Iterative solutions to the steady state density matrix for optomechanical systems

We present a sparse matrix permutation from graph theory that gives stable incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems. This reordering is efficient, adding little overhead to the computation, and results in a marked reduction in both memory and runtime requirements compared to other solution methods, with performance gains increasing with system size. Either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance. This reordering optimizes the condition number of the approximate inverse, and is the only method found to be stable at large Hilbert space dimensions. This allows for steady state solutions to otherwise intractable quantum optomechanical systems.Comment: 10 pages, 5 figure

Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered

High-growth firms: introduction to the special section

High-growth firms (HGFs) have attracted considerable attention recently, as academics and policymakers have increasingly recognized the highly skewed nature of many metrics of firm performance. A small number of HGFs drives a disproportionately large amount of job creation, while the average firm has a limited impact on the economy. This article explores the reasons for this increased interest, summarizes the existing literature, and highlights the methodological considerations that constrain and bias research. This special section draws attention to the importance of HGFs for future industrial performance, explores their unusual growth trajectories and strategies, and highlights the lack of persistence of high growth. Consequently, while HGFs are important for understanding the economy and developing public policy, they are unlikely to be useful vehicles for public policy given the difficulties involved in predicting which firms will grow, the lack of persistence in high growth levels, and the complex and often indirect relationship between firm capability, high growth, and macro-economic performance

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.Comment: 44 page

Significant elastic anisotropy in Ti1−x_{1-x}Alx_xN alloys

Strong compositional-dependent elastic properties have been observed theoretically and experimentally in Ti$_{1-x}$Al$_x$ N alloys. The elastic constant, C$_{11}$, changes by more than 50% depending on the Al-content. Increasing the Al-content weakens the average bond strength in the local octahedral arrangements resulting in a more compliant material. On the other hand, it enhances the directional (covalent) nature of the nearest neighbor bonds that results in greater elastic anisotropy and higher sound velocities. The strong dependence of the elastic properties on the Al-content offers new insight into the detailed understanding of the spinodal decomposition and age hardening in Ti$_{1-x}$Al$_x$N alloys.Comment: 3 figures, 3 page