Wikipedia:Reference desk/Archives/Mathematics/2013 September 28

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September 28

TVS's and Dual Space Relation

I know that, in general, the dual space of a topological vector space (more specific context = Banach Spaces) is not isomorphic its dual space. Are there any results that measure the degree to which they fail to be isomorphic (in any sense)? More interestingly, any results that link this to obstructions derived from the original space? Phoenixia1177 (talk) 06:05, 28 September 2013 (UTC)

More specifically, given a category of topological v. spaces over the complex numbers, is C ever an injective object; and if not, can the failure of Hom(-,C) to be an exact functor be used to characterize the relation between A, A*, and A** in that category? That A->A*->A** gives a natural isomorphism in the finite algebraic case, and so the functor is exact, seems like it should have some extension to the tvs case, and failures there. Thank you for any help:-)Phoenixia1177 (talk) 07:14, 28 September 2013 (UTC)

Reflexive space and Stereotype space might help.John Z (talk) 18:46, 28 September 2013 (UTC)

Looking for a word that means 'constantize'

Resolved

Hi all, what words exist to describe taking a multi-variate function and approximating some of the arguments into a single argument? Possibly even removing an argument or a set of arguments entirely and replacing them with a constant value that is the expected average for the removed argument(s). A few other explanations of the idea:

In realtime computer graphics there is a great term - "Baking" - (also known as "lightmaps") which means exactly what I'm thinking of in the context of shadows and ambient occlusion. Instead of rendering the lighting effects in realtime, the effects are pre-rendered with average lighting conditions and those pre-rendered images are then blitted onto the static texture images, and not computed in realtime. This is valuable to reduce the computation requirements during realtime rendering. The term baking is great, but I've never seen it used outside of that context and I'm sure there is a good generic way to describe it.

Considering the idea of Taylor polynomials, you could say what I'm interested in is a name for the 1st term when $n=0$ (equal to $f(a)$), but a slightly more generic word to describe finding something like this. I know of no verb form for the word 'constant', but anything similar is what I'm after.

There is also a really solid interpretation of this idea in probability theory. It could be explained as taking a conditional probability $P(A|C)$ and attempting to remove the condition to find $P(A)$. Taking the posterior probability and determining the prior probability.

You could also describe this word as what it is called to take a binary function and attempt to combine the arguments or approximate one of the arguments as a constant somehow such as to force it to be nullary.

An example of how this comes up for me is if, given a dataset, you are interested in determining who-knows-what based on the data. Instead of looking at every bit of data in the dataset, one can simply take the average value (effectively losing data) and then using that average to make any determination. This process of ignoring the rest of the data, and 'compressing' it into a single argument is exactly what I'm interested in.

Thank you for any insight regarding this. I encounter this problem on a regular basis and it kills me when I can't express myself. There has to be some kind of generic term for this, right? Thanks.

-- lulzmango (talk) 07:02, 28 September 2013 (UTC)

Sorts of words you'l get for that are instance instantiate or instantiation, currying, slicing, fixing, or constraining. Dmcq (talk) 09:59, 28 September 2013 (UTC)

Thank you, Dmcq. Most of those words are in the right vain. (currying is not relevant though, and I'm not sure what 'slicing' means here) --lulzmango (talk) 02:13, 30 September 2013 (UTC)

Slicing is applied more to matrices and there means to select entries in a particular row or column. For instance if you have figures about food and its nutrition then particular slices might be for cucumbers or salt. A selection would be another word. By the way, vein not vain. Dmcq (talk) 07:51, 30 September 2013 (UTC)

My guess is that you are talking about texture baking. In physics this would be a kind of perturbative approximation. In perturbation theory, one calculates a solution to a problem and uses it to find an approximate solution to a slightly different problem. In the computer graphics case, you solve one scene and use that solution to approximately solve a new scene in which say a new object is introduced. The scene affects the lighting of the new object, but we assume that the object has only a small, nay negligible, effect on all the other scene elements, so that they stay unchanged. This sort of technique is also used to incorporate expensive radiosity effects into raytraced scenes. A radiosity solution was first computed, then baked into the textures for the purposes of raytracing. It allows the incorporation of diffuse reflections into a render without having to recompute the radiosity solution for every scene. --Mark viking (talk) 12:18, 28 September 2013 (UTC)

Thank you very much, Mark. Pertubation theory is new to me and I'm finding it very interesting. --lulzmango (talk) 02:13, 30 September 2013 (UTC)

I would call this a restriction of the multivariate function to a lower-dimensional set, although there is also a reparametrization involved. This verbiage applies when replacing an argument with any known expression: for instance, taking $y=2x$ in $f(x,y):\mathbb {R} ^{2}\to \mathbb {R}$ yields ${\hat {f}}(t):=f(g(t))$ where $g(t):=(t,2t)$. Sometimes one just writes $f(t)$ since the change of argument count (or, non-rigorously, even renaming an argument!) implies the reparametrization. --Tardis (talk) 03:28, 1 October 2013 (UTC)

Tardis, thank you, that is a solid way of looking at it. --lulzmango (talk) 23:18, 1 October 2013 (UTC)