Wikipedia:Reference desk/Archives/Mathematics/2007 May 31

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May 31

F - test

How can I test the hypothesis that the standard deviations of two series are equal if they are not normally distributed? I cannot use f-test, can I?--202.52.234.140 05:46, 31 May 2007 (UTC)

Actually yes, you can use the F-Test assuming we are talking about the same F-Test. First, if you have only two data sets you are wishing to compare, then you need to use the T test to determine if they are significantly different. The F-Test is simply a verification of the validity of the T-Test's results. The F-Test I refer too here is just F = σ₁² ÷ σ₂². Then you take this value and compare it to the F-Test table. If F_calc is greater than F_table then T-Test not valid because the sample sets have different levels of variability. However, most commonly the F-test refers to Analysis of variance. They share the same F Table but are completely different tests. You don't need ANOVA for this one.

Your original question though was can we even do these tests on skewed data sets. There are actually ways of measuring "skewness" and certain levels of skewness imply certain statistical restrictions. I refer you to http://jalt.org/test/bro_1.htm for more information on skewness. I think though that if both of your data sets have similar skew, then comparing them by way of the T-Test should be fine as long as the variances are not significantly different (F-Test).Mrdeath5493 17:33, 31 May 2007 (UTC)

Deficiency

I went to a maths lacture a while back about navigation on curved surfaces and a bit came up about the deficiency of 3D objects, e.g. the deficiency of a cube is 4 pi radians, half a radian at each vertex. Could someone please point me to an article(s) that have further info on this? Cheers Algebra man 11:13, 31 May 2007 (UTC)

• Also while I'm at it, I asked a questions a while back about spherical triangles. I was told that as the triangle becomes smaller, its sum of internal angles approaches 180. Is there any maximum to the sum? Algebra man 11:21, 31 May 2007 (UTC)

  This has an amusing answer, with a subtlety. We draw a spherical triangle, as small as we like. Then we declare the outside to be the inside! (We can do this on a sphere.) If we do not let the triangle shrink to a point, the angle sum has an upper bound, but not a maximum. --KSmrq^T 12:11, 31 May 2007 (UTC)

(About the triangle on the sphere.) Consider your very small triangle. Most of the sphere is outside of it and only a small part is inside. The sum of angles on the inside is 180°, as you said. If the triangle grows, the sum of interior angles will grow too, so let the triangle grow very large, until most of the sphere is inside it and only a small part is outside. Clearly, the sum of angles on the outside now is 180° and at each vertex, the sum of the interior angle and exterior angle is 360°. Therefore the sum of angles on the inside is 3×360°-180°=900°. At least I think you can reason this way. —Bromskloss 12:16, 31 May 2007 (UTC)

Observe

• tetrahedron, 4 verticies, deficiency at each point pi, total deficiency 4 pi
• octahedron, 6 verticies, deficiency at each point 2 pi/3, total deficiency 4 pi
• cube, 8 verticies, deficiency at each point pi/2, total deficiency 4 pi
• icosohedron, 12 verticies, deficiency at each point pi/3, total deficiency 4 pi
• dodecahedron, 20 verticies, deficiency at each point pi/5, total deficiency 4 pi

so they are all the same. Indeed it just depends on the Euler characteristic of the surface, this is a linked to the Gauss–Bonnet theorem. --Salix alba (talk) 12:47, 31 May 2007 (UTC)

What is deficiency, really? —Bromskloss 12:52, 31 May 2007 (UTC)

I thing the approprate article in wikipedia is Defect (geometry). You can view it as a simplicial version of the Gaussian curvature. --Salix alba (talk) 13:34, 31 May 2007 (UTC)

Ah, thanks. —Bromskloss 13:52, 31 May 2007 (UTC)

Algebra man:Purely as a matter of interest, was this the talk given by Alan Beardon at the Cambridge maths open days on 28/4 and 5/5? Algebraist 13:55, 31 May 2007 (UTC)

Sorry for the delay in answering your primary question. What you probably want is Descartes' theorem on angle deficiency, from

• Descartes, René, "Progymnasmata de solidorum elementis", in Oeuvres de Descartes, vol. X, pp. 265–276

Unfortunately, I have been unable to locate a Wikipedia article, nor to find an online source. The best I could come up with is Ed Sandifer's MAA column How Euler Did It for July 2004, "V, E, and F, part 2". You might also search the web for "turtle geometry" and "discrete geometry". --KSmrq^T 14:28, 31 May 2007 (UTC)

These are all great answers, thanks. I looked at Euler characteristic and saw Euler's Formula, which reminded me that it said somewhere that this was adapted for 2D shapes. Is this true and if so what is it? Algebra man 14:44, 31 May 2007 (UTC)

Please paraphrase the question. What was adapted for 2D shapes? Is what true, and what is what? —Tamfang 01:50, 1 June 2007 (UTC)

Sorry I didn't see this bit earlier. I meant I had heard that while Euler's formula is for 3D objects I thought a similar one had been found for 2D shapes. By is this true I was asking if this 2D formula existed and by what I meant what is the formula - if it does exist - for 2D shapes. Algebra man 17:06, 1 June 2007 (UTC)

I get it now. I didn't find any such phrase as "adapted for 2D shapes", but $N_{0}-N_{1}+N_{2}$ is the Euler characteristic for a 2D figure, including the surface of a 3D object; where $N_{j}$ means number of j-dimensional elements, respectively vertices (0), edges (1), faces (2). For 3D figures, e.g. the surface of a polychoron or a map of the various empires in Star Trek, the Euler characteristic is $N_{0}-N_{1}+N_{2}-N_{3}$. Do you see the pattern? Alternate plus and minus, in order of dimensionality. If you really meant to ask for the Euler characteristic of the boundary of a 2D object, it fits the pattern: $N_{0}-N_{1}$. —Tamfang 00:59, 4 June 2007 (UTC)

Also Salix alba mentioned that it depends on the Euler Characteristic. Is this to say that all 3D objects with the same Euler Characteristic have the same deficiency? Algebra man 14:52, 31 May 2007 (UTC)

Well, "all" is hard to talk about, and by the way, usually mathematicians would describe objects like this (surfaces) as 2-dimensional since we are only interested in the surface of the object. But the answer is yet -- this is a consequence of the Gauss-Bonnet theorem, which relates the total curvature of an surface to its Euler characteristic. In the case of polyhedra, with flat sides, what you're doing is concentrating all the curvature of the surface at the vertices (note that every point on a polyhedron has a full 360° around it except for the vertices -- this is one way to measure curvature). Thus to calculate the total curvature (which by Gauss-Bonnet is the Euler characteristic up to a constant) you just need to measure how far short of 360° we come at each vertex, and this is the deficiency. Tesseran 16:41, 31 May 2007 (UTC)

I know this is getting slightly off topic but can someones explain what is meant by up to a constant in the above explanation as well as why 'all is hard to talk about' again from the above Algebra man 16:45, 31 May 2007 (UTC)

up to a constant means that the total curvature is not in fact the Euler characteristic, but rather the Euler char. times a constant (which happens to be $2\pi$). In other circumstances, "up to a constant" is used to mean up to adding a constant: for example, the indefinite integral of a function is only defined up to a constant. I think what was meant by '"all" is hard to talk about' is that "all 3D objects" is a very large class indeed, which includes some utterly horrible entities. Clearly we need to at least restrict to objects for which the Euler characteristic can be defined, for starters. Algebraist 19:16, 31 May 2007 (UTC)

What is the deficiency of a cylinder? I was told that it is not a curved surface and therefore must have a deficiency. Algebra man 09:01, 1 June 2007 (UTC)

By cylinder I assume you mean with caps on both ends (I'd call this a cylindrical can to be unambiguous I guess). The cylindrical can has edge singularities as opposed to vertex singularities. Deficiency at a vertex is a number, and the corresponding concept on a curved surface is curvature, which can then be integrated over the surface (as in the Gauss-Bonnet theorem). The corresponding concept for an edge singularity is a quantity that you integrate along the edge. This quantity is the sum $k_{1}+k_{2}$ of the (signed) curvatures of the edge as viewed from one side or the other. (Here we're talking about the curvature of the edge as a curve; see the "Curvature of Plane Curves" section of curvature. Signed means that "curving away" gets a negative sign, so for example if the "edge" is no edge at all but just a curve in the plane, then $k_{1}=-k_{2}$ and the quantity is zero.)

What this all means is that there's a version of the Gauss-Bonnet theorem which adds on the vertex deficiency and the edge singularity "deficiency" that I've defined above:

$\int _{M}K\;dA+\int _{E}(k_{1}+k_{2})\;ds+\sum _{V}d_{v}+\int _{\partial M}k_{g}\;ds=2\pi \chi (M),$

(Here, E = the edges, V = the vertices, and d_v is the deficiency of the vertex v.) I think Chapter 9 of "Minimal Networks" by Ivanov and Tuzhlin is a reference for the above. (The bit about edge singularities can also just be deduced directly from the Gauss-Bonnet theorem by splitting the surface up into pieces.)

If all you wanted was the "total deficiency" of the cylindrical can, then it's $4\pi$, like all shapes that are topologically the sphere (by the Gauss-Bonnet theorem).Kfgauss 09:54, 1 June 2007 (UTC)

OK could someone please explain to me how we know that the deficiency of a cylinder is $4\pi$ - in sligtly simpler language than the above as I am only an amateur mathematician but by no means an idiot - without using the fact that a cylinder is homeomorphic to a sphere. Algebra man 11:46, 1 June 2007 (UTC)

Consider a regular prism with n sides. At each vertex you have the sum of angles 2·(π/2) + (1 − 2/n)·π = 2π − 2/n·π, so the deficiency is 2/n·π. There are 2n vertices, so the total deficiency is 2/n·π · 2n = 4π. That value does not depend on n and holds for limit n → ∞, which transforms the prism into a cylinder. --CiaPan 12:20, 1 June 2007 (UTC)

Right I think I understand. So is the final sentence basically saying in this context a cylinder is a prism with an infinite amount of sides? Algebra man 12:48, 1 June 2007 (UTC)

Nope. But I'm not sure if I can explain it properly – I'm not that good in English as I am in maths. The cylinder is a 'limit shape', something which the prism approaches when n grows; however none of this sequence of prisms is a cylinder, they just differ less and less. This is exactly same as polygons and circle – as the number of sides grows, the regular n–gon shape approaches the circle, but the circle is not a polygon, just a limit of polygon sequence. --CiaPan 13:50, 1 June 2007 (UTC)

Mathematics

Is mathematics a language or science? Some people say "the science of mathematics" or "mathematics is a science of numbers and their operations . . ." Other people say that mathematics is a language because people can translate from polar coordinates to non-polar Cartesian coordinates. 69.218.238.123 19:29, 31 May 2007 (UTC)

When people say "the science of mathematics", they are using the word "science" in a very general sense, meaning "knowledge", not "science" as we usually think of it. That said, mathematics is neither language nor science, but a separate field of study. --Tugbug 20:44, 31 May 2007 (UTC)

"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense." - V. I. Arnold.[1] Fredrik Johansson 20:57, 31 May 2007 (UTC)

I disagree with both of the previous comments. I have regularly heard of mathematics referred to as the 'pure language', one argument being that it is thought to be the language we will use to converse with aliens. Also I feel that physics is part of mathematics not the other way around. You need mathematics in physics - look at any number of equations for starters - however the reverse is not true. There are areas of mathematics which currently have no physical application, one such example being number theory and delving even deeper for example the distribution of primes, therefore maths cannot be a part of physics. I have also heard of mathematics being called more a set of tools for other subjects and not a subject in its own right - I feel the second part is inherently wrong but the first is true, maths is a fundamental tools in science, economics, technology etc. Algebra man 21:57, 31 May 2007 (UTC)

I also hear occasionally that mathematics is the language of science. x42bn6 Talk Mess 22:15, 31 May 2007 (UTC)

Let's not forget that in a very rigorous way, much of mathematics is just a highly formalized language. There are rules about how to string together symbols, and then there other rules on how to change these strings of symbols. However, actually doing mathematics takes a lot of creativity and thought, and some occasions it is impossible to translate the results into mere symbol manipulation. nadav (talk) 22:27, 31 May 2007 (UTC)

"The untrained man reads a paper on natural science and thinks: ‘Now why couldn't he explain this in simple language.’ He can't seem to realize that what he tried to read was the simplest possible language – for that subject matter. In fact, a great deal of natural philosophy is simply a process of linguistic simplification – an effort to invent languages in which half a page of equations can express an idea which could not be stated in less than a thousand pages of so-called ‘simple’ language." —Thon Taddeo in A Canticle for Leibowitz (Tamfang 01:52, 1 June 2007 (UTC))

To quote the very first sentence of Science, "In the broadest sense, science (from the Latin scientia, 'knowledge') refers to any systematic methodology which attempts to collect accurate information about the shared reality and to model this in a way which can be used to make reliable, concrete and quantitative predictions about events, past, present, and future, in line with observations." By that broad definition mathematics can be considered science, since mathematics is a systematic means to collect information about mathematical realities and to construct conjectures and predictions that hopefully can be proven true. Mathematics uses mathematical language and symbology as a means to efficiently collect that information in a way that helps make information as univerisally understandable and precise as possible. Thus the language of mathematics is a primary tool by which the science of mathematics is facilitated. Just my opinion. Dugwiki 22:43, 31 May 2007 (UTC)

Mathematics is a game. Or rather a system of games with definite rules. 202.168.50.40 22:44, 31 May 2007 (UTC)

What I dislike about the conception of mathematics as science is that it is essentially a Platonic approach, that is, it assumes that there is a higher truth that is being slowly discovered. It ignores the fact that much mathematics is invented in order to reflect ideas that interest us. In physics, you have to make sure your model conforms to reality, but in mathematics, you can just invent new ideas, and people will study them as long as they're interesting or shed new light on old problems. Thus in some ways, math is more like art as opposed to science because it allows people to pursue aesthetics instead of just modeling reality. nadav (talk) 23:10, 31 May 2007 (UTC)

In response, I should point out that I didn't state that math is working toward a "higher truth". In fact there are an uncountably infinite number of true statements that are unprovable, or even finitely definable, meaning that no amount of mathematical investigation will be able to uncover even the smallest sliver of everything that's true. Also, a main difference between an asthetic art and science is that there is ultimately a black and white correct answer to mathematical questions. A mathematical conjecture can only ultimately ever be either true, false or axiomatically independent. Contrast that to art where it would normally make no sense to declare a work of art "true" or "false" or "right" or "wrong". Finally note that mathematics and mathematical ideas are a part of reality, and thus it doesn't make sense to say that math doesn't "model reality". Not all mathematical axiom systems model physical spacetime, but they are nonetheless part of the entire reality that includes the mental reality of consistent, logical ideas. Dugwiki 15:57, 1 June 2007 (UTC)

From the responses above, it is said that mathematics is an art, language, science, and a game. So, which one is it exactly? 69.218.238.123 23:45, 31 May 2007 (UTC)

For further reading on the topic, I think your best bet for now is the article Mathematics. This question is just beyond the scope of this mere reference desk. nadav (talk) 23:52, 31 May 2007 (UTC)

It is what it is: like foo in some ways, unlike foo in other ways, where foo is any of the set {art, language, science, game} among others. To seek a neat classification of all concepts is folly. —Tamfang 02:28, 1 June 2007 (UTC)

Is dance about beauty, or self-expression? Do people write textbooks because they want the money, or like to write, or like to teach? Different people (and groups, and schools of thought, etc) will give different answers, and most of them will claim everyone else is wrong, and/or "entitled to their opinion". The problem is, math is a catchall word for a wide variety of different pursuits, none of which interest every single person who has an interest in math. For instance, I have little in common with the people who grow up to become accountants. Nothing about number-crunching appeals to me, yet it does to them, and we both do math. To describe what math is overall, you'd have to describe it in terms of its meaning to each group and the trends among those meanings, not in terms of a single definition. The Mathematics article mentions one or two, the people here have mentioned a few others. There are many, many more. For instance, mathematics to a lot of people is a time-consuming way to balance their checkbook. Math to a competitive Mathlete is an exciting sport. Math to the people at Los Alamos was a voyage of discovery, a portal to an amazing world where atoms explode, and a way of defending their homeland against annihilation. None of these contradicts the others, unless you assume math has to be all things to all people. The other half of your question, and I think it does have two parts, is what system is used to achieve those goals - is math like science, the empirical empire, or more like language in that it's entirely composed of manipulable symbols? Once again, it is each depending on why you're trying to describe it in the first place. If you're trying to make the case that math has real-world applications or some inherent claim to truth, it's useful to be aware of its roots in and contributions to both natural sciences and pure logic. If you're trying to make the case that math has a lot in common with language, it's certainly not hard to point out its roots in and contributions to that as well - the symbols and words used to describe mathematical topics are clearly a well-constructed language distilled from the one we usually use, and they have influenced our natural language in turn. Just listen to someone describing person or point "A" in an anecdote, and you see one thing we've given back. Long story short(er), it's as complex and multifaceted as any other major category - cuisine, art, sports, you name it. Black Carrot 05:42, 1 June 2007 (UTC)

We have an article on mathematics as a language. With regard to the question "is mathematics a science ?", this has been discussed several times, at length, at Talk:Mathematics. I think the general consensus, as reflected in the Mathematics article, is that mathematics is a science but not a physical science. Gandalf61 13:03, 1 June 2007 (UTC)

All of these questions are controversial, and will probably always be. I would say that mathematics does in fact have empirical content. If you assume, for example, that the typical set theoretic picture is an accurate description of real objects existing independently of our reasoning about them, then it follows that it will not be possible to derive a contradiction from certain formal axioms. Thus that picture is, at least to some extent, empirically falsifiable. Moreover, if these objects really exist, then we do in fact have the linguistic tools to identify them (this gets a little technical for a refdesk discussion, but basically you take the maximum possible thing at every level of the von Neumann hierarchy), so if sets are real objects, then for example the continuum hypothesis is either really true or really false, even if we don't know which and also know that the accepted axiomatic framework is incapable of deciding it. So what this boils down to is that I disagree with nadav, though I probably would have agreed with him at the time I was applying to graduate programs in mathematics. --Trovatore 21:23, 2 June 2007 (UTC)