The closure of periodic functions and sums of waves with incommensurable periods

Consider the space {C(\mathbb R)} of all bounded continuous functions {f\colon \mathbb R\to\mathbb R}, with the uniform norm {\|f\| = \sup |f|}. Let {P} be its subset that consists of all periodic continuous functions: recall that {f} is periodic if there exists {T>0} such that {f(x+T)=f(x)} for all {x\in \mathbb R}.

The set {P} is not closed in the topology of {C(\mathbb R)}. Indeed, let {d(x) = \mathrm{dist}\,(x, \mathbb Z)} be the distance from {x} to nearest integer. The function {d} is periodic with {T=1}. Therefore, each sum of the form {\displaystyle \sum_{k=0}^n 2^{-k} d(2^{-k} x)} is periodic with {T=2^n}. Hence the sum of the infinite series {\displaystyle f(x) = \sum_{k=0}^\infty 2^{-k} d(2^{-k} x) } is a uniform limit of periodic functions. Yet, {f} is not periodic, because {f(0)=0} and {f(x)>0 } for {x\ne 0} (for every {x\ne 0} there exists {k} such that {2^{-k}x} is not an integer).

anti-blanc
Uniform limit of periodic functions

The above example (which was suggested to me by Yantao Wu) is somewhat similar to the Takagi function, which differs from it by the minus sign in the exponent: {\displaystyle T(x) = \sum_{k=0}^\infty 2^{-k} d(2^{k} x) }. Of course, the Takagi function is periodic with period {1}.

blankmange
Takagi function

Do we really need an infinite series to get such an example? In other words, does the set {\overline{P}\setminus P} contain an elementary function?

A natural candidate is the sum of trigonometric waves with incommensurable periods (that is, the ratio of periods must be irrational). For example, consider the function {g(x) = \cos (x) + \cos (\sqrt{2}x)} whose graph is shown below.

sum_cos
Looks somewhat periodic, does not it?

Since {g(0)=2} and {g(x) < 2} for all {x\ne 0}, the function {g} is not periodic. Its graph looks vaguely similar to the graph of {f}. Is {g} a uniform limit of periodic functions?

Suppose {h\colon \mathbb R\to\mathbb R} is a {T}-periodic function such that {\|h-g\|<\epsilon}. Then {h(0) > 2-\epsilon}, hence {h(nT)>2-\epsilon} for all {n\in \mathbb Z}, hence {g(nT) > 2- 2\epsilon }. By the definition of {g} this implies {\cos (nT) > 1-2\epsilon} and {\cos (\sqrt{2}nT) > 1-2\epsilon} for all {n\in \mathbb Z}. The following lemma shows a contradiction between these properties.

Lemma. If a real number {t} satisfies {\cos nt > -1/2} for all {n\in \mathbb Z}, then {t} is an integer multiple of {2\pi}.

Proof. Suppose {t} is not an integer multiple of {2\pi}. We can assume {0 < t < \pi} without loss of generality, because {t} can be replaced by {t - 2\pi k} to get it in the interval {(0, 2\pi)} and then by {2\pi - t} to get it in {(0, \pi)}. Since {\cos t > -1/2}, we have {t\in (0, 2\pi/3)}. Let {k} be the smallest positive integer such that {2^k t \ge 2\pi/3}. The minimality of {k} implies {2^{k-1} t < 2\pi/3}, hence {2^k t \in [2\pi/3, 4\pi/3)}. But then {\cos (2^k t) \le -1/2}, a contradiction. {\quad \Box}

The constant {-1/2} in the lemma is best possible, since {\cos (2n\pi/3)\ge -1/2} for all {n\in \mathbb Z}.

Returning to the paragraph before the lemma, choose {\epsilon=3/4} so that {1-2\epsilon = -1/2}. The lemma says that both {T} and {\sqrt{2} T} must be integer multiples of {2\pi}, which is impossible since they are incommensurable. This contradiction shows that {\|g-h\|\ge 3/4} for any periodic function {h}, hence {g} is not a uniform limit of periodic functions.

The above result can be stated as {\mathrm{dist}(g, P) \ge 3/4}. I guess {\mathrm{dist}(g, P)} is actually {1}. It cannot be greater than {1} since {|g(x)-\cos x|\le 1} for all {x}. (Update: Yantao pointed out that the density of irrational rotations implies the distance is indeed equal to 1.)

Note: the set {\overline{P}} is a proper subset of the set of (Bohr / Bochner / uniform) almost periodic functions (as Yemon Choi pointed out in a comment). The latter is a linear space while {\overline{P}} is not. I was confused by the sentence “Bohr defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm” on Wikipedia. To me, a trigonometric polynomial is a periodic function of particular kind. What Bohr called Exponentialpolynom is a finite sum of the form {\sum a_n e^{\lambda_n x}} where {\lambda_n} can be any real numbers. To summarize: the set considered above is the closure of {P} while the set of almost periodic functions is the closed linear span of {P}. The function {g(x)=\cos (x) + \cos(\sqrt{2} x)} is an example of the latter, not of the former.

Points of maximal curvature

In Calculus I students are taught how to find the points at which the graph of a function has zero curvature (that is, the points of inflection). The points of maximal curvature are usually not discussed. This post attempts to rectify this omission.

The (signed) curvature of {y=f(x)} is

{\displaystyle \kappa = \frac{y''}{(1+(y')^2)^{3/2}}}

We want to maximize the absolute value of {\kappa}, whether the function is positive or negative there (so both maxima and minima of {\kappa} can be of interest). The critical points of {\kappa} are the zeros of

{\displaystyle \kappa' = \frac{y''' (1+(y')^2) - 3 y' (y'')^2 }{(1+(y')^2)^{5/2}}}

So we are lead to consider a polynomial of the first three derivatives of {y}, namely {p := y''' (1+(y')^2) - 3 y' (y'')^2 }.

Begin with some simple examples:

{y = x^2} has {p = -24x} so the curvature of a parabola is maximal at its vertex. No surprise there.

{y = x^3} has {p = 6(1 - 45x^4)}, indicating two symmetric points of maximal curvature, {x\approx \pm 0.386}, pretty close to the point of inflection.

x3
y = x^3

 

{y=x^4} has {p = 24 x (1 - 56 x^6)}. This has three real roots, but {x=0} actually minimizes curvature (it vanishes there).

x4
y = x^4

More generally, {y=x^n} with positive integer {n} yields {\displaystyle p = n(n-1)x^{n-3} (n - 2 - (2n^3-n^2) x^{2n-2})} indicating two points {\displaystyle x = \pm \left(\frac{n-2}{2n^3-n^2} \right)^{1/(2n-2)}} which tend to {\pm 1} as {n} grows.

The graph of a polynomial of degree {n} can have at most {n-2} points of zero curvature, because the second derivative vanishes at those. How many points of maximal curvature can it have? The degree of expression {p} above is {3n-5} but it is not obvious whether all of its roots can be real and distinct, and also be the maxima of {|\kappa|} (unlike {x=0} for {y=x^4}). For {n=2} we do get {3n-5 = 1} point of maximal curvature. But for {n=3}, there can be at most {2} such points, not {3n-5 = 4}. Edwards and Gordon (Extreme curvature of polynomials, 2004) conjectured that the graph of a polynomial of degree {n} has at most {n-1} points of maximal curvature. This remains open despite several partial results: see the recent paper Extreme curvature of polynomials and level sets (2017).

A few more elementary functions:

{y = e^x} has {p = e^x(1-2e^{2x})}, so the curvature is maximal at {x=-\log(2)/2}. Did I expect the maximum of curvature to occur for a negative {x}? Not really.

exp
y = exp(x)

{y=\sin x} has {p = (\cos 2x - 3)\cos x}. The first factor is irrelevant: the points of maximum curvature of a sine wave are at its extrema, as one would guess.

{y = \tan x} has {p = -6\tan^8(x) - 16\tan^6(x) - 6\tan^4(x) + 8\tan(x)^2 + 4} which is zero at… ahem. The expression factors as

{\displaystyle p = -2(\tan^2 x+1)(3\tan^6(x) + 5\tan^4(x) - 2\tan^2(x) - 2)}

Writing {u = \tan^2(x)} we can get a cubic equation in {u}, but it is not a nice one. Or we could do some trigonometry and reduce {p=0} to the equation {8 - 41\cos 2x + \cos 6x =0}. Either way, a numerical solution is called for: {x \approx \pm 0.6937} (and {+\pi n} for other periods).

tan
y = tan(x)

Index of questions

This is a meta-post which collects links to other posts on this blog with (sometimes implicit) questions that were left unanswered there. This does not necessarily mean that nobody has an answer, just that I did not have one when writing the post. The collection is in reverse chronological order.

Branching

Multiple kinds of branching here. First, the motorsport content has been moved to formula7.blog. Two blogs? Well, it became clear that my Stack Exchange activity, already on hiatus since 2018, is not going to resume (context: January 14, January 15, January 17). But typing words in boxes is still a hobby of mine.

There may be yet more branching in the knowledge market space, with Codidact and TopAnswers attempting to rise from the ashes of Stack Exchange. (I do not expect either project to have much success.)

Also, examples of branching in complex analysis are often limited to the situations where any two branches differ either by an additive constant like {\log z} or by a multiplicative constant like {z^p}. But different branches can even have different branch sets. Consider the dilogarithm, which has a very nice power series in the unit disk:

{\displaystyle f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2} = z + \frac{z^2}{4} + \frac{z^3}{9} + \frac{z^4}{16} + \cdots}

The series even converges on the unit circle {|z|=1}, providing a continuous extension there. But this circle is also the boundary of the disk of convergence, so some singularity has to appear. And it does, at {z=1}. Going around this singularity and coming back to the unit disk, we suddenly see a function with a branch point at {z=0}, where there was no branching previously.

What gives? Consider the derivative:

{\displaystyle f'(z) = \sum_{n=1}^\infty \frac{z^{n-1}}{n} = -\frac{\log (1-z)}{z}}

As long as the principal branch of logarithm is considered, there is no singularity at {z} since {\log(1-0) = 0} cancels the denominator. But once we move around {z=1}, the logarithm acquires a multiple of {2\pi i }, and so {f'} gets an additional term {cz^{-1}}, and integrating that results in logarithmic branching at {z=0}.

Of course, this does not even begin the story of the dilogarithm, so I refer to Zagier’s expanded survey which has a few branch points itself.

Thus the dilogarithm is one of the simplest non-elementary functions one can imagine. It is also one of the strangest. It occurs not quite often enough, and in not quite an important enough way, to be included in the Valhalla of the great transcendental functions—the gamma function, Bessel and Legendre- functions, hypergeometric series, or Riemann’s zeta function. And yet it occurs too often, and in far too varied contexts, to be dismissed as a mere curiosity. First defined by Euler, it has been studied by some of the great mathematicians of the past—Abel, Lobachevsky, Kummer, and Ramanujan, to name just a few—and there is a whole book devoted to it. Almost all of its appearances in mathematics, and almost all the formulas relating to it, have something of the fantastical in them, as if this function alone among all others possessed a sense of humor.

Fundamental limits as differences of reciprocals

The three fundamental limits (as they were taught to me) are

{\displaystyle \lim_{x\to 0} \frac{e^x-1}{x} = \lim_{x\to 0} \frac{\log(1+x)}{x} = \lim_{x\to 0} \frac{\sin x}{x} = 1}

(exponential, logarithmic, trigonometric). Usually they come with pictures like

standard-pic
exp(x) – 1 and x

which show that the graph of the numerator in each limit, say {y=f(x)}, is indeed close to the line {y=x}.

But there are so many different degrees of “close”: for example, {0.0103\approx 0.01} but the reciprocals of these numbers differ by almost {3}. As a stress-test of the approximation {f(x)\approx x}, let us consider the behavior of {\displaystyle \frac{1}{f(x)} - \frac{1}{x}} for each of the fundamental limits.

Exponential limit

Expressed as a difference: {\displaystyle \frac{1}{e^x - 1} - \frac{1}{x}}

expm1
1/(exp(x)-1) – 1/x

Always negative, which is obvious once we recall that the graph of {e^x-1} lies above its tangent line. The graph has central symmetry about {(0, -1/2)} which is not as obvious but is an easy algebra exercise. Since the asymptote on the right is {y=0}, the other asymptote is {y=-1}. This looks a lot like the logistic function {1/(1+\exp(-x))} … but it is not, because the logistic function approaches its asymptotes exponentially fast, while our function does so at the rate {1/x}.
For comparison, here is the logistic curve {\displaystyle y = \frac{1}{1+\exp(-x/3)} - 1} (in green) scaled to match the behavior at {0}.

expm1-compare
Too slow to be logistic

This could be potentially useful if one needs a logistic-type function that behaves like a rational function at infinity. Simply using a rational function for this purpose would not do: it cannot have two distinct horizontal asymptotes.

Logarithmic limit

{\displaystyle \frac{1}{\log(x+1)} - \frac{1}{x}}

log1p
1/log(1+x) – 1/x

This one is always positive, and has a vertical asymptote at {x=-1} in addition to the horizontal asymptote at {0}. At a glance it may look like shifted/scaled hyperbola {y=1/x}. Indeed, {y = (1+x/3)/(2+x)} is a decent approximation to it, shown in green below.

log1p-compare
Not really a hyperbola

Trigonometric limit

{\displaystyle \frac{1}{\sin x} - \frac{1}{x}}

sin
1/sin(x) – 1/x

Unlike in the previous cases, the difference of reciprocals vanished at {0} due to the approximation {\sin x \approx x} being of higher order: the error term is cubic rather than quadratic. The graph looks like the tangent function but it cannot be exactly that since the nearest vertical asymptotes are {x=\pm \pi} rather than {x=\pm \pi/2}. (Not to mention other reasons such as non-periodicity.) A stretched and rescaled tangent, namely {\displaystyle \frac{1}{3}\tan \frac{x}{2}}, sort of fits:

sin-compare
Like a tangent?

Bonus limit

{\tan x \approx x}, with the reciprocal difference being {\displaystyle \frac{1}{\tan x} - \frac{1}{x}}.

tan
1/tan(x) – 1/x

The limit {\displaystyle \lim_{x\to 0} \frac{\tan x}{x} = 1} adds nothing new compared to the previous one, since {\cos 0=1}. But the difference of reciprocals is another story. For one thing, the principal error term for it is twice as large as for the sine limit: {-x/3} versis {x/6}. Accordingly, the graph looks more like {\displaystyle -\frac{2}{3}\tan \frac{x}{2}}.

tan-compare
Not really like a tangent

Taylor series

All of the above differences have derivatives of all orders at the origin, which is not easy to prove with the standard calculus tools. Complex analysis makes the situation clear: the reciprocal of a holomorphic function with a zero at {z=0} can be expanded into a Laurent series, and subtracting {1/z} eliminates the principal part of that series, leaving a convergent power series, i.e., another holomorphic function. Let us take a look at these series:

{\displaystyle \frac{1}{e^x - 1} - \frac{1}{x} = - \frac{1}{2} + \frac{x}{12} - \frac{x^{3}}{720} + \frac{x^{5}}{30240} - \frac{x^{7}}{1209600} + \frac{x^{9}}{47900160} -\cdots }

Nice, the coefficients have alternating signs and all of them have numerator {1}… oh no, next term is {\displaystyle - \frac{691 x^{11}}{1307674368000}}. The signs do continue to alternate. Apart from the constant term, only odd powers of {x} appear, according to the central symmetry noted above. The coefficient of {x^{n-1}} in this series is {B_n/n!} where {B_n} is the {n}th Bernoulli number. These are the “modern” Bernoulli numbers, with {B_1 = -1/2} rather than {1/2}.

{\displaystyle \frac{1}{\log(x+1)} - \frac{1}{x} = \frac{1}{2} - \frac{x}{12} + \frac{x^{2}}{24} - \frac{19 x^{3}}{720} + \frac{3 x^{4}}{160} - \frac{863 x^{5}}{60480} + \cdots }

Also alternating but not omitting even powers, and not decaying nearly as fast as the coefficients of the previous series. (Of course: this function has a singularity at {-1} while the nearest singularities of the exponential thing are at {\pm 2\pi i}.)  These are Gregory coefficients, which according to Wikipedia “often appear in works of modern authors who do not recognize them”. I would not recognize them either without OEIS.

{\displaystyle \frac{1}{\sin x} - \frac{1}{x} = \frac{x}{6} + \frac{7 x^{3}}{360} + \frac{31 x^{5}}{15120} + \frac{127 x^{7}}{604800} + \frac{73 x^{9}}{3421440} + \cdots }

These are all positive for some reason.

{\displaystyle \frac{1}{\tan x} - \frac{1}{x} = - \frac{x}{3} - \frac{x^{3}}{45} - \frac{2 x^{5}}{945} - \frac{x^{7}}{4725} - \frac{2 x^{9}}{93555} - \cdots }

These are all negative for some reason.

Discrete maximum principle for polynomials

The polynomially convex hull of a compact set {K\subset \mathbb C} is defined as the set of all points {z\in \mathbb C} such that the inequality {|p(z)|\le \sup_K |p|} (a form of the maximum principle) holds for every polynomial {p}. For example, the polynomially convex hull of a simple closed curve is the union of that curve with its interior region. In general, this process fills up the holes in the set {K}, resulting in the complement of the unbounded connected component of {\mathbb C\setminus K}.

We can recover the usual convex hull from this construction by restricting {p} to the polynomials of first degree. Indeed, when {p} is linear, the set {|p|\le M} is a closed disk, and we end up with the intersection of all closed disks that contain {K}. This is precisely the convex hull of {K}.

What if we restrict {p} to the polynomials of degree at most {n}? Let’s call the resulting set the degree-{n} convex hull of {K}, denoted {K_n}. By definition, it is contained in the convex hull and contains the polynomially convex hull. To exactly compute {K_n} for general {K} appears to be difficult even when {n=2}.

Consider finite sets. When {K} has at most {n} points, we have {K_n=K} because there is a polynomial of degree {n} whose zero set is precisely {K}. So, the first nontrivial case is of {K} having {n+1} points. Let us write {K=\{z_0, \dots, z_n\}}.

Depending on the location of the points, {K_n} may be strictly larger than {K}. For example, if {K} consists of the vertices of a regular {(n+1)}-gon, then {K_n} also contains its center. Here is why. By a linear transformation, we can make sure that {K=\{\zeta^k\colon k=0, \dots, n\}} where {\zeta = \exp(2\pi i/(n+1))}. For {j=1, \dots, n} we have {\sum_{k=0}^n \zeta^{kj} = (\zeta^{(n+1)j}-1)/(\zeta^j - 1) = 0}. Hence, for any polynomial {p} of degree at most {n}, the sum {\sum_{k=0}^n p(\zeta^k)} is equal to {(n+1)p(0)}. This implies {|p(0)|\le \max_{0\le k\le n}|p(\zeta^k)|}, a kind of a discrete maximum principle.

A more systematic approach is to use the Lagrange basis polynomials, that is {L_j(z) = \prod_{k\ne j} (z-z_k)/(z_j-z_k)}, which satisfy {L_j(z_k) = \delta_{jk}}. Since {p = \sum_j p(z_j)L_j} for any polynomial of degree at most {n}, it follows that {z\in K_n} if and only if {\left|\sum_j c_j L_j(z)\right|\le \max |c_j| } holds for every choice of scalars {c_0, \dots, c_n}. The latter is equivalent to the inequality {\sum_j |L_j(z)|\le 1}.

This leads us to consider the function {S=\sum_j |L_j|}, the sum of the absolute values of the Lagrange basis polynomials. (Remark: S is called a Lebesgue function for this interpolation problem.) Since {\sum_j L_j\equiv 1}, it follows that {S\ge 1} everywhere. By construction, {S=1} on {K}. At a point {z\notin K}, the equality {S(z)=1} holds if and only if {\arg L_j(z)} is the same for all {j}.

In the trivial case {K=\{z_0, z_1\}}, the function {S(z) = (|z-z_0|+|z-z_1|)/|z_0-z_1|} is equal to {1} precisely on the linear segment with endpoints {z_0, z_1}. Of course, this only repeats what we already knew: the degree-1 convex hull is the ordinary convex hull.

If {K=\{x_0, \dots, x_n\}} with {x_0<\cdots<x_n} real and {n\ge 2}, then {K_n=K}. Indeed, if {x\in K_n\setminus K}, then {x} lies in the convex hull of {K}, and therefore {x_{j-1}<x<x_j} for some {j}. The basis polynomial {L_j} is positive at {x}, since it is equal to {1} at {x_j} and does not vanish outside of {K}. Since a polynomial changes its sign at every simple zero, it follows that {L_{j+1}(x) < 0}. Well, there is no {L_{j+1}} if {j=n}, but in that case, the same reasoning applies to {L_{j-2}(x)<0}. In any case, the conclusion is that {\arg L_k(x)} cannot be the same for all {k}.

At this point one might guess that the vertex set of a regular polygon is the only kind of finite sets that admit a nontrivial discrete maximum principle for polynomials. But this is not so: the vertices of a rhombus work as well. Indeed, if {K=\{a, -a, ib, -ib\}} with {a, b>0}, then {L_j(0)>0} for all {j}, hence {S(0)=1}.

The vertices of a non-square rectangle do not work: if {K} is the set of these vertices, the associated function {S} is strictly greater than 1 on the complement of {K}.

Are there any other finite sets that support a discrete maximum principle for polynomials?

Institutions ranked by the number of AMS Fellows, 2020 edition

Only those with the count of 4 or greater are included. Not counting the deceased. Considering CUNY as a single institution. Source of data.

Top 10 changes compared to 2019 ranking: MIT takes sole possession of the 3rd place with Berkeley dropping into 4th, UIUC rises from 8th to 6th, Princeton drops from 6th to 8th, Stanford rises from 11th to 9th, Wisconsin-Madison drops from 8th to 10th, Illinois-Chicago rises from 13th to 10th. Due to ties, the “top 10” are actually top 12.

Honorable mention: Texas A&M rises from 19th to 16th. Having once set “top 20” as the goal of their “Vision 2020” campaign, they achieved it at least by this measure.

  1. Rutgers The State University of New Jersey New Brunswick : 44
  2. University of California, Los Angeles : 39
  3. Massachusetts Institute of Technology : 35
  4. University of California, Berkeley : 32
  5. University of Michigan : 31
  6. Cornell University : 26
  7. University of Illinois, Urbana-Champaign : 26
  8. Princeton University : 25
  9. Stanford University : 24
  10. Brown University : 23
  11. University of Illinois at Chicago : 23
  12. University of Wisconsin, Madison : 23
  13. New York University, Courant Institute : 22
  14. University of California, San Diego : 22
  15. University of Texas at Austin : 22
  16. Texas A&M University : 21
  17. University of Chicago : 21
  18. The City University of New York : 20
  19. University of Washington : 20
  20. Stony Brook University : 19
  21. University of Minnesota-Twin Cities : 19
  22. Purdue University : 17
  23. University of California, Santa Barbara : 17
  24. University of Pennsylvania : 17
  25. Duke University : 16
  26. Indiana University, Bloomington : 16
  27. Ohio State University, Columbus : 16
  28. University of Maryland : 16
  29. Georgia Institute of Technology : 15
  30. Northwestern University : 15
  31. Pennsylvania State University : 15
  32. University of California, Irvine : 14
  33. University of Utah : 14
  34. Johns Hopkins University, Baltimore : 13
  35. University of British Columbia : 12
  36. Boston University : 11
  37. Harvard University : 11
  38. University of California, Davis : 11
  39. University of Notre Dame : 11
  40. University of Toronto : 11
  41. Eidgenössische Technische Hochschule Zürich (ETH Zürich) : 10
  42. University of North Carolina at Chapel Hill : 10
  43. University of Virginia : 10
  44. Vanderbilt University : 10
  45. Brandeis University : 9
  46. Columbia University : 9
  47. Institute for Advanced Study : 9
  48. University of Oregon : 9
  49. Michigan State University : 8
  50. Rice University : 8
  51. Tel Aviv University : 8
  52. University of Georgia : 8
  53. University of Nebraska-Lincoln : 8
  54. University of Southern California : 8
  55. California Institute of Technology : 7
  56. Ecole Polytechnique Fédérale de Lausanne (EPFL) : 7
  57. Microsoft Research : 7
  58. North Carolina State University : 7
  59. University of Oxford : 7
  60. University of Rochester : 7
  61. Williams College : 7
  62. Carnegie Mellon University : 6
  63. Imperial College : 6
  64. Louisiana State University, Baton Rouge : 6
  65. The Hebrew University of Jerusalem : 6
  66. Université Pierre et Marie Curie (Paris VI) : 6
  67. University of Arizona : 6
  68. Harvey Mudd College : 5
  69. Northeastern University : 5
  70. Temple University : 5
  71. Université Paris-Diderot : 5
  72. University of California, Riverside : 5
  73. University of Colorado, Boulder : 5
  74. Virginia Polytechnic Institute and State University : 5
  75. Boston College : 4
  76. Florida State University : 4
  77. NYU Polytechnic School of Engineering : 4
  78. Norwegian University of Science and Technology : 4
  79. Rutgers The State University of New Jersey Newark : 4
  80. Université Paris-Sud (Paris XI) : 4
  81. University of California, Santa Cruz : 4
  82. University of Cambridge : 4
  83. University of Connecticut, Storrs : 4
  84. University of Missouri-Columbia : 4
  85. University of Tennessee, Knoxville : 4
  86. University of Warwick : 4
  87. Washington University : 4
  88. Weizmann Institute of Science : 4
  89. Yale University : 4
  90.