Eigenvalue Attraction

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix $M$. If $M$ is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when $M$ is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.Comment: v1:15 pages, 12 figures, 1 Matlab code. v2: very minor changes, fixed a reference. v3: 25 pages, 17 figures and one Matlab code. The results have been extended and generalized in various ways v4: 26 pages, 10 figures and a Matlab Code. Journal Reference Added. http://link.springer.com/article/10.1007%2Fs10955-015-1424-

Stability of Periodically Driven Topological Phases against Disorder

In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These nonequilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder similar to equilibrium topological phases. However, proving the existence of such topological phases in a general setting is an open problem. We propose a universal effective theory that leverages on modern free probability theory and ideas in random matrices to analytically predict the existence of the topological phase for finite driving frequencies and across a range of disorder. We find that, depending on the strength of disorder, such systems may be topological or trivial and that there is a transition between the two. In particular, the theory predicts the critical point for the transition between the two phases and provides the critical exponents. We corroborate our results by comparing them to exact diagonalizations for driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang models and find excellent agreement. This Letter may guide the experimental efforts for exploring topological phases.Comment: 5 pages + 9 pages supplementary material. 4 Figure

Awareness and perception of phishing variants from Policing, Computing and Criminology students in Canterbury Christ Church University

This study focuses on gauging awareness of different phishing communication students in the School of Law, Policing and Social Sciences and the School of Engineering, Technology and Design in Canterbury Christ Church University and their perception of different phishing variants. There is an exploration of the underlying factors in which students fall victim to different types of phishing attacks from questionnaires and a focus group. The students’ perception of different types of phishing variants was varied from the focus group and anonymised questionnaires. A total of 177 respondents participated in anonymised questionnaires in the study. Students were asked a mixture of scenario-based questions on different phishing attacks, their awareness levels of security tools that can be used against some phishing variants, and if they received any phishing emails in the past. Additionally, 6 computing students in a focus group discussed different types of phishing attacks and recommended potential security countermeasures against them. The vulnerabilities and issues of anti-phishing software, firewalls, and internet browsers that have security toolbars are explained in the study against different types of phishing attacks. The focus group was with computing students and their knowledge about certain phishing variants was limited. The discussion within the focus group was gauging the computing students' understanding and awareness of phishing variants. The questionnaire data collection sample was with first year criminology and final year policing students which may have influenced the results of the questionnaire in terms of their understanding, security countermeasures, and how they identify certain phishing variants. The anonymised questionnaire awareness levels on different types of phishing fluctuated in terms of lack of awareness on certain phishing variants. Some criminology and policing students either did not know about phishing variants or had limited knowledge about different types of phishing communication, security countermeasures, the identifying features of a phishing message, and the precautions they should take against phishing variants from fraudsters