Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements

International audienceWe consider non-absorbing inhomogeneous media represented by some refraction index. We have developed a method to reconstruct, from far-field measurements, the shape of the areas where the actual index differs from a reference index. Following the principle of the Factorization Method, we present a fast reconstruction algorithm relying on far field measurements and near field values, easily computed from the reference index. Our reconstruction result is illustrated by several numerical test cases

Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues.Comment: 12 pages, 4 figure

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples

The reaction Δ+N→N+N+ϕ\Delta+N\to N+N+\phi in ion-ion collisions

We study the threshold $\phi$-meson production in the process $\Delta+N\to N+N+\phi$, which appears as a possible important mechanism in high energy nuclei-nuclei collisions. The isotopic invariance of the strong interaction and the selection rules due to P-parity and total angular momentum result in a general and model independent parametrization of the spin structure of the matrix element in terms of three partial amplitudes. In the framework of one-pion exchange model these amplitudes can be derived in terms of the two threshold partial amplitudes for the process $\pi+N\to N+\phi$. We predict the ratio of cross sections for $\phi-$meson production in $pp$- and $\Delta N$-collisions and the polarization properties of the $\phi$-meson, in $\Delta+N\to N+N+\phi$, as a function of a single parameter, which characterizes the relative role of transversal and longitudinal $\phi$-meson polarizations in the process $\pi+N\to N+\phi$.Comment: 10 pages 3 figure

Multimorbidity in bipolar disorder and under-treatment of cardiovascular disease: a cross sectional study

Background: Individuals with serious mental disorders experience poor physical health, especially increased rates of cardiometabolic morbidity and premature morbidity. Recent evidence suggests that individuals with schizophrenia have numerous comorbid physical conditions which may be under-recorded and under-treated but to date very few studies have explored this issue for bipolar disorder. Methods:We conducted a cross-sectional analysis of a dataset of 1,751,841 registered patients within 314 primary-care practices in Scotland, U.K. Bipolar disorder was identified using Read Codes recorded within electronic medical records. Data on 32 common chronic physical conditions were also assessed. Potential prescribing inequalities were evaluated by analyzing prescribing data for coronary heart disease (CHD) and hypertension. Results: Compared to controls, individuals with bipolar disorder were significantly less likely to have no recorded physical conditions (OR 0.59, 95% CI 0.54-0.63) and significantly more likely to have one physical condition (OR 1.27, 95% CI 1.16-1.39), two physical conditions (OR 1.45, 95% CI 1.30-1.62) and three or more physical conditions (OR 1.44, 95% CI 1.30-1.64). People with bipolar disorder also had higher rates of thyroid disorders, chronic kidney disease, chronic pain, chronic obstructive airways disease and diabetes but, surprisingly, lower recorded rates of hypertension and atrial fibrillation. People with bipolar disorder and comorbid CHD or hypertension were significantly more likely to be prescribed no antihypertensive or cholesterol-lowering medications compared to controls, and bipolar individuals with CHD or hypertension were significantly less likely to be on 2 or more antihypertensive agents. Conclusions: Individuals with bipolar disorder are similar to individuals with schizophrenia in having a wide range of comorbid and multiple physical health conditions. They are also less likely than controls to have a primary-care record of cardiovascular conditions such as hypertension and atrial fibrillation. Those with a recorded diagnosis of CHD or hypertension were less likely to be treated with cardiovascular medications and were treated less intensively. This study highlights the high physical healthcare needs of people with bipolar disorder, and provides evidence for a systematic under-recognition and under-treatment of cardiovascular disease in this group

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Meson-Meson Scattering in the Quark Model: Spin Dependence and Exotic Channels

We apply a quark interchange model to spin-dependent and exotic meson-meson scattering. The model includes the complete set of standard quark model forces, including OGE spin-orbit and tensor and scalar confinement spin-orbit. Scattering amplitudes derived assuming SHO and Coulomb plus linear plus hyperfine meson wavefunctions are compared. In I=2 pi pi we find approximate agreement with the S-wave phase shift from threshold to 1.5 GeV, where we predict an extremum that is supported by the data. Near threshold we find rapid energy dependence that may reconcile theoretical estimates of small scattering lengths with experimental indications of larger ones based on extrapolation of measurements at moderate kpi^2. In PsV scattering we find that the quark-quark L*S and T forces map into L*S and T meson-meson interactions, and the P-wave L*S force is large. Finally we consider scattering in J^PC-exotic channels, and note that some of the Deck effect mechanisms suggested as possible nonresonant origins of the pi_1(1400) signal are not viable in this model.Comment: 51 pages, 10 figures, uses epsf.sty epsfig.st