一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

Drinfeld Realization of Twisted Quantum Affine Algebras

The quantum affine algebra has two realizations, the usual Drinfeld–Jimbo definition and a new Drinfeld realization given by Drinfeld. In this article, we use the adjoint action to prove that these two realizations are isomorphic for the twisted quantum affine algebra.     

A new approach to monotone iterative technique for solution of differential equations

We discuss a monotone interative technique for initial value problems. The novelty of the approach is the use of g-monotonicity, i.e. of monotonicity of the ratio of two functions. This is in contrast to the more usual notion of monotonicity     

A note on ratio and product type estimators

Summary The use of ratio and product estimators for the mean of a finite population is well known. This paper proposes transformed estimators obtained through parametric linear combination of the ratio, or product, and the usual unbiased estimator of the mean for any sample design. To the first degree of approximation, the proposed estimators have smaller mean square error than that of the ratio, product and the usual unbiased estimator, for suitable choice of the parameter. The superiority of the proposed estimators over others for small samples has been studied empirically.     

PROGRESSING WAVE SOLUTIONS TO QUASI-LINEAR SYSTEMS MIXED PROBLEMS

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On the Maximal Accuracy of Linear Optimal Systems

A Hilbert-space development of maximal accuracy and input squaring-downoperators, both topics of interest in optimal control and linearmultivariable design, is presented. The special theory of compact self-adjoint Hilbert-space operatorsenables a canonical form to be developed for the optimal valueof the quadratic cost function; further its asymptotic behaviourcan be investigated by introducing a scalar parameter into thecontrol weighting term. If the spectral theory is then linkedto the usual orthogonal-range space/null-space Hilbert-spacedecomposition, the theorems for the asymptotic value of theoptimal cost and the definition for maximal accuracy have aparticularly simple form. The relevance of these results to the assessment of the effectivenessof subsets of control inputs and the use of squaring-down operatorsis discussed. The link with the usual maximal-accuracy problemfor time-invariant state-space triples is also examined.     

Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples

We analyze heuristic worked-out examples as a tool for learning argumentation and proof. Their use in the mathematics classroom was motivated by findings on traditional worked-out examples, which turned out to be efficient for learning algorithmic problem solving. The basic idea of heuristic worked-out examples is that they encourage explorative processes and thus reflect explicitly different phases while performing a proof. We tested the hypotheses that teaching with heuristic examples is more effective than usual classroom instruction in an experimental classroom study with 243 grade 8 students. The results suggest that heuristic worked-out examples were more effective than the usual mathematics instruction. In particular, students with an insufficient understanding of proof were able to benefit from this learning environment.     

A priori bound for the supremum of solutions of stable stochastic differential equations

An upper bound for the solutions of stochastic differential equations for which the drift satisfies a usual stability assumption is given. This bound, valid for any time, only needs to make use of explicit constants which can be obtained from the coefficients of these stochastic differential equations. The same kind of result is obtained for Lyapounov functions.     

Differential geometry and the red blood cell

The usual mathematical expressions for the shape of the red blood cell, whether in rectangular or polar coordinates, are very complicated and difficult to use analytically. We have found that by using parametric equations and Jacobean elliptic functions to describe the cell things are much easier. The usual differential geometric concepts can be adapted to find interesting new formulas for ideas like Fick's Law, etc. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A CR Poincaré inequality on the complex sphere

The main purpose of this paper is to prove a CR Poincaré inequality with sharp exponent on the sphere in complex space. We use the complex tangential gradient on the sphere instead of the usual Laplace-Beltrami gradient on the sphere.     

On an expansion of nondeterministic finite automata

We consider a possible generalization of nondeterministic finite automata. The goals of this consideration are: to apply some obtained algorithms for various problems of minimization of classical nondeterministic automata; to use such automata for describing practical anytime algorithms for the same problems of minimization; to simplify some proofs for algorithms of simplification for usual nondeterministic automata.     

Analysis of three-component soil structures

Impregnated soil blocks were collected from a tillage trial at Condobolin, New South Wales. The object of the trial was to examine the consequences of performing the first tillage operation of the season on different dates. It was apparent that the soil broke up on tillage into the usual range of aggregate sizes plus a dust which was presumably composed of micro-aggregates. This was apparent on sections through the blocks which showed three components: voids, aggregates and dust.A statistical method for quantifying soil structure, developed earlier, is extended to include three components. Since this involves a large number of transition probabilities, two approximate simplified methods were tested. One of these, here called the ‘grey-state method’, could be widely applied in structure studies using automatic scanning devices where features brighter or darker than some adjustable threshold level are detected.The results show that the most recently tilled plot had the coarsest structure, but that the earlier two treatments were not significantly different from each other. The usual trend of decreasing aggregate size and decreasing porosity with increasing depth was not observed. Instead, there appeared to be some inversion of the usual layering of tilled soil which is attributed to the use of a motor-driven rod-weeder.     

Making Use of Incomplete Observations for Regression in Bivariate Normal Model

Two estimates of the regression coefficient in bivariate normal distribution are considered: the usual one based on a sample and a new one making use of additional observations of one of the variables. They are compared with respect to variance. The same is done for two regression lines. The conclusion is that the additional observations are worth using only when the sample is very small.     

Direct consequences of the basic Ballot Theorem

We use only the classic basic ballot result and simple combinatorial arguments to derive the distributions of the first passage time and the number of visits in the usual random walk model.     

on O-simple semirings,semigroup semirings,and two kinds of division semirings

In this paper we consider O-simple semirings S, where O denotes the multiplicative zero of S, which may be in particular the additive neutral o of S at the same time. In this context we give some statements on matrix semirings and introduce contracted semigroup semirings in §3, a matter of interest of its own. We further use our results to compare the usual concept of division semirings with a new one introduced in [18], and we show that a corresponding theorem in [18] is in general only valid for division semirings in the usual meaning. Dedicated to E.S. Lyapin on his 70th birthday     

Single-path routing of stochastic flows in networks

The paper revisits a very simple network routing and dimensioning model, with two specific assumptions: the traffic amounts to be routed are Gaussian random variables, and each commodity must use one single route in the network. The need to control congestion leads naturally to probabilistic constraints. The impact of stochastic assumptions on solution algorithms is investigated, when compared to the usual deterministic case.     

Inequalities for unitarily invariant norms and bilinear matrix products

We give several criteria that are equivalent to the basic singular value majorization inequality (1.1) that is common to both the usual and Hadamard products. We then use these criteria to give a unified proof of the basic majorization inequality for both products. Finally, we introduce natural generalizations of the usual and Hadamard products and show that although these generalizations do not satisfy the majorization inequality, they do satisfy an important weaker inequality that plays a role in establishing their submultiplicativity with respect to every unitarily invariant norm.     

Commutative subalgebras of the corona

We use methods of noncommutative functional analysis to extend the range of the usual functional calculus, for certain subalgebras of the corona. In particular, we construct corona projections with interesting properties.

     

The Riemann approach to stochastic integration using non-uniform meshes

In this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by Riemann's approach. However in this approach we use non-uniform meshes instead of the usual uniform meshes.     

Nonparametric regression with martingale increment errors

We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski’s selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called “stability”, which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a β-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one.