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## Orthogonal graphs of characteristic 2 and their automorphisms

### Science in China Series A: Mathematics (2009-02-01) 52: 361-380 , February 01, 2009

The (singular) orthogonal graph *O*(2ν + δ, *q*) over a field with *q* elements and of characteristic 2 (where ν ⩾ 1, and δ = 0, 1 or 2) is introduced. When ν = 1, *O*(2 · 1, *q*), *O*(2 · 1 + 1, *q*) and *O*(2 · 1 + 2, *q*) are complete graphs with 2, *q* + 1 and *q*^{2} + 1 vertices, respectively. When ν ⩾ 2, *O*(2ν + δ, *q*) is strongly regular and its parameters are computed. *O*(2ν + 1, *q*) is isomorphic to the symplectic graph *Sp*(2ν, *q*). The chromatic number of *O*(2ν + ν, *q*) except when δ = 0 and ν is odd is computed and the group of graph automorphisms of *O*(2ν + δ, *q*) is determined.

## A general form of Gelfand–Kazhdan criterion

### Manuscripta Mathematica (2011-09-01) 136: 185-197 , September 01, 2011

We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group, which consist of generalized functions on the group. As an application, we state and prove a Gelfand–Kazhdan criterion for a real reductive group in very general settings.

## Fiducial inference in the pivotal family of distributions

### Science in China Series A (2006-01-01) 49: 410-432 , January 01, 2006

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.

## Analyzing the general biased data by additive risk model

### Science China Mathematics (2017-04-01) 60: 685-700 , April 01, 2017

This paper proposes a unified semiparametric method for the additive risk model under general biased sampling. By using the estimating equation approach, we propose both estimators of the regression parameters and nonparametric function. An advantage is that our approach is still suitable for the lengthbiased data even without the information of the truncation variable. Meanwhile, large sample properties of the proposed estimators are established, including consistency and asymptotic normality. In addition, the finite sample behavior of the proposed methods and the analysis of three groups of real data are given.

## On “Problems on von Neumann Algebras by R. Kadison, 1967”

### Acta Mathematica Sinica (2003-07-01) 19: 619-624 , July 01, 2003

A brief summary of the development on Kadison's famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.

## Perovskite-like mixed oxides La2-x (Sr, Th) x CuO4±λ

### Science in China Series A: Mathematics (1997-11-01) 40: 1210-1215 , November 01, 1997

Two groups of mixed oxides La_{2-x}Th_{x}CuO_{4±λ} (0.0⊖x⊖0.4) and La_{2-x}Sr_{x}CuO_{4±β}(0.0⊖*x* ⊖1.0) were prepared. Their crystal structures were studied with XRD and IR spectra, etc. Meanwhile, the average valence of Cu ions and nonstoichiometric oxygen (⊖) was measured through chemical analyses. Catalysis of the abovementioned mixed oxides was investigated in phenol hydroxylation, good results were obtained for some mixed oxides, and found that the catalysis of these mixed oxides have close relation with their defect structure and composition. A radical substitution mechanism was also proposed for this catalytic reaction.

## Relative entropy between quantum ensembles

### Periodica Mathematica Hungarica (2009-12-01) 59: 223-237 , December 01, 2009

Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart.

## Cryptanalysis of Dual RSA

### Designs, Codes and Cryptography (2017-04-01) 83: 1-21 , April 01, 2017

In 2007, Sun et al. (IEEE Trans Inf Theory 53(8):2922–2933, 2007) presented new variants of RSA, called Dual RSA, whose key generation algorithm outputs two distinct RSA moduli having the same public and private exponents, with an advantage of reducing storage requirements for keys. These variants can be used in some applications like blind signatures and authentication/secrecy. In this paper, we give an improved analysis on Dual RSA and obtain that when the private exponent is smaller than
$$N^{0.368}$$
, the Dual RSA can be broken, where *N* is an integer with the same bitlength as the modulus of Dual RSA. The point of our work is based on the observation that we can split the private exponent into two much smaller unknown variables and solve a related modular equation on the two unknown variables and other auxiliary variables by making use of lattice based methods. Moreover, we extend this method to analyze the common private exponent RSA scheme, a variant of Dual RSA, and obtain a better bound than previous analyses. While our analyses cannot be proven to work in general, since we rely on some unproven assumptions, our experimental results have shown they work in practice.

## Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium

### Science China Mathematics (2011-12-01) 54: 2693-2712 , December 01, 2011

This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (PEC) boundary condition, employing the energy method. To this end, we first establish some new energy identities for the 2D Maxwell equations in a lossy medium with a PEC boundary condition. Then by making use of these energy identities, it is proved that the FDTD scheme and its time difference scheme are stable in the discrete *L*^{2} and *H*^{1} norms when the CFL condition is satisfied. It is shown further that the solution to both the FDTD scheme and its time difference scheme is second-order convergent in both space and time in the discrete *L*^{2} and *H*^{1} norms under a slightly stricter condition than the CFL condition. This means that the solution to the FDTD scheme is superconvergent. Numerical results are also provided to confirm the theoretical analysis.

## Fixed Design Nonparametric Regression with Truncated and Censored Data

### Acta Mathematicae Applicatae Sinica (2003-06-01) 19: 229-238 , June 01, 2003

###
*Abstract*

In this paper we consider a fixed design model in which the observations are subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covariate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator. Applications include trimmed mean and quantile function estimators.