A space-time certified reduced basis method for quasilinear parabolic partial differential equations

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

The hierarchy of totally α‎-c.a. degrees

This chapter investigates the hierarchy of totally α‎-c.a. degrees. It demonstrates precisely when this hierarchy collapses (Theorem 3.6) and refines this hierarchy when one considers totally less than α‎-c.a. degrees (Theorem 3.25). The chapter then looks at uniform versions of the classes. The array computable degrees were a uniform version of the totally ω‎-c.a. degrees, in that the previous chapter took a single computable bound on the mind-change function of approximations of functions in the degree. This chapter finds the right formulation that generalises this to define uniformly totally α‎-c.a. degrees, and shows (Theorem 3.20) how they fit in the hierarchy.

BOUNDS FOR THE TORSION OF ELLIPTIC CURVES OVER EXTENSIONS WITH BOUNDED RAMIFICATION

Let E be a semi-stable elliptic curve defined over ℚ, and fix N ≥ 2. Let KN/ℚ be a maximal algebraic Galois extension of ℚ whose ramification indices are all at most N. We show that there exists a computable bound B(N), which depends only on N and not on the choice of E/ℚ, such that the size of E(KN) Tors is always at most B(N).

On the complexity of Gödel's proof predicate

The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

Transducer Placement for Broadband Active Vibration Control Using a Novel Multidimensional QR Factorization

This paper advances the state of the art in the selection of minimal configurations of sensors and actuators for active vibration control with smart structures. The method extends previous transducer selection work by (1) presenting a unified treatment of the selection and placement of large numbers of both sensors and actuators in a smart structure, (2) developing computationally efficient techniques to select the best sensor-actuator pairs for multiple unknown force disturbances exciting the structure, (3) selecting the best sensors and actuators over multiple frequencies, and (4) providing bounds on the performance of the transducer selection algorithms. The approach is based on a novel, multidimensional extension of the Householder QR factorization algorithm applied to the frequency response matrices that define the vibration control problem. The key features of the algorithm are its very low computational complexity, and a computable bound that can be used to predict whether the transducer selection algorithm will yield an optimal configuration before completing the search. Optimal configurations will result from the selection method when the bound is tight, which is the case for many practical vibration control problems. This paper presents the development of the method, as well as its application in active vibration control of a plate.

A Bound on the Rate of Convergence for the Discrete Gibbs Sampler

We give a computable bound on the rate of convergence of the occupation measure for the Gibbs sampler to the stationary distribution.

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms, and: Multiplicative relations in number fields: Corrigenda and addenda

We indicate a number of qualifications and amendments that are necessary so as to correct the statements and proofs of the theorems in our papers “Computing the effectively computable bound in Baker's inequality for linear forms in logarithms”, 15 (1976), 33–57, and its sequel “Multiplicative relations in number fields”, 16 (1977), 83–98, and remark on recent observations that would yield yet sharper results.

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

21.—Truncation Error in the Solution of Integral Equations

SynopsisA way is suggested of modifying the kernel of an integral equation of the second kind so that truncation of the algebraic system of equations corresponding to the new kernel is subject to less error than that for the original kernel. A computable bound for the error is derived.