User andreas rüdinger - MathOverflow

Answer by Andreas Rüdinger for Newton equations, second order equation and (im)possible motions

2013-03-26T22:10:28Z

Regarding your question "What would be allowed additionally if the equations were of 3. or higher order?" I would like to mention that the equation you get in classical physics if you consider the effect of an electron's own electromagnetic on itself involves a time derivative of the acceleration, cf. e.g. equation (1) in http://www.philsoc.org/1962Spring/1526transcript.html, and especially remark (III). If I remember it right, more can find in the standard book on classical electrodynamics, Jackson, Classical Electrodynamics.

Answer by Andreas Rüdinger for sequences with a fractal dimension

2013-01-06T21:09:46Z

Interestingly, fractal dimensions of the human DNA sequence have been analyzed and different embeddings considered: http://biocomplexity.indiana.edu/jglazier/docs/papers/20_DNA_Analysis.pdf.

Answer by Andreas Rüdinger for Does Physics need non-analytic smooth functions?

2012-11-27T21:29:36Z

One example seems for me important and obvious, so I'm wondering, why no one has posted it so far: The time evolution of the state vector in quantum mechanics (Schrödinger's equation) is $|\psi(t)\rangle = \exp (-i \frac{H t}{\hbar}) \quad |\psi(0) \rangle$ and thus non-analytic in Plancks constant $\hbar$. This has important consequences in semiclassics ($\hbar \to 0$).

Minimal basis of set of positive integers

2010-05-30T11:16:50Z

Let $A$ be a set of positive integers and $A+A = \{a_1 + a_2 | a_1,a_2 \in A \}$. If $A+A$ contains all positive integers, $A$ is called a basis (of order 2) of the set of positive integers. A basis $A$ is called a minimal basis, if no proper subset of $A$ is a basis.

E.g. the set of all numbers with only "0"s and "1"s in ternary representation is a (even minimal) basis. This is somehow the discrete analogon to the well known fact that the sumset of two Cantor ternary sets is the closed interval [0,2].

I'm looking for a 'smallest' minimal basis of the set of positive integers ('smallest' in the sense of the lexikographic order - also other orders would be interesting, e.g. the order induced by sum of inverse squares of the elements of the basis).

There are a lot of articles about minimal asymptotic basis of the set of positive integers ("asymptotic" meaning that only every \textit{sufficiently large} natural number is the sum of two elements of the basis). But I'm struggling to get or find my question above answered.

ADDENDUM: Sorry for not having been clear, and thank you for your comments. I hope the following remarks will clarify things a bit: a) Yes, I consider only order 2. b) To make my intention clearer, lets consider the finite set $M_n = \{0, 1, 2, 3, \ldots, n-1\}$ and ask for a basis $A$ such that $M_n \subset A + A$. E.g. $A = \{0,1,3,4,9,10,12,13\}$ is a basis of $M_{27}$ (this is just the ternary construction mentionned above). In the finite case I'm looking for the basis with the smallest number of elements and in case of a tie (same number of elements), I prefer the basis which comes later in lexicographic order (i.e. I would prefer $\{0,1,3,4,10\}$ to $\{0,1,3,4,9\}$. My intention with the question was to ask this problem not for finite $M_n$, but for $M_{\infty} = \mathbb{N}$ (sorry also for the confusion whether to include "0" or not). Since the cardinality of the "ternary basis" scales with $n^{2/3}$ and the cardinality ofthe proposed basis of odd numbers is $n/2$, I regard the "ternary basis" as a better one. Perhaps the problem is still not well defined and/or there is no such "best" basis. This would also be helpful for me to know.

Social Reading Platform for Math or LaTeX texts

2010-12-05T17:47:38Z

Social reading is considered to be one of the big trends that could be catalysing learning by reading. Features could include:

Highlighting or annotating paragraphs or single steps in a proof for yourself (ok, this is not yet social reading), e.g. in a proof you could add to the sentence "Therefore we have ..." of the original work a more detailled explanation, thus "zooming in".
Publishing those of your own annotations you consider helpful for others
Reading all or a selection of the other reader's published annotations via a "show helpful annotations" button
Thumbing up other's annotations.
Asking questions to specific paragraphs or steps in a proof.
Answering questions asked by others and thumbing up questions and answers.
The original text would always keep the same, only being annotated

Thus all the insights or questions you have while reading a paper or a textbook could be shared with others and learning could be far more efficient. Authors of textbooks could take into account the annotations/questions/answers, thus optimizing their text in future editions (or developing different versions or an annotated version).

Questions: a) Is there any tool that can provide these features, e.g. LaTeX-tools or pdf-tools? b) Would the Math overflow be a good starting point to build such a platform? c) Does anybody know of such a platform that is already in place?

(Edit: Changed to community wiki)

Example for column rank $\neq$ row rank

2010-11-25T19:51:21Z

The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank $\neq$ row rank. i.e. can one find a $2 \times 3$-(block)matrix with real $2\times 2$-matrices as elements, which has different column and row ranks?

Answer by Andreas Rüdinger for fibonacci series mod a number

2010-11-07T17:19:14Z

Perhaps Elsenhans, Jahnel, "The Fibonacci sequence modulo $p^2$ – An investigation by computer for $p < 10^{14}$" http://www.uni-math.gwdg.de/tschinkel/gauss/Fibon.pdf will be interesting for you. There are sections about the algorithm.

Noninteger iterates of functions: How to get ODE from flow at a given time?

2010-05-30T15:47:55Z

Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, you get as solution the flow $\Phi(x_0,t):=x(t)$. To give a trivial example: If $f(x)=x$, then $\Phi(x_0,t)=x_0 \exp(t)$.

Now, I'm not interested in the trajectories for a given initial condition, that is in $\Phi(x_0,t)$ with $x_0$ fixed and $t$ variable; but in the map $x_0 -> \Phi(x_0, t)$ for a fixed $t$ (say $t=1$).

Given the function $f$, you can easily (at least in principle, by solving the ODE) get the function $\Phi(\cdot, 1)$. There are a lot of theorems about existence and uniqueness of this problem and analytical and numerical algorithms.

But how can one get $f$ out of $\Phi(\cdot, 1)$? Is this a well posed problem? Are there any theorems?

This problem is closely related to "interpolating" the $n$-fold functional iterates of $g$ (with $g^{[0]} = \mathrm{Id}, g^{[1]} = g, g^{[2]} = g \circ g, g^{[n+m]} = g^{[n]} \circ g^{[m]}$ for $n,m \in \mathbb{N}$) from $n \in \mathbb{N}$ to real values. If such an interpolation succeeds, on can get the ODE out of the flow $\Phi(\cdot, 1)$ by determining $\Phi(\cdot, 1)^{[\alpha]}$ for small $\alpha >0$. I have done some calculation, that give results, but lack in rigor.

For noninteger iterates of functions, a classical reference is http://www.math-inst.hu/~p_erdos/1960-07.pdf.

Answer by Andreas Rüdinger for Random linear recurrence relations

2010-09-16T19:02:32Z

Please see the paper "Random Fibonacci Sequences" by Clement Sire: http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106457v1.pdf where you find weak and strong disorder expansions and also analytical results. (I worked together with Clement Sire about 15 years ago.)

Graphical representation of mathematical structures (in the spirit of unified modeling language)

2010-08-14T22:33:02Z

In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are about ten diagrams for different views on the system. These diagrams have tremendously improved the ability to construct large systems by large teams, as one can look at the system at different levels and as one can profit from visualization. Furthermore the UML models contains all the system's information needed for implementation.

I'm wondering whether a similar method could be useful for doing (and commuicating about) mathematical structures and theories. Often so many definitions are built one upon the other and so many properties are introduced that it seems to be difficult to have an overview of the "architecture" of a theory.

For example one could have one sort of diagram showing how the mathematical structures are build form each other (e.g. a field built by two groups - with the respective axioms "inherited" - and further "compatibility conditions" between them). In priciple you would be able to track back all structures to the "mother structures" algebraic structure, order structure, topolological structure.) Interestingly the object oriented paradigm used by UML, i.e. encapsulating attributes and methods into classes is somehow similar to the categorical approach in mathematics (encapsulating objects and morphisms).

Another sort of diagram could represent a (part of a) theory by an annotated graph, the nodes/edges of which are the definitions, theorems and proofs and by navigating you see exactly which property/definition is used at which point.

I apologize for the "fuzziness" of the question but I feel a discussion about a sort of visual notation / graphical representation in mathematics could be of interest (perhaps the categorical viewpoint with its unifying force and its diagramms is already what can be achieved, but I think there could be other, complementary ways). Does anybody know of attempts in this direction? Would you consider such a graphical representation of mathematical structures (in addition to the standard, more linear way of representing things) helpful for communication in research and/or in education?

Answer by Andreas Rüdinger for 2- and 3-body problems when gravity is not inverse-square

2010-08-18T18:49:31Z

Regarding question 1):

a) In addition to the potentials for which all bound orbits are closed there are further central force potentials with analytic solutions, please cf.
http://en.wikipedia.org/wiki/Classical_central-force_problem#Central_forces_with_exact_solutions (Sorry, not quite what you were asking for [I read too fast], but perhaps interesting nevertheless.)

b) Perhaps this paper on "Multiparticle systems with a particle-interaction potential homogeneous of degree α = −2" by Borisov et al. is interesting for you: http://ics.org.ru/doc?pdf=1359&dir=e. ("Some new integrable and superintegrable systems generalizing the classical ones are also described.").

Answer by Andreas Rüdinger for Measure 0 sets on the line with Hausdorff dimension 1

2010-08-18T18:31:41Z

Perhaps the following charts "Universal measure zero sets with full Hausdorff dimension" http://mat.fsv.cvut.cz/Zindulka/papers/opava.pdf are of interest for you.

Answer by Andreas Rüdinger for Spectrum of the sum of generators for irrational rotation algebra

2010-08-17T18:59:18Z

Perhaps it is helpful for you to know that you can find papers on $u + v + u^{\dagger} + v^{\dagger}$ also by looking for "Harper equation", "Discrete mathieu equation" or "Hofstadter butterfly".

Here's an example of the butterfly. Hoftstadter found the (rough) structure of the butterfly in 1976 by looking at a model for Bloch electrons (i.e. electrons in a periodic structure) in a magnetic field. The irrationality $\theta$ represents essentially the magnetic flux through a unit cell of the lattice. (As for rational $p/q$ there is a translation symmetry one find $q$ "bloch bands", which touch at $E=0$ for pair $q$.)

I spent a part of my PhD thesis (no math, but renormalization from a more physical/heuristical point of view) on the multifractal properties of the spectrum for irrational values and gave some estimations on the minimal and maximal multifractal dimensions for quadratic irrationalities. If you're interested, here's a paper of mine: http://iopscience.iop.org/0305-4470/30/1/009.

Answer by Andreas Rüdinger for Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice

2010-08-16T20:16:42Z

If you look for "cover time of graph" you will find a lot of references, cf. e.g. "Jonasson, Schramm, ON THE COVER TIME OF PLANAR GRAPHS, Elect. Comm. in Probab. 5 (2000) 85-90, http://www.emis.de/journals/EJP-ECP/_ejpecp/ECP/include/getdocbfb7.pdf. In this paper you find the following result by Zuckermann (for details pls. cf the paper):

If $G$ is a finite portion of a $d$-dimensional lattice $\mathbb{Z}$ with $n$ vertices, the expected cover time (to visit all vertices) is $\Theta(n^2)$ for $d=1$, $\Theta(n (\log n)^2)$ for $d=2$, and $\Theta(n \log n)$ for $d \ge 3$.

Answer by Andreas Rüdinger for What is the right notion of self-dual (two-dimensional) percolation in R^4?

2010-08-04T21:55:42Z

The paper "PLAQUETTES, SPHERES, AND ENTANGLEMENT" by GEOFFREY R. GRIMMETT AND ALEXANDER E. HOLROYD does not deal with the self-dual problem, but nevertheless could be of interest for you. If shows that "The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d − 1)sphere and encloses the origin." Here's the link: http://www.statslab.cam.ac.uk/~grg/papers/sphere8.pdf

Answer by Andreas Rüdinger for A differential equation

2010-08-04T18:13:36Z

This seems to be a Bernoulli differential equation. Please cf. http://en.wikipedia.org/wiki/Bernoulli_differential_equation for the solution (in your case $n= \frac{\gamma}{\gamma-1}$).

Criteria for boundedness of power series

2010-06-04T20:23:13Z

Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$. Can one give necessary and sufficient criteria the sequence of the coefficients $(a_n)$ has to meet in order for $f$ to be bounded on $\mathbb{R}$? (Let's disregard the trivial case that $a_0$ is the only non-zero coefficient and let's call a sequence "function-bounded" if the power series is bounded.) Criteria for boundedness seem to be far more difficult to obtain than the usual criteria for convergence of a power series, here some remarks:

a) A necessary condition for $\sum_n a_n x^n$ to be bounded is that there are infinitely many non-zero coefficients which change sign infinitely many times.

b) The boundedness of $f$ is an "unstable" property of the sequence of coefficients: any non-zero change in any finite subset (except $a_0$) will destroy boundedness. Thus the linear subspace of all function-bounded sequences is rather "sparse" in the vector space of all sequences representing convergent power series.

c) On the other hand, the linear subspace of all function-bounded sequences contains at least all power series of functions that can be written as $\cos \circ h$ with $h$ an entire, real-analytic function, and the algebraic span of these functions. One could conjecture that this space is already the space of all bounded functions that can be represented as power series[EDIT: seems to be refuted, cf. comment below]. And perhaps this could be a starting point for deducing the criteria.

EDIT (conjecture added): Is is true, that every power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $\sum_{n=0}^{\infty} \epsilon_n a_n x^n, \quad \epsilon_n \in {\pm1}$ that is bounded for all real $x$?
Example: One can modifify the signs of the power series of the exponential function $\sum_{n=0}^{\infty} x^n/n!$ pretty easily to a bounded power series by $\epsilon_n = +1$ for $n = 0 or 1 \mod 4$ and $\epsilon_n = -1$ for $n = 2 or 3 \mod 4$, yielding the function $\sin(x) + \cos(x)$. (One can modify the signs pretty easily a bit further such that the power series is not only bounded on the real axis, but also on the imaginary axis - but this is not the question here). I have neither succeeded in finding a counterexample nor in prooving this conjecture.

EDIT2: Thanks for the nice counterexample. I would like to improve the conjecture as follows: Define a power series $\sum_{n=0}^{\infty} a_n x^n$ as nondominant, if for all $x \in \mathbb{R}$ the absolute value of every term $a_kx^k$ is smaller or equal than the sum of the absolute values of all the other terms: $|a_kx^k| \le \sum_{n \neq k} |a_n x^n|$. The improved conjecture is: Is is true, that every nondominant power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $\sum_{n=0}^{\infty} \epsilon_n a_n x^n, \quad \epsilon_n \in {\pm1}$ that is bounded for all real $x$?

Limit Distribution of Topological Form of Polyhedra with Large Number of Edges

2010-05-29T09:17:37Z

Consider the set of all topologically inequivalent polyhedral graphs with $k$ edges, the number of which is given by Sloan sequence A002840 (1,0,1,2,2,4,12,22,58,158,448)).

Now define a topological form parameter as $\beta:= (\text{number of vertices}=v)/(k+2)$ and consider the distribution of the polyhedral graphs with $k$ edges as a function of $\beta$. Due to duality the distribution is symmetric about $\beta=1/2$. Due to the fact that for a planar graph $k \le 3v-6$, the distribution vanishes outside the interval $\beta \in [1/3, 2/3]$.

Now a natural question is whether this distribution tends to a limiting distribution when the number of edges tends to infinity. Is it known whether such a limiting distribution exists - or will it be singular, i.e. concentrated with smaller and smaller variance around $\beta=1/2$, as numerical data seem to suggest? Is there any nontrivial limit distribution theorem by means of rescaling?

EDIT: Some numerical data can be found under http://www.numericana.com/data/polycount.htm. Using these data gives the following values for the standard deviation of the distribution $p(\beta)$ as a function of the number of edges: $\sigma(k=21) = 0.029895922, \sigma(k=27) = 0,027943943, \sigma(k=33) = 0,02625827$.

Answer by Andreas Rüdinger for Question on consecutive integers with similar prime factorizations

2010-07-20T21:56:58Z

Very rough heuristic of the $k$th such $n$ (prime signature twin) as a function of $k$:

1st assumption: The dominant effect is given by primes of the signature $(1,1)$, i.e. by semiprimes.

2nd assumption: The number of semiprimes below $n$ is given by $\pi_2(n) \sim \frac{n}{\ln n} \ln \ln n$, cf. http://en.wikipedia.org/wiki/Almost_prime

Then the probability density of the semiprimes is approximately given by $f_2(n) \sim \frac{\ln \ln n}{\ln n}$.

3rd assumption: The semiprimes are independently distributed.

Then the number of semiprime twins below $N$ is given by $\int^{N} f_2(x)^2 dx$.

Thus the number of prime signature twins is rougly given by $(\frac{\ln \ln n}{\ln n})^2$ (a better value or an asymptotic formula can be obtained by evaluating the integral.)

Thus the $k$th prime signature twin is roughly given byh $n = k \cdot (\frac{\ln k}{\ln \ln k})^2$. For $k = 200$ (as in the figure cited by the OP) this gives approximately a slope of 8,8, the same order of magnitude as in the figure.

Of course this very rough calculation can be improved in various ways.

Answer by Andreas Rüdinger for Can I derive the Boltzmann distribution by an invariance argument?

2010-07-11T21:19:55Z

For me the clearest derivation of the Boltzmann distribution is by maximizing the entropy $\sum n_i \ln(n_i)$ unter the constraint of constant total energy $\sum n_i E_i = \text{const.}$ and constant total particle number $\sum n_i = \text{const.}$. The Lagrange multiplicator for the first constraint gives $\beta$. You can immediately see that a shift of the energies does not change the distribution.

Sum f(p) over all primes convergent with sum f(n) over all natural numbers divergent?

2010-07-06T21:01:14Z

The sum $\sum_{n=1}^{\infty} 1/n^{s}$ is convergent for all real $s>1$ and diverges for all real $s \le 1$. The same holds for the sum $\sum_{p \ prime} 1/p^{s}$. Thus, for the functions $f(n)= 1/n^s, s \in \mathbb{R}$ the sum $\sum_{n=1}^{\infty}f(n)$ shows the same convergence behaviour as the sum $\sum_{p \ prime}f(p)$.

The same holds, if I'm not mistaken, for the functions $f(n)= 1/(n (\ln n)^s), s \in \mathbb{R}$ (both for $n \in \mathbb{N}$ and for primes convergence iff $s>1$).

Question: Is there a real monotonic function $f$ such that $\sum_{n=1}^{\infty}f(n)$ diverges whereas sum $\sum_{p \ prime}f(p)$ converges? (The monotony requirement is for preventing 'artificial' solutions that single out the primes (as e.g. $f(n) = 2^n$ if $n$ is prime; $f(n)=n$ if $n$ is not prime)).

Can an entire non-constant function be bounded on only a finite set of lines through the origin?

2010-06-27T21:41:47Z

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= const., r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).

What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant?

From intuition I would expect only finitely many directions. Is this correct?

(Picard's second theorem says that in any open set containing $\infty$ every value with possibly a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.)

Triangle centers as power sum minimizing points: Which is the locus of all these points?

2010-05-29T22:58:17Z

Some of the classical triangle centers can be expressed as solutions to minimization problems: Given a triangle $A_1, A_2, A_3$ define $d_i, i=1,2,3$ to be the distance of a given point $P$ to $A_i$, and $f_q$ as the sum of the $q$-th power of these distances: $f_q = \sum_{i=1}^3 d_i^q$ . I'm looking for the point $P$ which minimizes $f_q$. For $q=1$ this is the Steiner point, for $q=2$ the centroid, for $q \to \infty$ the circumcenter, for $q \to 0$ the point where the product of distances is minimized. An obvious question is to find the curve of all these points for reasonably general $q$ (e.g. $q \in \mathbb{R}_{>0}$). However, in the ressources for triangle centers, as e.g. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html this problem seems not to be considered.

EDIT:

1) I would like to restrict the problem to $q \ge 1$ since then $f_q$ is convex and a unique minimum is guaranteed.

2) I would like to add a generalization of the question: Consider all continuous functions $f(d_1,d_2,d_3)$ that a) are invariant under permutations of $d_1, d_2, d_3$ and b) have a unique minimum. What can be said about the locus of all these minima?

Answer by Andreas Rüdinger for How well does a truncated fourier expansion of a stepfunction perform near the expansionpoint

2010-06-18T18:56:42Z

Isn't it just the Gibbs phenomenom you are asking for? If there is a discontinuity (as for the Heaviside function) the discontinuity in the partial Fourier series does not die out for $n \to \infty$ , but will be about 18% larger than the discontinuity in the original function. This does not give you the precise value for 5 or 6 modes, but tells you that for Heaviside function the overshot will be 0,18 for $n \to \infty$, which will give you the order of magnitude also for 5 or 6 modes. (Please see also http://en.wikipedia.org/wiki/Gibbs_phenomenon.)

Answer by Andreas Rüdinger for The Jacobi Identity for the Poisson Bracket

2010-06-16T18:54:42Z

Perhaps one of the following references is helpful for you: a) page 12 of http://sundoc.bibliothek.uni-halle.de/habil-online/04/04A736/t3.pdf (unfortunately in German language), b) http://arxiv.org/PS_cache/physics/pdf/0210/0210074v1.pdf.

Osculating conics and cubics and beyond

2010-06-13T16:52:34Z

The osculating circle at a point of a smooth plain curve can be obtained by considering three points on the curve and the circle defined by them. When the three points approach $P$, the circle becomes the osculating circle at $P$.

Generalizing this method to a conic defined by five points on a curve, one obtains an 'osculating conic' at each point of a smooth plain curve. It is pretty straight forward to show that this osculating conic is an ellipse (resp. parabola, hyperbola) if $y'' y^{(4)} > \frac{5}{3}y'''^2$ (resp. =, <).

Thus one can classify points on a plain curve, e.g. all points on $y=e^x$ are hyperbolic (which is not obvious to the 'naked eye'). The mentionned criterion is an affine differential invariant.

I'm wondering why these 'osculating conics' seem to be relativley unknown (would give a nice textbook example connecting elementary calculus and linear algebra - you can find the criterion above in one handwritten page, starting with the taylor expansion) and whether there are any applications (generalization of evolute; can you get the curve back from the foci curve of the osculating conics)?

Furthermore it would be interesting to know whether there are any results for 'osculating cubics' (nine points define a cubic plain curve) or 'osculating quartics' (fourteen points define a quartic plain curve) or for 'osculating quadrics' (would in general yield another classification of points on a surface than the usual elliptic/parabolic/hyperbolic one).

Answer by Andreas Rüdinger for Quadratic Diophantine equations solver

2010-06-10T21:18:24Z

Have you had a look on this tutorial: http://reference.wolfram.com/mathematica/tutorial/DiophantineReduce.html ? To take different values n < n_max into account, a simple loop could work.

Answer by Andreas Rüdinger for Unit triangles with vertices on circles

2010-06-05T10:20:10Z

I know that the following remark does not connect in an obvious way to your question, but perhaps it is nevertheless helpful: Every simple closed curve has an inscribed unilateral triangle (in fact so many inscribed unilateral triangles that the set of vertices is dense in the curve), cf. e.g. theorem D and E in http://www.webpages.uidaho.edu/~markn/squares/ and the respective proofs.

Answer by Andreas Rüdinger for Examples of common false beliefs in mathematics.

2010-06-01T20:51:57Z

A common misbelief for the exponential of matrices is $AB=BA \Leftrightarrow \exp(A)\exp(B) = \exp(A+B)$ . While the one direction is of course correct: $AB=BA \Rightarrow \exp(A)\exp(B) = \exp(A+B)$ , the other direction is not correct, as the following example shows: $A=\begin{pmatrix} 0 & 1 \\ 0 & 2\pi i\end{pmatrix}, B=\begin{pmatrix} 2 \pi i & 0 \\ 0 & -2\pi i\end{pmatrix} $ with $AB \neq BA \text{ and} \exp(A)=\exp(B) = \exp(A+B) = 1$ .

Answer by Andreas Rüdinger for How can one best link a circle to a straight line by an arc of continuously varying curvature?

2010-05-31T18:40:26Z

In road and railway construction a curve called clothoid is often used to join circles and straight lines because its curvature varies linearly with its arc length.

Comment by Andreas Rüdinger

2012-11-28T19:20:08Z

@ Willie: Many thanks, Willie!

Comment by Andreas Rüdinger

2011-09-30T20:26:11Z

Could a natural definition of fractal be "not of integer dimension" (whatever definition of dimension you take)?

Comment by Andreas Rüdinger

2010-12-27T21:20:29Z

@Mark. Thank you. Fixed.

Comment by Andreas Rüdinger

2010-12-05T19:15:36Z

Thank you. I have changed to community wiki.

Comment by Andreas Rüdinger

2010-07-06T21:44:43Z

Thank you, how could I have overlooked this?

Comment by Andreas Rüdinger

2010-06-27T10:47:52Z

Many thanks for that remark. I had thought a bit about it before asking the question the way I did, but I had not continued thinking along these lines, because I'm mainly interested in a "real analysis" solution, and it is obviously not true that f is bounded if and only if all zeroes (in Weierstrass factorization) are real. But I will rethink about approaching the problem via Weierstrass.

Comment by Andreas Rüdinger

2010-06-15T21:53:46Z

Regarding (1) of the answer above: I'm sorry, but I do not fully understand why the fact that a mapping between metric spaces is continuous if and only if the preimage of an open set is open should justify the predominace of open sets in topology vs. closed sets. It is also true that a mapping between metric spaces is continuous if and only if the preimage of a closed set is closed. So this does not yield a pedagogic motivation for using open sets in my opinion.

Comment by Andreas Rüdinger

2010-06-15T21:49:22Z

I'm sorry, but I do not fully understand why the fact that a "mapping between metric spaces is continuous if and only if the inverse image of an open set is open" should justify the predominace of open sets in topology vs. closed sets. It is also true that a mapping between metric spaces is continuous if and only if the inverse image of a closed set is closed. So this does not yield a pedagogic motivation for using open sets in my opinion.

Comment by Andreas Rüdinger

2010-05-30T21:14:36Z

Many thanks for your comments. Sorry for not having been clear. I have just added an addendum above, trying to make clearer, what I wanted to ask.