Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups

Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squares distance $$\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}.$$ Connections of the results to different pure and applied areas are discussed

Polarization and frequency disentanglement of photons via stochastic polarization mode dispersion

We investigate the quantum decoherence of frequency and polarization variables of photons via polarization mode dispersion in optical fibers. By observing the analogy between the propagation equation of the field and the Schr\"odinger equation, we develop a master equation under Markovian approximation and analytically solve for the field density matrix. We identify distinct decay behaviors for the polarization and frequency variables for single-photon and two-photon states. For the single photon case, purity functions indicate that complete decoherence for each variable is possible only for infinite fiber length. For entangled two-photon states passing through separate fibers, entanglement associated with each variable can be completely destroyed after characteristic finite propagation distances. In particular, we show that frequency disentanglement is independent of the initial polarization status. For propagation of two photons in a common fiber, the evolution of a polarization singlet state is addressed. We show that while complete polarization disentanglement occurs at a finite propagation distance, frequency entanglement could survive at any finite distance for gaussian states.Comment: 2 figure

Cyclin F Is Degraded during G2-M by Mechanisms Fundamentally Different from Other Cyclins

Cyclin F, a cyclin that can form SCF complexes and bind to cyclin B, oscillates in the cell cycle with a pattern similar to cyclin A and cyclin B. Ectopic expression of cyclin F arrests the cell cycle in G2/M. How the level of cyclin F is regulated during the cell cycle is completely obscure. Here we show that, similar to cyclin A, cyclin F is degraded when the spindle assembly checkpoint is activated and accumulates when the DNA damage checkpoint is activated. Cyclin F is a very unstable protein throughout much of the cell cycle. Unlike other cyclins, degradation of cyclin F is independent of ubiquitination and proteasome-mediated pathways. Interestingly, proteolysis of cyclin F is likely to involve metalloproteases. Rapid destruction of cyclin F does not require the N-terminal F-box motif but requires the COOH-terminal PEST sequences. The PEST region alone is sufficient to interfere with the degradation of cyclin F and confer instability when fused to cyclin A. These data show that although cyclin F is degraded at similar time as the mitotic cyclins, the underlying mechanisms are entirely distinct

Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups

Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squares distance $$\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}.$$ Connections of the results to different pure and applied areas are discussed

Relation Between First Arrival Time and Permeability in Self-Affine Fractures with Areas in Contact

We demonstrate that the first arrival time in dispersive processes in self-affine fractures are governed by the same length scale characterizing the fractures as that which controls their permeability. In one-dimensional channel flow this length scale is the aperture of the bottle neck, i.e., the region having the smallest aperture. In two dimensions, the concept of a bottle neck is generalized to that of a minimal path normal to the flow. The length scale is then the average aperture along this path. There is a linear relationship between the first arrival time and this length scale, even when there is strong overlap between the fracture surfaces creating areas with zero permeability. We express the first arrival time directly in terms of the permeability.Comment: EPL (2012)

Diffusive Evolution of Stable and Metastable Phases II: Theory of Non-Equilibrium Behaviour in Colloid-Polymer Mixtures

By analytically solving some simple models of phase-ordering kinetics, we suggest a mechanism for the onset of non-equilibrium behaviour in colloid-polymer mixtures. These mixtures can function as models of atomic systems; their physics therefore impinges on many areas of thermodynamics and phase-ordering. An exact solution is found for the motion of a single, planar interface separating a growing phase of uniform high density from a supersaturated low density phase, whose diffusive depletion drives the interfacial motion. In addition, an approximate solution is found for the one-dimensional evolution of two interfaces, separated by a slab of a metastable phase at intermediate density. The theory predicts a critical supersaturation of the low-density phase, above which the two interfaces become unbound and the metastable phase grows ad infinitum. The growth of the stable phase is suppressed in this regime.Comment: 27 pages, Latex, eps