On the number of harmonic frames

There is a finite number $h_{n,d}$ of tight frames of n distinct vectors for ${\mathbb{C}}^d$ which are the orbit of a vector under a unitary action of the cyclic group ${\mathbb{Z}}_n$. These cyclic harmonic frames (or geometrically uniform tight frames) are used in signal analysis and quantum information theory, and provide many tight frames of particular interest. Here we investigate the conjecture that $h_{n,d}$ grows like $n^{d-1}$. By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimate$h_{n,d}\approx \frac{n^d}{\phi (n)}\ge n^{d-1},n\to \infty .$ By using a group theoretic approach, we also give some exact formulas for $h_{n,d}$, and estimate the number of cyclic harmonic frames up to projective unitary equivalence.     

A formula about W-operator and its application to Hurwitz number

$W$-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the $W$-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the $W$-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between $W$-operator $W\left(\left[d\right]\right)$ and the central elements $K_{1^{n?d}d}$ in $?S_n$. Based on the relation we find, we give another proof about a differential equation of the generating function of $d$-Hurwitz number.     

Tensor products of n-complete algebras

If A and B are n- and m-representation finite k-algebras, then their tensor product $\mathrm{\Lambda }=A{?}_{k}B$ is not in general $(n+m)$-representation finite. However, we prove that if A and B are acyclic and satisfy the weaker assumption of n- and m-completeness, then Λ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve d-representation finiteness in general, but it does preserve d-completeness. As a corollary, we get the necessary condition for Λ to be $(n+m)$-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi–Yau property.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

The space of lines in cyclic covers of projective space

Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ${\mathbb{P}}^n$ ramified over a general hypersurface $X\subset P^n$ of degree md. In this paper we study the space $F(Y)$ of lines in Y and show that it is smooth of dimension $2(n-1)-d(m-1)$ if $md>2n-3$ and $2(n-1)-d(m-1)\ge 0$. When $2(n-1)=d(m-1)$, our result gives a formula on the number of m-contact order lines of X (see Definition 1.2).     

Minimum degree of 3-graphs without long linear paths

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least $2d$ (Dirac, 1952). In this article, we give an extension of Dirac’s result to hypergraphs. We determine asymptotic lower bounds of the minimum degrees of 3-graphs to guarantee linear paths of specific lengths, and the lower bounds are tight up to an error term depending only on the lengths of the paths.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

On the centre of mass of a random walk

For a random walk $S_n$ on ${\mathbb{R}}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n=n^{?1}{\sum }_{i=1}^n{S}_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d\ge 2$. In the transient case we show that $G_n$ has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.     

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and $m,n,d$ positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are $m\times n$ matrices over R and two matrices A and B are adjacent if and only if $0<\mathrm{rank}(A-B)<d$. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.