On a matrix inequality related to the distillability problem

We investigate the distillability problem in quantum information in \bbC^d\ox\bbC^d. A special case of the problem has been reduced to proving a matrix inequality when $d=4$. We investigate the inequality for two families of non-normal matrices. We prove the inequality for the first family with $d=4$ and two special cases of the second family with $d\ge4$. We also prove the inequality for all normal matrices with $d>4$.Comment: 19 pages, comments are welcom

Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

We investigate the Castelnuovo--Mumford regularity and the multiplicity of the toric ring associated to a three-dimensional Ferrers diagram. In particular, in the rectangular case, we are able to provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanied pair of Ferrers diagrams under some mild conditions, and bound the Castelnuovo--Mumford regularity for more general cases.Comment: 22 pages, 2 figures and comments are welcom

Development of Bond-Order Potentials for Body-Centered-Cubic Transition Metals and Their Application in atomistic Studies of Plastic Properties Mediated by Dislocations

ABSTRACT DEVELOPMENT OF BOND-ORDER POTENTIALS FOR BODY-CENTERED-CUBIC TRANSITION METALS AND THEIR APPLICATION IN ATOMISTIC STUDIES OF PLASTIC PROPERTIES MEDIATED BY DISLOCATIONS Yi-Shen Lin Professor Vaclav Vitek Bond-order potentials (BOPs), based on the tight binding (TB) approach for the evaluation of bonding, are an real-space method. They are eminently suitable for atomistic simulations of extended defects in transition metals in which the bonding is mixed nearly free electron and covalent. The latter requires a rigorous quantum mechanical treatment performed within the TB. In this Thesis, new BOPs were developed for non-magnetic BCC transition metals, V, Nb, Ta, Cr, Mo and W, as well as for the ferromagnetic Fe. In these BOPs, bond integrals used in the bond part of the cohesive energy were directly extracted from DFT calculations employing a projection formalism and a physically more transparent functional form was established for the repulsive part of the cohesive energy. In the ferromagnetic Fe, the magnetism was introduced through the Stoner’s model of the itinerant magnetism. In the bond part of the cohesive energy only d bonds are included explicitly but the screening of these bonds by the surrounding s electrons needs to be taken into account. This is particularly important when studying atomic arrangements in which the deviation from the ideal BCC lattice is very localized and inhomogeneous. The developed BOPs show an excellent transferability to various atomic environments which was tested by calculating energies of alternative crystal structures, vacancy formation energies, transformation paths, phonon spectra and -surfaces, all of which allow for direct comparisons with experiments and/or DFT based calculations. Moreover, we show that with slight variation of the number of d electrons used in BOPs, they are suitable for the atomistic studies involving self-interstitial atoms. This is essential if the BOPs are to be used in studies of the radiation damage. Employing these BOPs, the core structures and glide of ½\u3c111\u3e screw dislocations, which govern the plastic deformation in BCC metals, were investigated using a variety of applied stress tensors. These simulations reveal a breakdown of the Schmid law that has two aspects. First is the so-called twinning-antitwinning asymmetry of the critical resolved shear stress, which has been known for a long time and occurs even when the applied stress is the pure shear stress parallel to the Burgers vector. The second relates to core transformations induced by the shear stress components perpendicular to the Burgers vector. Our simulations suggest that the latter may explain the anomalous slip found in a number of BCC transition metals, which has been known for a long time but its full understanding is still elusive. Most importantly, the calculations employing BOPs suggest significantly different anomalous slip for group 5 (V, Nb and Ta) and group 6 (Mo and W) metals, which is observed but has never been explained based on the standard continuum theory of dislocations. Finally, a very simple formalism was proposed for the development of BOPs for binary homogeneous substitutional alloys that leads to a good agreement with DFT calculations of basic structural and mechanical properties. Studies of ½[111] screw dislocations in Ta-W alloy confirm the applicability of this formalism to investigation of the effect of substitutional alloying on the dislocation glide