Related papers: On the number of harmonic frames

We exhibit finite cyclic group representations for relation algebras $57_{65}$ and $63_{65}$. As a consequence, of the ten symmetric integral RAs on four atoms having at least one flexible atom, all are now known to have a representation…

This work continues the study of the properties of finitely constrained groups of binary tree automorphisms in terms of their Hausdorff dimension. We prove that there are exactly $2^{2d-3}$ finitely constrained groups of binary tree…

Let $n$ be a natural number larger than two. Let $D_{2n}=\langle r,s : r^{n}=s^{2}=e, srs=r^{n-1} \rangle$ be the Dihedral group, and $\kappa $ an $n$-dimensional unitary representation of $D_{2n}$ acting in $\mathbb{C}^n$ as follows.…

Fix a dimension $d\ge 2$, and let $T_n$ be a random $d$-dimensional determinantal hypertree on $n$ vertices. We prove that \[\frac{\log|H_{d-1}(T_n,\mathbb{Z})|}{{{n\choose {d}}}}\] converges in probability to a constant $c_d$, which…

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set…

We construct and study finitely presented groups with quadratic Dehn function (QD-groups) and present the following applications of the method developed in our recent papers. (1) The isomorphism problem is undecidable in the class of…

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ is the subspace of $C(G)$, spanned by the…

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in…

For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is…

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this…

The symmetric group S_n acts as a reflection group on CP^{n-2} (for $n\geq 3$) . Associated with each of the $\binom{n}{2}$ transpositions in S_n is an involution on CP^{n-2} that pointwise fixes a hyperplane--the mirrors of the action. For…

We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$ is the number of ways to subdivide the $d$-dimensional hypercube into $n$ rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \& Dotsenko…

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that…

We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and if so returns a set of representatives of the twisted conjugacy classes.…

By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of…

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly…

We prove that if G is an (n,d,lambda)-graph (a d-regular graph on n vertices, all of whose non-trivial eigenvalues are at most lambda) and the following conditions are satisfied: 1. d/lambda >= (log n)^{1+epsilon} for some constant…

The asymptotic study of the conjugacy classes of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of…

In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds…