One of the most important axioms in analyzing voting systems is that of "neutrality", which stipulates that the system should treat all candidates symmetrically. Even though this doesn't always directly apply (such as in primary systems or those with intentional incumbent protection), it is extremely important both in theory and practice.If the voting systems in […]
This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We […]
By Hilbert’s theorem 90, if K is a cyclic number field with Galois group generated by g, then any element of norm 1 can be written as a/g(a). This gives […]
We describe a natural way to continuously extend arithmetic functions that admit a Ramanujan expansion and derive the conditions under which such an extension exists. In particular, we show that […]
Peg solitaire is a popular one person board game that has been played in many countries on various board shapes. Recently, peg solitaire has been studied extensively in two colors on […]
In 1932, Tarski conjectured that a convex body of width 1 can be covered by planks, regions between two parallel hyperplanes, only if the total width of planks is at least 1. In 1951, Bang proved the conjecture of Tarski. In this work we study the polynomial version of Tarski's plank problem. We note that […]
In this talk we link discrete Markov spectrum to geometry of continued fractions. As a result of that we get a natural generalization of classical Markov tree which leads to an efficient computation of Markov minima for all elements in generalized Markov trees.
I will explain how to apply presentations of algebras (together with some classical results from non-commutative algebra) to obtain some 5 polynomial invariants telling us when two pairs of 2x2 […]
As $\lambda$ runs through all integer partitions, the set of Schur functions $\{s_{\lambda}\}_\lambda$ forms a basis in the ring of symmetric functions. Hence the rule $$s_{\lambda}s_{\mu}=\sum c_{\lambda,\mu}^{\gamma} s_{\gamma}$$ makes sense […]
Suppose you are given a data set that can be viewed as a nonnegative integer-valued function defined on a finite set. A natural question to ask is whether the data […]
An elliptic curve $ E: y^2 + a_1 \, x \, y + a_3 \, y = x^3 + a_2 \, x^2 + a_1 \, x + a_6 $ is […]
The existence of a set $A\subset \N_0$ of positive upper Banach density such that $A-A:=\{m-n:m, n\in A, m>n\}$ does not contain a set of the form $S-S$ with $S$ a […]
