Wikipedia:Reference desk/Archives/Mathematics/2011 April 7

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April 7

Pairwise distinct

What are "pairwise distinct" columns in a matrix? Widener (talk) 02:24, 7 April 2011 (UTC)

"Pairwise distinct" means that no two are the same. —Bkell (talk) 04:10, 7 April 2011 (UTC)

Widener - in context, is a single column described as "pairwise distinct", or are two columns being compared and described as "pairwise distinct" from one another ? Gandalf61 (talk) 08:10, 7 April 2011 (UTC)

The elements of an indexed family are pairwise distinct if no pair of elements of the family consists of identical elements. If the cardinality of the family is 0 or 1, then there are no pairs, and thus no pair consists of identical elements, and so the elements of the family are indeed pairwise distinct. Bo Jacoby (talk) 15:45, 7 April 2011 (UTC).

Gsndalf61 - You're asked to construct a matrix that has four pairwise distinct columns (and which satisfies other conditions). I've figured out what it means now. Widener (talk) 21:23, 7 April 2011 (UTC)

Does anyone ever speak of a family of elements being distinct in a non-pairwise manner? I can see sense in speaking of a family of "pairwise disjoint" sets (as opposed to the weaker condition that the intersection of all the sets must be empty), and a family of "pairwise relatively prime" numbers (as opposed to the common gcd of all numbers being unity), or even a "pairwise linearly independent" set of vectors (but don't ask me what use this one would have). But does pairwiseness add any semantics to "distinct"? There seems to be only one halfway reasonable way to generalize ≠ from 2 to n operands. –Henning Makholm (talk) 16:24, 7 April 2011 (UTC)

Well, if you just say "distinct", it might be possible to read it as "not all columns are the same". Maybe not a very likely reading. --Trovatore (talk) 16:29, 7 April 2011 (UTC)

Good point. This does even feel structurally similar to my other examples. Thanks. –Henning Makholm (talk) 17:03, 7 April 2011 (UTC)

I don't think it's an unlikely reading at all - it's how I'd read it. Would you not describe (1,2,3) and (1,2,4) as being distinct from each other? --Tango (talk) 15:59, 9 April 2011 (UTC)

Would you describe (1,2,3), (1,2,4), and (1,2,3) as being distinct? They are certainly not all the same, so Trovatore's "unlikely" reading would call them district. On the other hand, they are clearly not pairwise distrinct, because (1,2,3) and (1,2,3) are the same. –Henning Makholm (talk) 19:14, 9 April 2011 (UTC)

Convergence to zero

Suppose $|\pi -x_{j+1}|=(\tan ^{2}c_{j})|\pi -x_{j}|$ for some $c_{j}$ between $x_{j}$ and $\pi$. Show that $\lim _{j\to \infty }|\pi -x_{j}|=0$ if $x_{0}$ is sufficiently close to $\pi$.
The best I can say is that $|\pi -x_{j+1}|<|\pi -x_{j}|$ for $x_{0}$ sufficiently close to $\pi$ (such that $\tan ^{2}c_{j}<1$) but that's not a proof that it converges to zero. Widener (talk) 09:44, 7 April 2011 (UTC)

Hint: $\tan \pi =0$ and tan is continuous, so for x sufficiently close to π, $\tan ^{2}c<1/2$. -- Meni Rosenfeld (talk) 11:01, 7 April 2011 (UTC)

So that means $c_{j}$ approaches π as j approaches infinity so $\lim _{j\to \infty }(\tan ^{2}c_{j})|\pi -x_{j}|=0$. — Preceding unsigned comment added by Widener (talk • contribs) 11:30, 7 April 2011 (UTC)

I don't understand this argument. That $c_{j}$ approaches π follows from $x_{j}\to \pi$, not the other way around. -- Meni Rosenfeld (talk) 12:10, 7 April 2011 (UTC)

Well, thanks for your help Meni. I'll have to think about this some more. Widener (talk) 22:12, 7 April 2011 (UTC)

Graph Ramsey number

I am trying to establish the following bound given as an exercise in my book: $r(K_{m}+{\overline {K_{n}}},K_{p}+{\overline {K_{q}}})\leq {\tbinom {m+p-1}{m}}n+{\tbinom {m+p-1}{p}}q$. Here r(G,H) denotes the Graph Ramsey number of G and H. Can anyone suggest a way to prove this? Thanks-Shahab (talk) 11:34, 7 April 2011 (UTC)

Shahab, I don't know if you're still interested in the answer to this question, but if you are, can you please clarify the notation? I am interpreting $r(K_{m}+{\overline {K_{n}}},K_{p}+{\overline {K_{q}}})$ to mean the smallest integer k such that if the edges of $K_{k}$ are colored red and blue, then there is either a red $K_{m}$ and a vertex-disjoint red ${\overline {K_{n}}}$, or a blue $K_{p}$ and a vertex-disjoint blue ${\overline {K_{q}}}$. Is that right? If so, what is meant by a red ${\overline {K_{n}}}$? I would interpret that to refer to simply any collection of n vertices, regardless of how the edges between them are colored. If that's the right interpretation, then it seems to me that you can say $r(K_{m}+{\overline {K_{n}}},K_{p}+{\overline {K_{q}}})\leq {\tbinom {m+p+2}{m-1}}+\max\{n,q\}$, since we know $r(K_{m},K_{p})\leq {\tbinom {m+p+2}{m-1}}$, and then you just add on a few more vertices that you don't really care about in order to take care of the ${\overline {K_{n}}}$ or ${\overline {K_{q}}}$. On the other hand, if a red ${\overline {K_{n}}}$ is supposed to mean a blue $K_{n}$, then I don't think $r(K_{m}+{\overline {K_{n}}},K_{p}+{\overline {K_{q}}})$ is well defined, since a complete graph of any order with all edges colored red has neither (a red $K_{m}$ and a blue $K_{n}$) nor (a blue $K_{p}$ and a red $K_{q}$). Or perhaps there is a third possible meaning that I am overlooking. —Bkell (talk) 09:21, 10 April 2011 (UTC)

Ah, in fact, if a red ${\overline {K_{n}}}$ just means any n vertices, then we can say $r(K_{m}+{\overline {K_{n}}},K_{p}+{\overline {K_{q}}})\leq \max\{{\tbinom {m+p+2}{m-1}},m+n,p+q\}$: If ${\tbinom {m+p+2}{m-1}}$ is at least as large as $m+n$ and $p+q$, then after we find a red $K_{m}$ or a blue $K_{p}$ in it there will be enough remaining vertices for the red ${\overline {K_{n}}}$ or the blue ${\overline {K_{q}}}$, and otherwise we just need to add a few extra vertices to make up the difference. —Bkell (talk) 19:28, 10 April 2011 (UTC)

Curvature

On reading about the curvature of a complex function $f(z)$ I have hit a section that I don't understand and was hoping that someone here may be able to help me out. Essentially, the text I'm reading says that to evaluate the curvature of $f(z)$, we write $f(z)$ in terms of the angle $\phi =arg(z)$ to give a representation $\mathbf {x} =\mathbf {x} (\phi )$ of the curve $f(C_{r})$, where $C_{r}$ is the circle radius r, centred at the origin, in terms of $\phi$:

$x(\phi )=Re[f(z(\phi ))]=Re[f(\phi )]$

$y(\phi )=Im[f(z(\phi ))]=Im[f(\phi )]$

where $\mathbf {x} (\phi )=(x(\phi ),y(\phi ))$.

From this, I am meant to show that the curvature |k| is given by

$|k|={\frac {|\mathbf {x'} \times \mathbf {x''} |}{{|\mathbf {x'} |}^{3}}}$

where

$\mathbf {x'} =(Re[{\frac {df(z(\phi ))}{d\phi }}],Im[{\frac {df(z(\phi ))}{d\phi }}])$

$\mathbf {x''} =(Re[{\frac {d^{2}f(z(\phi ))}{d{\phi }^{2}}}],Im[{\frac {d^{2}f(z(\phi ))}{d{\phi }^{2}}}])$

The reason I am confused is because I am unsure as to how to evaluate the expression for mod k. Somehow I am taking the cross product of two vectors that I am explicitly told are two dimensional. What meaning does this have and how do I go about computing mod k? Thanks. asyndeton talk 14:20, 7 April 2011 (UTC)

They must mean (a,b)×(c,d) = ad-bc. I'm sure I have seen this two-dimensional usage somewhere, though our cross product article does not mention it. Intuitively one can just imagine that there were a third dimension to take an ordinary cross product in, and then note that when the factors are in the xy-plane, there is only one nontrivial component of the product. (More abstractly, we can identify the 3D cross product with the antisymmetric tensor product -- up to vector space isomorphism -- and generalize from there).

In the complex plane, this works out to $z\times w={\tfrac {1}{2i}}({\bar {z}}w-{\bar {w}}z)\in \mathbb {R}$, for whatever that is worth. –Henning Makholm (talk) 16:49, 7 April 2011 (UTC)

OK, that does seem to be the best, and probably only, way of interpreting it. Thanks for the help. Any suggestions on how I might go about showing the following?

$|k|={\frac {|\mathbf {x'} \times \mathbf {x''} |}{{|\mathbf {x'} |}^{3}}}$

I've tried plunging into the algebra and that hasn't helped. Thanks. asyndeton talk 21:49, 7 April 2011 (UTC)

A fairly extensive derivation is given here [1]. If you compare equation 13 to your equation, it should be fairly easy where to go from there.

I've seen that derivation before but the problem is that I'm unsure how to unify the fact that in my problem, x and y are parameterised by $\phi$, which is also the tangential angle, but in that derivation they are parameterised by t and the tangential angle is still $\phi$. Since they also compute the derivative ${\frac {d\phi }{dt}}$, this is certainly not some simple change of variable from t to $\phi$. How do I rectify the two? asyndeton talk 14:02, 8 April 2011 (UTC)

The curvature measures the angular velocity of the tangent vector. I think that's what dφ/dt means. You have a curve parametrised by t; which we can assume to be arc length. Now, take a fixed unit vector in the plane, say v. By definition the dot product of v with the unit tangent vector T is the cosine of the angle between v and T. That is, v⋅T(t) = cos(φ(t)). Differentiate this identity with respect to t to give

${\mathbf {v} }\cdot \left[\kappa (t){\mathbf {N} }(t)\right]=-{\dot {\varphi }}(t)\sin(\varphi (t))\,.$

Since cos(φ+π/2) = −sin(φ) for all φ and N is obtained from T by a rotation through π/2 radians; it follows that

$\kappa (t)={\dot {\varphi }}(t)\,,$

which means that the curvature measures the angular velocity of T. (That was an answer to what I thought was the question. If not, then run it by me again and see if I get get it the second time.) Fly by Night on Tour (talk) 18:27, 8 April 2011 (UTC)

P.S. The most geometric characterisation of curvature is terms of the radius of curvature. Away from inflection points, there is a unique circle having at least three point contact with the curve at that point, the so-called osculating circle. (For more than three point contact, see vertex point.) The radius of this circle is the radius of curvature, and the reciprocal of the radius of curvature is the curvature. At inflections, a curve has at least three point contact with its tangent line. You can think of that as the osculating circle whose centre has "gone to infinity". That means the radius of curvature is infinite, and the curvature is zero. Fly by Night on Tour (talk) 18:40, 8 April 2011 (UTC)

Thank you for trying but I don't think that's answered my question. Your answer still uses the parameter t, with $\phi$ as a function of t. My question is simply how to modify the derivation of the curvature given here [2] so that I have x$(\phi )$ and $y(\phi )$, as my question demands in the above extract from it, instead of $x(t)$ and $y(t)$, which mathworld uses. The reason I can't just do it it myself is because it is more subtle than a straight change of variable from to $\phi$ due to the presence of terms such as ${\frac {d\phi }{dt}}$ in the mathworld derivation, which I don't know how to amend. asyndeton talk 22:19, 8 April 2011 (UTC)

MathWorld says that φ is the tangent slope, so it's just the same as my φ. I'm not sure it's possible, or useful to do it the other way. For example, consider the curve y = sin(x). Here –1 ≤ φ ≤ 1, and there are infinitely many curve points for each choice of φ. In this case they have the same curvatures because of the symmetry, but in general they won't. You'll end up with a one-to-many function from "tangent slope" to curve curvature. In case of a complex curve, consider z(θ) = θ·e^i·h(θ). This will be a one-to-infinity function. It's a really interesting problem; don't be me wrong. I just don't think the answer is too simple. Fly by Night on Tour (talk) 23:07, 8 April 2011 (UTC)

your phi is not the same as the phi in the mathworld derivation, yours is merely an input parameter. Your phi corresponds to the mathworld t, and the mathworld phi corresponds to the radius of curvature. You'll notice that if you write f[φ]=x[φ]+iy[φ] where x and y are real functions with real arguments, and then change all of your φ to t, without loss of generality you can reproduce the mathworld derivation.

That would make sense but I don't see at what point, in my initial question, I begin writing a $\phi$ that can be replaced by a t. It all seems to flow pretty soundly with, in my opinion at least, no obvious place where I can just start to write t. Could you identify for me the point at which I can start writing t instead of $\phi$ and why not earlier? asyndeton talk 12:02, 9 April 2011 (UTC)

What's so difficult about replacing all of your phi's with t's? Variable names are interchangeable; conventions for what to use when are just conventions. –Henning Makholm (talk) 14:28, 9 April 2011 (UTC)

Alernatively, keep your phis as they are, and instead rename Mathworld's phi to theta and Mathworld's t to phi. –Henning Makholm (talk) 14:36, 9 April 2011 (UTC)

The problem is that it won't work. Both the derivation given on mathworld and the above extract from my problem discuss $\phi$ as an angle but then my problem proceeds to use some $\phi$, which appears to me to be indistinguishable from the angle $\phi$, as a parameterisation of the curve whereas mathworld uses t. If I just replace all of my '$\phi$'s with 't's, I will end up with exactly the same problem: mathworld is using two variables where I am only using one. asyndeton talk 14:41, 9 April 2011 (UTC)

Your phi has nothing to do with Mathworld's one. Your phi is a parameter, the exact same thing as mathworld's t. Mathworld's phi is the direction of the tangent to the curve that is traced out as a function of your phi. The only thing they have in common is that you happened to choose the same Greek letter for one thing as Mathworld happens to use for another thing. More precisely,

$\phi _{\text{mathworld}}={\text{arg}}\left({\frac {d}{d\phi _{\text{asyndeton}}}}f(z(\phi _{\text{asyndeton}}))\right)$

–Henning Makholm (talk) 14:51, 9 April 2011 (UTC)

OK, this is going to take a bit of getting my head around. Thank you for your help and patience and please excuse my ignorance. asyndeton talk 14:16, 10 April 2011 (UTC)

More on "triangular graphs"

Following up on one of my questions above: We considered the set of all 15 unordered pairs of members of the 6-member set {a, b, c, d, e, f}. Regard each such pair as a vertex of a graph, and put an edge between any two pairs that have a member in common. Thus, each vertex has eight neighbors. This gets us the graph labeled T₆ at this page. Or so that page seems to tell us. My question: Is there an efficient way to match the unordered pairs with the vertices in the picture on that page? Michael Hardy (talk) 19:24, 7 April 2011 (UTC)

I suppose yes. Here's an attempt. First, choose a triangle from the picture and call its vertices {a,b}, {a,c}, {b,c} (you can do this for symmetry). Secondly find all vertices that are common neighbours of {a,b} and {a,c} but are not equal to {b,c}, these must be {a,d}, {a,e}, … {a,k}, and you can assign the names for them in any way to choose for symmetry again. Thirdly take any vertex not named yet (so it can't have a as a member), and find the two of its neighbours that have a as a member: if these are eg. {a,x} and {a,y} then you must call that vertex {x,y}. Finally verify that you've got a graph isomorphism. – b_jonas 09:06, 8 April 2011 (UTC)

On second thought, this simple method won't work for the picture of T_7 on that page because you can't make out the edges. – b_jonas 09:08, 8 April 2011 (UTC)

I see. I guess I was being lazy this time. It's not clear that that gives a felicitous way of associating alphabetical order with the way the graph is laid out, nor even that such a thing is possible, but I can reshuffle the labels afterwards.

Thanks. Michael Hardy (talk) 12:48, 8 April 2011 (UTC)

"Straight" line on a curved surface

Hi, say I have an arbitrary (sufficiently well-behaved) surface in 3D space. I want draw a line on it such that, for every point P on the line, the projection of the line onto a plane tangent to the surface at P is "as straight as possible" -- perhaps in the sense that the curvature is zero, or second derivative is zero, or something like that. Is this the same thing as a geodesic? 86.181.204.196 (talk) 22:03, 7 April 2011 (UTC).

I found this question confusing until I realized that by "line" you didn't mean "straight line", but rather a curve.

Differential geometry is something I'm only marginally acquainted with, and your specifications (perhaps in the sense that the curvature is zero, or....) are not too precise, but it does sound as if you're talking about geodesics. But don't take my comment as the final word. Michael Hardy (talk) 03:12, 8 April 2011 (UTC)

Later I came across this from Mathworld: "The normal vector to any point of a geodesic arc lies along the normal to a surface at that point." I think this is describing the same "straight as possible" property as I have in mind?? 86.181.206.30 (talk) 11:19, 8 April 2011 (UTC)

Think of the problem the other way: if you project a straight line through the origin of the tangent plane at a point, you get a curve on the surface. Hence "as straight as possible" can only mean "is a straight line" (in the tangent plane). But these will not, in general, be geodesics. An ellipsoid is a counterexample. Sławomir Biały (talk) 14:28, 8 April 2011 (UTC)

That's if you're only looking at one P at a time. If instead we ask for "the strongest approximation to "straight" that one can reasonably require to hold at all P for a general embedded surface", then I think the only answer must be that the curve is a geodesic, satisfying the property that its projection onto the tangent plane at any point of the curve has curvature 0 at the tangent point. (But that's just geometric intuition speaking -- caveat emptor!) –Henning Makholm (talk) 16:56, 8 April 2011 (UTC)

Ah, I thought the question was about a particular point. If we impose that the curvature of the projection into each tangent plane is zero, then the curve is a geodesic since by introducing an orthogonal basis adapted to the tangent space, it's easy to see that the normal to the curve must be normal to the tangent plane (by Frenet-Serret). So that is the right condition. Sławomir Biały (talk) 17:05, 8 April 2011 (UTC)

I think your idea of "as straight as possible" relates to geodesic curvature. Geodesics are curves in a surface for which the geodesic curvature is zero at each point. (Such points are also known as geodesic inflections.) Fly by Night on Tour (talk) 18:59, 8 April 2011 (UTC)

I think the geodesic curvature at P is the curvature of the projection of the curve onto the tangent plane at P. Our article doesn't seem to mention this very intuitive characterization, though. Sławomir Biały (talk) 19:24, 8 April 2011 (UTC)

Yeah, you're right there. Maybe we should add something. In fact, I noticed that all of the elementary differential geometry of plane curves, space curves and surfaces is lacking a lot. I don't even think that plane curves have their own coverage. There defiantly isn't any geometrical motivation either. Shall we mention this at the mathematics project page? Fly by Night on Tour (talk) 22:41, 8 April 2011 (UTC)