An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice {\Lambda}. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.Comment: 32 pages, LaTeX, minor corrections are mad

Spin Polaron Effective Magnetic Model for La_{0.5}Ca_{0.5}MnO_3

The conventional paradigm of charge order for La_{1-x}Ca_xMnO_3 for x=0.5 has been challenged recently by a Zener polaron picture emerging from experiments and theoretical calculations. The effective low energy Hamiltonian for the magnetic degrees of freedom has been found to be a cubic Heisenberg model, with ferromagnetic nearest neighbor and frustrating antiferromagnetic next nearest neighbor interactions in the planes, and antiferromagnetic interaction between planes. With linear spin wave theory and diagonalization of small clusters up to 27 sites we find that the behavior of the model interpolates between the A and CE-type magnetic structures when a frustrating intraplanar interaction is tuned. The values of the interactions calculated by ab initio methods indicate a possible non-bipartite picture of polaron ordering differing from the conventional one.Comment: 21 pages and 8 figures (included), Late

Understanding Selectivity of Mesoporous Silica-Grafted Diglycolamide-Type Ligands in the Solid-Phase Extraction of Rare Earths

Rare earth elements (REEs) and their compounds are essential for rapidly developing modern technologies. These materials are especially critical in the area of green/sustainable energy; however, only very high-purity fractions are appropriate for these applications. Yet, achieving efficient REE separation and purification in an economically and environmentally effective way remains a challenge. Moreover, current extraction technologies often generate large amounts of undesirable wastes. In that perspective, the development of selective, reusable, and extremely efficient sorbents is needed. Among numerous ligands used in the liquid-liquid extraction (LLE) process, the diglycolamide-based (DGA) ligands play a leading role. Although these ligands display notable extraction performance in the liquid phase, their extractive chemistry is not widely studied when such ligands are tethered to a solid support. A detailed understanding of the relationship between chemical structure and function (i.e., extraction selectivity) at the molecular level is still missing although it is a key factor for the development of advanced sorbents with tailored selectivity. Herein, a series of functionalized mesoporous silica (KIT-6) solid phases were investigated as sorbents for the selective extraction of REEs. To better understand the extraction behavior of these sorbents, different spectroscopic techniques (solid-state NMR, X-ray photoelectron spectroscopy, XPS, and Fourier transform infrared spectroscopy, FT-IR) were implemented. The obtained spectroscopic results provide useful insights into the chemical environment and reactivity of the chelating ligand anchored on the KIT-6 support. Furthermore, it can be suggested that depending on the extracted metal and/or structure of the ligand and its attachment to KIT-6, different functional groups (i.e., C= O, N-H, or silanols) act as the main adsorption centers and preferentially capture targeted elements, which in turn may be associated with the different selectivity of the synthesized sorbents. Thus, by determining how metals interact with different supports, we aim to better understand the solid-phase extraction process of hybrid (organo)silica sorbents and design better extraction materials

Neel probability and spin correlations in some nonmagnetic and nondegenerate states of hexanuclear antiferromagnetic ring Fe6: Application of algebraic combinatorics to finite Heisenberg spin systems

The spin correlations \omega^z_r, r=1,2,3, and the probability p_N$ of finding a system in the Neel state for the antiferromagnetic ring Fe(III)6 (the so-called `small ferric wheel') are calculated. States with magnetization M=0, total spin 0<=S<=15 and labeled by two (out of four) one-dimensional irreducible representations (irreps) of the point symmetry group D_6 are taken into account. This choice follows from importance of these irreps in analyzing low-lying states in each S-multiplet. Taking into account the Clebsch--Gordan coefficients for coupling total spins of sublattices (SA=SB=15/2) the global Neel probability p*_N can be determined. Dependencies of these quantities on state energy (per bond and in the units of exchange integral J) and the total spin S are analyzed. Providing we have determined p_N(S) etc. for other antiferromagnetic rings (Fe10, for instance) we could try to approximate results for the largest synthesized ferric wheel Fe18. Since thermodynamic properties of Fe6 have been investigated recently, in the present considerations they are not discussed, but only used to verify obtained values of eigenenergies. Numerical results re calculated with high precision using two main tools: (i) thorough analysis of symmetry properties including methods of algebraic combinatorics and (ii) multiple precision arithmetic library GMP. The system considered yields more than 45 thousands basic states (the so-called Ising configurations), but application of the method proposed reduces this problem to 20-dimensional eigenproblem for the ground state (S=0). The largest eigenproblem has to be solved for S=4; its dimension is 60. These two facts (high precision and small resultant eigenproblems) confirm efficiency and usefulness of such an approach, so it is briefly discussed here.Comment: 13 pages, 7 figs, 5 tabs, revtex

Magnetic translation groups in an n-dimensional torus

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus.Comment: 29 pages, LaTeX2e, title changed, re-organized, to be published in Journal of Mathematical Physic

Spray‐Dried Mesoporous Mixed Cu‐Ni Oxide@Graphene Nanocomposite Microspheres for High Power and Durable Li‐Ion Battery Anodes

Exfoliated graphene‐wrapped mesoporous Cu‐Ni oxide (CNO) nanocast composites are developed using a straightforward nanostructure engineering strategy. The synergistic effect of hierarchical mesoporous CNO nanobuilding blocks that are homogeneously wrapped by graphene nanosheets (GNSs) using a rapid spray drying technique effectively preserves the electroactive species against the volume changes resulting from the charge/discharge process. Owing to the intriguing structural/morphological features arising from the caging effect of exfoliated graphene sheets, these 3D/2D CNO@GNS nanocomposite microspheres are promising as high‐performance Li‐ion battery anode materials. They exhibit unprecedented electrochemical behavior, such as high reversible specific capacity (initial discharge capacities exceeding 1700 mAh g−1 at low 0.1 mA g−1, stable 850 and 730 mAh g−1 at 1 and 5 mA g−1 after 800 and 1300 cycles, respectively, and higher than 400 mAh g−1 at very high current density of 10 mA g−1 after more than 2000 cycles), excellent coulombic efficiency and long‐term stability (more than 3000 cycles with >55% capacity retention) at high current density that are remarkable compared to most transition metal oxides and nanocomposites prepared by conventional techniques. This simple, yet innovative, material design is inspiring to develop advanced conversion materials for Li‐ion batteries or other energy storage devices

Geometric entropy, area, and strong subadditivity

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a density matrix with non zero entropy. This geometric entropy is believed to be deeply related to the entropy of black holes. Indeed, previous calculations in the context of quantum field theory, where the result is actually ultraviolet divergent, have shown that the geometric entropy is proportional to the area for a very special type of subsets. In this work we show that the area law follows in general from simple considerations based on quantum mechanics and relativity. An essential ingredient of our approach is the strong subadditive property of the quantum mechanical entropy.Comment: Published versio

HIRDES - The High-Resolution Double-Echelle Spectrograph for the World Space Observatory Ultraviolet (WSO/UV)

The World Space Observatory Ultraviolet (WSO/UV) is a multi-national project grown out of the needs of the astronomical community to have future access to the UV range. WSO/UV consists of a single UV telescope with a primary mirror of 1.7m diameter feeding the UV spectrometer and UV imagers. The spectrometer comprises three different spectrographs, two high-resolution echelle spectrographs (the High-Resolution Double-Echelle Spectrograph, HIRDES) and a low-dispersion long-slit instrument. Within HIRDES the 102-310nm spectral band is split to feed two echelle spectrographs covering the UV range 174-310nm and the vacuum-UV range 102-176nm with high spectral resolution (R>50,000). The technical concept is based on the heritage of two previous ORFEUS SPAS missions. The phase-B1 development activities are described in this paper considering performance aspects, design drivers, related trade-offs (mechanical concepts, material selection etc.) and a critical functional and environmental test verification approach. The current state of other WSO/UV scientific instruments (imagers) is also described.Comment: Accepted for publication in Advances in Space Researc

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing

Analysis of Agglomerative Clustering

The diameter $k$-clustering problem is the problem of partitioning a finite subset of $\mathbb{R}^d$ into $k$ subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of $k$) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming that the dimension $d$ is a constant, we show that for any $k$ the solution computed by this algorithm is an $O(\log k)$-approximation to the diameter $k$-clustering problem. Our analysis does not only hold for the Euclidean distance but for any metric that is based on a norm. Furthermore, we analyze the closely related $k$-center and discrete $k$-center problem. For the corresponding agglomerative algorithms, we deduce an approximation factor of $O(\log k)$ as well.Comment: A preliminary version of this article appeared in Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS '11), March 2011, pp. 308-319. This article also appeared in Algorithmica. The final publication is available at http://link.springer.com/article/10.1007/s00453-012-9717-