Related papers: One class genera of lattice chains over number fie…
Let $\Lambda$ be a lattice in $\R^n$, and let $Z\subseteq \R^{m+n}$ be a definable family in an o-minimal structure over $\R$. We give sharp estimates for the number of lattice points in the fibers $Z_T={x\in \R^n: (T,x)\in Z}$. Along the…
We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification…
The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice…
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…
If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give…
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating…
We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We…
In this note we prove that the Chermak-Delgado lattice of a ZM-group is a chain of length $0$. A similar conclusion is obtained for all dihedral groups $D_{2m}$ with $m\neq 4$.
This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze…
We prove that the number of symmetric chain decompositions of the Boolean lattice $2^{[n]}$ is $$\left(\frac{n}{2e}+o(n)\right)^{2^n}.$$ Furthermore, the number of symmetric chain decompositions of the hypergrid $[t]^n$ is…
We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…
We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices,…
For a positive integer n, we denote by SUB (resp., SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (resp., P of length at most n). We prove the following…
We show that several classes of ordered structures (namely, convex linear orders, layered permutations, and compositions) admit first-order logical limit laws.
We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements.…
We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly…
A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and…
A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the $L^2$ norm all lattices of dimension $n$ are standard if…