Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions

This paper concerns the reconstruction of possibly complex‐valued coefficients in a second‐order scalar elliptic equation that is posed on a bounded domain from knowledge of several solutions of that equation. We show that for a sufficiently large number of solutions and for an open set of corresponding boundary conditions, all coefficients can be uniquely and stably reconstructed up to a well‐characterized gauge transformation. We also show that in some specific situations, a minimum number of such available solutions equal to $I_n = {1 \over 2}n(n + 3)$ is sufficient to uniquely and globally reconstruct the unknown coefficients. This theory finds applications in several coupled‐physics medical imaging modalities including photo‐acoustic tomography, transient elastography, and magnetic resonance elastography. © 2013 Wiley Periodicals, Inc.     

The energy of dislocation dipoles

In conventional dislocation theory based on the concept of Volterra dislocations, the energy of coplanar dislocation dipoles cannot be obtained without introducing an ill-defined ad hoc annihilation distance. When the dislocations are described more realistically by Peierls dislocations, absolute values for the dipole energy can be obtained from physical arguments. It is found that when dislocations in a dipole come close, they may not annihilate each other but the dipole can collapse when the dislocations have reached a critical separation of several lattice constants.     

Pseudo-differential operators with involutive double characteristics

Suppose G is a bounded C²domain in IR ⁿ, n ? 2 . We exam¬ine the regularity at the boundary of solutions to a class of quasi-linear elliptic equations having continuous boundary values ? . If ? has a modulus of continuity β , we give a modulus of continunity for the solution which depends on β and the generalized mean curvature of ?G . When the order of non-uniformity of the equation is between 0 and 1 , no curvature condition on ?G is needed.     

On Modelling and Pricing Rainfall Derivatives with Seasonality

Abstract

We are interested in pricing rainfall options written on precipitation at specific locations. We assume the existence of a tradeable financial instrument in the market whose price process is affected by the quantity of rainfall. We then construct a suitable ‘Markovian gamma’ model for the rainfall process which accounts for the seasonal change of precipitation and show how maximum likelihood estimators can be obtained for its parameters.

We derive optimal strategies for exponential utility from terminal wealth and determine the utility indifference price of the claim. The method is illustrated with actual measured data on rainfall from a location in Kenya and spot prices of Kenyan electricity companies.     

Chemical potential of a nonionic surfactant in solution

The chemical potential of a surfactant in solution can be calculated from the Gibbs adsorption equation when the surface excess of the surfactant and the surface tension of the solution as a function of surfactant concentration are known. We have investigated a solution of the nonionic surfactant 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) in the polar solvent 3-hydroxypropionitrile at concentrations below and above the critical micelle concentration (cmc). Neutral impact collision ion scattering spectroscopy was applied for the direct measurement of the surface excess of POPC as a function of concentration. The Gibbs adsorption equation was applied in conjunction with surface tension measurements to evaluate the chemical potential and the activity coefficients of POPC, respectively. We find that the solution shows ideal behavior up to the cmc and that the chemical potential remains constant at concentrations larger than the cmc.     

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.     

High-efficiency ultrasmall polarization converter in InP membrane

An ultrasmall (<10 μm length) polarization converter in InP membrane is fabricated and characterized. The device relies on the beating between the two eigenmodes of chemically etched triangular waveguides. Measurements show a very high polarization conversion efficiency of >99% with insertion losses of <-1.2 dB at a wavelength of 1.53?μm. Furthermore, our design is found to be broadband and tolerant to dimension variations.     

Determination of Space-Time Structures from Gravitational Perturbations

We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel to and return. We apply the result to two concrete examples when the source fields are scalar fields (i.e., Einstein–scalar field equations) and electromagnetic fields (i.e., Einstein-Maxwell equations). © 2019 Wiley Periodicals, Inc.     

Arithmetic equivalence for function fields, the Goss zeta function and a generalisation

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gaßmann equivalence (a purely group theoretical property), but this statement can fail if the degree exceeds the characteristic. We introduce a ‘Teichmüller lift’ of the Goss zeta function and show that equality of such is always the same as Gaßmann equivalence.     

[4 + 2]-Cycloadditionen mit inversem Elektronenbedarf,XVII. Oxepin und 2,7-Dimethyloxepin als Dienophile bei Diels-Alder-Cycloadditionen mit inversem Elektronenbedarf

[4 + 2] Cycloadditions with Inverse Electron Demand, XVII. Oxepin and 2,7-Dimethyloxepin as Dienophiles in Diels-Alder Cycloadditions with Inverse Electron Demand The cycloaddition reactions of oxepin ( 1 ) with the tetrazine 3 and the triazine 12 yield the dihydrooxepino[4,5-d]pyridazine 5 and the oxepino[4,5-c]pyridine 13 , resp., via [2 + 4] cycloaddition as well as the phthalazine 9 and the isoquinoline 14 , resp., most probably via [6 + 4] cycloaddition. On the contrary, 2,7-dimethyloxepin ( 15 ) only gives the products 16 and 20 , which result from [2 + 4] cycloaddition. Oxidation of 16 by silver(I) oxide leads to the oxepino[4,5-d]pyridazine 17 , which as well as 20 in hydrochloric acid rearranges to the pseudoazulenes 19 and 21 , resp.