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Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture.

Introduction: Monosaccharide compositions analysis (MCA) is indispensable for structural characterisations and structure–activity relationships of plant polysaccharides. Objectives: To develop a concise and direct MCA method, we established a quantitative analysis of the multi‐monosaccharaides by single marker (QAMS) by high‐performance anion‐exchange chromatography with pulsed‐amperometric detection (HPAEC‐PAD) method. Methodology: A stable and reproducible HPAEC‐PAD method for simultaneous determination of aldoses, ketoses and uronic acids (i.e., l‐arabinose, d‐xylose, d‐ribose, l‐rhamnose, d‐fucose, d‐mannose, d‐glucose, d‐galactose, d‐fructose, d‐glucuronic acid and d‐galacturonic acid) was established by systematic optimisation of stationary phases, column temperatures and elution programmes. On this basis, the QAMS method was proposed through comprehensive investigations of relative correction factor (RCF) variations under different influencing factors, for example, sample concentrations, flow rates, and column temperatures. Results: Using rhamnose as an internal reference standard, the contents of the other monosaccharide components in polysaccharides from Panax quinquefolium L. and Achyranthes bidentata Bl. samples were simultaneously determined by QAMS, and there was no significant difference between the results from the QAMS and external standard method (t test, P > 0.520). In addition, a MCA fingerprinting of 30 batches of P. quinquefolium polysaccharide was established by HPAEC‐PAD, and six common peaks were assigned and determined. Conclusions: The established HPAEC‐PAD‐QAMS method was successfully applied to the MCA of polysaccharides from P. quinquefolium and A. bidentata after optimisation of hydrolysis conditions. HPAEC‐PAD‐QAMS was proposed and established for MCA of plant polysaccharides for the first time. This study aimed to develop a concise and direct monosaccharide compositional analysis method. A quantitative analysis of the multi‐monosaccharaides by single marker (QAMS) by high‐performance anion‐exchange chromatography with pulsed‐amperometric detection (HPAEC‐PAD) method, was proposed for the simultaneous determination of neutral and acidic monosaccharides without derivatisation. It was successfully applied to analyse the composition of monosaccharides released from Panax quinquefolium polysaccharide (PQP) and Achyranthes bidentata polysaccharide (ABP). Additionally, a monosaccharide compositional fingerprinting of PQP was established. [ABSTRACT FROM AUTHOR]

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$ , let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$. We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline {n})$ for all $\overline {n}\in \mathbb {Z}_m$ , where m is an even integer. [ABSTRACT FROM AUTHOR]

The aim of this work was to extract, isolate, and purify polysaccharides from the heads of Hypomesus olidus and to evaluate their anticoagulant activities and anticoagulant mechanisms. Response surface methodology was used to optimize the extraction conditions for ultrasonic‐assisted extraction of polysaccharides from the heads of Hypomesus olidus. The optimal extraction conditions consisted of ultrasonic power of 275 W, ultrasonic time of 50 min, and solid–liquid ratio of 5 ml/g, giving the yield of crude polysaccharides (GYT) of 7.73 ± 0.042%. Three polysaccharide fractions, GYT‐1, GYT‐2, and GYT‐3 were purified from GYT by using DEAE‐cellulose‐52 column and Sephadex G‐100 column for anticoagulant activities. The results showed that two doses (2 and 4 mg/ml) of GYT‐1 and GYT‐3 could significantly prolong (p <.01) in partial thromboplastin time (APTT) (2.19 and 2.37 times, 2.22 and 2.44 times, respectively) and thrombin time (TT) (2.39 and 2.46 times, 2.44 and 2.80 times, respectively) compared with normal control. In particular, GYT‐3 had stronger anticoagulant activity than GYT‐1, and it was composed of arabinose, fructose, glucose, and lactose with molar ratios of 0.595:1: 2.026:0.273. However, GYT‐2 had no anticoagulant activity (p >.05). In addition, anticoagulation mechanism of polysaccharides from the heads of Hypomesus olidus (GYT‐3) was evaluated. The results showed that the anticoagulant activity of GYT‐3 was based on their binding with antithrombin AT‐III. And the inhibitory effects of GYT‐3 on factor IIa and Xa were related to the concentration of AT‐III in plasma. This study may provide a new and promising anticoagulant drug. [ABSTRACT FROM AUTHOR]