Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Real and Complex Analysis

Functions of One Complex Variable

A Tutorial on Computable Analysis

This paper considers the problem of routing and rebalancing a shared fleet of autonomous (i.e., self-driving) vehicles providing on-demand mobility within a capacitated transportation network, where congestion might disrupt throughput. We model the problem within a network flow framework and show that under relatively mild assumptions the rebalancing vehicles, if properly coordinated, do not lead to an increase in congestion (in stark contrast to common belief). From an algorithmic standpoint, such theoretical insight suggests that the problems of routing customers and rebalancing vehicles can be decoupled, which leads to a computationally-efficient routing and rebalancing algorithm for the autonomous vehicles. Numerical experiments and case studies corroborate our theoretical insights and show that the proposed algorithm outperforms state-of-the-art point-to-point methods by avoiding excess congestion on the road. Collectively, this paper provides a rigorous approach to the problem of congestion-aware, system-wide coordination of autonomously driving vehicles, and to the characterization of the sustainability of such robotic systems.

http://www.roboticsproceedings.org/rss12/p32.pdf

Routing autonomous vehicles in congested transportation networks: structural properties and coordination algorithms

Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of non-trivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliably in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers.In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of complex functions f(z)=z2+c for |c|

The computational complexity of some julia sets

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

The alternation hierarchy for Turing machines with a space bound between loglog and log is infinite. That applies to all common concepts, especially a) to two-way machines with weak space-bounds, b) to two-way machines with strong space-bounds, and c) to one-way machines with weak space-bounds. In all of these cases the ∑
 k
 -and II
 k
 -classes are not comparable fork>-2. Furthermore the ∑
 k
 -classes are not closed under intersection and the II
 k
 -classes are not closed under union. Thus these classes are not closed under complementation. The hierarchy results also apply to classes determined by an alternation depth which is a function depending on the input rather than on a constant.

The alternation hierarchy for sublogarithmic space is infinite

Let Csc and Cwc be classes of the semi-computable and weakly computable real numbers, respectively, which are discussed by Weihrauch and Zheng [12]. In this paper we show that both Csc and Cwc are not closed under the total computable real functions of finite length on some closed interval, although such functions map always a semi-computable real numbers to a weakly computable one. On the other hand, their closures under general total computable real functions are the same and are in fact an algebraic field. This field can also be characterized by the limits of computable sequences of rational numbers with some special converging properties.

Weakly Computable Real Numbers and Total Computable Real Functions

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore how the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Robert Rettinger

Papers

The computational complexity of some julia sets

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

The alternation hierarchy for sublogarithmic space is infinite

Weakly Computable Real Numbers and Total Computable Real Functions

Weak computability and representation of reals