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arithmetic progressions in self-similar sets

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Frontiers of Mathematics in China 2019年第5期14卷 957-966页

作者： Lifeng XI Kan JIANG Qiyang PEI Department of Mathematics Ningbo UniversityNingbo 315211China

Given a sequence{bi}i=1^n and a ratio λ∈(0,1),let E=Ui=1^n(λE+bi)be a homogeneous self-similar set.In this paper,we study the existence and maximal length of arithmetic progressions in E.Our main idea is from the m... 详细信息

关键词： Self-similar sets arithmetic progression (AP) β-expansions

Averages of shifted convolution sums for arithmetic functions

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Frontiers of Mathematics in China 2019年第1期14卷 123-134页

作者： Miao LOU School of Mathematics Shandong University Jinan 250100 China

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolu... 详细信息

关键词： Average shifted convolution sum arithmetic function

The arithmetic fundamental lemma: An update

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Science China Mathematics 2019年第11期62卷 2409-2422页

作者： Wei Zhang Department of Mathematics Massachusetts Institute of Technology

This is an expository article on the recent progress on the arithmetic fundamental lemma conjecture,based largely on Zhang(2019). Beside stating the local conjecture, we will present three global intersection problems... 详细信息

关键词： arithmetic fundamental lemma arithmetic Gan-Gross-Prasad conjecture Gross-Zagier formula relative trace formula

Efficient implementation of quantum arithmetic operation circuits

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Science China(Physics,Mechanics & Astronomy) 2021年第1期64卷 116-117页

作者： Heng Fan Institute of Physics Chinese Academy of SciencesBeijing100190China CAS Center for Excellence in Topological Quantum Computation University of Chinese Academy of SciencesBeijing100190China Songshan Lake Material Laboratory Dongguan523000China

Realization of robust quantum computing is a tremendous task.On one hand,one must reduce errors as many as possible using various means[1-3];on the other hand,one may make large error rates tolerable via fault-tolerant implementation.In achieving tolerant against noises,the cost to pay is the increased number of gates used.Therefore,an efficient implementation of fault-tolerant gates is crucial.As already evidenced,Clifford gates with a T gate can form a universal set of fault-tolerant gates.

关键词： quantum arithmetic Efficient

HUNGARIAN SEMIGROUPS, THE arithmetic OF GENERALIZED RENEWAL SEQUENCES AND SEMI-P-FUNCTIONS

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Chinese Annals of Mathematics,Series B 1990年第2期11卷 127-137页

作者：梁之舜黄之瑞 Department of Mathematics Zhongshan UniversityGuangzhou Department of Mathematics Zhongshan University Guangzhou

In this paper the authors discuss the additional conditions under which a Hungarian semigroup is possessed of the fundamental Delphic properties. It is proved that both the positive generalized renewal sequence semigroup and the tame semi-p-function semigroup are Hungarian semigroups and possessed of the fundamental Delphic properties. Then the arithmetic properties of these two classes of the special semigroups are studied respectively.

关键词： semigroup renewal arithmetic erior possessed hereditary algebra Kendall 落兀 uniformly

A New arithmetic Coding System Combining Source Channel Coding and MAP Decoding

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Journal of Shanghai Jiaotong university(Science) 2007年第4期12卷 442-447页

作者：逄玉叶孙军王嘉 Dept.of Electronic Eng. Shanghai Jiaotong Univ.

A new arithmetic coding system combining source channel coding and maximum a posteriori decoding were proposed. It combines source coding and error correction tasks into one unified process by introducing an adaptive forbidden symbol. The proposed system achieves fixed length code words by adaptively adjusting the probability of the forbidden symbol and adding tail digits of variable length. The corresponding improved MAP decoding metric was derived. The proposed system can improve the performance. Simulations were performed on AWGN channels with various noise levels by using both hard and soft decision with BPSK modulation.The results show its performance is slightly better than that of our adaptive arithmetic error correcting coding system using a forbidden symbol.

关键词： arithmetic code (AC) forbidden symbol joint source channel coding (JSCC) maximum a posteriori (MAP) decoding

An Adaptive Joint Source/Channel Coding Using Error Correcting arithmetic Codes

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Journal of Shanghai Jiaotong university(Science) 2007年第4期12卷 448-453页

作者：刘军清逄玉叶孙军 Dept.of Electronic Eng. China Three Gorges Univ. Dept.of Electronic Eng. Shanghai Jiaotong Univ.

An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcting in a single process, with superior performance compared with traditional separated techniques. The concept of adaptiveness is applied not only to the source model but also to the amount of coding redundancy. In addition, an improved branch metric computing algorithm and a faster sequential searching algorithm compared with the system proposed by Grangetto were proposed. The proposed system is tested in the case of image transmission over the AWGN channel, and compared with traditional separated system in terms of packet error rate and complexity. Both hard and soft decoding were taken into account.

关键词： arithmetic coding joint source channel coding maximum a posteriori (MAP) forbidden symbol

The divisor problem for arithmetic progressions

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Chinese Science Bulletin 1995年第4期40卷 265-267页

作者：李红泽 Department of Mathematics Shandong University Jinan 250100 China

Let n≥1 and r≥2 be integers and let dr（n） denote the number of ordered r-tuples （n1,…,nr） of natural numbers for which multiply from 1≤j≤r （nj）=n For （a,q）=1,define Dr（X,q,a）=sum from n≤X n≡a（modg） ... 详细信息

关键词： arithmetic progression divisor problem.

A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

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Journal of Systems Science & Complexity 2019年第1期32卷 185-204页

作者： WANG Yu XIA Bican LMAM & School of Mathematical Sciences Peking University

Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method ?rst constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al.(2017). For each root returned by the homotopy continuation method, which is a complex approximation of some(complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and ?nally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.

关键词： Homotopy continuation interval arithmetic polynomial system real variety