Dispersive Bounds on the Shape of B -> D^(*) l nu Form Factors

Dispersive constraints on the shape of the form factors which describe the exclusive decays B -> D^(*) l nu are derived by fully exploiting spin symmetry in the ground-state doublet of heavy-light mesons. The analysis includes all twenty B^(*) -> D^(*) semileptonic form factors. Heavy-quark symmetry, with both short-distance and 1/m_Q corrections included, is used to provide relations between the form factors near zero recoil. Simple one-parameter functions are derived, which describe the form factors in the semileptonic region with an accuracy of better than 2%. The implications of our results for the determination of |V_cb| are discussed.Comment: 32 pages, 6 figure

Experiment and Theory in Computations of the He Atom Ground State

Extensive variational computations are reported for the ground state energy of the non-relativistic two-electron atom. Several different sets of basis functions were systematically explored, starting with the original scheme of Hylleraas. The most rapid convergence is found with a combination of negative powers and a logarithm of the coordinate s = r_{1}+ r_{2}. At N=3091 terms we pass the previous best calculation (Korobov's 25 decimal accuracy with N=5200 terms) and we stop at N=10257 with E = -2.90372 43770 34119 59831 11592 45194 40444 ... Previous mathematical analysis sought to link the convergence rate of such calculations to specific analytic properties of the functions involved. The application of that theory to this new experimental data leaves a rather frustrating situation, where we seem able to do little more than invoke vague concepts, such as ``flexibility.'' We conclude that theoretical understanding here lags well behind the power of available computing machinery.Comment: 15 page

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results

Finite element approximation for the fractional eigenvalue problem

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.Comment: 20 pages, 6 figure