Abstract: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning. Among the topics dealt with are: classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The book is self-contained, and yet sufficiently detailed to provide an introduction to research in this area.

Abstract: Consideration is given to the problems of filtering and smoothing for linear systems in an H/sup infinity / setting, i.e. the plant and measurement noises have bounded energies (are in L/sub 2/), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered: in one case the initial condition is assumed known; in the other case it is not known, but the initial condition, the plant, and the measurement noise are in some weighted ball of R/sup n/*L/sub 2/. Both finite-horizon and infinite-horizon cases are considered. The authors present necessary and sufficient conditions for the existence of estimators (both filters and smoothers) that achieved a prescribed performance bound and develop algorithms that result in performance within the bounds. They also present the optimal smoother. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which time-varying and time-invariant linear systems can be considered with equal ease. >

Abstract: The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H/sub infinity / bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin. All controllers satisfying a suboptimal stability margin are characterized, and explicit state-space formulas are given. >

Abstract: For linear systems described by $\dot x(t) = Ax(t) + Bu(t)$, where A generates a semigroup on the state space X and B is an unbounded operator, some necessary as well as some sufficient conditions are given for B to be admissible, i.e., for any t, the state $x(t)$ should be in X and should depend continuously on the input $u \in L^p $. This approach begins with an axiomatic description of such a system in terms of a functional equation. The results are applied to the wave equation on a bounded interval.

Abstract: The Fourier algebra A(G) of a locally compact group G is the space of matrix coefficients of the regular representation, and is the predual of the yon Neumann algebra VN(G) generated by the regular representation of G on L 2 (G). A multiplier m of A (G) is a bounded operator on A (G) given by pointwise multiplication by a function on G, also denoted m. We say m is a completely bounded multiplier ofA (G) if the transposed operator on VN(G) is completely bounded (definition below). It may be possible to find a net ofA (G)-functions, (m i : ie I) say, such that mi tends to

Abstract: We study the asymptotic behavior of solutions to a system of recursions u in+1 = Qi[mu n], i = 1, ..., k. The vector operator Q has the origin theta and a positive vector beta as fixed points and is defined for vector-valued functions bounded between theta and gamma where gamma greater than or equal to beta. In addition, Q is order-preserving, commutes with translation, and is continuous in the topology of uniform convergence on compact subsets. Let theta less than or equal to pi much less than beta, and suppose that for all pi much less than alpha much less than beta, Q(n) alpha]----beta as n----infinity. If u0 much greater than pi on a sufficiently large ball and has bounded support, then un propagates with a speed c*(xi) in the direction of the unit vector xi as n----infinity. In certain cases, c*(xi) can be calculated explicitly. The results generalize those of a scalar equation studied by Weinberger.

Abstract: The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d > 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

Abstract: The robustness of adaptive controllers with respect to uncertainty is examined. The uncertainties include bounded input disturbances, unknown and time-varying plant parameters, and unmodeled dynamics. A simple example shows instability of a recent manipulator control scheme in the presence of bounded disturbances. The adaptive laws for updating the controller parameters are modified so that instabilities are counteracted and robustness is guaranteed. >

Abstract: Instance-based representations have been applied to numerous classification tasks with some success Most of these applications involved predicting a symbolic class based on observed attributes This paper presents an instance-based method for predicting a numeric value based on observed attributes We prove that, given enough instances, if the numeric values are generated by continuous functions with bounded slope, then the predicted values are accurate approximations of the actual values We demonstrate the utility of this approach by comparing it with a standard approach for value prediction The instance-based approach requires neither ad hoc parameters nor background knowledge

Abstract: The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p<2), and find sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T). Using a Harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions. Uniqueness of solutions is shown if u0 belongs to L1loc(RN), but is left open in the case that u0 is merely a locally bounded measure. The results are closely related to papers by Aronson-Caffarelli, Benilan-Crandall-Pierre, and Dahlberg-Kenig about the porous medium equation ut = Δum. The proofs are different and allow one to generalize some of the above results to equations with variable coefficients.

Abstract: This paper derives a feedback controlf(t), ‖f(t)‖E≦r,r>0, which forces the infinite-dimensional control systemdu/dt=Au+Bf, u(0)=u o ≠H to have the asymptotic behavioru(t)→0 ast→∞ inH. HereA is the infinitesimal generator of aC o semigroup of contractionse At on a real Hilbert spaceH andB is a bounded linear operator mapping a Hilbert space of controlsE intoH. An application to the boundary feedback control of a vibrating beam is provided in detail and an application to the stabilization of the NASA Spacecraft Control Laboratory is sketched.

Abstract: The authors consider the construction of navigation functions on configuration spaces whose geometric expressiveness is rich enough for navigation amidst real-world obstacles. They describe a general methodology which extends the construction of navigation functions on sphere worlds to any smoothly deformable space. According to this methodology, the problem of constructing a navigation function is reduced to the construction of a transformation mapping a given space into its model sphere world. The transformation must satisfy certain regularity conditions guaranteeing invariance of the navigation function properties. The authors demonstrate this idea by constructing navigation functions on star worlds: n-dimensional star shaped subsets of E/sup n/ punctured by any finite number of smaller disjoint n-dimensional stars. This construction yields automatically a bounded torque feedback control law which is guaranteed to guide the robot to destination point from almost every initial position without hitting any obstacle. >

Abstract: Supported by the National Science Foundation during 1986–1988 at the Mathematical Sciences Research Institute, Berkeley, and at the Institute for Advanced Study, Princeton.

Abstract: A theory satisfies the k-variable property if every first-order formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a model of temporal logic satisfies the k -variable property for some k if and only if there exists a finite basis for the temporal connectives over that model. We give a model-theoretic method for establishing the k -variable property, involving a restricted Ehrenfeucht-Fraisse game in which each player has only k pebbles. We use the method to unify and simplify results in the literature for linear orders. We also establish new k -variable properties for various theories of bounded-degree trees, and in each case obtain tight upper and lower bounds on k . This gives the first finite basis theorems for branching-time models of temporal logic.

Abstract: The problem of developing a control law which can force the output of a linear time-varying plant to track the output of a stable linear time-invariant reference model is discussed. It is shown that the standard model reference controller, used for linear time-invariant plants, cannot guarantee zero tracking error in general when the plant is time-varying. A new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time-varying plants with known parameters. When the time-varying plant parameters are unknown but vary slowly with time, it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean. The assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori. Such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters. >

Abstract: Two results on interactive proof systems with two-way probabilistic finite-state verifiers are proved. The first is a lower bound on the power of such proof systems if they are not required to halt with high probability on rejected inputs: it is shown that they can accept any recursively enumerable language. The second is an upper bound on the power of interactive proof systems that halt with high probability on all inputs. The proof method for the lower bound also shows that the emptiness problem for one-way probabilistic finite-state machines is undecidable. In the proof of the upper bound some results of independent interest on the rate of convergence of time-varying Markov chains to their halting states are obtained.<>

Abstract: We introduce a graceful approach to probabilistic inference called bounded conditioning. Bounded conditioning monotonically refines the bounds on posterior probabilities in a belief network with computation, and converges on final probabilities of interest with the allocation of a complete resource fraction. The approach allows a reasoner to exchange arbitrary quantities of computational resource for incremental gains in inference quality. As such, bounded conditioning holds promise as a useful inference technique for reasoning under the general conditions of uncertain and varying reasoning resources. The algorithm solves a probabilistic bounding problem in complex belief networks by breaking the problem into a set of mutually exclusive, tractable subproblems and ordering their solution by the expected effect that each subproblem will have on the final answer. We introduce the algorithm, discuss its characterization, and present its performance on several belief networks, including a complex model for reasoning about problems in intensive-care medicine.

Abstract: In this paper we prove that, for every positive integer k, there exists a contractible bounded domain Ω in ℝN with N≥3, where the problem (*) (see Introduction) has at least k solutions.

Abstract: This paper considers the analysis and synthesis of control systems subject to two types of disturbance signals: signals with bounded power spectral density and signals with bounded power. The resulting control problem involves minimizing a mixed H2 and H∞ norm of the system. It is shown that the controller shares a separation property similar to those of pare H2 or H∞. controller. It is also shown that the mixed problem reduces naturally to H2 and H∞ problem in special cases. Some necessary and sufficient conditions are obtained for the existence of a solution to the mixed problem. Explicit state space formulae are given for the optimal controllers.

Abstract: Because games and game-like phenomena occur naturally in a computational setting, it is natural to formulate many problems in Computer Science in terms of games. In order to understand their complexity, various models of computation have been developed which reflect the game-like properties of such problems. These models include the alternating Turing machines of Chandra, Kozen, and Stockmeyer (CKS81), the games against nature of Papadimitriou (PAP83), the Arthur-Merlin games of Babai (BAB85), and the interactive proof systems of Goldwasser, Micali, and Rackoff (GMR85). We unify and extend the work on these game-like models of computation. We define a new computational model of two person games, called a probabilistic game automaton. Three important features of games are included in the definition: randomness, secrecy and limited power for the players. Probabilistic game automata are defined as language acceptors, where the input is accessible to both players. We prove a number of results on the complexity of some classes of languages accepted by special types of game automata encompassed by our model. To state these results precisely, we use a consistent notation throughout the dissertation. Some of these results are summarized here. In our notation, we let UP, (UC) denote the class of two-person games with unbounded two-sided error and partial information (complete information), where one player plays randomly. Hence, UC refers to games against known nature and UP refers to games against unknown nature. We show that ATIME(poly(n)) = UC--TIME(poly(n)) = UP--TIME(poly(n)) and ASPACE(poly(n)) = UC--SPACE(poly(n)) $\subseteq$ UP--SPACE(log((n))), where ATIME and ASPACE refer to alternating Turing machines and poly(n) is any polynomial function of n. Here and in the results below we assume that the space and time bounds are deterministically constructible. We also prove a number of new results on the power of the space bounded analogues of Arthur-Merlin games and interactive proof systems. We denote these by BC and BP respectively, for probabilistic games with bounded error with complete and partial information, respectively. Our main results are that ASPACE(poly(n)) = BC--SPACE(poly(n)) $\subseteq$ BP--SPACE(log(n)). As a consequence, any language recognizable in deterministic exponential time has an interactive proof which uses only logarithmic space.

Abstract: 1. Non-degeneracy in the perturbation theory of integrable dynamical systems Helmut Riissmann 2. Infinite dimensional inverse function theorems and small divisors J. A. G. Vickers 3. Metric Diophantine approximation of quadratic forms S. J. Patterson 4. Symbolic dynamics and Diophantine equations Caroline Series 5. On badly approximable numbers, Schmidt games and bounded orbits of flows S. G. Dani 6. Estimates for Fourier coefficients of cusp forms S. Raghavan and R. Weissauer 7. The integral geometry of fractals K. J. Falconer 8. Geometry of algebraic continued fractals J. Harrison 9. Chaos implies confusion Michel Mendes France 10. The Riemann hypothesis and the Hamiltonian of a quantum mechanical system J. V. Armitage.

Abstract: A framework for algebraic specification of abstract data types is introduced. It involves so-called unified algebras, where sorts are treated as values, so that operations can be applied to sorts as well as to the elements that they classify. An institution for unified algebras is defined and shown to be liberal. However, the ordinary forgetful functor does not forget any values in unified algebras, so the usual data constraints do not have any models. A more forgetful functor is introduced and used to define so-called bounded data constraints, which have the expected models. >