geometric shape, which is a definition I'm not seeing on this page. Mathematically, this function should take in a vector V and return the point on the surface of the geometry that results in the largest value when you dot V with every point on the geometry's surface.


 * As examples: the support function of a sphere would always return normal(V) * radius; the support function of a box would always return one of the eight box vertices; the support function of a capsule/sphyl would always be a point on one of the two endcaps.


 * In this context, I believe the support function for non-convex shapes is undefined. Does anyone with a stronger math background than me want to confirm or correct any of this stuff, or point out which article it belongs in? I'd like to make sure that I correctly link the GJK article there for the relevant information. --Kyle Davis 18:40, 29 November 2005 (UTC)

Essential Support?
Can someone sanity check me on this? The quotes "smallest closed subset F of X such that f = 0 μ-almost everywhere outside F" and "largest open set on which f = 0 μ-almost everywhere" seem unsourced. Are they true? Certainly an arbitrary union of open sets is open (or equivalently the arbitrary intersection of closed sets is closed), but it is not clear to me that an arbitrary union of sets such that f = 0 a.e. will also be a set for which f = 0 a.e.

I checked the citation "Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. p. 13. ISBN 978-0821827833." and they do not characterize it this way, I found it for free here. They define $\Omega^\sim$ to be the set of all open sets such that f = 0 a.e., and then defines the complement of the essential support to be the union over $\Omega^\sim$. But they do not imply that the complement of the essential support is itself in $\Omega^\sim$, while the article here on wiki does make this claim.

I'm suspicious the claim is untrue but even if true it certainly seems unjustified and unsourced. Thoughts?

help
I clicked this page thinking it was were I could get some SUPPORT for a math question!

qua! qua!
While this is clearly a special point, it is more precise to say that the transform qua distribution has singular support {0} [...]

Qua? --Abdull 04:41, 13 June 2006 (UTC)

Support of a probability distribution
I don't like this sentence: "In particular, in probability theory, the support of a probability distribution is the closure of the set of possible values of a random variable having that distribution." In probability distribution it is defined better: "The support of a distribution is the smallest closed set whose complement has probability zero." --130.94.162.64 00:57, 21 June 2006 (UTC)


 * To add one more thing:
 * A point x is said to belong to the support of a distribution function F if for every $$ \epsilon\ > 0$$, we have:
 * $$ F(x + \epsilon\ ) + F(x - \epsilon\ ) > 0 $$. —The preceding unsigned comment was added by Musically ut (talk • contribs) 13:10, 4 February 2007 (UTC).

Explanation?
I've taken some analysis, but can't follow this article. It opens
 * 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. The most common situation occurs when X is a topological space (such as the real line) and f is a continuous function.

Staying with real-valued functions (e.g., f:R&rarr;R) it isn't clear to me if X is a set of real numbers, of real-valued functions, or what. That is, is this saying that the support, S, is defined as
 * $$S = \{ x \in X | f(x) \neq 0 \}$$

or is it something else? —Ben FrantzDale 13:40, 5 July 2006 (UTC)

Definitions?
"In mathematics, the support of a function is, in general, the set of points where the function is not zero. More specifically, a support of a function f from a set X to the real numbers R is a subset Y of X such that f(x) is zero for all x in X that are not in Y." The "in general" definition is narrower than the "[m]ore specifically" definition! (According to the latter, a function has several supports; according to the former, it has exactly one, which is the smallest possible one according to the latter definition.) Isn't that backwards? -- Army1987!! 09:11, 10 July 2008 (UTC)


 * I've deleted the second sentence. Clearly someone was confused. Michael Hardy (talk) 21:01, 18 August 2008 (UTC)


 * The article is still inconsistent in the definition: In "Formulation", it is said that a superset of a support is again a support, contradicting the Introduction's definition.--Roentgenium111 (talk) 22:48, 21 April 2009 (UTC)

Examples?
How about some examples, instead of talking in generalities all the time?

--212.139.126.204 (talk) 14:54, 18 July 2009 (UTC)

Compact support
I am not sure whether I understood something wrong or if I found a (widespread) mistake. Let us assume: Can any experienced mathematician correct either me or the article, please? --FDominec (talk) 19:43, 26 October 2009 (UTC)
 * We have a smooth function that is nonzero in the interval (-x, x) and zero in (-inf, -x) and (x, inf). As it is smooth, its value at -x or x has to be equal to its limit in this point, which is 0 by definition. Therefore, this function is obviously nonzero only at the open interval (-x, x).
 * A compact subset of real numbers has to be bounded.
 * Therefore, any such function can never have compact support, as there would have to be a discontinuity at -x, x.
 * (This can be probably extended to any smooth function.)


 * This doesn't seem right. First of all, support has nothing to do with continuity. It is simply the "closure" of a set where a function is nonzero. In other words, a function that is only nonzero on (-x, x) has a compact support, since it is nonzero on the open interval and therefore its support, the closure of the interval, is compact. -- Taku (talk) 22:46, 26 October 2009 (UTC)


 * I see, there is In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set. in the support article. As far as I can remember, we considered the support not to be a closure of anything at the school, just the set of nonzero points. Nevertheless, it should be rewritten more clearly. --FDominec (talk) 14:34, 27 October 2009 (UTC)


 * Yes it could be better worder. Is this one definition (it's one or the other depending on context), or two competing definitions (some authors define it one way, other the other)? I'm guessing the latter, given what you say and that Wolfram Mathworld only gives the closure version. In that case, which definition has been chosen for this Wikipedia article? Dependent Variable (talk) 13:38, 2 March 2011 (UTC)

Definition/source
"In mathematics, the support of a function is the set of points where the function is not zero-valued, or the closure of that set.[1]:678".

In "[1]:678" it states:
 * "The support of a continuous function u is the closure of the set {x | u(x) =/= 0} and is denoted by the symbol supp(u)."

Thus the given source doesn't source wp's definition, as "function" and "continuous function" is something different. So either wp's definition has to be changed or another source has to be given. Futhermore "set [...] not zero-valued, or the closure of that set" partly implies that one can choice (maybe based on different definitions) which set it is or that the author/wp doesn't know which set it is. Accourding to the mentioned source it's always the closure of that set (though the closure of that set might be the same as that set).-Yodonothav (talk) 10:37, 26 November 2012 (UTC)

Vanishing at infinity
I think this sentence, from the Compact Support section, is problematic: "Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are functions of bounded support and therefore vanish at infinity (and negative infinity)." While technically it is true that such functions vanish at infinity, it is a weak statement—they vanish considerably sooner than infinity.

I initially thought that "and therefore [the functions] vanish at infinity" was intended as justification for why functions of bounded support also have compact support. I realized my mistake as soon as I clicked "vanish at infinity" and read the definition. There is no further mention of vanishing at infinity, so I think that it is distracting to bring it up at all. Does anyone else feel the same way? I don't usually edit Wikipedia (except to correct grammar or spelling), so I'm not sure what to do about this. — Preceding unsigned comment added by 137.22.232.216 (talk) 21:58, 17 January 2013 (UTC)

Definition
I have changed "and" to "or" in the definition. The definition doesn't seem to make sense with an "and" in it. There can be several meanings of "support", but (within one chosen meaning) "the support" has to be either ... or ... Bj norge (talk) 09:39, 2 July 2013 (UTC)

Confusion
There seems to be a contradiction between the definition at the top (support "is the set of points where the function is not zero-valued, or the closure of that set" - taking into account the definition of closure) and the later statement that "any superset of a support is also a support". If the term is used with many different meanings then this should be made clear from the beginning in such a way that the article is self-consistent. — Preceding unsigned comment added by 192.76.7.215 (talk) 14:39, 26 September 2013 (UTC)


 * I made some edits to address some of the points above, but I agree that there's still some confusion between a support for a function and the support. Maybe this is due, in part, to the term "support" being used in different ways in different areas? Reader634 (talk) 21:09, 12 January 2014 (UTC)
 * I reworked the section on "Formulation" since (lacking references) I don't believe it's usual to define supports so that "any superset of a support is also a support". Reader634 (talk) 07:47, 14 January 2014 (UTC)

question/suggestion
My immediate reaction to the current intro paragraph is to ask the next classic big ‘W’: why? Why is it all but those that map to zero? What is the purpose?? A good overview intro should answer the questions what, why, where and possibly who and do all those things in simple terms. — Preceding unsigned comment added by 2600:387:6:805:0:0:0:B1 (talk) 15:03, 13 August 2019 (UTC)