On T-spaces and related concepts and results

This paper is a brief survey of the T-space concept and related results. A connection between T-spaces and so-called varieties of pairs, analogous to the connection between T-ideals and varieties of algebras, is established. The concepts of A-equivalency and T-equivalency are introduced, after which some applications are considered.     

Limit Cycle Stability

A method for determining the stability of limit cycles in non-linearsystems is presented. It is based on the describing functionmethod used in engineering, and is especially suitable for usewith single-loop feedback systems, though it can be used inits present form with autonomous sets of ordinary differentialor difference equations which can be transformed to single-loopfeedback form. The stability test uses a successive approximationmethod which is shown to be convergent; explicit error boundsare not given but a feature of the method is that it is apparentwhen a sufficiently high order approximation is being used.From a practical point of view, the stability criterion's mainadvantage appears to be that the nonlinear part of the systemis not greatly restricted—discontinuities and multiplebranches cause no difficulties, as evidenced by an example givenwhere the nonlinearity is a relay with hysteresis. Unlike earlierlimit cycle stability tests using describing functions, thisone includes its own reliability guide and allows a better approximationto be used if the current one is not good enough.     

巴氏空间中随机元的极限理论及其应用

本文综述了巴拿赫空间中随机元的极限理论及其应用的一些结果，主要内容为：极限理论的某些新结果；等周方法的优化技巧；巴氏空间几何的概率方法；在经验过程研究中的应用。重点是近十年来有关问题研究的一些进展。     

对一道习题的再思考

对一道典型数列极限题,采用类比的方法,从离散型到连续型,从低维到高维进行推广,得到了几个推广命题.     

The second important Limit

We know the first important limit is lim x→0 sinx/x=1, and the second important limit is lim n→∞ （1＋1/n）^n=e. Although it is more important and useful than the first one, it is still named ＂the second important limit＂ because it was discovered after the first one.     

The second important Limit

Example 4 lim△r→0 ln（1＋△x/x）/△x/x =？     

Limit Cycle-Strange Attractor Competition

To understand the competition between what are known as limit cycle and strange attractor dynamics, the classical oscillators that display such features were coupled and studied with and without external forcing. Numerical simulations show that, when the Duffing equation (the strange attractor prototype) forces the van der Pol oscillator (the limit cycle prototype), the limit cycle is destroyed. However, when the van der Pol oscillator is coupled to the Duffing equation as linear forcing, the two traditionally stable steady states are destabilized and a quasi-periodic orbit is born. In turn, this limit cycle is eventually destroyed because the coupling strength is increased and eventually gives way to strange attractor or chaotic dynamics. When two van der Pol oscillators are coupled in the absence of external periodic forcing, the system approaches a stable, nonzero steady state when the coupling strengths are both unity; trajectories approach a limit cycle if coupling strengths are equal and less than 1. Solutions grow unbounded if the coupling strengths are equal and greater than 1. Quasi-periodic solutions give way to chaos as the coupling strength increases and one oscillator is strongly coupled to the other. Finally, increasing the nonlinearity in both the oscillators is stabilizing whereas increasing the nonlinearity in a single oscillator results in subcritical instability.     

Limit computable integer parts

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every ${r \in R}$ , there exists an ${i \in I}$ so that i ?? r < i?+?1. Mourgues and Ressayre (J Symb Logic 58:641?C647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is ${\Delta^0_2(R)}$ ??limit computable relative to R. We show that there is a maximal Z-ring ${I \subseteq R}$ which is ${\Delta^0_2(R)}$ . However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ??77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie??s ideas, we produce a real closed field R with a Z-ring ${I \subseteq R}$ such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class ${\Delta^0_2}$ . We know that certain subclasses of ${\Delta^0_2}$ are not sufficient. We show that for each ${n \in \omega}$ , there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n.     

Uniform-in-Bandwidth Functional Limit Laws

We provide uniform-in-bandwidth functional limit laws for the increments of the empirical and quantile processes. Our theorems, established in the framework of convergence in probability, imply new sharp uniform-in-bandwidth limit laws for functional estimators. In particular, they yield the explicit value of the asymptotic limiting constant for the uniform-in-bandwidth sup-norm of the random error of kernel density estimators. We allow the bandwidth to vary within the complete range for which the estimators are consistent.     

Approximate Central Limit Theorems

We refine the classical Lindeberg–Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory.     

一个强极限定理

设是可列非齐次马氏链，本文通过利用［１］中提出的在Ｗｉｅｎｅｒ概率空间的一种实现，而给出了一个对任意可列非齐次马氏链普遍成立的强极限定理。     

The Discounted Limit Theorems

The large deviation theorems, exponential inequalities and a non-uniform estimate of the Berry–Esséen theorem in a discounted version are proved.Dedicated to Professor Vytautas Statulevičius on the occasion of his 75th birthday.     

Inductive Limit Violates Quasi-cocommutativity

We show that the inductive limit of a certain inductive system of quasi-cocommutative C^*-bialgebras is not quasi-cocommutative. This implies that the category of quasi-cocommutative C^*-bialgebras is not closed with respect to the inductive limit.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.     

Strong Limit Power Functions

A new C^*-algebra of strong limit power functions is proposed. The Gelfand space of the C^*-algebra is endowed with an Abelian compact group structure. As applications of this, Fourier analysis and the Bochner-Fejér approximation are carried out for a strong limit power function. Finally, the functions are extended to more general cases and their properties are investigated in that settings.     

Limit theorems for mergesort

Central and local limit theorems (including large deviations) are established for the number of comparisons used by the standard top-down recursive mergesort under the uniform permutation model. The method of proof utilizes Dirichlet series, Mellin transforms, and standard analytic methods in probability theory. © 1996 John Wiley & Sons, Inc.     

A Nonconventional Local Limit Theorem

Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as \(N\rightarrow \infty \) of probabilities having the form \(P\{ S_N=k\}\) where \(S_N=\sum ^N_{n=1}F(\xi _n)\) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables \(\{\xi _j\}\). Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form \(S_N=\sum _{n=1}^NF(\xi _n,\xi _{2n}, \ldots ,\xi _{\ell n})\) which continues the recent line of research studying various limit theorems for such expressions.     

Synchronization of Coupled Limit Cycles

A unified approach to the analysis of synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented by a discussion of numerical simulations of a compartmental model of a neuron.