Fractional Tikhonov regularization for linear discrete ill-posed problems

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse of AA ^T as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization.     

Linearized Krylov subspace Bregman iteration with nonnegativity constraint

Numerical Algorithms - Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they...     

High-pressure,high-resolution NMR probe working at 400 MHZ

Abstract

A probe for obtaining high-resolution multinuclear NMR spectra at elevated pressures with a Bruker AM-400 spectrometer is described. The probe is designed for pressures up to 200 MPa and has been used between -40 to 150°C. We obtain routinely a resolution of about 1 Hz for proton (400 MHz) spectra using deuterium as an internal field lock. This probe is easily interchangeable with a commercial probe. We also describe a simple sample tube made of a 5 mm commercial NMR tube and a machinable glass cap with a total volume of about 1 cm³.     

From holes to sponge at irradiated Ge surfaces with increasing ion energy—an effect of defect kinetics?

We show that hole patterns and sponge-like layers at irradiated Ge surfaces originate from the same driving force, namely the kinetics of ion beam induced defects in the amorphous Ge surface layer. Ge hole patterns reported earlier for irradiation with low energy (5 keV) Ga⁺ ions were reproduced for low energy Bi⁺ but also for Ge⁺ self-irradiation, which proves that the dominating driving force for morphology evolution cannot originate from the implanted impurities. At higher ion energies the well-known formation of sponge-like Ge surface layers after heavy ion irradiation was found for Bi⁺ irradiation and Ge⁺ self-irradiation, also. The transition from smooth surfaces via hole patterns to sponge-like morphologies with increasing ion energies was studied in detail. A model based on the kinetics of ion beam induced defects was developed and implemented in 3D kinetic Monte Carlo simulations, which reproduce the transition from hole patterns to sponge-like layers with increasing ion energy.     

Measurement of diffusion coefficients using a quick echo split NMR imaging technique

A new method for the ultrafast generation of diffusion-weighted images is reported. The technique combines a quick echo split NMR imaging sequence with the principle of Stejskal and Tanner. It allows to determine the diffusion constant with nearly the same accuracy as the conventional spin-echo technique, requiring only a fraction of the time. The determined values for water doped with 1 g Cu(NO₃)₂ per liter of H₂O and pure acetone were D_water = (1.95 ± 0.02) × 10⁻⁹ m²/s and D_acetone = (4.05 ± 0.02) × 10⁻⁹ m²/s at 18.5°C. They are in good agreement both with literature and our own reference measurements using a diffusion-weighted spin-echo sequence. In addition, the temperature dependence of D_water was measured in the range of 18.5–45.9°C and a good correspondence with reported data was found.     

Asymptotic Two-Point Approximations by Rational Function

Continuous functions in 0 ≤ s ≤ ∞ with asymptotic power expansions both for t → +0 and t → +∞ are approximated by rational functions, being generalized Padé approximants both for s → +0 and s → +∞. The existence and the uniform convergence of such approximants are proved. As examples, piecewise rational functions, square roots of rational functions, and a modified error function are considered.     

On the Combinatorial Foundations of Regge-Calculus

Lipschitz-Killing curvatures of piecewise flat spaces are combinatorial analogues of Lipschitz-Killing curvatures of Riemannian manifolds. In the following paper rigorous combinatorial representations and proofs of all basic results for Lipschitz-Killing curvatures not using analytic arguments are given. The principal tools for an elementary representation of Regge calculus can be developed by means of basic properties of dihedral angles.