DSpace@MIT
http://dspace.mit.edu:80
The DSpace@MIT digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 21 Jan 2017 11:40:28 GMT2017-01-21T11:40:28ZEigenvalue Attraction
http://hdl.handle.net/1721.1/106582
Eigenvalue Attraction
Movassagh, Ramis
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.
Tue, 01 Dec 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/1065822015-12-01T00:00:00ZEditorial introduction
http://hdl.handle.net/1721.1/106581
Editorial introduction
Borst, Sem; Proutiere, Alexandre; Shah, Devavrat
We are pleased to present this special issue “Recent Trends in the Mathematics of Wireless Communication Networks: Algorithms, Models, and Methods.” Wireless communication systems have experienced a spectacular expansion over the last few decades, now providing the predominant means of Internet access. The capacity of these systems is constrained by a set of scarce resources such as radio frequencies, transmit power, and time slots. Information theory offers a powerful mathematical framework to understand how these transmission resources should be allocated so as to maximize the capacity at the physical layer, yielding valuable insights for the design of efficient schemes for, e.g., modulation, coding, and power control. Typically, however, information-theoretic models pertain to idealized scenarios: They do not account for random user behavior and dynamics at higher network layers; the practical application-specific performance requirements are largely ignored, and algorithmic implementation constraints are usually not considered. Designing systems while systematically addressing all of these aspects has posed major challenges in the last few decades. The vital need for wireless networks with significantly better performance has rejuvenated research activities toward tackling these challenges.
Sat, 01 Sep 2012 00:00:00 GMThttp://hdl.handle.net/1721.1/1065812012-09-01T00:00:00ZFaddeev–Jackiw Hamiltonian reduction for free and gauged Rarita–Schwinger theories
http://hdl.handle.net/1721.1/106580
Faddeev–Jackiw Hamiltonian reduction for free and gauged Rarita–Schwinger theories
Dengiz, Suat
We study the Faddeev–Jackiw symplectic Hamiltonian reduction for 3 + 1-dimensional free and Abelian gauged Rarita–Schwinger theories that comprise Grassmannian fermionic fields. We obtain the relevant fundamental brackets and find that they are in convenient forms for quantization. The brackets are independent of whether the theories contain mass or gauge fields, and the structures of constraints and symplectic potentials largely determine characteristic behaviors of the theories. We also note that, in contrast to the free massive theory, the Dirac field equations for free massless Rarita–Schwinger theory cannot be obtained in a covariant way.
Sat, 01 Oct 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1065802016-10-01T00:00:00ZTensionless superstrings: view from the worldsheet
http://hdl.handle.net/1721.1/106579
Tensionless superstrings: view from the worldsheet
Chakrabortty, Shankhadeep; Parekh, Pulastya; Bagchi, Arjun
In this brief note, we show that the residual symmetries that arise in the analysis of the tensionless superstrings in the equivalent of the conformal gauge is (a trivial extension of) the recently discovered 3d Super Bondi-Metzner-Sachs algebra, discussed in the context of asymptotic symmetries of 3d Supergravity in flat-spacetimes. This helps us uncover a limiting approach to the construction of the tensionless superstring from the point of view of the worldsheet, analogous to the one we had adopted earlier for the closed tensionless bosonic string.
Sat, 01 Oct 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1065792016-10-01T00:00:00Z