The addition to the ergodic theorem on probability MV-algebras with product

 This article deals with the prototype of the strong law of large numbers, with individual ergodic theorem on probability MV-algebras with product. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday This research is supported by Grant 1/9056/02 1991 Mathematics Subject Classification: 28E10     

An ergodic theorem for filtering with applications to stability

Ergodic properties of the signal–filtering pair are studied for continuous time finite Markov chains, observed in white noise. The obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions. The Furstenberg–Khasminskii formula is derived for the top Lyapunov exponent of the Zakai equation and is used to estimate the stability index of the filter.     

The inclusion-exclusion principle for IF-events

The probabilistic version of the inclusion-exclusion principle is generalized for IF-events. Two versions of the generalized formula, corresponding to different t-conorms applied for defining the union of IF-events are shown.     

A strong direct product theorem for quantum query complexity

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma—computing the parity of k copies of f with fewer than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.     

The ergodic theory of cellular automata

Ergodic theory is the study of how a dynamical system transforms the information encoded in an invariant probability measure. This article reviews the major recent results in the ergodic theory of cellular automata.     

The martingale convergence theorem, with tears

The convergence rate of the martingale convergence theorem is explored, and it is shown that this rate is not robust as a function of the random variable being estimated with a sequential scheme. These results are then extended to a family of operators which generalize conditional expectation. It is shown that in both of these cases the convergence rate can be arbitrarily slow.     

An ergodic AIMD algorithm with application to high-speed networks

In this article we propose a version of the Additive-Increase Multiplicative-Decrease (AIMD) algorithm that provides a suitable basis to develop congestion control protocols that can be deployed in both conventional and high-speed communication networks. Our algorithm retains many of the properties of the standard AIMD algorithm. However, unlike other non-standard AIMD algorithms, our scheme can be shown to be ergodic under very general assumptions.     

On the ergodic theory of cellular automata

When properly viewed, the transition rule of a cellular automaton becomes a mapF from a set to itself. The set may be made a probability space. Sufficient conditions are given to ensure thatF be measure-preserving and ergodic. Some geometric consequences of ergodicity are noted.     

An experiment with the Boyer-Moore theorem prover: A proof of Wilson's theorem

This paper describes the use of the Boyer-Moore theorem prover in mechanically generating a proof of Wilson's theorem: for any prime p, (p-1)! and p-1 are congruent modulo p. The input to the theorem prover consists of a sequence of three function definitions and forty-two propositions to be proved. The proofs generated by the system are based on a library of lemmas relating to list manipulation and number theory, including Fermat's theorem.     

The last player theorem

Game trees are an important model of decision-making situations, both in artificial intelligence and decision analysis, but many of the properties of game trees are not well understood. One of these properties is known as biasing: when a minimax search is done to an odd search depth, all moves tend to look good, and when it is done to an even search depth, all modes tend to look bad.One explanation sometimes proposed for biasing is that whenever a player makes a move his position is ‘strengthened’, and that the evaluation function used in the minimax search reflects this. However, the mathematical results in this paper suggest that biasing may instead be due to the errors made by the evaluation function.     

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.     

Interaction with the Boyer-Moore theorem prover: A tutorial study using the arithmetic-geometric mean theorem

There are many papers describing problems solved using the Boyer-Moore theorem prover, as well as papers describing new tools and functionalities added to it. Unfortunately, so far there has been no tutorial paper describing typical interactions that a user has with this system when trying to solve a nontrivial problem, including a discussion of issues that arise in these situations. In this paper we aim to fill this gap by illustrating how we have proved an interesting theorem with the Boyer-Moore theorem prover: a formalization of the assertion that the arithmetic mean of a sequence of natural numbers is greater than or equal to their geometric mean. We hope that this report will be of value not only for (non-expert) users of this system, who can learn some approaches (and tricks) to use when proving theorems with it, but also for implementors of automated deduction systems. Perhaps our main point is that, at least in the case of Nqthm, the user can interact with the system without knowing much about how it works inside. This perspective suggests the development of theorem provers that allow interaction that is user oriented and not system developer oriented. This research was supported in part by ONR Contract N00014-94-C-0193. The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the Office of Naval Research, or the U.S. government.     

On the conditional expectation on probability MV-Algebras with product

 In this paper the conditional expectation is defined and a variant of the martingale convergence theorem on probability MV-algebras with product is proved.     

The Lyapunov–Malkin theorem and stabilization of the unicycle with rider

This paper analyzes stabilization of a nonholonomic system consisting of a unicycle with rider. It is shown that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider's limb.     

Central limit theorem and large deviations of the fading Wyner cellular model via product of random matrices theory

We apply the theory of products of random matrices to the analysis of multi-user communication channels similar to the Wyner model, which are characterized by short-range intra-cell broadcasting. We study fluctuations of the per-cell sum-rate capacity in the non-ergodic regime and provide results of the type of the central limit theorem (CLT) and large deviations (LD). Our results show that CLT fluctuations of the per-cell sum-rate C _m are of order \(1/\sqrt m \), where m is the number of cells, whereas they are of order 1/m in classical random matrix theory. We also show an LD regime of the form P(|C _m ? C| > ?) ≤ e ^?mα with α = α(?) > 0 and C = \(\mathop {\lim }\limits_{m \to \infty } \) C _m , as opposed to the rate \(e^{ - m^2 \alpha } \) in classical random matrix theory.     

Quadratic stability in the circle theorem or positivity theorem

In this technical note, we study the quadratic stability problem for linear time-invariant multi-input, multi-output systems which have time-varying sector-bounded or positive real uncertainty feedback. It is known that the circle theorem and positivity theorem are sufficient conditions for robust stability, for sector-bounded and positive real feedback, respectively. We show that the circle theorem is necessary and sufficient condition for quadratic stability when the feedback is sector-bounded, and that the positivity theorem is necessary and sufficient condition for quadratic stability when the feedback is positive-real.     

A theorem on the Lyapunov matrix equation

Given the Lyapunov matrix equationA'P + PA + 2sigmaQ = 0where σ is some positive scalar, a necessary and sufficient condition for the real parts of the eigenvalues ofAto be less than -σ is thatP - Qis negative definite. The condition provides an upper bound to the solution of the Lyapunov matrix equation and is useful in the design of minimum-time or minimum-cost linear control systems.