Codes within codes

Summary We prove coding theorems for list codes for compound channels and for codes within codes. The theorems imply corresponding results for what are usually called simply codes, which are list codes where one is the maximum decoding list length permitted to the decoder. The Bergmans coding theorem for degraded channels ([4], Theorem 15.2.1) and the positive part of the Wyner-Ziv theorem ([4], Theorem 13.2.1) are easy consequences of our results.Research supported by the National Science Foundation under Grant No. MCS 78-02148     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Infinite distance transitive graphs of finite valency

We determine all infinite distance transitive graphs of finite valency, thereby proving a conjecture of C. D. Godsil. The proof makes heavy use of a theorem of M. J. Dunwoody concerning cuts of infinite graphs. In section 1 there is a rough analysis of the appearance of such graphs, and in section 2 we state and apply Dunwoody’s theorem. The proof is completed in section 3.     

Deforming abelian SU(2)-representations of knot groups

The aim of this paper is to generalize a theorem of C. D. Frohman and E. P. Klassen ([FK91]) concerning deformations of abelian SU(2)-representations of knot groups into non-abelian representations. The proof of our main theorem makes use of a generalization of a result of X.-S. Lin ([Lin92]) which should be interesting in itself. Received: December 1, 1997     

Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

The theory of algebraically closed non‐Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning, unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stabilization of interconnected stochastic systems

The aim of this work is to prove that if the equilibrium solution of a nonlinear control stochastic system is locally asymptotically stable in probability by means of a continuous state feedback law, then the resulting stochastic system obtained by adding an integrator is also locally asymptotically stable in probability by means of a smooth, except possibly at the equilibrium solution, state feedback law. This result extends to the stabilization of stochastic systems a result proved by Tsinias [9] for deterministic systems. In our proof, we make use of the stochastic version of Artstein's theorem established in [4]     

Inverse coding theorem for infinite-dimensional quantum channels

In this paper we make a conjecture about the quantum capacity of an infinite-dimensional quantum channel. The proof of the inverse theorem is given based on definitions and properties of the coherent information in the infinite-dimensional case.     

LDPC codes from Singer cycles

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes-characterised by admitting a sparse parity check matrix-with good correction capabilities. In the present paper the orbits of subspaces of a finite projective space under the action of a Singer cycle are investigated. The incidence matrix associated to each of these structures yields an LDPC code in a natural manner.     

一个线性代数定理证明的新方法

实对称矩阵合同相似于对角矩阵这一结论是实二次型理论中的一个基本定理．本文给出了证明这一定理的新方法，该方法简洁、清晰、通俗易懂．     

Hypothesis testing for arbitrarily varying source

Hypothesis testing for arbitrarily varying source (AVS), which is to decide between the two hypotheses for the varying behavior of the distribution of AVS, is considered in this paper. We determine the best asymptotic exponent of the second kind of error probability when the first kind of error probability is fixed. This result generalizes the well-known lemma of Stein in statistics. As a corollary, Strassen's coding theorem for AVS is obtained.Supported by the Young Teacher Foundation of Chinese Educational Ministry and by the National Natural Science Foundation of China.     

Chinese interest in right-angled triangles

This paper traces briefly the discovery of the Pythagorean theorem and the contradictory views about the origin of the famous problem. The earliest evidence of the theorem goes back to a cuneiform text of the Old Babylonian period (c. 1900 to 1600 B.C.). In the East, the theorem is found to have been applied in the Sulba-sutras (c. 800 to 500 B.C.) by the Vedic Hindus, but with no geometric proof given by them. In China, a discussion of the theorem is found in the Chou-pi suan-ching (c. 5th century B.C. to 3rd century B.C.). Chao Chün-ch'ing (c. 3rd century A.D.), the first to comment on the text, gave an algebraic treatment of the theorem involving quadratic equations. This ‘proof’ of the theorem is also found in Bhaskara II's work in the 12th century A.D., giving rise to the question of possible transmission. Finally, this paper surveys Chinese interest in right-angled and similar triangles in three other old Chinese texts, namely, the Chiu-chang suan-shu, Hai-tao suan-ching and Chang Ch'iu-chien suan-ching, and concludes that Chinese interest in the right-angled triangles was mainly practical, for the purposes of surveying.     

An Algebraic Proof of Barlet's Join Theorem

In this article we give an algebraic proof of Barlet's join theorem, using an improved version of an algorithm for computing the join that was presented in (J. Dalbec and B. Sturmfels,in“Invariant Methods in Discrete and Computational Geometry, Curaçao, 1994,” pp. 37–58, Kluwer Academic, Dordrecht, 1995). The improvements in the algorithm consist of proving the correctness of the algorithm in large prime characteristics and giving explicit formulas for the functionsp(q,L) that were left unspecified in the original. The first three sections are largely expository; the fourth presents the modified algorithm and the proof of the main theorem. The last section gives an example of the algorithm; unfortunately, the complexity of the process makes it difficult to compute anything without resorting to tricky shortcuts, as we do in the example.     

Authority and whole-class proving in high school geometry: The case of Ms. Finley

Students’ experiences with proving in schools often lead them to see proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas. We investigated how authority manifested in whole-class proving episodes within Ms. Finley’s high school geometry classroom. We designed a coding scheme that helped us identify the proving actions and interactions that occurred during whole-class proving and how Ms. Finley and her students contributed to those processes. By considering the authority over proof initiation, proof construction, and proof validation, the episodes illustrate how whole-class proving interactions might relate to students’ potential development (or maintenance) of authoritative proof schemes. In particular, the authority of the teacher and textbook limited students’ opportunities to engage collectively in proving and sometimes allowed invalid arguments to be accepted in the public discourse. We offer suggestions for research and practice with respect to authority and proof instruction.     

概率计算中易混淆问题的教法研究

首先从具体实例提出了物品检验概率计算中一类易混淆问题.然后从理论上证明了无放回逐次抽取中两种概率计算方法的合理性,其定理证明过程充分揭示了这类检验方法与其它相关抽取方式之间的关系并强调了最根本的分析问题思路.最后,研究了文中所得到的一般性结论的适用范围,从而使得这类易混淆问题的讲解变得清晰条理.     

关于连续变换的遍历性质的一些注记

设(X,d)是一个紧的距离空间,T是(X,d)上的连续变换.利用平均遍历定理证明了:对任意的x∈X,1/n sum from i=0 to n-1 f(T~i x)在C(X)上收敛.该结果是连续变换的Birkhoff型个别遍历定理的推广.由此结果研究了T的其它遍历性质,特别,不依赖深刻的Choquet积分表示定理,给出了遍历分解定理的一个较为简单而直接的证明.     

Optimal transmission of “television type” gaussian signals through a noiseless feedback channel

Using the methods and results of the theory of conditionally Gaussian filtering of stochastic processes and fields, an optimal scheme of “television type” signal transmission through a noiseless feedback channel is constructed under the usual power conditions, the signals being evolutionary Gaussian fields θ={θ_t(x)),tε[0,T),xεDεRⁿ . Explicit representations for optimal coding and decoding functionals, which are optimal in the sense of a special square criterion, and the expression for the error of signal reproduction are given.     

A generalization of max flow—min cut

We present a theorem which generalizes the max flow—min cut theorem in various ways. In the first place, all versions of m.f.—m.c. (emphasizing nodes or arcs, with graphs directed or undirected, etc.) will be proved simultaneously. Secondly, instead of merely requiring that cuts block all paths, we shall require that our general cuts be weight assignments which meet paths in amounts satisfying certain lower bounds. As a consequence, our general theorem will not only include as a special case m.f.—m.c., but also includes the existence of integral solutions to a transportation problem (with upper bounds on row and column sums) and its dual.     

有理插值(m/n)_f方程组的性质与作用

1引言 记Pn为次数不超过n的一元多项式函数类,约定零多项式的次数为-∞,即deg(0)=一∞;记Rm,n为分子属于Pm,分母属于Pn＼{0}的一元有理函数类.在[1-5]的基础上,文[6]引进了有理插值问题的(m-n)f方程组,其为经典(m/n)f方程组的一种等价变换.由于变换之后,使得参数之间地位相同,并且在个数上也与空间自由度一致,因此成为分析有理插值的一个有力工具.文[7]利用(m-n)f方程组,讨论了有理插值的基本特征,给出并证明了关于基本特征的基本关系定理.文[8]则在此基础上解决了有理插值的适定性问题.     

Asymptotically Constant Solutions of Functional Difference Systems

We consider the functional difference system ( A ) j x i ( n )= f i ( n ; X ), 1 h i h k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ] Z . We give sufficient conditions for ( A ) to have solutions that approach specified constant vectors as n M X . Some of the results guarantee only that the solutions are defined for n sufficiently large, while others are global. The proof of the main theorem is based on the Schauder-Tychonoff theorem. Applications to specific quasi-linear systems are included.     

On regular rings and annihilators

Let F_n stand for the distribution of a normalized sum of n independent random variables with common distribution H. In [6] we assumed the restricted convergence. and obtain an analogous result. The method of proof is considerably different, in particular a very recent continuation theorem (lemma 3.2) for infinitely divisible distributions is needed.