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We study a random matching problem on closed compact $2$-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers $n$ and $m=m(n)$ of points, asymptotically equivalent as $n$ goes to infinity, the optimal transport plan between the two empirical measures $\mu^n$ and $\nu^{m}$ is quantitatively well-approximated by $\big(\mathrm{Id},\exp(\nabla h^{n})\big)_\#\mu^n$ where $h^{n}$ solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Amp\`ere equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the $\alpha$-mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure., Comment: Comments very welcome!

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phase-transition" in the rates when $d=2+1/H$, akin to the Ajtai-Koml\'os-Tusn\'ady theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on $\mathbb{R}^d$., Comment: Comments very welcome

We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments., Comment: Comments very welcome

The magnetism in the inherently nanolaminated ternary MAX-phase Cr$_{2}$GeC is investigated by element-selective, polarization and temperature-dependent, soft X-ray absorption spectroscopy and X-ray magnetic circular dichroism. The measurements indicate an antiferro-magnetic Cr-Cr coupling along the $c$-axis of the hexagonal structure modulated by a ferromagnetic ordering in the nanolaminated $ab$-basal planes. The weak chromium magnetic moments are an order of magnitude stronger in the nanolaminated planes than along the vertical axis. Theoretically, a small but notable, non-spin-collinear component explains the existence of a non-perfect spin compensation along the $c$-axis. As shown in this work, this spin distortion generates an overall residual spin moment inside the unit cell resembling that of a ferri-magnet. Due to the different competing magnetic interactions, electron correlations and temperature effects both need to be considered to achieve a correct theoretical description of the Cr$_{2}$GeC magnetic properties., Comment: 10 pages, 6 figures, 2 tables

This is a critical review of MAX-phase carbides and nitrides from an electronic-structure and chemical bonding perspective. This large group of nanolaminated materials is of great scientific and technological interest and exhibit a combination of metallic and ceramic features. These properties are related to the special crystal structure and bonding characteristics with alternating strong M-C bonds in high-density MC slabs, and relatively weak M-A bonds between the slabs. Here, we review the trend and relationship between the chemical bonding, conductivity, elastic and magnetic properties of the MAX phases in comparison to the parent binary MX compounds with the underlying electronic structure probed by polarized X-ray spectroscopy. Spectroscopic studies constitute important tests of the results of state-of-the-art electronic structure density functional theory that is extensively discussed and are generally consistent. By replacing the elements on the M, A, or X-sites in the crystal structure, the corresponding changes in the conductivity, elasticity, magnetism and other materials properties makes it possible to tailor the characteristics of this class of materials by controlling the strengths of their chemical bonds., Comment: 46 Pages, 23 Figures, 6 Tables

The anisotropy in the electronic structure of the inherently nanolaminated ternary phase Cr$_{2}$GeC is investigated by bulk-sensitive and element selective soft x-ray absorption/emission spectroscopy. The angle-resolved absorption/emission measurements reveal differences between the in-plane and out-of-plane bonding at the (0001) interfaces of Cr$_{2}$GeC. The Cr $L_{2,3}$, C $K$, and Ge $M_{1}$, $M_{2,3}$ emission spectra are interpreted with first-principles density-functional theory (DFT) including core-to-valence dipole transition matrix elements. For the Ge $4s$ states, the x-ray emission measurements reveal two orders of magnitude higher intensity at the Fermi level than DFT within the General Gradient Approximation (GGA) predicts. We provide direct evidence of anisotropy in the electronic structure and the orbital occupation that should affect the thermal expansion coefficient and transport properties. As shown in this work, hybridization and redistribution of intensity from the shallow $3d$ core levels to the $4s$ valence band explain the large Ge density of states at the Fermi level., Comment: 8 pages, 5 figures, http://iopscience.iop.org/article/10.1088/0953-8984/27/41/415501/meta;jsessionid=871D247E3FCB99A5EEB41A656054C246.c1

The magnetic properties, electronic band structure and Fermi surfaces of the hexagonal Cr2GeC system have been studied by means of both generalized gradient approximation (GGA) and the +U corrected method (GGA+U). The effective U value has been computed within the augmented plane-wave theoretical scheme by following the constrained density functional theory formalism of Anisimov et al. [1991 Phys. Rev. B 45, 7570]. On the basis of our GGA+U calculations, a compensated anti-ferromagnetic spin ordering of Cr atoms has been found to be the ground state solution for this material, where a Ge-mediated super-exchange coupling is responsible for an opposite spin distribution between the ABA stacked in-plane Cr-C networks. Structural properties have also been tested and found to be in good agreement with the available experimental data. Topological analysis of Fermi surfaces have been used to qualitatively address the electronic transport properties of Cr2GeC and found an important asymmetrical carrier-type distribution within the hexagonal crystal lattice. We conclude that an appropriate description of the strongly correlated Cr-d electrons is an essential issue for interpreting the material properties of this unusual Cr-based MAX-phase., Comment: 13 pages, 10 pictures

Nanolaminated materials exhibit characteristic magnetic, mechanical, and thermoelectric properties, with large contemporary scientific and technological interest. Here, we report on the anisotropic Seebeck coefficient in nanolaminated Ti3SiC2 single-crystal thin films and trace the origin to anisotropies in element-specific electronic states. In bulk polycrystalline form, Ti3SiC2 has a virtually zero Seebeck coefficient over a wide temperature range. In contrast, we find that the in-plane (basal ab) Seebeck coefficient of Ti3SiC2, measured on single-crystal films has a substantial and positive value of 4-6 muV/K. Employing a combination of polarized angle-dependent x-ray spectroscopy and density functional theory we directly show electronic structure anisotropy in inherently nanolaminated Ti3SiC2 single-crystal thin films as a model system. The density of Ti 3d and C 2p states at the Fermi level in the basal ab-plane is about 40 % higher than along the c-axis. The Seebeck coefficient is related to electron and hole-like bands close to the Fermi level but in contrast to ground state density functional theory modeling, the electronic structure is also influenced by phonons that need to be taken into account. Positive contribution to the Seebeck coefficient of the element-specific electronic occupations in the basal plane is compensated by 73 % enhanced Si 3d electronic states across the laminate plane that give rise to a negative Seebeck coefficient in that direction. Strong phonon vibration modes with three to four times higher frequency along the c-axis than along the basal ab-plane also influence the electronic population and the measured spectra by the asymmetric average displacements of the Si atoms. These results constitute experimental evidence explaining why the average Seebeck coefficient of Ti3SiC2 in polycrystals is negligible over a wide temperature range., Comment: 13 pages, 6 figures; http://prb.aps.org/abstract/PRB/v85/i19/e195134

The electronic structures of epitaxially grown films of Ti3AlC2, Ti3SiC2 and Ti3GeC2 have been investigated by bulk-sensitive soft X-ray emission spectroscopy. The measured high-resolution Ti L, C K, Al L, Si L and Ge M emission spectra are compared with ab initio density-functional theory including core-to-valence dipole matrix elements. A qualitative agreement between experiment and theory is obtained. A weak covalent Ti-Al bond is manifested by a pronounced shoulder in the Ti L-emission of Ti3AlC2. As Al is replaced with Si or Ge, the shoulder disappears. For the buried Al and Si-layers, strongly hybridized spectral shapes are detected in Ti3AlC2 and Ti3SiC2, respectively. As a result of relaxation of the crystal structure and the increased charge-transfer from Ti to C, the Ti-C bonding is strengthened. The differences between the electronic structures are discussed in relation to the bonding in the nanolaminates and the corresponding change of materials properties., Comment: 15 pages, 8 figures

The electronic structure in the new transition metal carbide Ti4SiC3 has been investigated by bulk-sensitive soft x-ray emission spectroscopy and compared to the well-studied Ti3SiC2 and TiC systems. The measured high-resolution Ti L, C K and Si L x-ray emission spectra are discussed with ab initio calculations based on density-functional theory including core-to-valence dipole matrix elements. The detailed investigations of the Ti-C and Ti-Si chemical bonds provide increased understanding of the physical properties of these nanolaminates. A strongly modified spectral shape is detected for the buried Si monolayers due to Si 3p hybridization with the Ti 3d orbitals. As a result of relaxation of the crystal structure and the charge-transfer from Ti (and Si) to C, the strength of the Ti-C covalent bond is increased. The differences between the electronic and crystal structures of Ti4SiC3 and Ti3SiC2 are discussed in relation to the number of Si layers per Ti layer in the two systems and the corresponding change of materials properties., Comment: 12 pages, 7 figures, 1 table

The anisotropy of the electronic structure of ternary nanolaminate V2GeC is investigated by bulk-sensitive soft x-ray emission spectroscopy. The measured polarization-dependent emission spectra of V L2,3, C K, Ge M1 and Ge M2,3 in V2GeC are compared to those from monocarbide VC and pure Ge. The experimental emission spectra are interpreted with calculated spectra using ab initio density-functional theory including dipole transition matrix elements. Different types of covalent chemical bond regions are revealed; V 3d - C 2p bonding at -3.8 eV, Ge 4p - C 2p bonding at -6 eV and Ge 4p - C 2s interaction mediated via the V 3d orbitals at -11 eV below the Fermi level. We find that the anisotropic effects are high for the 4p valence states and the shallow 3d core levels of Ge, while relatively small anisotropy is detected for the V 3d states. The macroscopic properties of the V2GeC nanolaminate result from the chemical bonds with the anisotropic pattern as shown in this work., Comment: 15 pages, 6 figures, 1 table, http://link.aps.org/doi/10.1103/PhysRevB.78.035117

The electronic structure of nanolaminate Ti2AlN and TiN thin films has been investigated by bulk-sensitive soft x-ray emission spectroscopy. The measured Ti L, N K, Al L1 and Al L2,3 emission spectra are compared with calculated spectra using ab initio density-functional theory including dipole transition matrix elements. Three different types of bond regions are identified; a relatively weak Ti 3d - Al 3p bonding between -1 and -2 eV below the Fermi level, and Ti 3d - N 2p and Ti 3d - N 2s bonding which are deeper in energy observed at -4.8 eV and -15 eV below the Fermi level, respectively. A strongly modified spectral shape of 3s states of Al L2,3 emission from Ti2AlN in comparison to pure Al metal is found, which reflects the Ti 3d - Al 3p hybridization observed in the Al L1 emission. The differences between the electronic and crystal structures of Ti2AlN and TiN are discussed in relation to the intercalated Al layers of the former compound and the change of the materials properties in comparison to the isostructural carbides., Comment: 18 pages, 7 figures; http://link.aps.org/doi/10.1103/PhysRevB.76.195127