On Optimal Frame Conditioners

A (unit norm) frame is scalable if its vectors can be rescaled so as to result into a tight frame. Tight frames can be considered optimally conditioned because the condition number of their frame operators is unity. In this paper we reformulate the scalability problem as a convex optimization question. In particular, we present examples of various formulations of the problem along with numerical results obtained by using our methods on randomly generated frames.Comment: 11 page

Activity Related Variations of High-Degree p-Mode Amplitude, Width and Energy in Solar Active Regions

We study the properties of high degree p-mode oscillations in flaring and dormant ARs and compare them with those in corresponding quiet regions (QRs) to find the association of mode parameters with magnetic and flare related activities. Our analysis of several flaring and dormant ARs, showed strong association of mode amplitude, width and energy with magnetic and flare activities although their changes are combined effects of foreshortening, filling factor, magnetic activity, flare activity, and measurement uncertainties. We find that the largest decrease in mode amplitude and background power of an AR are caused by the angular distance of the AR from the solar disc centre. After correcting the mode parameters for foreshortening and filling factor, we find that the mode amplitude of flaring and dormant ARs are smaller than in corresponding QRs, and decreases with increasing MAI suggesting a larger mode power suppression in ARs with stronger magnetic fields. The mode widths in ARs are larger than in corresponding QRs and increase with MAI, indicating shorter lifetimes of modes in ARs than in QRs. The largest decrease (increase) in mode amplitude (mode width) of dormant ARs is found in the five minute frequency band. The average mode energy of both the flaring and dormant ARs are smaller than in their corresponding QRs, decreasing with increasing MAI. But the average mode energy decrease rate in flaring ARs are smaller than in dormant ARs. Also, the increase in mode width rate in dormant (flaring) ARs is followed by decrease (increase) in amplitude variation rate. Furthermore, inclusion of mode corrections for MAI shows that mode amplitude and mode energy of flaring ARs increase with FI while mode width shows an opposite trend. The increase (decrease) in mode amplitude (width) is larger in the five minute and higher frequency band. Increase in width variation rate is ..

Regio- and Stereoselective Ruthenium Catalyzed Hydrovinylation of 1,3-Dienes: Application to the Generation of a 20S-Steroidal Sidechain

The addition of ethylene to 1,3-dienes and 1-vinylcycloalkenes, catalyzed by two ruthenium complexes, proceeds in a regioselective fashion to afford 3-methyl-1,4-dienes as products. For a steroidal-based 1-vinylcycloalkene, the addition is stereospecific, giving a product with a 20(S) configuration

Highly Cooperative Tetrametallic Ruthenium-μ-Oxo-μ-Hydroxo Catalyst for the Alcohol Oxidation Reaction

The tetrametallic ruthenium-oxo-hydroxo-hydride complex {[(PCy3)(CO)RuH]4(μ4-O)(μ3-OH)(μ2-OH)} (1) was synthesized in two steps from the monomeric complex (PCy3)(CO)RuHCl (2). The tetrameric complex 1 was found to be a highly effective catalyst for the transfer dehydrogenation of alcohols. Complex 1 showed a different catalytic activity pattern toward primary and secondary benzyl alcohols, as indicated by the Hammett correlation for the oxidation reaction of p-X-C6H4CH2OH (ρ = −0.45) and p-X-C6H4CH(OH)CH3 (ρ = +0.22) (X = OMe, CH3, H, Cl, CF3). Both a sigmoidal curve from the plot of initial rate vs [PhCH(OH)CH3] (K0.5 = 0.34 M; Hill coefficient, n = 4.2 ± 0.1) and the phosphine inhibition kinetics revealed the highly cooperative nature of the complex for the oxidation of secondary alcohols

Uncovering Hierarchical Structure in Social Networks using Isospectral Reductions

We employ the recently developed theory of isospectral network reductions to analyze multi-mode social networks. This procedure allows us to uncover the hierarchical structure of the networks we consider as well as the hierarchical structure of each mode of the network. Additionally, by performing a dynamical analysis of these networks we are able to analyze the evolution of their structure allowing us to find a number of other network features. We apply both of these approaches to the Southern Women Data Set, one of the most studied social networks and demonstrate that these techniques provide new information, which complements previous findings.Comment: 17 pages, 5 figures, 5 table

A study of energy concentration and drain in incompressible fluids

In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if $u$ is a regular solution on a time interval $[0,T)$ and if $u \in L^rL^\infty$ for some $r\geq \frac{2}{N}+1$, where $N$ is the dimension of the fluid, then the energy at the time $T$ cannot concentrate on a set of Hausdorff dimension samller than $N - \frac{2}{r-1}$. The same holds for solutions of the three-dimensional Navier-Stokes equation in the range $5/3<r<7/4$. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T-t)^{1-\d}, for any \d>0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.Comment: an update of the previous versio

Nonexistence of self-similar singularities for the 3D incompressible Euler equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of compact support for the vorticit