Abstract:
Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.

Abstract:
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.

Abstract:
We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of $P_m(a)$. This approach enables us to give a combinatorial interpretation of the log-concavity of $P_m(a)$ which was conjectured by Moll and confirmed by Kauers and Paule.

Abstract:
We introduce the structure of vacillating Hecke tableaux, and establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions by using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis and Yong. Linked partitions arise in free probability theory. Motivated by the Hecke insertion algorithm, we define a Hecke diagram as a Young diagram possibly with a marked corner. A vacillating Hecke tableau is defined as a sequence of Hecke diagrams subject to certain addition and deletion of rook strips. The notion of a rook strip was introduced by Buch in the study of the Littlewood-Richardson rule for stable Grothendieck polynomials. A rook strip is a skew Young diagram with at most one square in each row and column. We show that the crossing number and the nesting number of a linked partition can be determined by the maximal number of rows and the maximal number of columns of the diagrams in the corresponding vacillating Hecke tableau. The proof relies on a theorem due to Thomas and Yong concerning the lengths of the longest strictly increasing and the longest strictly decreasing subsequences in a word. This implies that the crossing number and the nesting number have a symmetric joint distribution over linked partitions, confirming a conjecture of de Mier. We also prove a conjecture of Kim which states that the crossing number and the nesting number have a symmetric joint distribution over the front representations of partitions.

Abstract:
The influence of bulk inversion asymmetry in [001] and [013] grown HgTe quantum wells is investigated theoretically. The bulk inversion asymmetry leads to an anti-crossing gap between two zero-mode Landau levels in a HgTe topological insulator, i.e., the quantum well with inverted band structure. It is found that this is the main contribution to the anti-crossing splitting observed in recent experimental magneto spectroscopic measurements. The relevant optical transitions involve different subbands, but the electron-electron interaction induced depolarization shift is found to be negligibly small. It is also found that the splitting of this anti-crossing only depends weakly on the tilting angle when the magnetic field is tilted away from the perpendicular direction to the quantum well. Thus, the strength of bulk inversion asymmetry can be determined via a direct comparison between the theoretical calculated one-electron energy levels and experimentally observed anti-crossing energy gap.

Abstract:
We investigate the efficiency of electrical manipulation on two-dimensional topological insulators by considering a lateral potential superlattice on the system. The electronic states under various conditions are examined carefully. It is found that the dispersion of the mini-band and the electron distribution in the potential well region display an oscillatory behavior as the potential strength of the lateral superlattice increases. The probability of finding an electron in the potential well region can be larger or smaller than the average as the potential strength varies. This indicates that the electric manipulation efficiency on two-dimensional topological insulators is not as high as expected, which should be carefully considered in designing a device application that bases on two-dimensional topological insulators. These features can be attributed to the coupled multiple-band nature of the topological insulator model. In addition, it is also found that these behaviors are not sensitive to the gap parameter of the two-dimensional topological insulator model.

Abstract:
We report the exact wave functions for the eigen state of a disk-shaped two dimensional topological insulator. The property of the edge state whose energy lies inside the bulk gap is studied. It is found that the edge state energy is affected by the radius of the disk. For a fixed angular momentum index, there is a critical disk radius below which there exists no edge state. The value of this critical radius increases as the angular momentum index increases. In the limit of large disk radius, the energy of the edge state approaches a limiting value determined by the system parameters and independent of the angular momentum index. The derivation from this limiting value is inversely proportional to the radius with a coefficient proportional to the angular momentum index. In the general case, the energy differences between two edge states with adjacent angular momentum indexes are not equal. The exact and analytical wave functions also facilitates the investigation of electronic state in other structures of the two dimensional topological insulator.

Abstract:
The electronic structure of InAs/AlSb/GaSb quantum wells embedded in AlSb barriers and in the presence of a perpendicular magnetic field is studied theoretically within the $14$-band ${\bf k}\cdot{\bf p}$ approach without making the axial approximation. At zero magnetic field, for a quantum well with a wide InAs layer and a wide GaSb layer, the energy of an electron-like subband can be lower than the energy of hole-like subbands. As the strength of the magnetic field increases, the Landau levels of this electron-like subband grow in energy and intersect the Landau levels of the hole-like subbands. The electron-hole hybridization leads to a series of anti-crossing splittings of the Landau levels. The energies of some Landau level transitions and their corresponding transition strengthes are calculated. The magnetic field dependence of some dominant transitions is shown with their corresponding initial-states and final-states indicated. This information should be useful in analyzing an experimentally measured magneto-optical spectrum. At high magnetic fields, multiple transitions due to the initial-state splitting can be observed. The dominant transitions at high fields can be roughly viewed as two spin-split Landau level transitions with many electron-hole hybridization induced splittings. The energy separations between the dominant transitions may decrease or increase versus the magnetic field locally, or may be almost field independent. The separations can be tuned by changing the width of InAs layer or the width of middle AlSb layer. When the magnetic field is tilted, the electron-like Landau level transitions show additional anti-crossing splittings due to the subband-Landau level coupling.

Abstract:
We present an effective algorithm for detecting feature curves on point sets. Based on the local surface fitting method, our algorithm first compute the curvatures and principal directions of each point of point sets. The algorithm then extracts potential feature points according to the biggist principal curvature of the point, and evaluates the principal directions of the detected points. By projecting the points onto the principal axes of their neighborhoods, the potential feature points are smoothed. Using the principal directions with each optimized point, feature curves are generated by polyline growing along the principal directions of feature points. The results indicate that our algorithm is sensitive to both sharp and smooth feature curves of point set, and it supports multi-resolution extraction of features.

Abstract:
Fullerenes have several advantages as potential materials for organic spintronics. Through a theoretical first-principles study, we report that fullerene C$_{60}$ adsorption can induce a magnetic reconstruction in a Ni(111) surface and expose the merits of the reconstructed C$_{60}$/Ni(111) \emph{spinterface} for molecular spintronics applications. Surface reconstruction drastically modifies the magnetic properties at both sides of the C$_{60}$/Ni interface. Three outstanding properties of the reconstructed structure are revealed, which originate from reconstruction enhanced spin-split $\mathrm{\pi}$$-$d coupling between C$_{60}$ and Ni(111): 1) the C$_{60}$ spin polarization and conductance around the Fermi level are enhanced simultaneously, which can be important for read-head sensor miniaturization; 2) localized spin-polarized states appear in C$_{60}$ with a spin-filter functionality, and 3) magnetocrystalline anisotropic energy and exchange coupling in the outermost Ni layer are reduced enormously. Surface reconstruction can be realized simply by controlling the annealing temperature in experiments.