Non-holonomicity of the sequence $\log 1, \log 2, \log 3, ...$
Martin Klazar
Gerhold提出并有条件地证明了（log n：n= 1，2，.）并不是一个完整的序列Flajolet，Gerhold和Salvy给出了一个证明使用分析机器。我们给出一个简单的证明。
Gerhold conjectured and proved conditionally that (log n: n=1,2,...) is not a holonomic sequence. Flajolet, Gerhold and Salvy gave a proof using an analytic machinery. We give a simple proof.
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We answer a question of Calkin and Wilf concerning the maximal order of Stern's diatomic sequence. Specifically, we prove that $$\limsup_{n\to\infty}\frac{a(n)}{φ^{\log_2 n}}=\frac{φ^{\log_2 3}}{\sqrt{5}},$$ where $φ=(\sqrt{5}+1)/2$ is the golden ratio. This improves on previous results given by Berlekamp, Conway, and Guy, who showed that the limit value was bounded above by 1.25, and by Calkin and Wilf, who showed that the exact value was in the interval $[(φ/{\sqrt{5}})({3}/{2})^{\log_2φ},(φ+1)/{\sqrt{5}}].$
Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation
令${n_k}$为递增缺项序列，即，$n_{k+1}/n_k>1+r$换一些$r>0$。1987年，P.Erdos提出了整数图$G$的色数，其中两个整数$a,b$通过边相连当且仅当它们的差$|a-b|$在序列${n_k}$中. Y. Katznelson发现了一个连接到一个丢番图逼近问题（也是由于鄂尔多斯）：存在$x$在$(0,1)$，使所有的倍数$n_j x$至少距离$δ(x)>0$从一组整数。Katznelson用$Cr^{-2}|\log r|$限制了$G$的色数.我们应用Lovász局部引理对某些$x$建立了$δ(x)>cr|\log r|^{-1}$，这意味着$G$的色数至多为$Cr^{-1} |\log r|$ .这是尖锐的对数因子。
Horizontal and vertical generating functions and recursion relations have been investigated by Comtet for triangular double sequences. In this paper we investigate the horizontal and vertical log-concavity of triangular sequences assigned to polynomials which show up in combinatorics, number theory and physics. This includes Laguerre polynomials, the Pochhammer polynomials, the D'Arcais and Nekrasov--Okounkov polynomials.
Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\geq1$ and $\varepsilon>0$ the interval $[x,x+C_{\varepsilon}\exp((C+\varepsilon)\log x/\log\log x)]$ contains an element of…
An order-$s$ Davenport-Schinzel sequence over an $n$-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length $s+2$. The main problem is to determine the maximum length of such a sequence, as a function of $n$ and $s$. When $s$ is fixed this problem has been settled but when $s$ is a function of $n$, very little is known about the extremal function $λ(s,n)$ of such sequences. In this paper we give a new recursive construction of Davenport-Schinzel sequences that is based on dense 0-1 matrices avoiding large all-1 submatrices (aka Zarankiewicz's Problem.) I…
Log-Concave Sequences in Coding Theory
该研究将对数凹序列概念引入编码理论，定义了对数凹线性码，证明了Hamming码、Reed-Muller码等经典码类的对数凹性，并给出了MDS码为对数凹的参数条件，为编码理论研究提供了新方向。
Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases
该研究提出有理函数的合并对数凹性概念，构造相关显式函数并研究单峰序列的临界点，同时将其应用于理想玻色-费米气体，揭示了金属比临界点对应的自由能相变。
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