Zhi‐Wei Sun

On Two New Kinds of Restricted Sumsets

Han Wang, Zhi‐Wei Sun

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets$$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1}\ \text{for}\ 1\le i

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Undecidability on Diophantine equations over Z[i] with 20 unknowns

Yuri Matiyasevich, Zhi‐Wei Sun

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Screening of Characteristic Metabolites in Bee Pollen from Different Floral Sources Based on High-Resolution Mass Spectrometry

Lanhua Liu, Zhi‐Wei Sun, Run Zhang, Siqi He et autres

Bee pollen is a natural nutrient substance collected by bees from plants. Its metabolites have been extensively studied, yet the characteristic metabolites of bee pollen from different floral sources have not been clearly identified. In this study, we collected four types of …

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On Diophantine equations over ℤ[ i ] with 52 unknowns

Yuri Matiyasevich, Zhi‐Wei Sun

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Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns

Yuri Matiyasevich, Zhi‐Wei Sun

It is known that Hilbert's Tenth Problem over the Gaussian ring $\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}$ is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{20})=0$ (with integer coefficients) …

Evaluation of a determinant involving Legendre symbols

Chen-kai Ren, Zhi‐Wei Sun

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv …

On Some Determinants Arising from Quadratic Residues

Chen-Kai Ren, Zhi‐Wei Sun

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Some Determinants Involving Binary Forms

Yue-Feng She, Zhi‐Wei Sun

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Fast converging irrational series for $ L(2,(\frac d\cdot))$

Zhi‐Wei Sun, Yajun Zhou

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $π$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we …

New series involving binomial coefficients (III)

Zhi‐Wei Sun

We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3π}2.$$ Motivated by Galois theory, we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type, and apply it to find …

Integrated volatilomic profiles and chemometrics provide new insights into the aroma differences of volatile compounds in filler tobacco leaves of six grades

Mingzhu Zhang, Dongfeng Guo, Zhi‐Wei Sun, Guanglong Wu et autres

As an economically important crop, the quality grade of filler tobacco leaves (FTLs) is a key factor determining their commercial value. To investigate the sources and variations in aroma among different quality grades of filler tobacco leaves (DGFTLs), the volatile components of …

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A family of polynomials and related congruences and series

Zhi‐Wei Sun

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac x{2x-1}\left(1+\left(\frac{1-4x^2}p\right)\right)\pmod p $$ for any odd prime $p$ and integer $x\not\equiv1/2\pmod p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also formulate some …