The supremum of autoconvolutions, with applications to additive number theory

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \| f \|_1^2 / I. This improves the previous bound of 0.591389 \| f \|_1^2 / I. Consequently, we improve the known bounds on several related number-theoretic problems. For a subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.Comment: 17 pages. to appear in IJ

A reinterpretation of set differential equations as differential equations in a Banach space

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting

Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments

Results of ASTM round robin testing for mode 1 interlaminar fracture toughness of composite materials

The results are summarized of several interlaboratory 'round robin' test programs for measuring the mode 1 interlaminar fracture toughness of advanced fiber reinforced composite materials. Double Cantilever Beam (DCB) tests were conducted by participants in ASTM committee D30 on High Modulus Fibers and their Composites and by representatives of the European Group on Fracture (EGF) and the Japanese Industrial Standards Group (JIS). DCB tests were performed on three AS4 carbon fiber reinforced composite materials: AS4/3501-6 with a brittle epoxy matrix; AS4/BP907 with a tough epoxy matrix; and AS4/PEEK with a tough thermoplastic matrix. Difficulties encountered in manufacturing panels, as well as conducting the tests are discussed. Critical issues that developed during the course of the testing are highlighted. Results of the round robin testing used to determine the precision of the ASTM DCB test standard are summarized

Financial intermediaries, markets and growth

We build a model in which financial intermediaries provide insurance to households against a liquidity shock. Households can also invest directly on a financial market if they pay a cost. In equilibrium, the ability of intermediaries to share risk is constrained by the market. This can be beneficial because intermediaries invest less in the productive technology when they provide more risk-sharing. Our model predicts that bank-oriented economies should grow slower than more market-oriented economies, which is consistent with some recent empirical evidence. We show that the mix of intermediaries and market that maximizes welfare under a given level of financial development depends on economic fundamentals. We also show the optimal mix of two structurally very similar economies can be very different. --Financial Intermediaries,Risk Sharing,Finance and Growth,Comparing Financial Systems

The ROMES method for statistical modeling of reduced-order-model error

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability