User matthias beck - MathOverflow

Answer by matthias beck for Lower bound for constrained ordered partitions (i.e., compositions)?

2013-01-19T07:20:58Z

The geometry behind your problem is that you intersect the $m$-dimensional cube $[0,N]^m$ with the $(m-1)$-dimensional simplex given by the constraints $a_j \ge 0$ and $a_1 + \cdots + a_m = n$. Your question ("$n$ in a certain range") is a bit vague, but from this geometric point of view, the power $m$ of your lower bound seems to be too big. I could certainly get you a lower bound $c \cdot N^{m-1}$ if you choose the range for $n$ carefully enough.

Answer by matthias beck for Video lectures of mathematics courses available online for free

2011-02-05T20:43:13Z

Federico Ardila's (full-semester) courses on polytopes, combinatorial commutative algebra, Coxeter groups, combinatorial Hopf algebras, and matroid theory. They include lecture videos and lecture notes. See http://math.sfsu.edu/federico/teaching.html

Now they are also on YouTube here: polytopes, combinatorial commutative algebra, Coxeter groups, combinatorial Hopf algebras, and matroid theory.

Answer by matthias beck for Showing that a family of polynomials has positive and real roots.

2012-08-01T03:43:10Z

This paper by P. Bränden and this paper by M. Visontai & N. Williams give a somewhat general approach to proving real rootedness of polynomials (especially those coming up combinatorially).

Answer by matthias beck for Example of Mobius Inversion on Integer Partition Poset

2011-12-09T02:27:18Z

If you mean the poset you get with the relation "refinement" then its Möbius function is not known (see Exercise 122 in Chapter 3 of Stanley's Enumerative Combinatorics, 2nd edition).

Answer by matthias beck for Using slides in math classroom

2011-11-04T18:37:30Z

I use a hybrid version for some of my classes which take place in a room that allows this: I use computer slides (and animations, computations, etc.) and the board. I learned this from my colleague Serkan Hosten, and it works really well in some classes. E.g., I use slides for definitions and theorems (including the relevant ones from the previous lecture) but then work out examples and proofs on the board. This has the obvious advantage of spending time on exactly the items that need time and just the right pauses to get digested, but it also has nice side effects: e.g., the statement of the theorem will stay on the screen even if I'll have to clean the board.

Answer by matthias beck for non negative integer solutions : Diophantine Equations

2011-11-03T19:42:41Z

There's a detailed discussion of your function $d(n;a_1, \dots, a_k$) in Chapters 1 & 8 of a book I wrote with Sinai Robins (and both chapters contain further pointers to the literature, including the Frobenius problem).

Answer by matthias beck for Open source mathematical software.

2011-03-17T03:03:18Z

... and probably polytopes.

I recommend polymake for general polytope computations, LattE for lattice-point enumeration, Normaliz for computations related to lattice polytopes, and 4ti2 for Gröbner basis computations.

Answer by matthias beck for Generalizations of the Birkhoff-von Neumann Theorem

2011-02-11T23:13:35Z

Dave Perkinson and coauthors have studied sub-polytopes of the Birkhoff-von Neumann polytope, in the sense that they consider convex hulls of the permutation matrices corresponding to certain subgroups of $S_n$. See, e.g., his paper with Jeff Hood for the case $A_n$.

Answer by matthias beck for Counting integral points of a polytope in R^3 (the c_1 coefficient of Ehrhart polynomial)

2010-11-19T06:18:18Z

Jamie Pommersheim gave a general formula for $c_1$ in his 1993 paper in Math. Ann. for the case of a tetrahedron. So if you can easily triangulate your polytope, this might be useful.

If you're "only" interested in bounds, you might want to look at the Ehrhart series instead, i.e., the generating function of the Ehrhart polynomial. This is a rational function with denominator (1-x)^4, so the information encoded in the numerator is the same information encoded in the Ehrhart polynomial (in fact, it's a linear transformation). A famous theorem of Stanley states that the numerator coefficients of the Ehrhart series are nonnegative, and of course this gives you inequalities among the coefficients of the Ehrhart polynomial. For more inequalities of this type, check out Alan Stapledon's work (e.g., http://front.math.ucdavis.edu/0904.3035 or http://front.math.ucdavis.edu/0801.0873).

(And thanks for your kind words, Andres.)

Answer by matthias beck for Rational Dilates of Integral Convex Polytopes

2010-11-19T05:59:14Z

R. Diaz, S. Robins, and I studied your question for the inequality $b_1 x_1 + \dots + b_d x_d \le t$ for integral $t$ (which gives a rational polytope) in http://front.math.ucdavis.edu/math.NT/0204035. The case where $t$ is truly a rational variable is more complicated. http://front.math.ucdavis.edu/1006.5612 is a starting point...

Answer by matthias beck for What's your favorite equation, formula, identity or inequality?

2010-08-21T06:43:50Z

Pick's theorem $A = I + \frac 1 2 B - 1$, where $A$, $I$, and $B$ are the area, number of interior integer points, and number of boundary integer points, respectively, of a polygon with vertices on the integer lattice. Picks identity is fascinating because it computes a continuous quantity completely discretely. (Of course, this is not quite correct, since we have quite a discrete requirement about the vertices of the polygon.) Also, the "1" is not an accident, but the Euler characteristic of the polygon (and so there are various natural extensions of Pick's theorem).

Answer by matthias beck for Math puzzles for dinner

2010-06-24T05:21:18Z

(I learned this problem from Persi Diaconis.) A deck of $n$ different cards is shuffled and laid on the table by your left hand, face down. An identical deck of cards, independently shuffled, is laid at your right hand, also face down. You start turning up cards at the same rate with both hands, first the top card from both decks, then the next-to-top cards from both decks, and so on. What is the probability that you will simultaneously turn up identical cards from the two decks? What happens as $n \to \infty$? And does the answer for small $n$ (say, $n=7$) differ greatly from $n=52$?

Comment by matthias beck

2012-01-14T17:37:55Z

This is an exercise, and this unsuitable for MO.

Comment by matthias beck

2011-05-04T15:26:27Z

This follows from a finite-Fourier-series version of Cauchy-Schwartz. The earliest reference I'm aware of is H. Rademacher, Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956) 445-463. With some work, one can obtain even better bounds (see, e.g., <a href="front.math.ucdavis.edu/math.NT/… paper with S. Robins and S. Zacks</a>).

Comment by matthias beck

2011-03-08T00:58:04Z

If your parameters are real, there are no simple formulas. The best results in this case are bounds (for the exact counting problem) given in a series of papers by Stephen Yau (UIC) and coauthors.

Comment by matthias beck

2011-03-03T06:56:09Z

It depends what you mean by "exact"... the two papers I mentioned are the state of the art, as far as I know.

Comment by matthias beck

2010-12-28T22:40:03Z

Math circles are all over the map, both location-wise and with respect to age groups. mathcircles.org has more than enough info for anyone interested...

Comment by matthias beck

2010-12-15T03:16:13Z

The above-mentioned twiddla gives you a free "Pro account" (which allows you, e.g., to store your white board sessions) if you have an email address ending with .edu -- see the bottom of twiddla.com/Pricing.aspx for details.

Comment by matthias beck

2010-08-22T06:05:33Z

@ Robin: I'm not responsible for the title... :) But there is certainly a connection here.