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	1. Theory of scissor products and applications
	ZHU Yong-Wen
	Mathematics 18 November 2020
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	Abstract:In this paper, the concept of scissor products is introduced and its fundamental properties are discussed. Combining with the smaller-than-smaller carry method, scissor products can be used in the numerical calculation to form a new and systematic rapid calculation method for the multiplication of integers, which is parallel to but superior to the well-known rapid calculation method of Shi Fengshou. The advantages of the new theory lie in the following two aspects: (1) the scissor products can be understood and remembered very easily with the help of the ordinary $9\times 9$ multiplication table; (2) the smaller-than-smaller carry method makes carrying very easy. Our theory of scissor products can be applied to the rapid multiplication in two ways, in which we use or do not use the virtual carry method respectively.
	TO cite this article:ZHU Yong-Wen. Theory of scissor products and applications[OL].[18 November 2020] http://en.paper.edu.cn/en_releasepaper/content/4753020


	2. A Siegel-Weil Formula for Unitary Groups Over a Function Field
	XIONG Wei,WANG Ying
	Mathematics 27 March 2020
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	Abstract:In this paper a Siegel-Weil formula for the dual pair $(U(1,1),U(V))$ over a function field is established, where V is a hermitian space over a function field of dimension greater than 2. This formula says that the integral of a theta series over the unitary group associated to $V$ is equal to some Eisenstein series.
	TO cite this article:XIONG Wei,WANG Ying. A Siegel-Weil Formula for Unitary Groups Over a Function Field[OL].[27 March 2020] http://en.paper.edu.cn/en_releasepaper/content/4751391

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	3. Stochastic stability of stochasticdifferential systems with semi-Markovian switching
	HOU Zhenting,ZHANG Zhuo,HOU Xianmin
	Mathematics 04 May 2017
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	Abstract:In this paper, we investigates the stability of switched stochastic differential systems whose switching signal is a semi-Markovian process. Firstly, we transform this kind of systems to Markovprocesses and define their generators under suitable condition.Secondly, we give some sufficient conditions for their exponentialstability.
	TO cite this article:HOU Zhenting,ZHANG Zhuo,HOU Xianmin. Stochastic stability of stochasticdifferential systems with semi-Markovian switching[OL].[ 4 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4732346


	4. Some results of Diophantine approximation by unlike powers of primes
	Liu Zhixin
	Mathematics 29 April 2017
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	Abstract:Let $k$ be an integer with $kgeq 6$. Suppose that $lambda_1, cdots, lambda_5$ be non-zero real numbers not all of the same sign, satisfying that$lambda_1/lambda_2$ be irrational, and $eta$ be a real number.In this paper, for any $arepsilon>0$, we consider the inequality$$left\|lambda_1p_1+lambda_2p_2^2+lambda_3p_3^3+lambda_4p_4^4+lambda_5p_5^k+eta ight\|<(max p_j)^{-sigma(k)+arepsilon}$$has infinitely many solutions in prime variables $p_1, cdots, p_5$, whereegin{eqnarray}sigma(k)=egin{cases}1/64, &k=6, 7, 8,cr1/80, &k=9, 10,cr3/256, &k=11, 12,cr1/(2^{[(k-1)/6]+5}), &13 leq k leq 48, cr3/(8k^2+8k+48), &kgeq 49.end{cases}end{eqnarray}Our result gives an improvement of the recent result.Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximationby unlike powers of primes.
	TO cite this article:Liu Zhixin. Some results of Diophantine approximation by unlike powers of primes[OL].[29 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4729579


	5. Small prime solutions of a nonlinear equation
	Liu Zhixin
	Mathematics 29 April 2017
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	Abstract:Let $a_1, cdots, a_4$ be non-zero integers and $n$ any integer.Suppose that $a_1, cdots, a_4$ and $n$ satisfy some related conditions.In this paper we prove that\(i) if $a_j$ are not all of the same sign, then theequation $a_1p_1+a_2p_2^2+a_3p_3^2+a_4p_4^2=n$ has prime solutions satisfying${ m max} {p_1, p_2^2, p_3^2, p_4^2} ll \|n\|+ extrm{max}{\|a_j\|}^{14+arepsilon}$;\(ii) if all $a_j$ are positive and $n gg extrm{max}{\|a_j\|}^{15+arepsilon}$, then the equation$a_1p_1+a_2p_2^2+a_3p_3^2+a_4p_4^2=n$ is soluble in primes $p_j$.
	TO cite this article:Liu Zhixin. Small prime solutions of a nonlinear equation[OL].[29 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4729576


	6. Sums of generators of ideals in residue class ring
	JI Chun-Gang,ZHANG Xin
	Mathematics 08 October 2016
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	Abstract:Let $K$ be an algebraic number field with $mathcal{O}_K$ the ring of integers and $mathfrak{n}$ a non-zero ideal of $mathcal{O}_K$.For any $a in mathcal{O}_K/mathfrak{n}$, we define $(mathcal{O}_K/mathfrak{n})^* cdot a$ as the orbit of $a$.Then we give explicitly which orbits are part of the union which constitutes the sumset of two given orbits.We also obtain an exact formula for the number of representations of $a in mathcal{O}_k/mathfrak{n}$ as a sum of two orbits in $mathcal{O}_k/mathfrak{n}$.
	TO cite this article:JI Chun-Gang,ZHANG Xin. Sums of generators of ideals in residue class ring [OL].[ 8 October 2016] http://en.paper.edu.cn/en_releasepaper/content/4706002


	7. Remarks on the maximum gap in binary cyclotomic polynomials
	JI Chun-Gang, ZHANG Bin
	Mathematics 02 June 2016
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	Abstract:Let $p<q$ be odd primes, and let$g(Phi_{pq})$ denote the maximum of the differences (gaps)between two consecutive exponents occurring in the $pq$-thcyclotomic polynomial $Phi_{pq}(x)$. In this note, we give asimple proof of the following result which was established by Honget al. [J. Number Theory, 132 (2012), pp. 2297-2315]$$g(Phi_{pq})=p-1.$$In addition, we show that the number of maximum gaps in$Phi_{pq}(x)$ is given by $2lfloorrac{q}{p} floor$.
	TO cite this article:JI Chun-Gang, ZHANG Bin. Remarks on the maximum gap in binary cyclotomic polynomials[OL].[ 2 June 2016] http://en.paper.edu.cn/en_releasepaper/content/4693263


	8. Difference of a generalized D. H. Lehmer number and its $m$-th power
	XU Zhe-Feng
	Mathematics 22 February 2014
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	Abstract:Let $n>2$, $mge 2$ be integers. For any integer $a$with $1le ale n, (a,n)=1$, there exist unique integer $b$ with$1le ble n$ such that $bequiv a^mpmod n$, denote it by$(a)_n$. If $a$ and $(a^m)_n$ are of opposite parity, we call $a$generalized D. H. Lehmer number. For the case $m=phi(n)-1$, it isthe classical D. H. Lehmer number. The main purpose of this paperis to study the difference $\|a-(a^m)_n\|$ and give an asymptoticformula for mean value$$mathop{mathop{{sum}'}_{a=1}^{[lambda n]}}_{2midleft(a+left(a^might)_night)}left\|a-left(a^m ight)_night\|^k$$for any nonnegative integer $k$ and real constant $lambdain(0,1]$.
	TO cite this article:XU Zhe-Feng. Difference of a generalized D. H. Lehmer number and its $m$-th power[OL].[22 February 2014] http://en.paper.edu.cn/en_releasepaper/content/4586790


	9. Representations of an integer as sum of a primitive root and a $k$-th residue
	XU Zhe-Feng
	Mathematics 22 February 2014
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	Abstract:Let $pge3$ be a prime, $n, k$ be integers with $1leqn, kleq p-1$. The asymptotic property of $N_k(n, p)$, the numberof representations of $n$ as sum of a primitive root and a $k$-thresidue modulo $p$, and the square mean value of the error term of$N_2(n, p)$ are studied in this short note.
	TO cite this article:XU Zhe-Feng. Representations of an integer as sum of a primitive root and a $k$-th residue[OL].[22 February 2014] http://en.paper.edu.cn/en_releasepaper/content/4586787


	10. On the integrality of the elementary symmetric functions ofa reciprocal high degree polynomial sequence
	HONG Shaofang, LUO Yuanyuan, QIAN Guoyou, WANG Chunlin
	Mathematics 23 December 2013
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	Abstract:Erd"{o}s and Niven in 1946 proved that for any positive integers $m$ and $d$,there are at most finitely many integers $n$ for which at least one of the elementarysymmetric functions of $1/m, 1/(m+d), ...,1/(m+(n-1)d)$ are integers. Recently, Wangand Hong refined this result by showing that if $ngeq 4$, then none of the elementary symmetric function of $1/m, 1/(m+d), ...,1/(m+(n-1)d)$ is an integer for any positiveintegers $m$ and $d$. In this paper, we show that none of the elementary symmetric functions of$1/f(1), 1/f(2), ...,1/f(n)$ is an integer except for $f(x)=x^{m}$ with $mgeq2$ beingan integer and $k=n=1$ and $f$ being a polynomial with nonnegative integer coefficients.
	TO cite this article:HONG Shaofang, LUO Yuanyuan, QIAN Guoyou, et al. On the integrality of the elementary symmetric functions ofa reciprocal high degree polynomial sequence[OL].[23 December 2013] http://en.paper.edu.cn/en_releasepaper/content/4577037

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