Instability in a delay-PDE model of earthquake occurrence

Summary Ductile deformation prior to brittle fracture in rocks causes fracture to take place with a time delay after the critical stress for fracture is reached, in the presence of an increasing load stress. We discuss the stability of a stochastic model of interactive earthquake occurrence under the influence of time delays resulting from the ductile process. A threshold for oscillatory behavior is found for large enough coupling and discrete time delays. The system is stable if the time delays are spread over a broad time interval, even for large coupling.     

A general converse theorem for mean‐value theorems in linear elasticity

We prove a rather general mean‐value formula in the theory of elasticity, which expresses the value of the displacement at the centre of a sphere in terms of certain combinations of integral averages over the sphere itself of the traction and the displacement. We also establish the corresponding converse to this mean‐value formula under minimal smoothness assumptions on the displacement. Copyright © 2006 John Wiley & Sons, Ltd.     

Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation

Product integration rules generalizing the Fej?r, Clenshaw-Curtis,and Filippi quadrature rules respectively are derived for integralswith trigonometric and hyperbolic weight factors. The Chebyshevmoments of the weight functions are found to be given by well-conditionedexpressions, in terms of hypergeometric functions ₀F₁. An a priori error estimator is discussed which is shown bothto avoid wasteful invocation of the integration rule and toincrease significantly the robustness of the automatic quadratureprocedure. Then, specializing to extended Clenshaw-Curtis (ECC) rules,three types of a posteriori error estimates are considered andthe existence of a great risk of their failure is demonstratedby large scale validation tests. An empirical error estimator,superseding them for slowly varying integrands, is found toresult in a spectacular increase in the output reliability. Finally, enhancements in the control of the interval subdivisionstrategy aiming at increasing code robustness is discussed.Comparison with the code DQAWO of QUADPACK, with about a hundredthousand solved integrals, is illustrative of the increasedrobustness and error estimate reliability of our computer codeimplementation of the ECC rules.     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

De novo design of a redox-active minimal rubredoxin mimic

Metal-binding sites in metalloproteins frequently occur at the interfaces of elements of secondary structure, which has enabled the retrostructural analysis of natural proteins and the de novo design of helical bundles that bind metal ion cofactors. However, the design of metalloproteins containing beta-structure is less well developed, despite the frequent occurrence of beta-conformations in natural metalloproteins. Here, we describe the design and construction of a beta-protein, RM1, that forms a stable, redox-active 4-Cys thiolate Fe(II/III) site analogous to the active site of rubredoxin. The protein folds into a beta-structure in the presence and absence of metal ions and binds Fe(II/III) to form a redox-active site that is stable to repeated cycles of oxidation and reduction, even in an aerobic environment.     

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments. The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed. R. Jones is partially supported by NSF Grant DMS-9531526. J. Rosenblatt is partially supported by NSF Grant DMS-9705228. M. Wierdl is partially supported by NSF Grant DMS-9801602.     

Reverse inequalities

There are reverse inequalities for square functions of differences arising in ergodic theory and differentiation of functions. For example, it is shown that if A_n is the usual average in ergodic theory, and (n_k∶k=1,2,3,...) is an increasing lacunary sequence with no non-trivial common divisor, then one has for any p, 1<p<∞, there is a constant C_p such that for all f∃ L_p(X),