Talk:Homogeneous function

[Untitled Section]
What the hell is x times gradient of f(x) supposed to mean, dot product?


 * It means that for a vector function f(x) that is homogenous of degree k, the dot production of a vector x and the gradient of f(x) evaluated at x will equal k * f(x). CodeLabMaster 12:12, 05 August 2007 (UTC)


 * Yes, as can be seen from the furmula under that one. I've added the dot and changed vector symbols to bold. mazi 18:04, 22 February 2006 (UTC)

The gradient is not even mentioned. Why not simply state the theorem in sum-of-derivative form? Even if a beginner digs through these comments, most likely looking up "gradient" will confuse them unnecessarily. — Preceding unsigned comment added by 140.112.177.67 (talk) 15:01, 21 January 2019 (UTC)

Not all homogeneous functions are differentiable
The article, before I changed it a moment ago, implied that all homogeneous functions are differentiable. Here's a counterexample: $$f\colon \mathbb R^2 \to \mathbb R$$, $$k=1$$, with $$f(x,y)= x$$ (if $$xy>0$$) or $$0$$ (otherwise). --Steve 03:23, 6 August 2007 (UTC)

Is the derivative theorem correct?
According to planetmath, the theorem about derivatives is not correct unless we replace "homogeneous" by "positive homogeneous" throughout. Their counterexample is wrong (I just submitted a correction on the site), but could that claim be correct? Does anyone have a reference, or a proof, or a proper counterexample?

Update: The person maintaining that planetmath page responded to my correction by taking away the counterexample but keeping the claim. Again, a reference, proof, or proper counterexample is needed to resolve this. --Steve 15:51, 5 October 2007 (UTC)

The result is correct for functions which are homogeneous of degree $$k $$. I've added the elementary proof of this result to the page (and merged "Other properties" with "Euler's theorem" as the proofs are very similar). Is the planetmath contributor worried about $$\alpha=0$$? Clearly the definition of homogeneous of degree $$k$$ for $$k < 0 $$ has to be modified so that the condition holds for all $$ \alpha \neq 0 $$, and I've just changed this too. Mark (talk) 16:31, 11 February 2008 (UTC)

Notation
Notation such as
 * $$\frac{\partial f}{\partial x_1} (\alpha \mathbf{x}) $$

can be confusing: Are we differentiating the expression with respect to the first component of $$\mathbf{x}$$ or do we mean the partial derivative of $$f$$ with respect to its first argument evaluated at the point $$\alpha \mathbf{x} $$?

It's therefore better to write
 * $$\frac{\partial f}{\partial x_1} (\alpha \mathbf{y}) $$:

now it's clear that $$x_1,\ldots, x_n$$ are the arguments of $$f$$ and we are differentiating with respect to the first argument of $$f$$ evaluated at the point $$\alpha \mathbf{y} $$.

I've cleaned up the notation in my proof of Euler's theorem accordingly. 91.21.25.30 (talk) 21:50, 11 February 2008 (UTC)


 * I agree. Sorry about my incorrect edit to that effect earlier, thanks for reverting :-) --Steve (talk) 18:00, 11 February 2008 (UTC)

the name
Euler's theorem? why the name, is he the 1st guy prove this? if yes, why don't we use his work as a reference? Jackzhp (talk) 17:29, 4 December 2008 (UTC)
 * Keep in mind the Euler lived in the 18th century and wrote mostly in Latin so not really a good reference for a modern audience. It might be worth adding the original work as a historical reference though.--RDBury (talk) 21:44, 18 April 2010 (UTC)

restriction of k for nth degree homogeneity?
What are the restrictions on k? Must k be contained within the domain of the vector space, reals, or what?

Thanks, Jgreeter (talk) 06:51, 16 May 2011 (UTC)


 * For arbitrary fields, k should be an integer. For the reals, it makes sense to define this notion also for real numbers k. For example, the square root is homogeneous with k=1/2.
 * But it seems to me that usually k is an integer. --Aleph4 (talk) 16:49, 16 May 2011 (UTC)

Note on change http://en.wikipedia.org/w/index.php?title=Homogeneous_function&diff=529079679&oldid=520543743
The Phi used in the equations, that is /varphi inside a &lt;math&gt; tag, was rendered differently in my browser (mobile Safari) than the &amp;phi used in the descriptions below (see Phi) -- so I switched the descriptions to use the clumsier but definitely-consistent &lt;math&gt; tag construction. 184.17.182.96 (talk) 06:26, 21 December 2012 (UTC)

Keep it simple please
People who look up homogeneous function may not necessarily understand what "ƒ : V → W is a function between two vector spaces over a field F" means; Likewise people who know what a Banach space are not likely to wonder "what the heck is a homogeneous function" and look it up in Wikipedia. Do not scare people away from math please. And foremost, be mindful Wikipedia is basically an encyclopedia; it's meant for ordinary people to look up stuff :) --Sahir 08:41, 8 December 2015 (UTC)

Typo in the introduction?
In the second line, should it say $$f(ax,y) = a^kf(x,y)$$ rather than $$f(ax,ty)=a^kf(x,y)$$? It looks like a typo but I don't know enough about the subject to edit. Mrdouglasweathers (talk) 12:35, 1 November 2016 (UTC)
 * This makes sense. I fixed it. --Erel Segal (talk) 13:16, 1 November 2016 (UTC)

degree 0 example?
How about an example of homogeneity of degree 0? Btyner (talk) 21:48, 1 September 2017 (UTC)

functions defined on positive cones or rays - and absolute homogeneity
The former is covered by the introduction, but that's it; later on, the square root is given as example, without discussing that it is not a function on a vector space. Absolute homogeneity is defined only later, but - being a defining property of norms - it could very well go into the introduction. (And: Zero removed? What about "[...] e.g. making matrix rank homogeneous of degree zero, once one omits the exceptional s=0".) — Preceding unsigned comment added by 193.90.163.192 (talk) 09:17, 6 December 2021 (UTC)