An induction theorem and nonlinear regularity models

A general nonlinear regularity model for a set-valued mapping $F:X\times R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping $F:X\rightrightarrows Y$.Comment: 28 page

Accelerating Dynamical Density Response Code on Summit and Its Application for Computing the Density Response Function of Vanadium Sesquioxide

This thesis details the process of porting the Eguiluz group dynamical density response computational platform to the hybrid CPU+GPU environment at the Summit supercomputer at Oak Ridge National Laboratory (ORNL) Leadership Computing Center. The baseline CPU-only version is a Gordon Bell-winning platform within the formally-exact time-dependent density functional theory (TD-DFT) framework using the linearly augmented plane wave (LAPW) basis set. The code is accelerated using a combination of the OpenACC programming model and GPU libraries -- namely, the Matrix Algebra for GPU and Multicore Architectures (MAGMA) library -- as well as exploiting the sparsity pattern of the matrices involved in the matrix-matrix multiplication. Benchmarks show a 12.3x speedup compared to the CPU-only version. This performance boost should accelerate discovery in material and condensed matter physics through computational means. After the hybrid CPU+GPU code has been sufficiently optimized, it is used to study the dynamical density response function of vanadium sesquioxide, and the results are compared with spectroscopic data from non-resonant inelastic X-ray scattering {NIXS} experiments

Magnetocaloric effect in nano- and polycrystalline manganite La0.7Ca0.3MnO3La_{0.7}Ca_{0.3}MnO_3

$La_{0.7}Ca_{0.3}MnO_3$ samples were prepared in nano- and polycrystalline forms by sol-gel and solid state reaction methods, respectively, and structurally characterized by synchrotron X-ray diffraction. The magnetic properties determined by ac susceptibility and dc magnetization measurements are discussed. The magnetocaloric effect in this nanocrystalline manganite is spread over a broader temperature interval than in the polycrystalline case. The relative cooling power of the poly- and nanocrystalline manganites is used to evaluate a possible application for magnetic cooling below room temperature.Comment: 6 pages, 5 (double) figures, 1 table, 16 references; submitted to Appl. Phys.

Some results of the Lipschitz constant of 1-Field on Rn\mathbb{R}^n

We study the relations between the Lipschitz constant of $1$-field and the Lipschitz constant of the gradient canonically associated with this $1$-field. Moreover, we produce two explicit formulas that make up Minimal Lipschitz extensions for $1$-field. As consequence of the previous results, for the problem of minimal extension by continuous functions from $\mathbb{R}^m$ to $\mathbb{R}^n$, we also produce analogous explicit formulas to those of Bauschke and Wang. Finally, we show that Wells's extensions of $1$-field are absolutely minimal Lipschitz extension when the domain of $1$-field to expand is finite. We provide a counter-example showing that this result is false in general.Comment: E.L.G. and T.V.P. are partially supported by the ANR (Agence Nationale de la Recherche) through HJnet projet ANR-12-BS01-0008-0