Related papers: Thirty-six Officers and their Code

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in $R^2$, this…

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of gcd(m,4) and the odd prime divisors of m. We show that |G|\le q(m)k^2/\phi(m) where \phi denotes Euler's totient function. This…

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be…

Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a…

We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X.…

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories,…

We show that there are infinitely many primes $p$ such that not only does $p + 2$ have at most two prime factors, but $p + 6$ also has a bounded number of prime divisors. This refines the well known result of Chen.

We propose a Law of Nature? Viz., Pure Regularity Occurs at Na\"ive Levels and Regularity has Affinity with Evenness. In a series of three papers, it was established that regular Euler graphs with only one type of (pure) cycles are…

Euler graphs with only one (two) type(s) of cycles under (mod 4) operation were studied in Part-I(II). Here we consider the class of Euler graphs with only three types of cycles under (mod 4). This gives rise to four cases viz., graphs…

There are fourteen fine gradings on the exceptional Lie algebra $\frak e_6$ over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that they are all.

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

This research announcement describes in very rough terms methods and a computer language under development, which can be used to prove the nonexistence of binary linear codes. Over a hundred new results have been obtained by the author. For…

An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise…

A recent breakthrough in computer-assisted mathematics showed that every set of $30$ points in the plane in general position (i.e., without three on a common line) contains an empty convex hexagon, thus closing a line of research dating…

Let $S$ be a set of $n\geq 7$ points in the plane, no three of which are collinear. Suppose that $S$ determines $n+1$ directions. That is to say, the segments whose endpoints are in $S$ form $n+1$ distinct slopes. We prove that $S$ is, up…

We prove new non-existence results for vectorial monomial Dillon type bent functions mapping the field of order $2^{2m}$ to the field of order $2^{m/3}$. When $m$ is odd and $m>3$ we show that there are no such functions. When $m$ is even…

We bring additional support to the conjecture saying that a rational cuspidal plane curve is either free or nearly free. This conjecture was confirmed for curves of even degree, and in this note we prove it for many odd degrees. In…

We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.