Spectral Lower Bounds for the Orthogonal and Projective Ranks of a Graph

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many spectral lower bounds for the chromatic number, $\chi$, are also lower bounds for $\xi$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $\chi$ are also lower bounds for the quantum chromatic number $\chi_q$. It is known that the quantum chromatic number and the orthogonal rank are incomparable. We conclude by proving an inertial lower bound for the projective rank $\xi_f$, and conjecture that a stronger inertial lower bound for $\xi$ is also a lower bound for $\xi_f$.

Hitting Time Theorems for Random Matrices

Starting from ann-by-nmatrix of zeros, choose uniformly random zero entries and change them to ones, one at a time, until the matrix becomes invertible. We show that with probability tending to one asn→ ∞, this occurs at the very moment the last zero row or zero column disappears. We prove a related result for random symmetric Bernoulli matrices, and give quantitative bounds for some related problems. These results extend earlier work by Costello and Vu [10].

Detectability and Locatability of Multiple Gross Errors in DEM Matching Algorithm

DEM matching algorithm without control points is the base for detecting deformation method, the deformation existing in DEMs is treated as gross error. The detectability and locatability of multiple gross errors is very critical according to the surveying error and reliability theory. The interactive observations number (ION) derived from gross error judgement matrix is adapted in this paper. The relationship between zero-column vectors in judge matrix and the rank of the coefficient matrix of DEM matching is discussed in theory, and prove none zero-column vectors exist with real date sets. ION Equal to the number of redundant observations. Experimental results show that DEM matching algorithm has the detectability and locatability of multiple gross errors, and the deformation can be correctly detected by robust estimators in prearranged confidence level.

Weak Pressure Gradient Approximation and Its Analytical Solutions

A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.