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Hao Pan

Publications and source records attributed to Hao Pan.

At least 127 records · Page 7Linked to original sources
arXiv2012

Restricted Sumsets in Finite Nilpotent Groups

Suppose that $A,B$ are two non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\dotplus B={a+b: a\in A, b\in B, a\neq b} $$ is at least $$\min{p(G),|A|+|B|-2},$$ where $p(G)$ denotes the least prime factor of $|G|$.

math.CO↗
arXiv2012

Factors of Some Lacunary q-Binomial Sums

In this paper, we prove a divisibility result for the lacunary $q$-binomial sum $$ \sum_{k\equiv r\pmod{c}}(-1)^kq^{\binom{k}{2}}\qbinom{n}{k}{q}\qbinom{(k-r)/c}{l}{q^{c}}. $$

math.NT↗
arXiv2011

On divisibility of sums of Apery polynomials

For any positive integers $m$ and $α$, we prove that $$\sum_{k=0}^{n-1}ε^k(2k+1)A_k^{(α)}(x)^m\equiv0\pmod{n}, $$ where $ε\in\{1,-1\}$ and $$ A_n^{(α)}(x)=\sum_{k=0}^n\binom{n}{k}^α\binom{n+k}{k}^αx^k.$$

math.NT↗
arXiv2010

On the Capability of Measurement-Based Quantum Feedback

As a key method in dealing with uncertainties, feedback has been understood fairly well in classical control theory. But for quantum control systems, the capability of measurement-based feedback control (MFC) has not been investigated systematically. In contrast to the control of classical systems where the measurement effect is negligible, the quantum measurement will cause a quantum state to collapse, which will inevitably introduce additional uncertainties besides the system initial uncertainty. Therefore, there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement, and thus a theoretical investigation of the capability of MFC is of fundamental importance. In this paper, inspired by both the Heisenberg uncertainty principle for quantum systems and the investigation of the feedback capability for classical systems, we try to answer the following three basic questions: (i) How to choose the measurement channel appropriately? (ii) Is the MFC still superior to the open loop control in dealing with the system information uncertainties? and (iii) What is the maximum capability or limitation of the MFC? These questions will be answered theoretically by establishing several theorems concerning the asymptotic reachability of eigenstates of a typical class of Hamiltonian control mode.

quant-ph↗
arXiv2010

On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{α(n),α(r)\}}$, where $α(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.

math.CO↗
arXiv2009

Some q-congruences related to 3-adic valuations

In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore $$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of the former congruence was conjectured by Guo and Zeng. In this paper we prove the Guo-Zeng conjecture and also give a $q$-analogue of the latter congruence.

math.NT↗
arXiv2009

A new extension of the Erdos-Heilbronn conjecture

Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\in A_1,...,x_n\in A_n, and x_i\not=x_j if i\not=j}. The result extends the Erdos-Heilbronn conjecture in a new way.

math.NT↗
arXiv2009

Some congruences for the second-order Catalan numbers

Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.

math.NT↗
arXiv2009

On the number of distinct prime factors of $nj+a^hk$

Let $ω(n)$ denote the number of distinct prime factors of $n$. Then for any given $K\geq 2$, small $ε>0$ and sufficiently large (only depending on $K$ and $ε$) $x$, there exist at least $x^{1-ε}$ integers $n\in[x,(1+K^{-1})x]$ such that $ω(nj\pm a^hk)\geq(\log\log\log x)^{{1/3}-ε}$ for all $2\leq a\leq K$, $1\leq j,k\leq K$ and $0\leq h\leq K\log x$.

math.NT↗