108 votes

Which high-degree derivatives play an essential role?

Given two sets $A$ and $B$ in $\mathbb{R}^n$, the Minkowski sum written $A+B$ is the set $\{a+b:a\in A,b\in B\}$. If $A$ and $B$ are convex subsets of $\mathbb{R}^2$ with real-analytic boundaries ...
79 votes

Mathematical conjectures on which applications depend

An important specific conjecture is that you cannot factor large integers fast. Many security systems for Internet and other transactions, depend on this.
63 votes

Which high-degree derivatives play an essential role?

Moser’s theorem in (1962) famously required estimates on the first 333 derivatives.
50 votes

Mathematical conjectures on which applications depend

It is an open problem to resolve a question formalized by G. Shephard in 1975: Q. Can the surface of every convex polyhedron be cut along edges and unfolded flat to one non-self-overlapping ...
42 votes

Which high-degree derivatives play an essential role?

There is a famous story that Richard Nixon once made use of the third time derivative to support his re-election, via a claim that the rate of increase of inflation was decreasing. http://www.ams.org/...
39 votes

Mathematical conjectures on which applications depend

The use of RSA for public-key encryption is widely believed to rely on the assumption that factoring is hard. Actually it relies on a stronger assumption than this, namely that the RSA problem is hard....
39 votes

Which high-degree derivatives play an essential role?

The error in Simpson's rule for integration is usually expressed in terms of the fourth derivative of the integrand.
39 votes

Which high-degree derivatives play an essential role?

In "classical (Euler-Bernoulli) beam theory" the motion of a beam is modelled by the 4th-order PDE $$ EI \frac{\partial^4 w}{\partial x^4} = -\mu \frac{\partial^2 w}{\partial t^2} + q. $$
35 votes

Mathematical conjectures on which applications depend

The Miller-Rabin primality test works very well in practice as a probabilistic algorithm for finding "practical" (not provable) primes in cryptography, but the algorithm would become an efficient ...
34 votes

On Mathematical Analysis of MathSciNet & MathOverflow

• Mathoverflow has been studied as a "complex network" in Social achievement and centrality in MathOverflow, by L.V. Montoya, A. Ma, and R.J. Mondragón. The analysis distinguishes degree centrality (...
Carlo Beenakker's user avatar
31 votes
Accepted

Why is persistent cohomology so much faster than persistent homology

There are several factors contributing to the improved performance of the algorithm reported in the paper; the use of cohomology is one, but there is also a computational shortcut involved, and the ...
Ulrich Bauer's user avatar
30 votes

Examples of theorems misapplied to non-mathematical contexts

This recent article is a striking example of debunking a misuse of mathematics in social sciences. In short, some diversity scholars had claimed to prove a "theorem" that diverse groups of less able ...
30 votes

Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

• Concerning question 2, you might want to take a look at Simulation of cubical particle packing under mechanical vibration (2016). The precise effect mentioned in the 2017 paper is not considered in ...
Carlo Beenakker's user avatar
30 votes

Which high-degree derivatives play an essential role?

Nonlinear solitonic wave equations often feature high($3^+$) order of derivatives. The most famous one may be the KdV equation: $\partial_t \phi + \partial_{xxx} \phi -6 \phi \partial_x \phi = 0$. ...
29 votes

Do bubbles between plates approximate Voronoi diagrams?

The soap froth has a dynamics that Voronoi diagrams lack. The two-dimensional network of soap bubbles evolves in time according to the area law $$\frac{dA}{dt}=k(n-6),\qquad\qquad(*)$$ where $A$ is ...
Carlo Beenakker's user avatar
25 votes

Which high-degree derivatives play an essential role?

In optimal transport, there is a quantity known as the MTW tensor [1] which depends on the fourth derivatives of the cost function. The regularity theory of transport depends in a crucial way on the ...
25 votes
Accepted

Importance of integral equations

One important point is that differential equations encode local behaviour of a system, while integral equations typically endcode global behaviour. Local behaviour is often easier to model and to ...
Jochen Glueck's user avatar
23 votes

On Mathematical Analysis of MathSciNet & MathOverflow

With regards to reputation on Stack Exchange, I did a very short analysis last year on the distribution of reputation on Stack Overflow. Thanks to the Stack Exchange Data Explorer, I can easily run ...
Glorfindel's user avatar
  • 2,743
22 votes

Mathematical conjectures on which applications depend

Although we have a proof now, I suspect grocers had intuitively stacked oranges in the most efficient way prior to the proof of Kepler's conjecture. I don't know for sure if any grocer was an applied ...
22 votes

Is the field of q-series 'dead'?

Jehanne Dousse has recently obtained a sought-after CNRS position. From what I remember seeing, she was inundated with job interviews. This should disprove the death of q-series, at least as far as ...
21 votes

On Mathematical Analysis of MathSciNet & MathOverflow

In 2010, Joseph F. Grcar published a paper in the Notices of the AMS entitled Topical Bias in Generalist Mathematics Journals. The paper analyzed data from 2000 to 2009 in Zentralblatt to investigate ...
Timothy Chow's user avatar
  • 78.1k
20 votes

Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

I doubt that a mathematically rigorous explanation of the phenomenon discovered in that paper exists using today's technology. While mathematical statistical mechanics is a well-developed field of ...
j.c.'s user avatar
  • 13.5k
20 votes

Mathematical conjectures on which applications depend

Navier Stokes equations are believed to be well-posed.
20 votes

Is the field of q-series 'dead'?

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given. On ...
18 votes

Which high-degree derivatives play an essential role?

The Kuramoto-Sivashinsky equation $$\partial_tu+\Delta^2u+\Delta u+\frac12|\nabla u|^2=0$$ where $\Delta$ is the Laplace operator (second order) was derived to model diffusive instabilities in a ...
17 votes

What "real life" problems can be solved using billiards?

Gregory Galperin invented billiard method of computing $\pi$, see Playing Pool With $\pi$ (The Number $\pi$ From A Billiard Point Of View) To calculate $\pi$, take two identical balls. Put one near a ...
Alexey Ustinov's user avatar
17 votes

Recent uses of applied mathematics in pure mathematics

If mathematical developments in physics count as "applied mathematics" there are many examples --- as requested by the OP here is a recent one (< 30 years old) and an older one: Gauge ...
17 votes

Importance of integral equations

In physics, the predominance of differential over integral equations is not that obvious. Any system with a "memory", where the response at a certain time depends on the state at earlier ...
Carlo Beenakker's user avatar
16 votes
Accepted

Do bubbles between plates approximate Voronoi diagrams?

There are two nice connections to generalizations of Voronoi diagrams that I'm aware of. Moukarzel showed that 2D soap bubbles are sectional multiplicative Voronoi partition (SMVP), i.e. 2D slices (...
j.c.'s user avatar
  • 13.5k

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