Related papers: The q-Queens Problem: One-Move Riders on the Recta…
The author has constructed and enumerated tours of knight having various magic properties on 4 x n and 6 x n boards. 16 magic tours of knight have been discovered on 4 x 18 board, 88 on 4 x 20 board, 464 on 4 x 22 board, 2076 on 4 x 24…
We provide the following result and its discrete equivalent: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist a point $p \in \mathbb{R}^{n-1}$ and a compact subset $S \subset…
Some preliminary results are reported on the equivalence of any n-queens problem with the roots of a Boolean valued quadratic form via a generic dimensional reduction scheme. It is then proven that the solutions set is encoded in the…
We apply periodic orbit theory to a quantum billiard on a torus with a variable number N of small circular scatterers distributed randomly. Provided these scatterers are much smaller than the wave length they may be regarded as sources of…
A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…
The structure of the semiclassical trace formula can be used to construct a quasi-classical evolution operator whose spectrum has a one-to-one correspondence with the semiclassical quantum spectrum. We illustrate this for marginally…
Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…
We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths.…
In this paper, we introduce polynomial time algorithms that generate random $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions with uniform probability. A $k$-noncrossing partition does not contain any $k$ mutually…
In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$…
We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of…
Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of…
The classical inner and outer billiards can be formulated in variational terms, with length and area as the respective generating functions. The other two combinations, ``inner with area'' and ``outer with length,'' are more recently…
We prove a new tableaux formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking…
The vicious random walker problem on a one dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the…
We develop a general mapping from given kink or pulse shaped travelling-wave solutions including their velocity to the equations of motion on one-dimensional lattices which support these solutions. We apply this mapping - by definition an…
We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces…
The main result in this paper is a non-existence Theorem of entire $Q_{n-1}$-translators in $\mathbb{R}^{n+1}$. In addition, an example of non-entire complete $Q_{n-1}$-translator has been found and a Tangential Principle for…
We generalize Rock Paper Scissors to complete directed graphs, or tournaments, on $n$ vertices. Properties of the mixed-strategy Nash equilibria of these tournaments are discussed, particularly those with Nash equilibria where all of the…
We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvatal and Erdos. We show that in the (m:b) clique game played on K_{N}, the complete graph on N vertices, Maker can achieve a K_{q} for q = (m/(log_{2}(b +…