On generalized Erdős–Ginzburg–Ziv constants of Cnr

Let $G$ be an additive finite abelian group with exponent $exp\left(G\right)=n$. For any positive integer $k$, the $k$th Erdős–Ginzburg–Ziv constant ${\mathsf{s}}_{kn}\left(G\right)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $kn$. It is easy to see that ${\mathsf{s}}_{kn}\left(C_n^r\right)\ge (k+r)n-r$ where $n,r\in \mathbb{N}$. Kubertin conjectured that the equality holds for any $k\ge r$. In this paper, we prove the following results:

• •[(1)] For every positive integer $k\ge 6$, we have ${\mathsf{s}}_{kn}\left(C_n^3\right)=(k+3)n+O\left(\frac{n}{lnn}\right).$
• •[(2)] For every positive integer $k\ge 18$, we have ${\mathsf{s}}_{kn}\left(C_n^4\right)=(k+4)n+O\left(\frac{n}{lnn}\right).$
• •[(3)] For $n\in \mathbb{N}$, assume that the largest prime power divisor of $n$ is $p^a$ for some $a\in \mathbb{N}$. Forany fixed $r\ge 5$, if $p^t\ge r$ for some $t\in \mathbb{N}$, then for any $k\in \mathbb{N}$ we have ${\mathsf{s}}_{k{p}^{t}n}\left(C_n^r\right)\le (k{p}^t+r)n+c_r\frac{n}{lnn},$where $c_r$ is a constant that depends on $r$.

Our results verify the conjecture of Kubertin asymptotically in the above cases.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.     

Pseudo Frobenius numbers

For a prime $p$, we call a positive integer $n$ a Frobenius $p$-number if there exists a finite group with exactly $n$ subgroups of order $p^a$ for some $a\ge 0$. Extending previous results on Sylow’s theorem, we prove in this paper that every Frobenius $p$-number $n\equiv 1(mod{p}^2)$ is a Sylow $p$-number, i. e., the number of Sylow $p$-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order $3^a$ for any $a\ge 0$.     

Common values of a class of linear recurrences

Let $\left(a_n\right),\left(b_n\right)$ be linear recursive sequences of integers with characteristic polynomials $A\left(X\right),B\left(X\right)\in \mathbb{Z}\left[X\right]$ respectively. Assume that $A\left(X\right)$ has a dominating and simple real root $\alpha$, while $B\left(X\right)$ has a pair of conjugate complex dominating and simple roots $\beta ,\stackrel{?}{\beta }$. Assume further that $\alpha ,\beta ,\alpha /\beta$ and $\stackrel{?}{\beta }/\beta$ are not roots of unity and $\delta =log\left|\beta \right|/log\left|\alpha \right|\in \mathbb{Q}$. Then there are effectively computable constants $c_0,c_1>0$ such that the inequality $|a_n?b_m|>{\left|a_n\right|}^{1?\left(c_0{log}^{2}n\right)/n}$holds for all $n,m\in {\mathbb{Z}}_{\ge 0}^2$ with $max\{n,m\}>c_1$. We present $c_0$ explicitly.     

Hilbert’s Nullstellensatz for analytic trigonometric polynomials

This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if ${\left\{f_j\right\}}_{j=1}^{n\ge 2}$ are analytic trigonometric polynomials without common zero in the finite complex plane $?$ then there are analytic trigonometric polynomials ${\left\{g_j\right\}}_{j=1}^{n\ge 2}$ obeying ${\sum }_{j=1}^{n\ge 2}{f}_j{g}_j=1$ in $?$, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on $?$.     

Decomposition of Schramm–Loewner evolution along its curve

We show that, for $\kappa \in (0,8)$, the integral of the laws of two-sided radial SLE${}_{\kappa }$ curves through different interior points against a measure with SLE${}_{\kappa }$ Green’s function density is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s natural length. We also show that, for $\kappa >0$, the integral of the laws of extended SLE${}_{\kappa }(?8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s capacity length restricted in that set. Another result is that, for $\kappa \in (4,8)$, if one integrates the laws of two-sided chordal SLE${}_{\kappa }$ curves through different force points on $\mathbb{R}$ against a measure with density on $\mathbb{R}$, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE${}_{\kappa }$ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.     

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.     

On a Riemann–Hilbert problem for the Fokas–Lenells equation

The solution of the initial value problem (IVP) for the Fokas–Lenells equation (FLE) was constructed in terms of the solution $M(x,t,k)$ of a 2 $\times$ 2 matrix Riemann–Hilbert problem (RHP) as $k\to \infty$, and the one-soliton solution of the FLE was derived based on this Riemann–Hilbert problem, in Lenells and Fokas (2009). However, in fact, the derivative with respect to $x$ of the solution of the FLE ($u_x(x,t)$) was recovered from the RHP as $k\to \infty$. In this paper, we construct the solution of the FLE in terms of the RHP as $k\to 0$, because the Lax pair of the FLE contains the negative order of the spectral variable $k$. We show that the one-soliton solution of the FLE obtained in this paper is the same as Lenells and Fokas (2009), but avoiding a complex integral.     

Blowup analysis for integral equations on bounded domains

Consider the integral equation

$f^{q?1}(x)=\underset{\mathrm{\Omega }}{\int }\frac{f(y)}{|x?y{|}^{n?\alpha }}dy,f(x)>0,x\in \stackrel{￣}{\mathrm{\Omega }},$

where $\mathrm{\Omega }?{\mathbb{R}}^n$ is a smooth bounded domain. For $1<\alpha <n$, the existence of energy maximizing positive solution in the subcritical case $2<q<\frac{2n}{n+\alpha }$, and nonexistence of energy maximizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are proved in [6]. For $\alpha >n$, the existence of energy minimizing positive solution in the subcritical case $0<q<\frac{2n}{n+\alpha }$, and nonexistence of energy minimizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{+}$ (in the case of $1<\alpha <n$), and the blowup behaviour of energy minimizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{?}$ (in the case of $\alpha >n$) are analyzed. We see that for $1<\alpha <n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for $\alpha >n$, different phenomena appear.     

On the growth factor upper bound for Aasen’s algorithm

Aasen’s algorithm factorizes a symmetric indefinite matrix $A$ as $A=P^{T}LT{L}^{T}P$, where $P$ is a permutation matrix, $L$ is unit lower triangular with its first column being the first column of the identity matrix, and $T$ is tridiagonal. In this note, we provide a growth factor upper bound for Aasen’s algorithm which is much smaller than that given by Higham. We also show that the upper bound we have given is not tight when the matrix dimension is greater than or equal to 6.     

Interior gradient estimate for steady flows of electrorheological fluids

In this paper, we consider the equations of stationary motion of electrorheological fluids in any dimension $n\ge 2$. We show that the first gradient of local solutions to the system has optimal $L^q$-regularity with some $q>1$ with respect to the one of the external force. This is achieved by comparison principle and a good $\lambda$-estimate.     

Zero-sum invariants on finite abelian groups with large exponent

Let $G$ be an additive finite abelian group with exponent $n$. Let $\mathsf{D}\left(G\right)$ be the Davenport constant of $G$, ${\mathsf{s}}_{kn}\left(G\right)$ the $k$th Erd?s–Ginzburg–Ziv constant of $G$, where $k$ is a positive integer. Recently, Gao, Han, Peng and Sun conjectured that ${\mathsf{s}}_{kn}\left(G\right)=kn+\mathsf{D}\left(G\right)?1$ holds if $k\ge ?\frac{\mathsf{D}\left(G\right)}{n}?$. Let $m,n$ be positive integers and $H$ an abelian $p$-group with $\mathsf{D}\left(H\right)\le p^n$. Let $G=H\oplus C_{m{p}^n}$. For any integer $k\ge 2$, we prove that ${\mathsf{s}}_{km{p}^n}\left(G\right)=(k+1)m{p}^n+\mathsf{D}\left(H\right)?2=km{p}^n+\mathsf{D}\left(G\right)?1.$ This verifies the above conjecture in this case. We also provide asymptotically tight bounds for zero-sum invariants $\mathsf{D}\left(G\right)$, ${\mathsf{s}}_{kn}\left(G\right)$ and $\eta \left(G\right)$ for a class of abelian groups with large exponent.     

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.