Implicit Möbius strip

Reading the Wolfram MathWorld™ article on the Möbius strip I came across an equation that was news to me.

After the familiar parametric form

\displaystyle    x=(1+s\cos(t/2))\cos t,\quad y=(1+s\cos(t/2))\sin t, \quad z = s\sin(t/2) \qquad (1)

it is said that “In this parametrization, the Möbius strip is therefore a cubic surface with equation…”

\displaystyle  y(x^2+y^2+z^2-1)-2z(x^2+y^2+x) =0 \qquad\qquad\qquad(2)

That’s a neat cubic equation for sure. But… wouldn’t the sign of the left hand side of (2) (call it F) distinguish the “positive” and “negative” sides of the surface, contrary to its non-orientability?

I went to SageMathCloud™ to check and got this plot of (2):

Implicit plot, looks complicated
Implicit plot, looks complicated

That’s… not exactly the Möbius strip as I know it. But it’s true that (1) implies (2): the cubic surface contains the Möbius strip. Here is the plot of (1) superimposed on the plot of (2).

Same with the Moebius band superimposed
Same with the Moebius band superimposed

The self-intersection arises where the “positive side” F > 0 suddenly transitions to negative F< 0 .

Trigonometric approximation and the Clenshaw-Curtis quadrature

Trying to approximate a generic continuous function on {[-1,1]} with the Fourier trigonometric series of the form {\sum_n (A_n\cos \pi nx+B_n\sin \pi nx)} is in general not very fruitful. Here’s such an approximation to {f(x)=\exp(x)}, with the sum over {n\le 4}:

Poor approximation to exp(x)
Poor approximation to exp(x)

It’s better to make a linear change of variable: consider {f(2x-1)} on the interval {[0,1]}, and use the formula for the cosine series. This results in {\exp(2x-1)}, which is approximated by the partial sum {\sum_{n=0}^4 A_n\cos \pi nx} of its cosine Fourier series as follows.

Better approximation after a change of variable
Better approximation after a change of variable

But one can do much better with a different, nonlinear change of variable. Consider {f(\cos x)} on the interval {[0,\pi]}, and again use the formula for the cosine series. This results in {\exp(\cos x)}, which is approximated by the partial sum {\sum_{n=0}^4 A_n\cos nx} of its cosine Fourier series as follows.

Excellent approximation after nonlinear change of variable
Excellent approximation after nonlinear change of variable

Yes, I can’t see any difference either: the error is less than {10^{-3}}.

The composition with cosine improves approximation because {f(\cos t)} is naturally a periodic function, with no jumps or corners in its graph. Fourier series, which are periodic by nature, follow such functions more easily.


A practical implication of this approximation of {f(\cos t)} is the Clenshaw-Curtis integration method. It can be expressed in one line:

\displaystyle    \int_{-1}^1 f(x)\,dx = \int_0^\pi f(\cos t)\sin t\,dt \approx \int_0^\pi \sum_{n=0}^N a_n \cos nt \sin t\,dt   = \sum_{n=0}^N \frac{(1+(-1)^n) a_n}{1-n^2}

The integral {\int_{-1}^1 f(x)\,dx} is approximated by summing {2a_{2k}/(1-4k^2)}, where {a_{2k}} are even-numbered cosine coefficients of {f(\cos t)}. In the example with {f(x)=\exp(x)} using just three coefficients yields

\displaystyle    \frac{2a_0}{1}+\frac{2a_2}{-3}+\frac{2a_4}{-15} \approx 2.350405

while the exact integral is {\approx 2.350402}.


At first this doesn’t look practical at all: after all, the Fourier coefficients are themselves found by integration. But since {f(\cos t)} is so close to a trigonometric polynomial, one can sample it at equally spaced points and apply the Fast Fourier transform to the result, quickly obtaining many coefficients at once. This is what the Clenshaw-Curtis quadrature does (at least in principle, the practical implementation may streamline these steps.)

Normalizing to zero mean or median

Pick two random numbers {x_1,x_2} from the interval {[0,1]}; independent, uniformly distributed. Normalize them to have mean zero, which simply means subtracting their mean {(x_1+x_2)/2} from each. Repeat many times. Plot the histogram of all numbers obtained in the process.

Two random numbers  normalized to zero mean
Two random numbers normalized to zero mean

No surprise here. In effect this is the distribution of {Y=(X_1-X_2)/2} with {X_1,X_2} independent and uniformly distributed over {[0,1]}. The probability density function of {Y} is found via convolution, and the convolution of {\chi_{[0,1]}} with itself is a triangular function.

Repeat the same with four numbers {x_1,\dots,x_4}, again subtracting the mean. Now the distribution looks vaguely bell-shaped.

Four random numbers  normalized to zero mean
Four random numbers normalized to zero mean

With ten numbers or more, the distribution is not so bell-shaped anymore: the top is too flat.

Ten random numbers  normalized to zero mean
Ten random numbers normalized to zero mean

The mean now follows an approximately normal distribution, but the fact that it’s subtracted from uniformly distributed {x_1,\dots,x_{10}} amounts to convolving the Gaussian with {\chi_{[0,1]}}. Hence the flattened top.


What if we use the median instead of the mean? With two numbers there is no difference: the median is the same as the mean. With four there is.

Four random numbers  normalized to zero median
Four random numbers normalized to zero median

That’s an odd-looking distribution, with convex curves on both sides of a pointy maximum. And with {10} points it becomes even more strange.

Ten random numbers  normalized to zero median
Ten random numbers normalized to zero median

Scilab code:

k = 10
A = rand(200000,k)
A = A - median(A,'c').*.ones(1,k)
histplot(100,A(:))

Lattice points in a disk

The closed disk of radius {72} has area {\pi \cdot 72^2\approx 16286}. But it happens to contain only {16241} points with integer coordinates. Here is a picture of one quarter of this disk.

Unusually few lattice points in the disk
Unusually few lattice points in the disk

The radius {72} is somewhat notable is that the discrepancy of {45} between the area and the number of integer points is unusually large. Here is the plot of the absolute value of the difference |area-points| as a function of integer radius {n}. The curve in red is {y = 1.858 r^{0.745}}, which is an experimentally found upper bound for the discrepancy in this range of {n}.

Radius from 1 to 100
Radius up to 100

On the scale up to {n=1000}, the upper bound is {4.902 r^{0.548}}, and the radii bumping against this bound are {449} and {893}. The exponent {0.548} begins to resemble the conjectural {0.5+\epsilon} in the Gauss circle problem.

Radius up to 1000
Radius up to 1000

Finally, over the interval {1000\le n\le 3000} the upper bound comes out as {6.607n^{0.517}}. The exponent {0.517} looks good.

Radius from 1000 to 3000
Radius from 1000 to 3000

This little numerical experiment in Matlab involved using the convex hull function convhull on log-log scale. The function identifies the vertices of the convex hull, which is a polygon. I pick the side of the polygon lying over the midpoint of the range; this yields a linear upper bound for the set of points. On the normal scale, this line becomes a power function. Matlab code is given below; it’s divided into three logical steps.

Find the difference between area and the number of lattice points
a = 1000; 
b = 3000;
R = a:b;
E = [];
for n = a:b
    [X,Y] = meshgrid(1:n, 1:n);
    pts = 4*n + 1 + 4*nnz(X.^2+Y.^2<=n^2);
    E = [E, max(1,abs(pts - pi*n^2))];
end
Pick a suitable side of log-log convex hull
ix = convhull(log(R), log(E));
k = numel(ix);
while (R(ix(k))<(a+b)/2)
    k = k-1;
end
Plot the result and output the parameters of the upper bound
R1 = R(ix(k)); E1 = E(ix(k));
R2 = R(ix(k+1)); E2 = E(ix(k+1));
b = log(E1/E2)/log(R1/R2);
a = E1/R1^b;
plot(R, E, '.');
hold on
plot(R, a*R.^b , 'r-');
axis tight
hold off
fprintf('a = %.3f, b = %.3f\n', a, b);

B-splines and probability

If one picks two real numbers {X_1,X_2} from the interval {[0,1]} (independent, uniformly distributed), their sum {S_2=X_1+X_2} has the triangular distribution.

Also known as the hat function
Also known as the hat function

The sum {S_3} of three such numbers has a differentiable probability density function:

Piecewise quadratic, C1 smooth
Piecewise quadratic, C1 smooth

And the density of {S_4=X_1+X_2+X_3+X_4} is smoother still: the p.d.f. has two
continuous derivatives.

This begins to look normal
This begins to look normal

As the number of summands increases, these distributions converge to normal if they are translated and scaled properly. But I am not going to do that. Let’s keep the number of summands to four at most.

The p.d.f. of {S_n} is a piecewise polynomial of degree {n-1}. Indeed, for {S_1=X_1} the density is piecewise constant, and the formula

\displaystyle  S_n(x) = \int_{x-1}^x S_{n-1}(t)\,dt

provides the inductive step.

For each {n}, the translated copies of function {S_n} form a partition of unity:

\displaystyle  \sum_{k\in\mathbb Z}S_n(x-k)\equiv 1

The integral recurrence relation gives an easy proof of this:

\displaystyle  \sum_{k\in\mathbb Z}\int_{x-k-1}^{x-k} S_{n-1}(t)\,dt = \int_{\mathbb R} S_{n-1}(t)\,dt = 1

And here is the picture for the quadratic case:

Piecewise quadratic partition of unity
Piecewise quadratic partition of unity

A partition of unity can be used to approximate functions by piecewise polynomials: just multiply each partition element by the value of the function at the center of the corresponding interval, and add the results.

Doing this with {S_2} amounts to piecewise linear interpolation: the original function {f(x)=e^{-x/2}} is in blue, the weighted sum of hat functions in red.

Using the function exp(-x/2)
PL interpolation of exp(-x/2)

With {S_4} we get a smooth curve.

Approximating exp(-x/2) with a cubic B-spline
Approximating exp(-x/2) with a cubic B-spline

Unlike interpolating splines, this curve does not attempt to pass through the given points exactly. However, it has several advantages over interpolating splines:

  • Is easier to calculate; no linear system to solve;
  • Yields positive function for positive data;
  • Yields monotone function for monotone data

The shortest circle is a hexagon

Let {\|\cdot\|} be some norm on {{\mathbb R}^2}. The norm induces a metric, and the metric yields a notion of curve length: the supremum of sums of distances over partitions. The unit circle {C=\{x\in \mathbb R^2\colon \|x\|=1\}} is a closed curve; how small can its length be under the norm?

For the Euclidean norm, the length of unit circle is {2\pi\approx 6.28}. But it can be less than that: if {C} is a regular hexagon, its length is exactly {6}. Indeed, each of the sides of {C} is a unit vector with respect to the norm defined by {C}, being a parallel translate of a vector connecting the center to a vertex.

Hexagon as unit disk
Hexagon as unit disk

To show that {6} cannot be beaten, suppose that {C} is the unit circle for some norm. Fix a point {p\in C}. Draw the circle {\{x\colon \|x-p\|=1\}}; it will cross {C} at some point {q}. The points {p,q,q-p, -p, -q, p-q} are vertices of a hexagon inscribed in {C}. Since every side of the hexagon has length {1}, the length of {C} is at least {6}.

It takes more effort to prove that the regular hexagon and its affine images, are the only unit circles of length {6}; a proof can be found in Geometry of Spheres in Normed Spaces by Juan Jorge Schäffer.

Nonlinear Closed Graph Theorem

If a function {f\colon {\mathbb R}\rightarrow {\mathbb R}} is continuous, then its graph {G_f = \{(x,f(x))\colon x\in{\mathbb R}\}} is closed. The converse is false: a counterexample is given by any extension of {y=\tan x} to the real line.

Closed graph, not continuous
Closed graph, not continuous

The Closed Graph Theorem of functional analysis states that a linear map between Banach spaces is continuous whenever its graph is closed. Although the literal extension to nonlinear maps fails, it’s worth noting that linear maps are either continuous or discontinuous everywhere. Hence, if one could show that a nonlinear map with a closed graph has at least one point of continuity, this would be a nonlinear version of the Closed Graph Theorem.


Here is an example of a function with a closed graph and an uncountable set of discontinuities. Let {C\subset {\mathbb R}} be a closed set with empty interior, and define

\displaystyle    f(x) = \begin{cases} 0,\quad & x\in C \\ \textrm{dist}\,(x,C)^{-1},\quad & x\notin C \end{cases}

For a general function, the set of discontinuities is an {F_\sigma} set. When the graph is closed, we can say more: the set of discontinuities is closed. Indeed, suppose that a function {f} is bounded in a neighborhood of {a} but is not continuous at {a}. Then there are two sequences {x_n\rightarrow a} and {y_n\rightarrow a} such that both sequences {f(x_n)} and {f(y_n)} converge but have different limits. Since at least one of these limits must be different from {f(a)}, the graph of {f} is not closed. Conclusion: a function with a closed graph is continuous at {a} if and only if it is bounded in a neighborhood of {a}. In particular, the set of discontinuities is closed.


Furthermore, the set of discontinuities has empty interior. Indeed, suppose that {f} is discontinuous at every point of a nontrivial closed interval {[a,b]}. Let {A_n = G_f \cap ([a,b]\times [-n,n])}; this is a closed bounded set, hence compact. Its projection onto the {x}-axis is also compact, and this projection is exactly the set {B_n=\{x\in [a,b] : |f(x)|\le n\}}. Thus, {B_n} is closed. The set {B_n} has empty interior, since otherwise {f} would be continuous at its interior points. Finally, {\bigcup B_n=[a,b]}, contradicting the Baire Category theorem.

Summary: for closed-graph functions on {\mathbb R}, the sets of discontinuity are precisely the closed sets with empty interior. In particular, every such function has a point of continuity. The proof works just as well for maps from {\mathbb R^n} to any metric space.


However, the above result does not extend to the setting of Banach spaces. Here is an example of a map {F\colon X\rightarrow X} on a Banach space {X} such that {\|F(x)-F(y)\|=1} whenever {x\ne y}; this property implies that the graph is closed, despite {F} being discontinuous everywhere.

Let {X} the space of all bounded functions {\phi \colon (0,1]\rightarrow\mathbb R} with the supremum norm. Let {(q_n)_{n=1}^\infty} be an enumeration of all rational numbers. Define the function {\psi =F(\phi )} separately on each subinterval {(2^{-n},2^{1-n}]}, {n=1,2,\dots} as

{\displaystyle    \psi(t) = \begin{cases} 1 \quad &\text{if } \phi (2^nt-1) > q_n \\   0 \quad &\text{if } \phi (2^n t-1)\le q_n\end{cases}}

For any two distinct elements {\phi_1,\phi_2} of {X} there is a point {s\in (0,1]} and a number {n\in\mathbb N} such that {q_n} is strictly between {\phi_1(s)} and {\phi_2(s)}. According to the definition of {F} this implies that the functions {F(\phi_1)} and {F(\phi_2)} take on different values at the point {t=2^{-n}(s+1)}. Thus the norm of their difference is {1}.


So much for Nonlinear Closed Graph Theorem. However, the space {X} in the above example is nonseparable. Is there an nowhere continuous map between separable Banach spaces such that its graph is closed?

Institutions ranked by the number of AMS Fellows: 2015 update

Adding 2015 Fellows changed a few things compared to 2014 rankings. UCLA and Rutgers rise from 2nd and 3rd to tie Berkeley for the first place. Syracuse loses nine places, dropping from #334(tie) to #343(tie).

Podium 231

  • 1. Rutgers The State University of New Jersey New Brunswick: 34
  • 1. University of California, Los Angeles: 34
  • 1. University of California, Berkeley: 34
  • 4. University of Michigan: 32
  • 5. Massachusetts Institute of Technology: 29
  • 6. University of Wisconsin, Madison: 23
  • 7. Cornell University: 22
  • 7. New York University, Courant Institute: 22
  • 7. University of Illinois, Urbana-Champaign: 22
  • 10. University of Texas at Austin: 21
  • 10. University of Chicago: 21
  • 10. Princeton University: 21
  • 13. University of Washington: 20
  • 14. University of California, San Diego: 19
  • 14. Stanford University: 19
  • 16. University of Pennsylvania: 17
  • 16. University of Minnesota-Twin Cities: 17
  • 18. University of California, Santa Barbara: 16
  • 18. Pennsylvania State University: 16
  • 18. Stony Brook University: 16
  • 18. Brown University: 16
  • 22. Purdue University: 15
  • 22. University of Maryland: 15
  • 24. University of California, Irvine: 14
  • 24. Duke University: 14
  • 26. Ohio State University, Columbus: 13
  • 26. Eidgenössische Technische Hochschule Zürich (ETH Zürich): 13
  • 26. University of Illinois at Chicago: 13
  • 29. Northwestern University: 12
  • 29. Georgia Institute of Technology: 12
  • 29. Johns Hopkins University, Baltimore: 12
  • 29. Texas A&M University: 12
  • 33. University of Toronto: 11
  • 33. Harvard University: 11
  • 33. Indiana University, Bloomington: 11
  • 36. University of North Carolina at Chapel Hill: 10
  • 36. University of British Columbia: 10
  • 36. The Graduate Center: 10
  • 36. Rice University: 10
  • 40. Vanderbilt University: 9
  • 41. University of Utah: 8
  • 41. Boston University: 8
  • 41. California Institute of Technology: 8
  • 41. University of Nebraska-Lincoln: 8
  • 41. Institute for Advanced Study: 8
  • 41. University of Notre Dame: 8
  • 47. University of Georgia: 7
  • 47. University of Southern California: 7
  • 47. University of Virginia: 7
  • 47. University of California, Davis: 7
  • 47. University of Oregon: 7
  • 47. Brandeis University: 7
  • 47. Microsoft Research: 7
  • 54. University of Oxford: 6
  • 54. Université Pierre et Marie Curie (Paris VI): 6
  • 54. University of Arizona: 6
  • 54. Carnegie Mellon University: 6
  • 54. Michigan State University: 6
  • 54. Columbia University: 6
  • 54. The Hebrew University of Jerusalem: 6
  • 54. Williams College: 6
  • 54. North Carolina State University: 6
  • 54. Tel Aviv University: 6
  • 64. University of Rochester: 5
  • 64. Lehman College: 5
  • 64. NYU Polytechnic School of Engineering: 5
  • 67. University of California, Riverside: 4
  • 67. Ecole Polytechnique Fédérale de Lausanne (EPFL): 4
  • 67. Université Paris-Diderot: 4
  • 67. Florida State University: 4
  • 67. Harvey Mudd College: 4
  • 67. University of Tennessee, Knoxville: 4
  • 67. Yale University: 4
  • 67. Louisiana State University, Baton Rouge: 4
  • 67. Northeastern University: 4
  • 67. Virginia Polytechnic Institute and State University: 4
  • 67. University of Colorado, Boulder: 4
  • 78. Tsinghua University: 3
  • 78. Norwegian University of Science and Technology: 3
  • 78. McGill University: 3
  • 78. University of Cambridge: 3
  • 78. University of Memphis: 3
  • 78. University of Melbourne: 3
  • 78. KU Leuven: 3
  • 78. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences: 3
  • 78. KTH Royal Institute of Technology: 3
  • 78. Københavns Universitet: 3
  • 78. The City College: 3
  • 78. Università degli Studi di Milano: 3
  • 78. University of Warwick: 3
  • 78. University of Connecticut, Storrs: 3
  • 78. Barnard College, Columbia University: 3
  • 78. Australian National University: 3
  • 78. Université Paris-Sud (Paris XI): 3
  • 78. Washington University: 3
  • 78. Weizmann Institute of Science: 3
  • 78. Mathematical Institute, Oxford University: 3
  • 98. Bar-Ilan University: 2
  • 98. Rheinische Friedrich-Wilhelms-Universität Bonn: 2
  • 98. American Mathematical Society: 2
  • 98. Emory University: 2
  • 98. Rutgers The State University of New Jersey Newark: 2
  • 98. Sapienza – Università di Roma: 2
  • 98. Shanghai Jiao Tong University: 2
  • 98. Smith College: 2
  • 98. Lund University: 2
  • 98. University of Florida: 2
  • 98. University of Freiburg: 2
  • 98. Steklov Institute of Mathematics of the Russian Academy of Sciences: 2
  • 98. University of Heidelberg: 2
  • 98. Boston College: 2
  • 98. Tata Institute of Fundamental Research: 2
  • 98. University of Iowa: 2
  • 98. University of Kansas: 2
  • 98. University of Kentucky: 2
  • 98. University of Manchester: 2
  • 98. Technical University of Denmark: 2
  • 98. University of Massachusetts, Amherst: 2
  • 98. Case Western Reserve University: 2
  • 98. Temple University: 2
  • 98. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV): 2
  • 98. Académie des sciences, Institut de France: 2
  • 98. University of Missouri-Columbia: 2
  • 98. CNRS, École Normale Supérieure de Lyon: 2
  • 98. University of New Hampshire: 2
  • 98. IDA Center for Communications Research: 2
  • 98. Imperial College: 2
  • 98. University of Oklahoma: 2
  • 98. The Fields Institute: 2
  • 98. University of Oslo: 2
  • 98. Indian Institute of Technology: 2
  • 98. Indiana University-Purdue University Indianapolis: 2
  • 98. Tulane University: 2
  • 98. Universidad Nacional Autónoma de México: 2
  • 98. Auburn University: 2
  • 98. Universität Wien: 2
  • 98. Université de Genève: 2
  • 98. Université de Montréal: 2
  • 98. Institut des Hautes Etudes Scientifiques (IHES): 2
  • 98. The City University of New York: 2
  • 98. Institute of Mathematics, University of Paderborn: 2
  • 98. University of Alabama at Birmingham: 2
  • 98. Oklahoma State University: 2
  • 98. Vienna University of Technology: 2
  • 98. University of Bristol: 2
  • 98. Oregon State University: 2
  • 98. Wayne State University: 2
  • 98. Dartmouth College: 2
  • 98. Wesleyan University: 2
  • 98. Brigham Young University: 2
  • 151. Silesian University of Technology: 1
  • 151. Pontifícia Universidade Católica do Rio de Janeiro: 1
  • 151. Carleton University: 1
  • 151. Grambling State University: 1
  • 151. Queen Mary, University of London: 1
  • 151. Queen’s University: 1
  • 151. Reed College: 1
  • 151. Rensselaer Polytechnic Institute: 1
  • 151. Research Institute for Mathematical Sciences, Johannes Kepler University Linz: 1
  • 151. Research Institute for Mathematical Sciences, Kyoto University: 1
  • 151. B.I.Verkin Institute for Low Temperature Physics and Engineering: 1
  • 151. American University: 1
  • 151. Roskilde University: 1
  • 151. Rutgers The State University of New Jersey Camden: 1
  • 151. Haverford College: 1
  • 151. Hillman University: 1
  • 151. Saint Petersburg State University: 1
  • 151. San Francisco State University: 1
  • 151. Hong Kong University of Science and Technology: 1
  • 151. Seoul National University: 1
  • 151. IBM Research: 1
  • 151. Aarhus University: 1
  • 151. Center for Communications Research, Princeton, New Jersey: 1
  • 151. Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences: 1
  • 151. Society for Industrial and Applied Mathematics (SIAM): 1
  • 151. Spelman College: 1
  • 151. St Louis University: 1
  • 151. St Olaf College: 1
  • 151. St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences: 1
  • 151. IMFUFA, Roskilde University: 1
  • 151. Centre for Quantum Geometry of Moduli Spaces (QGM), Aarhus University: 1
  • 151. Stevens Institute of Technology: 1
  • 151. Centre of Mathematics for Applications, University of Oslo: 1
  • 151. SUNY, Maritime College: 1
  • 151. Amgen: 1
  • 151. Centrum Wiskunde & Informatica and University of Amsterdam: 1
  • 151. Technion – Israel Institute of Technology: 1
  • 151. Technische Universität Berlin: 1
  • 151. Technische Universität Darmstadt: 1
  • 151. Institució Catalana de Recerca i Estudis Avançats (ICREA): 1
  • 151. Institut de Mecanique Celeste et de Calcul des Ephemerides (IMCCE): 1
  • 151. Chalmers University of Technology: 1
  • 151. The Abdus Salam International Centre for Theoretical Physics: 1
  • 151. The Chinese University of Hong Kong: 1
  • 151. Institut mathématique de Jussieu: 1
  • 151. Institut Universitaire de France: 1
  • 151. Chennai Mathematical Institute: 1
  • 151. Institute for Information Transmission Problems: 1
  • 151. Institute for Quantum Computing, Waterloo: 1
  • 151. Institute of Mathematical Sciences, The Chinese University of Hong Kong: 1
  • 151. The Institute for System Research of the Russian Academy of Sciences: 1
  • 151. The Institute of Mathematical Sciences, Chennai, India: 1
  • 151. The OEIS Foundation Incorporated: 1
  • 151. The University of British Columbia: 1
  • 151. The University of Liverpool: 1
  • 151. The University of Western Australia: 1
  • 151. Tomakomai National College of Technology: 1
  • 151. Institute of Mathematics, Academia Sinica, ROC: 1
  • 151. TU Dortmund University: 1
  • 151. Tufts University: 1
  • 151. Chern Institute of Mathematics, Nankai University: 1
  • 151. Univeristy of Edinburgh: 1
  • 151. Universidad Autónoma de Madrid: 1
  • 151. Universidad de Concepción: 1
  • 151. Universidad de Valladolid: 1
  • 151. Universidad del País Vasco: 1
  • 151. Instituto de Matemáticas Universidad Nacional Autónoma de México: 1
  • 151. Instituto de Matematica Pura e Aplicada (IMPA): 1
  • 151. Universität Bielefeld: 1
  • 151. Universität des Saarlandes: 1
  • 151. Universität Konstanz, Germany: 1
  • 151. Universität Regensburg: 1
  • 151. Instituto de Matematicas, Universidad de Talca: 1
  • 151. Universität Zürich: 1
  • 151. Université Bordeaux 1: 1
  • 151. Université de Caen: 1
  • 151. Jacobs University: 1
  • 151. Chinese Academy of Sciences: 1
  • 151. Université du Québec à Montréal: 1
  • 151. City University of Hong Kong: 1
  • 151. Université Paris-Nord (Paris XIII): 1
  • 151. Kent State University, Kent: 1
  • 151. King Abdullah University of Science and Technology: 1
  • 151. Universitat Politecnica de Catalunya: 1
  • 151. Universiteit Gent: 1
  • 151. Universiteit Leiden: 1
  • 151. Universiteit Utrecht: 1
  • 151. Universiteit van Amsterdam: 1
  • 151. University at Buffalo: 1
  • 151. University College London: 1
  • 151. University of Aberdeen: 1
  • 151. King’s College London: 1
  • 151. University of Alaska Fairbanks: 1
  • 151. King Saud University: 1
  • 151. University of Auckland: 1
  • 151. University of Basel: 1
  • 151. Kobe University: 1
  • 151. Korea Institute for Advanced Study: 1
  • 151. Cleveland State University: 1
  • 151. CMLA, École Normale Supérieure de Cachan: 1
  • 151. Kyoto University: 1
  • 151. Langorigami.com: 1
  • 151. Lawrence Berkeley National Laboratory: 1
  • 151. Lehigh University: 1
  • 151. CNRS and Université Paris-Sud (Paris XII): 1
  • 151. Loyola University of Chicago: 1
  • 151. University of Central Florida: 1
  • 151. Ludwig-Maximilians-Universität München (LMU München): 1
  • 151. University of Cincinnati: 1
  • 151. Baylor University: 1
  • 151. Macalester College: 1
  • 151. University of Edinburgh: 1
  • 151. CNRS, Institut de Mathématiques de Toulouse: 1
  • 151. Massey University: 1
  • 151. Math for America: 1
  • 151. University of Hawaii at Manoa: 1
  • 151. Mathematical Institute, Leiden University: 1
  • 151. University of Helsinki: 1
  • 151. University of Houston: 1
  • 151. Mathematical Institute, Linkoeping University: 1
  • 151. Collège de France: 1
  • 151. Mathematics Institute, Freiburg University: 1
  • 151. Max Planck Institute for Gravitational Physics (Albert Einstein Institute): 1
  • 151. Max Planck Institute for Mathematics in the Sciences: 1
  • 151. University of Leeds: 1
  • 151. University of Liverpool: 1
  • 151. Beijing Normal University: 1
  • 151. McMaster University: 1
  • 151. University of Massachusetts Boston: 1
  • 151. Bielefeld University: 1
  • 151. Croatian Academy of Sciences and Arts: 1
  • 151. Montana State University: 1
  • 151. University of Miami: 1
  • 151. Moravian College: 1
  • 151. University of Michoacan: 1
  • 151. University of Minnesota Rochester: 1
  • 151. University of Minnesota-Duluth: 1
  • 151. Mount Holyoke College: 1
  • 151. Muenster University: 1
  • 151. Nagoya University: 1
  • 151. University of Newcastle: 1
  • 151. Nanyang Technological University: 1
  • 151. University of New Mexico: 1
  • 151. University of New South Wales: 1
  • 151. University of Nice: 1
  • 151. National Science Foundation: 1
  • 151. University of North Carolina at Charlotte: 1
  • 151. National Security Agency: 1
  • 151. National Taiwan University: 1
  • 151. National Tsing Hua University, Taiwan: 1
  • 151. New Jersey Institute of Technology: 1
  • 151. New Mexico State University, Las Cruces: 1
  • 151. Amherst College: 1
  • 151. University of Pittsburgh: 1
  • 151. Nihon University: 1
  • 151. University of San Francisco: 1
  • 151. University of South Carolina: 1
  • 151. Arizona State University: 1
  • 151. Eötvös Loránd University: 1
  • 151. Northern Illinois Univeresity: 1
  • 151. University of Texas at Dallas: 1
  • 151. University of Texas at San Antonio: 1
  • 151. University of Tokyo: 1
  • 151. University of Toledo: 1
  • 151. Northrop Grumman Corporation: 1
  • 151. Ecole des hautes études en sciences sociales (EHESS): 1
  • 151. University of Valencia: 1
  • 151. University of Vermont: 1
  • 151. University of Victoria: 1
  • 151. Ateneo de Manila University: 1
  • 151. University of Warsaw: 1
  • 151. Åbo Akademi University: 1
  • 151. El Colegio Nacional: 1
  • 151. Albert-Ludwigs-Universitat: 1
  • 151. University of Wisconsin, Milwaukee: 1
  • 151. Utah State University: 1
  • 151. Open University, U.K.: 1
  • 151. Victoria University of Wellington: 1
  • 151. Austrian Academy of Sciences: 1
  • 151. Osaka University: 1
  • 151. Wake Forest University: 1
  • 151. Waseda University: 1
  • 151. Freie Universität Berlin: 1
  • 151. Philipps-Universität Marburg: 1
  • 151. Pitzer College: 1
  • 151. Pohang University of Science and Technology (POSTECH): 1
  • 151. Western Washington University: 1
  • 151. Westfälische Wilhelms-Universität Münster: 1
  • 151. Polish Academy of Sciences: 1
  • 151. Wolfram Research: 1
  • 151. Pomona College: 1
  • 151. York University: 1
  • 343. Syracuse University: 0

To create this list, I parsed the AMS page with the following JavaScript code:

var b = document.getElementById('amsContentDiv').children;
var inst = []; 
var j = -1; 
for (var i=10; i<b.length; i++) {
  if (b[i].tagName=='SPAN') {
    if (b[i].style['font-weight']=='bold') {
      j++; 
      inst[j] = {}; 
      inst[j].name = b[i].textContent; 
      inst[j].count = 0;
    } 
    else {
      inst[j].count++;
    }
  }
}
inst.sort(function(a,b) {return b.count-a.count});
var html = [];
var place = 1;
for (var i=0; i<inst.length; i++) {
  html.push('<li><strong>'+place+'.</strong> '+inst[i].name+': <em>'+inst[i].count+'</em></li>');
  if (inst[i+1] && inst[i+1].count<inst[i].count) {
    place = i+2;
  }
}

Completely monotone imitation of 1/x

I wanted an example of a function {f} that behaves mostly like {1/x} (the product {xf(x)} is bounded between two positive constants), but such that {xf(x)} does not have a limit as {x\rightarrow 0}.

The first thing that comes to mind is {(2+\sin(1/x))/x}, but this function does not look very much like {1/x}.

sin(1/x) makes it too wiggly
sin(1/x) makes it too wiggly

Then I tried {f(x)=(2+\sin\log x)/x}, recalling an example from Linear Approximation and Differentiability. It worked well:

I can't believe it's not a hyperbola!
I can’t believe it’s not a hyperbola!

In fact, it worked much better than I expected. Not only if {f'} of constant sign, but so are {f''} and {f'''}. Indeed,

\displaystyle    f'(x) = \frac{\cos \ln x - \sin \log x - 2}{x^2}

is always negative,

\displaystyle    f''(x) = \frac{4 -3\cos \log x + \sin \log x}{x^3}

is always positive,

\displaystyle    f'''(x) = \frac{10\cos \log x -12}{x^4}

is always negative. The sign becomes less obvious with the fourth derivative,

\displaystyle    f^{(4)}(x) = \frac{48-40\cos\log x - 10\sin \cos \ln x}{x^5}

because the triangle inequality isn’t conclusive now. But the amplitude of {A\cos t+B\sin t} is {\sqrt{A^2+B^2}}, and {\sqrt{40^2+10^2}<48}.

So, it seems that {f} is completely monotone, meaning that {(-1)^n f^{(n)}(x)\ge 0} for all {x>0} and for all {n=0,1,2,\dots}. But we already saw that this sign pattern can break after many steps. So let’s check carefully.

Direct calculation yields the neat identity

\displaystyle    \left(\frac{1+a\cos \log x+b\sin\log x}{x^n}\right)' = -n\,\frac{1+(a-b/n)\cos\log x+(b+a/n) \sin\log x}{x^{n+1}}

With its help, the process of differentiating the function {f(x) = (1+a\cos \log x+b\sin\log x)/x} can be encoded as follows: {a_1=a}, {b_1=b}, then {a_{n+1}=a_n-b_n/n} and {b_{n+1} = b_n+a_n/n}. The presence of {1/n} is disconcerting because the harmonic series diverges. But orthogonality helps: the added vector {(-b_n/n, a_n/n)} is orthogonal to {(a_n,b_n)}.

The above example, rewritten as {f(x)=(1+\frac12\sin\log x)/x}, corresponds to starting with {(a,b) = (0,1/2)}. I calculated and plotted {10000} iterations: the points {(a_n,b_n)} are joined by piecewise linear curve.

Harmonic spiral
Harmonic spiral

The total length of this curve is infinite, since the harmonic series diverges. The question is, does it stay within the unit disk? Let’s find out. By the above recursion,

\displaystyle    a_{n+1}^2 + b_{n+1}^2 = \left(1+\frac{1}{n^2}\right) (a_n^2+b_n^2)

Hence, the squared magnitude of {(a_n,b_n)} will always be less than

\displaystyle    \frac14 \prod_{n=1}^\infty \left(1+\frac{1}{n^2}\right)

with {1/4} being {a^2+b^2}. The infinite product evaluates to {\frac{\sinh \pi}{\pi}a\approx 3.7} (explained here), and thus the polygonal spiral stays within the disk of radius {\frac12 \sqrt{\frac{\sinh \pi}{\pi}}\approx 0.96}. In conclusion,

\displaystyle    (-1)^{n} \left(\frac{1+(1/2)\sin\log x}{x}\right)^{(n)} = n!\,\frac{1+a_{n+1}\cos\log x+b_{n+1} \sin\log x}{x^{n+1}}

where the trigonometric function {a_{n+1}\cos\log x+b_{n+1} \sin\log x} has amplitude strictly less than {1}. Since the expression on the right is positive, {f} is completely monotone.

The plot was generated in Sage using the code below.

a,b,c,d = var('a b c d')
a = 0
b = 1/2
l = [(a,b)]
for k in range(1,10000):
    c = a-b/k
    d = b+a/k
    l.append((c,d))
    a = c
    b = d
show(line(l),aspect_ratio=1)

2014 syllabus

This is a sample of what you could have learned by taking Calculus VII in 2014. One post from each month.