Achievable Angles Between two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach

The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensing matrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors separated by an angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Phi*u and Phi*v. Motivated by geometric interpretations of RIP and with the aid of the well-known law of cosines, we propose a plane geometry based formulation for the study of the considered problem. It is shown that all the RIP-induced norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a simple geometric diagram in the two-dimensional plane. This allows for a joint analysis of all the considered algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the maximal and minimal achievable angles are derived. Computer simulations confirm that the proposed solution is tighter than an existing algebraic-based estimate derived using the polarization identity. The obtained results are used to derive a tighter restricted isometry constant of structured sensing matrices of a certain kind, to wit, those in the form of a product of an orthogonal projection matrix and a random sensing matrix. Follow-up applications to three CS problems, namely, compressed-domain interference cancellation, RIP-based analysis of the orthogonal matching pursuit algorithm, and the study of democratic nature of random sensing matrices are investigated.Comment: submitted to IEEE Trans. Information Theor

Variational skinning of an ordered set of discrete 2D balls

This paper considers the problem of computing an interpolating envelope of an ordered set of 2D balls. By construction, the envelope is constrained to be C1 continuous, and for each ball, it touches the ball at a point and is tangent to the ball at the point of contact. Using an energy formulation, we derive differential equations that are designed to minimize the envelope’s arc length and/or curvature subject to these constraints. Given an initial envelope, we update the envelope’s parameters using the differential equations until convergence occurs. We demonstrate the method’s usefulness in generating interpolating envelopes of balls of different sizes and in various configurations

F-theory and Neutrinos: Kaluza-Klein Dilution of Flavor Hierarchy

We study minimal implementations of Majorana and Dirac neutrino scenarios in F-theory GUT models. In both cases the mass scale of the neutrinos m_nu ~ (M_weak)^2/M_UV arises from integrating out Kaluza-Klein modes, where M_UV is close to the GUT scale. The participation of non-holomorphic Kaluza-Klein mode wave functions dilutes the mass hierarchy in comparison to the quark and charged lepton sectors, in agreement with experimentally measured mass splittings. The neutrinos are predicted to exhibit a "normal" mass hierarchy, with masses m_3,m_2,m_1 ~ .05*(1,(alpha_GUT)^(1/2),alpha_GUT) eV. When the interactions of the neutrino and charged lepton sectors geometrically unify, the neutrino mixing matrix exhibits a mild hierarchical structure such that the mixing angles theta_23 and theta_12 are large and comparable, while theta_13 is expected to be smaller and close to the Cabibbo angle: theta_13 ~ theta_C ~ (alpha_GUT)^(1/2) ~ 0.2. This suggests that theta_13 should be near the current experimental upper bound.Comment: v2: 83 pages, 10 figures, references adde

Geometrically Induced Phase Transitions at Large N

Utilizing the large N dual description of a metastable system of branes and anti-branes wrapping rigid homologous S^2's in a non-compact Calabi-Yau threefold, we study phase transitions induced by changing the positions of the S^2's. At leading order in 1/N the effective potential for this system is computed by the planar limit of an auxiliary matrix model. Beginning at the two loop correction, the degenerate vacuum energy density of the discrete confining vacua split, and a potential is generated for the axion. Changing the relative positions of the S^2's causes discrete jumps in the energetically preferred confining vacuum and can also obstruct direct brane/anti-brane annihilation processes. The branes must hop to nearby S^2's before annihilating, thus significantly increasing the lifetime of the corresponding non-supersymmetric vacua. We also speculate that misaligned metastable glueball phases may generate a repulsive inter-brane force which stabilizes the radial mode present in compact Calabi-Yau threefolds.Comment: 47 pages, 7 figure

Status of the Electroforming Shield Design (ESD) project

The utilization of a digital computer to augment electrodeposition/electroforming processes in which nonconducting shielding controls local cathodic current distribution is reported. The primary underlying philosophy of the physics of electrodeposition was presented. The technical approach taken to analytically simulate electrolytic tank variables was also included. A FORTRAN computer program has been developed and implemented. The program utilized finite element techniques and electrostatic theory to simulate electropotential fields and ionic transport

From Liquid Structure to Configurational Entropy: Introducing Structural Covariance

We connect the configurational entropy of a liquid to the geometrical properties of its local energy landscape, using a high-temperature expansion. It is proposed that correlations between local structures arises from their overlap and, being geometrical in nature, can be usefully determined using the inherent structures of high temperature liquids. We show quantitatively how the high-temperature covariance of these local structural fluctuations arising from their geometrical overlap, combined with their energetic stability, control the decrease of entropy with decreasing energy. We apply this formalism to a family of Favoured Local Structure (FLS) lattice models with two low symmetry FLS's which are found to either crystallize or form a glass on cooling. The covariance, crystal energy and estimated freezing temperature are tested as possible predictors of glass-forming ability in the model system