Talk:Scale invariance

message: I've removed the expert tag after some alterations to clean up, correct and clarify. Still more to do, and I will return at a later point--Jpod2 15:28, 9 August 2006 (UTC)
thread_title: Untitled

message: OK, I have done what I suggested above. I think in a way you gave your tacit approval :) See what you think now. I have tried to explain that fractal dimensions *can* sometimes be critical exponents. This needs expansion, perhaps an explicit example.
ANyway, I think the ordering of the page is better now, though perhaps there will be more iterations. You could try to convince me that self-similarity/fractals can appear first, but we'd need to expand the stuff in fractals referring to physical processes (e.g. phase transitions and and critical exponents) since at the moment I think that is better understood in the CFT section.
I am reading more about the SLE approach to various 2d CFTs. I agree with your assertion that fractal dimensions in these cases can be related to the highest weight states in the CFT. I would not be shocked if similar things work for the 1d Potts model, though I would be kind of surprised if the same relations hold for CFTs in d>2. I see there is some literature in specific cases, particular by Fields medallists.... 
This is fascinating stuff, but I think we need to explain what CFTs are, and what fractals are (or at least introduce them both and point people to the right pages) first, and then perhaps introduce some of the connections between them. It would also be at home on the CFT and fractals pages, but in particular the CFT page needs a lot of work. If you want to we could make progress on that over the next few weeks---I will certainly start editing more on it when I get the chance, thoigh the thesis is pressing. All the best--Jpod2 09:27, 23 August 2006 (UTC)
thread_title: Reordering

message: I'm not sure this particularly belongs there in that context. Why does it? --Jpod2 21:58, 27 August 2006 (UTC)
thread_title: Boltzmann distribution

message: so what about a longer discussion at the beginning, introducing roughly four different areas of scale invariance
(1) scale-invariant functions in mathematics
(2) scale-invariant probability distributions for random processes
(3) scale-invariant classical field theories describing classical physics with no inherent length scale
(4) scale invariance of the fluctuations in phase transitions and QFT. Introduce idea of universality, copy some of text from your universality section. introduce quantum field theory/stat mech connection
then more on each of these in the sections we already have in the order above, but (a) with computer vision and fractals removed from the stochastic section and (b) the remainder of universality discussed after phase transitions.
I guess I am essentially moving stochastic before classical field theory, because maybe it follows on from the discussion of scale-invariant functions, and there is not so much of a direct link with CFT. also splitting universality into the intro and outro, and the rest you have copied into the universality article.
how does that sound? 
btw, I notice the ising model and potts model article don't seem to have much on the actual exponents, i might look into copying some of this stuff over--Jpod2 23:54, 27 August 2006 (UTC)
thread_title: new order of development

message: Could we qualify the domain of \lambda?  I'm concerned about the dilation, \lambda = 0.
Conr2286 03:19, 28 September 2007 (UTC)
thread_title:  Domain of <math>\lambda</math> 

message: "In physics, mathematics, statistics, and economics, scale invariance is a feature of objects or laws that does not change if scales of length, energy, or other variables, are multiplied by a common factor." is, as of Jan 7, 2015 the first sentence in this article. I have two issues with this sentence. First: describing scale invariance as what 'does not change' seems circular to me. Isn't it more correct to say scale invariance describes (or defines) a characteristic of (or feature of) objects or laws that doesn't change when the the scales of the independent variables change? That is, rather than describing IT as the characteristic that is invariant, isn't it be more correct that it describes that characteristic? Second, the definitions in the references require both the factor and its degree to be Real numbers. This should be made explicit, calling it a factor (coefficient) is a pretty weak constraint. Two other comments: the example of the invariance of a logarithmic spiral to scaling is both too complicated and too simple. It is too complicated in that I see no reason to use (1/b)ln(r/a) when ln(r) will do just as well. Why include two irrelevant parameters? Second, θ=ln(r) and θ=ln(λr) do not result in the same θ. Its obvious to anyone familiar with log functions that ln(λr)=ln(r)+ ln(λ)= ln(r)+ c. What needs more explanation, imho, is why the resulting curves are considered identical. I believe the statement "Allowing for rotations ..." doesn't sufficiently explain the point. How is it that if you "allow" rotations, then the fact that ln(r)=ln(λr)-c means θ is scale invariant?? (I also note that ln(r) and ln(λr) when λ is negative do not have any overlap in domains of r.) One other comment, it would be useful for two other simple examples to be included: one where the degree is fractional (irrational? transcendental?) and one with a positive integer coefficient, > 1.Abitslow (talk) 22:14, 7 January 2015 (UTC)
thread_title:  I think the first sentence is wrong.