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April 26

Wedge product

I am having difficulty with exterior products and differential forms. I have an expression like $\omega =dx_{1}\wedge dx_{3}\wedge dx_{9}$ and want to express this in terms of $dy_{1}\wedge \ldots \wedge dy_{k}$ and I have a linear relationship $dx_{i}=\sum M_{ij}dy_{j}$ where $M$ is a matrix. So in this case I have $\omega =\left(\sum M_{1j}dy_{j}\right)\wedge \left(\sum M_{3j}dy_{j}\right)\wedge \left(\sum M_{9j}dy_{j}\right)$. But I'm having difficulty expressing this in a nice terse way. Does Spivak deal with this? If so, I don't see it. Can anyone point to a better way of thinking about this? Thanks, Robinh (talk) 05:35, 26 April 2019 (UTC)

First, ω is a 3-form and dy₁∧...∧dy_k is a k-form, so unless k=3 they don't live in the same space. I'm assuming you want ω as an expression in dy_p∧dy_q∧dy_r, 1≤p<q<r≤k, which would have Choose(k, 3) terms and each coefficient would be a 3×3 determinant in the M_ij's. Not very terse so maybe you're better off leaving it as dx₁∧dx₂∧dx₃. --RDBury (talk) 18:17, 26 April 2019 (UTC)

(OP) thanks for this. The problem is, there are a lot of coefficients, each one of which is a pretty complicated expression. And things get worse if I have something like $dx_{1}\wedge dx_{3}\wedge dx_{9}+2dx_{2}\wedge dx_{4}\wedge dx_{5}$. All of which seems like an awful lot of work for what is conceptually very simple (in my case an orthogonal transformation of my coordinate system), which made me think I've missed something obvious. Maybe not though! Robinh (talk) 19:34, 26 April 2019 (UTC)

Given a linear transformation F from V to W there is a corresponding transformation ∧^k(F) from ∧^k(V) to ∧^k(W). (See Exterior algebra#Functoriality.) As you say this isn't hard conceptually but I think it would be hard to write it out in terms of coordinates. The matrix for ∧^k(F) would be Choose(m, k) by Choose(n, k) and I'm guessing each entry would be a k×k determinant. If F was a transformation of V and you picked a basis so that the matrix of F was diagonal then I imagine that things would simplify considerably. I'm not up on spectral theory though, so I'm not sure what the status is of orthogonal matrices with respect to being diagonalizable, Spectral theorem may help with that. --RDBury (talk) 23:56, 26 April 2019 (UTC)

Every orthogonal matrix is diagonalizable with unit-modulus eigenvalues. If M is diagonal, the sums above become considerably simpler.--Jasper Deng (talk) 08:07, 29 April 2019 (UTC)

Thanks for this Jasper. How does diagonalizability help here? If my matrix is $M=P^{-1}DP$ where 'D' is diagonal, how does this help me? Is 'P' of some form that differential substitution becomes simpler? Robinh (talk) 03:37, 30 April 2019 (UTC)

I meant if M is itself a diagonal matrix with respect to these bases of the dual spaces (i.e. spaces of 1-forms), in which case $dx_{i}=\lambda _{i}dy_{i}$.

It may also in general be useful to think of the exterior product as a quotient (by the two-sided ideal $x\otimes x$ for all x in the original tensor product space) of the tensor product of several copies of the dual space. The extension of the linear mapping induced by M to the tensor product is much more straightforward and you can pass to the quotient.--Jasper Deng (talk) 05:59, 30 April 2019 (UTC)

April 27

prime numerators and denominators.

Is there a name for fractions in which the numerator and denominator are both prime? 1/2, 3/7, 11/23, etc. 87.115.27.55 (talk) 09:08, 27 April 2019 (UTC)

I don't know, but regarding your first example, 1 is not prime. CodeTalker (talk) 15:08, 27 April 2019 (UTC)

If they are relatively prime then it is an Irreducible fraction. Bubba73 ^{You talkin' to me?} 01:38, 28 April 2019 (UTC)

April 29

difference in minimum circle for regular polyhedra vs complete coverage for same polyhedra.

Yeah, its a long title and would love a replacement...

For each of the 5 regular polyhedra, what is the difference in the radius of the smallest cylinder that it can pass through vs. the radius of the smallest cylinder that it can pass through in any orientation. For example for the cube, the smallest cylinder would have 4 parallel sides all on the edge of the cylinder, but a much larger cylinder would be needed if it went through with line through the opposite corner being parallel to the edge of the cylinder.Naraht (talk) 13:15, 29 April 2019 (UTC)

For those with central symmetry (i.e. excluding the simplex) I guess it's the difference between the circumradius (marked by a vertex) and the midradius (marked by the centre of an edge). —Tamfang (talk) 03:44, 30 April 2019 (UTC)

TamfangThanx. I'd put together the circumradius, but for the mid radius, do you end up with a full set of parallel edges able to go through the "straw" the way that you can with a cube?Naraht (talk) 14:08, 2 May 2019 (UTC)

April 30

Distance between two curves

If I have 2 arbitrary but smooth curves (equations would exist), how can I determine a point on each curve that results in the minimum distance between the two points.

If I knew one point, I could calculate the distance to each point on the other curve and look for a minimum. But I don't see how to do it with two variables.

For the purposes of this question, assume two circles with different centres and radii that do not intersect and do NOT use the fact that we can determine the points by constructing a line between the two centres.

Thanks -- SGBailey (talk) 10:44, 30 April 2019 (UTC)

• I will answer the question assuming you are new to optimization problems, as I guess from your assertion that you would know how to solve the problem if there was only one variable but not with two. Please accept my apologies and provide a tad more detail if that is not the case.

Denoting by $d(x,y)$ the distance between the point at coordinate x on the first curve and y on the second curve, the question is to find the minimum of that function of two variables. If we ignore for a moment that the function to minimize has a geometrical definition, we are facing a standard optimization problem. An overly simplified summary of the field is that it is extremely easy to find a minimum even for pathological functions but extremely hard to ensure it is not a local minimum even for well-behaved ones.

For an example of an optimization algorithm, see our decent article about gradient descent. For code libraries, Python's scipy.optimize implements all common optimization algorithms.

Now: can we take advantage of the additional information on d to make the optimization run faster/better? Probably. Is it worth it? You tell us, depending on runtime constraints and the like. Tigraan^{Click here to contact me} 12:51, 30 April 2019 (UTC)

• • The standard tool for problems of this sort is the method of Lagrange multipliers, particularly with multiple constraints: Lagrange_multiplier#Multiple_constraints. There the equation of one curve (involving variables with one set of names, like x and y) is one constraint, the equation of the second curve (involving an independent set of variables, like w and z) is the other constraint, and the distance (a function of all four variables) is the quantity being minimized. Often for distances it is easier to use the square of the distance. This reduces the problem to pure algebra, but it may or may not be tractable depending on your curves.
  • In some cases, it is also possible to treat this as two chained single-variable optimization problems: first optimize with respect to one variable, treating the other as a constant, then substitute and optimize with respect to the remaining variable. --JBL (talk) 13:26, 30 April 2019 (UTC)

In this case you can simplify things a bit by assuming that the tangent lines at the two points are parallel and both lines are perpendicular to the line connecting the two points. (Pretty sure Lagrange multipliers will tell you this eventually.) Note that the same condition applies to pairs of points at maximal distance and various local extrema, so you need another step to find which pair of points is the one you want. According to my calculations the resulting system would have 4 (non-linear) equations and 4 unknowns so working it out algebraically might be tricky. --RDBury (talk) 17:23, 30 April 2019 (UTC)

More specifically, if the curves are given by f(x, y) = 0 and g(x, y) = 0 then the points (x₁, y₁), (y₂, x₂) satisfy the simultaneous equations

$f(x_{1},y_{1})=g(x_{2},y_{2})={\begin{vmatrix}f_{x}(x_{1},y_{1})&f_{y}(x_{1},y_{1})&0\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\\end{vmatrix}}={\begin{vmatrix}g_{x}(x_{2},y_{2})&g_{y}(x_{2},y_{2})&0\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\\end{vmatrix}}=0$

--RDBury (talk) 18:49, 30 April 2019 (UTC)

We can simply assume that these two curves are specified parametrically as $\mathbf {x} _{1}(s_{1})$ and $\mathbf {x} _{2}(s_{2})$. Then the distance squared is

$D^{2}=(\mathbf {x} _{1}(s_{1})-\mathbf {x} _{2}(s_{2}))^{2}$.

Now stationary points can be found from the system

$\left\{{\begin{array}{lcr}{\frac {\partial D^{2}}{\partial s_{1}}}&=&2(\mathbf {x} _{1}(s_{1})-\mathbf {x} _{2}(s_{2})){\frac {d\mathbf {x} _{1}(s_{1})}{ds_{1}}}=0\\{\frac {\partial D^{2}}{\partial s_{2}}}&=&-2(\mathbf {x} _{1}(s_{1})-\mathbf {x} _{2}(s_{2})){\frac {d\mathbf {x} _{2}(s_{2})}{ds_{2}}}=0\end{array}}\right.$

Ruslik_Zero 19:28, 30 April 2019 (UTC)

I took the liberty of fixing your parentheses, but otherwise that's an analytic statement of the geometrical conditions on the tangent lines above, though I probably would have used a ⋅ to indicate dot product. I was thinking about it more geometrically but it works out nicer the way you did it. I should have mentioned it above but the cases where both points are the same point of intersection of the curves appear as solutions to the equations. The distance is then 0 but I'm not sure that's what the OP had in mind. --RDBury (talk) 13:09, 1 May 2019 (UTC)

Thx all. -- SGBailey (talk) 11:11, 2 May 2019 (UTC)

• @SGBailey and Tigraan: As an example for difficult judgement on simple curves let's consider a circle within an ellipse (or vice versa), being almost concentric. A small change in their relative position can switch the global minimum from one apex (a local minimum) to another.

Two concentric circles present another difficulty: they're equally distant from one another everywhere, but you (i.e., your algorithm) will never know if they were actually analyzed 'everywhere'; it's always possible there is some elbow on one of them, which was not detected bit which will make a sharp minimum. --CiaPan (talk) 20:48, 6 May 2019 (UTC)

???? The hypothesis of the question includes (in the very first sentence) the statement that equations for the curves are known. This could mean different things (I took it to mean an implicit equation, Ruslik took it to mean parametric equations), but all of them confer absolute knowledge of the entire curve. And there are three or four complete correct answers above your post. --JBL (talk) 21:57, 6 May 2019 (UTC)

May 2

Does anybody here know, what Skolem arithmetic (i.e. the first order theory of multiplication of natural numbers) looks like?

185.46.78.56 (talk) 08:04, 2 May 2019 (UTC)

The same user [[User:185.46.78.56]|185.46.78.56]]] ([[User talk:185.46.78.56]|talk]] · [[Special:Contribs/185.46.78.56]|contribs]]) asked the same question on my page. I must admit that I do not understand the question. I mostly know what is written in the wikipedia page. I also know how to decide whether a sentence holds or not/whether a formula is satisfiable, but it does not seems to answer the question. Thus I'd like to have more precision about what «looks like» mean. Arthur MILCHIOR (talk) 12:59, 2 May 2019 (UTC)

I meant, what are the axioms of Skolem arithmetic? Are they, all Peano's axioms, excluding those that contain the "+" symbol? Additionally, is the axiomatic system of Skolem arithmetic - finite? Is it complete? Unfortunately, the page about this arithmetic contains no hint on that... 185.46.78.56 (talk) 13:03, 2 May 2019 (UTC)

I suggest that you also post this question in the talk page of the article, instead of writing it in my talk page. Since you asked the question to me precisely, I'll answer. I've never seen Skolem arithmetic used in proof theory. I've only seen it used in model theory. Which means that I don't know whether anyone have given a set of axiom for this logic. Each time I've seen «Skolem arithmetic», it was always assumed that we all know what the natural numbers are, and that we wanted to study what can or can not be done with multiplication alone. Note that I used to be a model theorist, which means that I was strongly biased in the article I did read. However, a quick google search let me think that there are indeed no axiomatization of this logic. I should emphasize that Peano's axiom for multiplication uses the addition symbol. Thus you can't just omit the axioms of addition and have a consistent theory. What I would do, if I were to give an axiomatization, is to consider that a number $n$ is an infinite sequence $(a_{i})$ with finite support, such that $n=\prod _{i}p_{i}^{a_{i}}$, with $p_{i}$ the primes. Then I'll introduce a symbol $P_{p_{i}}$ for each prime number, and repeat the axiom of addition for each prime number. That is: $P_{p_{i}}(n)=P_{p_{i}}(m)\implies n=m$, $P_{p_{i}}(n)\neq 1$, $n\cdot 1=n$, $P_{p_{i}}(n\cdot m)=n\cdot P_{p_{i}}(m)$. But it leads to an infinite number of axiom and does not even allow to have anything similar to induction (because, in order to do induction correctly, your hypothesis would need to consider every single $P_{p_{i}}$, instead of considering only the successor symbol. Note that, I've got no citation for what I've just wrote. So it can't be used in an article. I hope at least it'll help your research. Arthur MILCHIOR (talk) 03:48, 3 May 2019 (UTC)

Thank you so much. I've adopted your suggestion and posted my new questions on the article talk page as well.

I wonder if Skolem himself wrote down any axiomatization for his arithmetic, unless he only discussed it from a model theorist's viewpoint - like you; In my view, the latter possibility does not seem to be reasonable, because he was a well known theorist of axiomatizations as well (who also contributed his "foundation axiom" to set theory, for example). I also wonder whether Skolem arithmetic can be axiomatized by a complete system, just as Presburger arithmetic (having its induction axiom scheme) can. Anyway, I'm very grateful for your important comments on this issue. 185.46.78.67 (talk) 07:08, 3 May 2019 (UTC)

I'm not an expert on model theory, but from a purely algebraic point of view we're talking about a free commutative monoid with countably infinite generators. Pretty sure the fundamental theorem of arithmetic would be one of the axioms. --RDBury (talk) 15:54, 3 May 2019 (UTC)

It says in the article that Skolem arithmetic is decidable, probably by quantifier arithmetic like Presburger arithmetic. It looks like it is just the multiplicative fragment of Peano arithmetic, while Presburger arithmetic is the additive fragment. It apparently shows up in complexity theory, which I hadn't seen before arXiv:1504.04181). 67.164.113.165 (talk) 09:15, 6 May 2019 (UTC)

But the multiplicative fragment of Peano arithmetic contains one axiom only: 0X=0. Could Skolem call it "arithmetic"? 185.46.78.26 (talk) 10:12, 6 May 2019 (UTC)

Peano arithmetic means the whole theory (i.e. including all the theorems, not just the axioms), and similarly for its fragments. The multiplicative fragment is basically all the sentences in the theory that don't use addition. Even the axioms (PA has some infinite axiom schemas) have a subset that don't use addition. But you should probably try a web search for something with a formal definition of Skolem arithmetic, if that is what you are looking for. I don't remember if you're the OP who asked last week about splitting PA into two decidable fragments that when combined give an undecidable theory, but this would seem to be an example. The person around here who really understood this stuff was CBM, but he seems to have given up on us :(. 67.164.113.165 (talk) 21:54, 6 May 2019 (UTC)

All axioms in the scheme you've mentioned, do use addition. As far as I know, PA has no axioms that don't use addition, except the axiom 0X=0, so I wonder how Skolem could build his multiplicative arithmetic using the multiplicative fragment of Peano arithmetic only. Unless he was only interested in the multiplicative theorems of Peano, and allowed to use additive axioms, but then I wonder if the name "multiplicative arithmetic" is justifiable (Yes I am the OP who asked last week about splitting PA into two complete fragments that when combined give an incomplete theory, but I didn't understand how this can be helpful). 185.46.78.64 (talk) 20:15, 7 May 2019 (UTC)

May 4

Flatness of a transfer function

In terms of a transfer function, what conditions must be satisfied to ensure the output amplitude is maximally flat. Not a homework question . I'm just curious.86.8.201.210 (talk) 23:28, 4 May 2019 (UTC)

Our Butterworth filter article, particularly the maximal flatness section, might (or might not) give some clues.catslash (talk) 14:53, 5 May 2019 (UTC)

May 7

Are there "interesting" theorems, that only use the addition operation, and that are only provable by induction?

Besides the well known "interesting" theorem which claims, that every natural number not being any sum of two identical natural numbers, is followed by a successor which is such a sum. 185.46.78.64 (talk) 20:28, 7 May 2019 (UTC)