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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.

## ESL-SELO: A robust image denoising algorithm with penalty

### Acta Mathematicae Applicatae Sinica, English Series (2017-07-01) 33: 753-770 , July 01, 2017

Robust image recovery methods have been attracted more and more attention in recent decades for its good property of tolerating system errors or measuring noise. In this paper, we propose a new robust method (ESL-SELO) to recover nosing image, which combine exponential loss function and seamless-L0 (SELO) penalty function to guarantee both accuracy and robustness of the estimator. Theoretical result showed that our method has a local optimal solution and good asymptotic properties. Finally, we compare our method with other methods in simulation which shows better robustness and takes much less time.

## Soft-control for collective opinion of weighted DeGroot model

### Journal of Systems Science and Complexity (2017-06-01) 30: 550-567 , June 01, 2017

The DeGroot model is a classic model to study consensus of opinion in a group of individuals (agents). Consensus can be achieved under some circumstances. But when the group reach consensus with a convergent opinion value which is not what we expect, how can we intervene the system and change the convergent value? In this paper a mechanism named soft control is first introduced in opinion dynamics to guide the group’s opinion when the population are given and evolution rules are not allowed to change. According to the idea of soft control, one or several special agents, called shills, are added and connected to one or several normal agents in the original group. Shills act and are treated as normal agents. The authors prove that the change of convergent opinion value is decided by the initial opinion and influential value of the shill, as well as how the shill connects to normal agents. An interesting and counterintuitive phenomenon is discovered: Adding a shill with an initial opinion value which is smaller (or larger) than the original convergent opinion value dose not necessarily decrease (or increase) the convergent opinion value under some conditions. These conditions are given through mathematical analysis and they are verified by the numerical tests. The authors also find out that the convergence speed of the system varies when a shill is connected to different normal agents. Our simulations show that it is positively related to the degree of the connected normal agent in scale-free networks.

## 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.

## A Dynamic Approach to Calculate Shadow Prices of Water Resources for Nine Major Rivers in China

### Journal of Systems Science and Complexity (2006-03-01) 19: 76-87 , March 01, 2006

China is experiencing from serious water issues. There are many differences among the Nine Major Rivers basins of China in the construction of dikes, reservoirs, floodgates, flood discharge projects, flood diversion projects, water ecological construction, water conservancy management, etc. The shadow prices of water resources for Nine Major Rivers can provide suggestions to the Chinese government. This article develops a dynamic shadow prices approach based on a multiperiod input–output optimizing model. Unlike previous approaches, the new model is based on the dynamic computable general equilibrium (DCGE) model to solve the problem of marginal long-term prices of water resources. First, definitions and algorithms of DCGE are elaborated. Second, the results of shadow prices of water resources for Nine Major Rivers in 1949–2050 in China using the National Water Conservancy input–holding–output table for Nine Major Rivers in 1999 are listed. A conclusion of this article is that the shadow prices of water resources for Nine Major Rivers are largely based on the extent of scarcity. Selling prices of water resources should be revised via the usage of parameters representing shadow prices.

## 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.

## Cooperation in two-stage games on undirected networks

### Journal of Systems Science and Complexity (2017-06-01) 30: 680-693 , June 01, 2017

In the paper, cooperative two-stage network games are studied. At the first stage of the game, players form a network, while at the second stage players choose their behaviors according to the network realized at the first stage. As a cooperative solution concept in the game, the core is considered. It is proved that some imputations from the core are time inconsistent, whereas one can design for them a time-consistent imputation distribution procedure. Moreover, the strong time consistency problem is also investigated.

## 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.

## Positive solutions of a Schrödinger equation with critical nonlinearity

### Zeitschrift für angewandte Mathematik und Physik ZAMP (2004-07-01) 55: 592-605 , July 01, 2004

We study the nonlinear Schröodinger equation
$$-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N},$$
with critical exponent
2^{*}= 2
*N*/(
*N*-2),
*N* ≥ 4,
where
*a* ≥ 0,
has a potential well. Using variational methods we
establish existence and multiplicity of positive solutions which
localize near the potential well for μ small and λ
large.