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Linear complexity problems of level sequences of Euler quotients and their related binary sequences

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science China(information sciences) 2016年第3期59卷 75-86页

作者： Zhihua NIU Zhixiong CHEN Xiaoni DU school of Computer Engineering and science Shanghai University school of mathematics Putian University College of mathematics and information science Northwest Normal University

The Euler quotient modulo an odd-prime power pr(r > 1) can be uniquely decomposed as a p-adic number of the form(up-1pr-1- 1)/pr≡ a0(u) + a1(u)p + ··· + ar-1(u)pr-1(mod pr), gcd(u, p) = 1, where0≤aj(u) 1. We firstly study certain arithmetic properties of the level sequences(aj(u))u≥0over Fp via introducing a new quotient. Then we determine the exact values of linear complexity of(aj(u))u≥0 and values of k-error linear complexity for binary sequences defined by(aj(u))u≥0.

关键词： cryptography Euler quotients Fermat quotients pseudorandom sequences binary sequences lin ear complexity k-error linear complexity

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

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Chinese Physics B 2014年第11期23卷 118-124页

作者：屈改珠张顺利李尧龙 Center for Nonlinear Studies Department of mathematics northwest university school of mathematics and information science Weinan normal university

In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.

关键词： nonlinear evolution equation quadratic operator invariant subspace method blow-up solution

A Filtration of Wick Algebra and Its Application to Quantum SDEs

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Acta Mathematica Sinica,English Series 2004年第6期20卷 999-1008页

作者： Cai Shi WANG Zhi Yuan HUANG school of mathematics and information science Northwest Normal University Department of mathematics Huazhong University of Science & Technology

A family of closed subalgebras, indexed by R (the set of real numbers), of the Wick algebra is constructed. Fundamental properties of the family are shown including the increasing property and the right–continuity. The notion of adaptedness to the family is defined for quantum stochastic processes in terms of generalized operators. The existence and uniqueness of solutions adapted to the family is established for quantum stochastic differential equations in terms of generalized operators.

关键词： White noise Wick algebra Filtration

Single-cell omics: experimental workflow, data analyses and applications

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science China Life sciences 2025年第1期68卷 5-102页

作者： Fengying Sun Haoyan Li Dongqing Sun Shaliu Fu Lei Gu Xin Shao Qinqin Wang Xin Dong Bin Duan Feiyang Xing Jun Wu Minmin Xiao Fangqing Zhao Jing-Dong J.Han Qi Liu Xiaohui Fan Chen Li Chenfei Wang Tieliu Shi Department of Clinical Laboratory the Affiliated Wuhu Hospital of East China Normal University (The Second People's Hospital of Wuhu City) Pharmaceutical Informatics Institute College of Pharmaceutical SciencesZhejiang University Key Laboratory of Spine and Spinal Cord Injury Repair and Regeneration (Tongji university) Ministry of EducationOrthopaedic DepartmentTongji HospitalBioinformatics DepartmentSchool of Life Sciences and TechnologyTongji University Frontier science Center for Stem Cells School of Life Sciences and TechnologyTongji University Translational Medical Center for Stem Cell Therapy and Institute for Regenerative Medicine Shanghai East HospitalBioinformatics DepartmentSchool of Life Sciences and TechnologyTongji University Research Institute of Intelligent Computing Zhejiang Lab Shanghai Research Institute for Intelligent Autonomous Systems Center for Single-cell Omics School of Public HealthShanghai Jiao Tong University School of Medicine National Key Laboratory of Chinese Medicine Modernization Innovation Center of Yangtze River DeltaZhejiang University Center for Bioinformatics and Computational Biology Shanghai Key Laboratory of Regulatory Biologythe Institute of Biomedical Sciences and School of Life SciencesEast China Normal University Beijing Institutes of Life science Chinese Academy of Sciences Peking-Tsinghua Center for Life sciences Academy for Advanced Interdisciplinary StudiesCenter for Quantitative Biology (CQB)Peking University Zhejiang Key Laboratory of Precision Diagnosis and Therapy for Major Gynecological Diseases Women's HospitalZhejiang University School of Medicine Key Laboratory of Advanced Theory and Application in Statistics and Data science-MOE School of StatisticsEast China Normal University

Cells are the fundamental units of biological systems and exhibit unique development trajectories and molecular features. Our exploration of how the genomes orchestrate the formation and maintenance of each cell, and control the cellular phenotypes of various organismsis, is both captivating and intricate. Since the inception of the first single-cell RNA technology, technologies related to single-cell sequencing have experienced rapid advancements in recent years. These technologies have expanded horizontally to include single-cell genome, epigenome,proteome, and metabolome, while vertically, they have progressed to integrate multiple omics data and incorporate additional information such as spatial sc RNA-seq and CRISPR screening. Single-cell omics represent a groundbreaking advancement in the biomedical field,offering profound insights into the understanding of complex diseases, including cancers. Here, we comprehensively summarize recent advances in single-cell omics technologies, with a specific focus on the methodology section. This overview aims to guide researchers in selecting appropriate methods for single-cell sequencing and related data analysis.

关键词： single-cell sequencing genome epigenome proteomics metabolomics multimodal spatial transcriptomics CRISPR screening

High-dimensional *** Problem over Short Intervals

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Acta Mathematica Sinica,English Series 2014年第2期30卷 213-228页

作者： Zhe Feng XU Tian Ping ZHANG Department of mathematics Northwest University College of mathematics and information science Shaanxi Normal University

Letk be a positive integer and n a nonnegative integer,0 〈 λ1,...,λk＋1 ≤ 1 be real numbers and w =（λ1,λ2,...,λk＋1）.Let q ≥ max{[1/λi ]：1 ≤ i ≤ k ＋ 1} be a positive integer,and a an integer coprime to *** by N（a,k,w,q,n） the 2n-th moment of（b1··· bk c） with b1··· bk c ≡ a（mod q）,1 ≤ bi≤λiq（i = 1,...,k）,1 ≤ c ≤λk＋1 q and 2（b1＋ ··· ＋ bk ＋ c）.We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N（a,k,w,q,n）,which generalized the result of *** we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N（a,k,w,q,n） in the case ofw =（1/2,1/2,...,1/2） and n = *** order to show our result is close to the best possible,the mean-square value of N（a,k,q） φk（q）/2k＋2and the mean value weighted by the high-dimensional Cochrane sum are studied too.

关键词： D. H. Lehmer problem short intervals trigonometric sum character sum Cochrane sum

Characterizing Bell nonlocality and EPR steering

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science China(Physics,Mechanics & Astronomy) 2019年第3期62卷 22-35页

作者： HuaiXin Cao ZhiHua Guo school of mathematics and information science Shaanxi Normal University

Bell nonlocality and Einstein-Podolsky-Rosen(EPR) steering are very important quantum correlations in composite quantum systems. Bell nonlocality of a bipartite state is observed in some local quantum measurements, while EPR steering was first observed by Schr o¨dinger in the context of famous EPR paradox. In this paper, we discuss the Bell nonlocality and EPR steering of bipartite states, including mathematical definitions and characterizations of these two quantum correlations, the convexity as well as the closedness of the sets of all Bell local states and all EPR unsteerable states, respectively. We also derive sufficient conditions for a state to be steerable; these conditions imply that Alice can steer Bob's state whenever Alice has two POV measurements such that the sets of Bob's normalized conditional states become two disjoint sets of pure states, or whenever she has one POV measurement such that Bob's normalized conditional states become a linearly independent set of pure states.

关键词： quantum state Bell nonlocality EPR steering composite quantum system quantum measurement

Polynomial AND homomorphic cryptosystem and applications

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science China(information sciences) 2020年第1期63卷 115-129页

作者： Shundong LI Sufang ZHOU Jiawei DOU Wenli WANG school of Computer science Shaanxi Normal University school of mathematics and information science Shaanxi Normal University

Homomorphic cryptosystems are fundamental and highly effective cryptographic primitives for addressing security problems arising in information processing, data analysis and data applications, particularly in secure cloud computing and secure multiparty computation. To privately compute functions such as E(x1 ∧···∧ xt), E(x1 ∨···∨ xt) and E[(x11 ∧···∧ xm1) ∨···∨(x1 n ∧···∧ xmn)](m disjunctive normal form(mD NF)) on E(x1),..., E(xt) and E(x11),..., E(xmn) without knowing the decryption key, Boolean homomorphic cryptosystems are necessary. Exploring new homomorphic cryptosystems to solve these problems is appealing, and is of high theoretical and practical significance. To solve problems such as these, we propose a polynomial AND homomorphic cryptosystem based on the ideal theory of abstract algebra; the scheme can be obtained from available multiplicatively homomorphic cryptosystems such as the El Gamal. We prove that the cryptosystem is semantically secure. This polynomial AND homomorphic cryptosystem is a highly effective tool for designing various cryptographic protocols. As examples, we demonstrate its applications to privately compute any DNF(i.e., P(X1,..., Xm) = E[(x11 ∧···∧xm1)∨···∨(x1 n ∧···∧xmn)] on ciphertexts E(xij) of xij without knowing the decryption key) and the intersection and union of certain private sets.

关键词： polynomial AND homomorphic cryptosystem semigroup ideal cloud computing security cryptography secure multiparty computation

Several teleportation schemes of an arbitrary unknown multi-particle state via different quantum channels

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Chinese Physics B 2013年第5期22卷 160-167页

作者：彭家寅莫智文 College of mathematics and Software science Sichuan Normal University school of mathematics and information science Neijiang Normal University

We first provide four new schemes for two-party quantum teleportation of an arbitrary unknown multi-particle state by using three-, four-, and five-particle states as the quantum channel, respectively. The successful probability and fidelity of the four schemes reach 1. In the first two schemes, the receiver can only apply one of the unitary transformations to reconstruct the original state, making it easier for these two schemes to be directly realized. In the third and fourth schemes, the sender can preform Bell-state measurements instead of multipartite entanglement measurements of the existing similar schemes, which makes real experiments more suitable. It is found that the last three schemes may become tripartite controlled teleportation schemes of teleporting an arbitrary multi-particle state after a simple modification. Finally, we present a new scheme for three-party sharing an arbitrary unknown multi-particle state. In this scheme, the sender first shares three three-particle GHZ states with two agents. After setting up the secure quantum channel, an arbitrary unknown multi-particle state can be perfectly teleported if the sender performs three Bell-state measurements, and either of two receivers operates an appropriate unitary transformation to obtain the original state with the help of other receiver＇s three single-particle measurements. The successful probability and fidelity of this scheme also reach 1. It is demonstrated that this scheme can be generalized easily to the case of sharing an arbitrary unknown multi-particle state among several agents.

关键词： quantum information quantum teleportation quantum state sharing multi-particle state

Quantum uncertainty relations of two generalized quantum relative entropies of coherence

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science China(Physics,Mechanics & Astronomy) 2018年第8期61卷 8-16页

作者： FuGang Zhang YongMing Li school of mathematics and information science Shaanxi Normal University school of Computer science Shaanxi Normal University

In this paper, we consider the quantum uncertainty relations of two generalized relative entropies of coherence based on two measurement bases. First, we give quantum uncertainty relations for pure states in a d-dimensional quantum system by making use of the majorization technique; these uncertainty relations are then generalized to mixed states. We find that the lower bounds are always nonnegative for pure states but may be negative for some mixed states. Second, the quantum uncertainty relations for single qubit states are obtained by the analytical method. We show that the lower bounds obtained by this technique are always positive for single qubit states. Third, the lower bounds obtained by the two methods described above are compared for single qubit states.

关键词： Tsallis relative entropy of coherence Renyi relative entropy of coherence quantum uncertainty relation the lower bound