On Inverses of the Dirac Comb

We determine tempered distributions which convolved with a Dirac comb yield unity and tempered distributions, which multiplied with a Dirac comb, yield a Dirac delta. Solutions of these equations have numerous applications. They allow the reversal of discretizations and periodizations applied to tempered distributions. One of the difficulties is the fact that Dirac combs cannot be multiplied or convolved with arbitrary functions or distributions. We use a theorem of Laurent Schwartz to overcome this difficulty and variants of Lighthill’s unitary functions to solve these equations. The theorem we prove states that double-sided (time/frequency) smooth partitions of unity are required to neutralize discretizations and periodizations on tempered distributions.

Thermodynamic properties of underdoped YBa2Cu3O6+x cuprates for several doping values

We report the thermodynamic properties of cuprate superconductors YBa2Cu3O[Formula: see text], with [Formula: see text] ranging from underdoped ([Formula: see text]) to optimally doped ([Formula: see text]) regions. We model cuprates as a boson–fermion gas mixture immersed in a layered structure, which is generated via a Dirac-comb potential applied in the perpendicular direction to the CuO2 planes, while the particles move freely in the other two directions. The optimal system parameters, namely, the planes’ impenetrability and the paired-fermion fraction, are obtained by minimizing the Helmholtz free energy in addition to fixing the critical temperature [Formula: see text] to its experimental value. Using this optimized scheme, we calculate the entropy, the Helmholtz free energy and the specific heat as functions of temperature. Additionally, some fundamental properties of the electronic specific heat are obtained, such as the normal linear coefficient [Formula: see text], the quadratic [Formula: see text] term and the jump height at [Formula: see text]. We reproduce the cubic [Formula: see text] term of the total specific heat for low temperatures. Also our multilayer model inherently brings with it the mass anisotropy observed in cuprate superconductors. Furthermore, we establish the doping value beyond which superconductivity is suppressed.

Multigrain indexing of unknown multiphase materials

A multigrain indexing algorithm for use with samples comprising an arbitrary number of known or unknown phases is presented. No a priori crystallographic knowledge is required. The algorithm applies to data acquired with a monochromatic beam and a conventional two-dimensional detector for diffraction. Initially, candidate grains are found by searching for crystallographic planes, using a Dirac comb convoluted with a box function as a filter. Next, candidate grains are validated and the unit cell is optimized. The algorithm is validated by simulations. Simulations of 500 cementite grains and ∼100 reflections per grain resulted in 99.2% of all grains being indexed correctly and 99.5% of the reflections becoming associated with the right grain. Simulations with 200 grains associated with four mineral phases and 50–700 reflections per grain resulted in 99.9% of all grains being indexed correctly and 99.9% of the reflections becoming associated with the right grain. The main limitation is in terms of overlap of diffraction spots and computing time. Potential areas of use include three-dimensional grain mapping, structural solution and refinement studies of complex samples, and studies of dilute phases.

Measures with locally finite support and spectrum

The goal of this paper is the construction of measures μ on Rn enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transform μ^ of μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.