Wikipedia:Reference desk/Archives/Mathematics/2012 November 11

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November 11

Calculus notation

What type of function is a "d" with a forward-slanted strikethrough, I think it is a type of differencial.

An example: d(U - TS) less than or equal to (d̸)w_expansion + (d̸)w_nonexpansion. Plasmic Physics (talk) 00:36, 11 November 2012 (UTC)

In thermodynamics the state functions have perfect differentials (d), and then there are differentials d̸ which are not perfect, meaning that they are not differentials of state functions. For example, in a fluid the increase of energy dE = TdS−PdV where T is temperature, dS is increase in entropy, P is pressure, and dV is increase in volume. Here dE and dS and dV are perfect differentials, because E and S and V are state functions. For reversible processes TdS=d̸ Q is the heat transferred to the fluid, and PdV=d̸ W is the work done by the fluid. But 'heat content' Q and 'work content' W are not state functions. Bo Jacoby (talk) 01:28, 11 November 2012 (UTC).

Explicit solution that gives the n^th Harmonic number

Does one exist without integrals or other such operations that require the further determination of a solution? --Melab±1 ☎ 03:07, 11 November 2012 (UTC)

I'm fairly certain the harmonic number function is not elementary. As such there is no "explicit solution" for it in the form you seem to want. If you're satisfied with an approximation, taking a few terms from the asymptotic expansion works well. -- Meni Rosenfeld (talk) 06:08, 11 November 2012 (UTC)

What does f ⊑ g mean?

I recently read a paper that used the notation "f ⊑ g" without defining the symbol "⊑". It looks like a kind of square "subset" symbol. f and g were functions. Does this notation mean anything to anyone? --Doradus (talk) 22:54, 11 November 2012 (UTC)

Could they have been partial functions? It often means "extends".--80.109.106.49 (talk) 00:42, 12 November 2012 (UTC)

Unicode Character 'SQUARE IMAGE OF OR EQUAL TO' (U+2291) – though this probably does not help much. — Quondum 01:06, 12 November 2012 (UTC)

Thanks for the tip on partial functions. When I get a chance, I'll re-read that passage and see if that explains it. --Doradus (talk) 18:46, 12 November 2012 (UTC)

Of course it would have been boring to tell us the title of the paper or even its topic. That would have made the question much too easy. Looie496 (talk) 03:33, 12 November 2012 (UTC)

Of course it would be silly just to ask for the information you want. Much better to post a sarcastic remark insinuating that every newcomer asking a question ought to know the right amount of context to provide. --Doradus (talk) 18:46, 12 November 2012 (UTC)

To me it just means "f is a subset of g", but I could be wrong. Perhaps they are used in Cauchy spaces, but then again, perhaps not. Is there a way to search for such a symbol encyclopedia-wide in Wikipedia math-notation Special:Search function? ~AH1 ^(discuss!) 04:01, 13 November 2012 (UTC)

Actually Wikipedia search turns up zero hits for that symbol. Perhaps the Search function has some trouble with unusual unicode characters? --Doradus (talk) 13:38, 13 November 2012 (UTC)

For anyone who is interested, the paper is this one: "Strictness and Binding-Time Analyses: Two for the Price of One". It doesn't seem to be available for free online. The sentence is:

A domain projection γ on a domain D is a continuous function γ: D → D such that (i) γ ⊑ ID, and (ii) γ ∘ γ = γ (idempotence).

I take it that "ID" is the identity function over D. (perhaps it should have been I_D?) --Doradus (talk) 13:49, 13 November 2012 (UTC)

(edit conflict) This is the unicode character U+2291 (Square Image Of Or Equal To) (see for example this website). The German Wikipedia lists it in the article Unicodeblock Mathematische Operatoren. According to page 5 of this pdf file it denotes some kind of relation. but I don't know to what mathematical objects this is applied It seems Doradus found something. -- Toshio Yamaguchi (tlk−ctb) 13:52, 13 November 2012 (UTC)

Seeing the word “domain”: ⊑ is commonly used to denote partial orders (domains) in domain theory. If D, E are domains, the order on the function space D → E is defined pointwise, i.e., f ⊑ g iff f(x) ⊑ g(x) for every x ∈ D. So, if ID is indeed the identity function as you suggest, then (i) translates to γ(x) ⊑_D x for every x, where ⊑_D is the order supplied by the definition of D being a domain.—Emil J. 14:30, 13 November 2012 (UTC)

I had a look at the paper, and indeed it seems to be talking about domain theory (with some category theory mixed in). However, the author is very reader-unfriendly when it comes to providing context, or definitions of the concepts he’s using (let alone notation).—Emil J. 14:53, 13 November 2012 (UTC)

That sounds like the definition of a retract (in the sense used in topology, not in the sense used in category theory, see retract (category theory) for that). — Tobias Bergemann (talk) 14:32, 13 November 2012 (UTC)

Terrific! Thanks again, everyone, for all the leads. --Doradus (talk) 20:59, 17 November 2012 (UTC)

Fibonacci numbers

Hello, could anyone explain me Fibonacci sequence in short? And what is the meaning? Thanks. Bennielove (talk) 23:43, 11 November 2012 (UTC)

The article Fibonacci numbers starts at a fairly basic level. Please let us know what information there you would like more information on.Naraht (talk) 00:12, 12 November 2012 (UTC)

1 + 0 is 1, 1 + 1 is 2, 2 + 1 is 3, 3 + 2 is 5, 5 + 3 is 8, 8 + 5 is 13, 13 + 8 is 21, 21 + 13 is 34, and so on. Placing these numerical values into the side lengths of adjacent squares gives you the Fibonacci spiral, which is found everywhere from Ammonites to pinecones to the fractal-like interactions of tropical cyclones and just everything else found in nature. See Golden ratio and natural computing for some direct applications. ~AH1 ^(discuss!) 04:04, 13 November 2012 (UTC)

Ahem no. Please read Fibonacci Flim-Flam to find most of these statements to be debunked myths. "Nature has many spiral forms. None of them are golden spirals. Many don't even come close. None of them are 'explained' by Fibonacci mathematics." --KnightMove (talk) 12:32, 14 November 2012 (UTC)