On a Computer- Aided Decomposition of the Complete Digraph into Orientations of K4-e with a Double Edge

The abstract is available as an Additional File

A Computer-Aided Decomposition of the Complete Digraph into Orientations of K4 − e with a Double Edge

The abstract is available as an Additional File

Classifying Generalized Symmetric Spaces for Unipotent and Semisimple Elements in SO(3,p)

In this project, we look at the Special Orthogonal group of 3x3 matrices over a finite field, denoted SO(3,p). In particular, we focus on classifying the generalized symmetric spaces, which are defined by an involution f:SO3,pSO3,p such that ffM=M for these matrices. We begin by explaining what types of involutions exist for our group, and once those involutions are established, we classify two important spaces: the Extended Symmetric Space R and the General Symmetric Space Q. We describe these spaces for the two isomorphy classes of involutions (building off of Benim et al.) through counting arguments, in which we split R and Q into unipotent and semisimple cases. Some counting arguments are established for the size of Ru, Qu, and Rss (unipotent matrices in R, unipotent matrices in Q, and semisimple matrices in R, respectively). Further progress can be made on verifying our other conjectures and generalizing our results to field extensions. Applications of our research can be seen in physics, where the SO(3,p) matrices are particularly effective at describing the effects of rotation and spin

Classifying Symmetric Spaces for SO(3,q)

Computer Graphics Rotational Matrices are also Special Orthogonal. Because special orthogonal matrices preserve size and shape, they are used to rotate objects in 3D space, and thus can be used by computers to animate objects in a 3D scene

A Computer-Aided Decomposition of the Complete Digraph into Orientations of K4 − e with a Double Edge

The abstract is available as an Additional File

Irreducibility and Galois Groups of Random Polynomials

In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin\u27s arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials

Eliminating interactions between non-neighboring qubits in the preparation of cluster states in quantum molecules

We propose a scheme to eliminate the effect of non-nearest-neighbor qubits in preparing cluster state with double-dot molecules. As the interaction Hamiltonians between qubits are Ising-model and mutually commute, we can get positive and negative effective interactions between qubits to cancel the effect of non-nearest-neighbor qubits by properly changing the electron charge states of each quantum dot molecule. The total time for the present multi-step cluster state preparation scheme is only doubled for one-dimensional qubit chain and tripled for two-dimensional qubit array comparing with the time of previous protocol leaving out the non-nearest-neighbor interactions.Comment: 5 pages, 4 figures, 2 table

Systematic study of Oxygen vacancy tunable transport properties of few-layer MoO3- x enabled by vapor-based synthesis

Bulk and nanoscale molybdenum trioxide (MoO3) has shown impressive technologically relevant properties, but deeper investigation into 2D MoO3 has been prevented by the lack of reliable vapor-based synthesis and doping techniques. Herein, the successful synthesis of high-quality, few-layer MoO3 down to bilayer thickness via physical vapor deposition is reported. The electronic structure of MoO3 can be strongly modified by introducing oxygen substoichiometry (MoO3- x), which introduces gap states and increases conductivity. A dose-controlled electron irradiation technique to introduce oxygen vacancies into the few-layer MoO3 structure is presented, thereby adding n-type doping. By combining in situ transport with core-loss and monochromated low-loss scanning transmission electron microscopy–electron energy-loss spectroscopy studies, a detailed structure–property relationship is developed between Mo-oxidation state and resistance. Transport properties are reported for MoO3- x down to three layers thick, the most 2D-like MoO3- x transport hitherto reported. Combining these results with density functional theory calculations, a radiolysis-based mechanism for the irradiation-induced oxygen vacancy introduction is developed, including insights into favorable configurations of oxygen defects. These systematic studies represent an important step forward in bringing few-layer MoO3 and MoO3- x into the 2D family, as well as highlight the promise of MoO3- x as a functional, tunable electronic material