arXiv subjects

Song Guo

Publications and source records attributed to Song Guo.

208 records · Page 12Linked to original sources

arXiv2017

A recursive algorithm for trees and forests

Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.

math.CO↗

arXiv2016

Proof of a recent conjecture of Z.-W. Sun

The polynomials $d_n(x)$ are defined by \begin{align*} d_n(x) &= \sum_{k=0}^n{n\choose k}{x\choose k}2^k. \end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \begin{align*} \sum_{k=0}^{p-1}\frac{2k\choose k}{4^k} d_k\left(-\frac{1}{4}\right)^2 &\equiv \begin{cases} 2(-1)^{\frac{p-1}{4}}x,&\text{if $p=x^2+y^2$ with $x\equiv 1\pmod{4}$,} 0,&\text{if $p\equiv 3\pmod{4}$,} \end{cases} [5pt] \sum_{k=0}^{p-1}\frac{2k\choose k}{4^k} d_k\left(-\frac{1}{6}\right)^2 &\equiv 0, \quad\text{if $p>3$,} [5pt] \sum_{k=0}^{p-1}\frac{2k\choose k}{4^k} d_k\left(\frac{1}{4}\right)^2 &\equiv \begin{cases} 0,&\text{if $p\equiv 1\pmod{4}$,} (-1)^{\frac{p+1}{4}}{\frac{p-1}{2}\choose \frac{p-3}{4}},&\text{if $p\equiv 3\pmod{4}$.} \end{cases} \sum_{k=0}^{p-1}\frac{2k\choose k}{4^k} d_k\left(\frac{1}{6}\right)^2 &\equiv 0, \quad\text{if $p>5$.} \end{align*} The $p\equiv 3\pmod{4}$ case of the first one confirms a conjecture of Z.-W. Sun, while the second one confirms a special case of another conjecture of Z.-W. Sun.

math.NT↗

arXiv2016

Technique Report: Near-Optimal Routing Protection for SDN Networks Using Distributed Markov Approximation

Software Defined Networking (SDN) brings numbers of advantages along with many challenges. One particular concern is on the control-plane resilience, while the existing protection approaches proposed for SDN networks mainly focus on data-plane. In order to achieve the carrier-grade recovery from link failures, we adopt the dedicated protection scheme towards finding optimal protection routing for control-plane traffic. To this end, we study a weighted cost minimization problem, in which the traffic load balancing and flow table rule placement are jointly considered when selecting protection paths for controller-switch sessions. Because this problem is known as NP-hard, we propose a Markov approximation based combinatorial optimization approach for routing protection in SDN control-plane, which produces near-optimal solution in a distributed fashion. We then extend our solution to an on-line case that can handle the single-link failure one at a time. The induced performance fluctuation is also analyzed with theoretical derivation. Extensive experimental results show that our proposed algorithm has fast convergence and high efficiency in resource utilization.

cs.DC↗

arXiv2016

Providing Probabilistic Robustness Guarantee for Crowdsensing

Due to its flexible and pervasive sensing ability, crowdsensing has been extensively studied recently in research communities. However, the fundamental issue of how to meet the requirement of sensing robustness in crowdsensing remains largely unsolved. Specifically, from the task owner's perspective, how to minimize the total payment in crowdsensing while guaranteeing the sensing data quality is a critical issue to be resolved. We elegantly model the robustness requirement over sensing data quality as chance constraints, and investigate both hard and soft chance constraints for different crowdsensing applications. For the former, we reformulate the problem through Boole's Inequality, and explore the optimal value gap between the original problem and the reformulated problem. For the latter, we study a serial of a general payment minimization problem, and propose a binary search algorithm that achieves both feasibility and low payment. The performance gap between our solution and the optimal solution is also theoretically analyzed. Extensive simulations validate our theoretical analysis.

cs.GT↗

arXiv2015

High-spin level structure of the neutron-rich nucleus 91Y

High-spin level structure of the neutron-rich nucleus 91Y has been reinvestigated via the 82Se(13C, p3n)91Y reaction. A newly constructed level scheme including several key levels clarifies the uncertainties in the earlier studies. These levels are characterized by the breaking of the Z=38 and N=56 subshell closures, which involves in the spin-isospin dependent central force and tensor force.

nucl-ex↗

arXiv2009

On Bialostocki's conjecture for zero-sum sequences

Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\sum_{k=1}^n a_k\equiv\sum_{k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $σ$ on {1,...,n} such that $\sum_{k=1}^n w_k a_{σ(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.

math.CO↗

arXiv2008

A variant of Tao's method with application to restricted sumsets

In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it can be used to prove the following extension of the Erdos-Heilbronn conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|{a+b: a\in A, b\in B, a-b not\in S}|\ge min {p,|A|+|B|-2|S|-1}$.

math.NT↗

arXiv2007

Mixed sums of squares and triangular numbers (II)

For an integer $x$ let $t_x$ denote the triangular number $x(x+1)/2$. Following a recent work of Z. W. Sun, we show that every natural number can be written in any of the following forms with $x,y,z\in\Z$: $$x^2+3y^2+t_z, x^2+3t_y+t_z, x^2+6t_y+t_z, 3x^2+2t_y+t_z, 4x^2+2t_y+t_z.$$ This confirms a conjecture of Sun.

math.NT↗

arXiv2007

Determination of the two-color Rado number for $a_1x_1+...+a_mx_m=x_0$

For positive integers $a_1,a_2,...,a_m$, we determine the least positive integer $R(a_1,...,a_m)$ such that for every 2-coloring of the set $[1,n]={1,...,n}$ with $n\ge R(a_1,...,a_m)$ there exists a monochromatic solution to the equation $a_1x_1+...+a_mx_m=x_0$ with $x_0,...,x_m\in[1,n]$. The precise value of $R(a_1,...,a_m)$ is shown to be $av^2+v-a$, where $a=min{a_1,...,a_m}$ and $v=\sum_{i=1}^{m}a_i$. This confirms a conjecture of B. Hopkins and D. Schaal.

math.CO↗

arXiv2005

On odd covering systems with distinct moduli

A famous unsolved conjecture of P. Erdos and J. L. Selfridge states that there does not exist a covering system {a_s(mod n_s)}_{s=1}^k with the moduli n_1,...,n_k odd, distinct and greater than one. In this paper we show that if such a covering system {a_s(mod n_s)}_{s=1}^k exists with n_1,...,n_k all square-free, then the least common multiple of n_1,...,n_k has at least 22 prime divisors.

math.NT↗