Loewy lengths of centers of blocks II

Let ZB be the center of a p-block B of a finite group with defect group D. We show that the Loewy length LL(ZB) of ZB is bounded by $\frac{|D|}{p}+p-1$ provided D is not cyclic. If D is non-abelian, we prove the stronger bound $LL(ZB)<\min\{p^{d-1},4p^{d-2}\}$ where $|D|=p^d$. Conversely, we classify the blocks B with $LL(ZB)\ge\min\{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.Comment: 9 pages, this paper incorporates arXiv:1611.0605

Ion radial diffusion in an electrostatic impulse model for stormtime ring current formation

Guiding-center simulations of stormtime transport of ring-current and radiation-belt ions having first adiabatic invariants mu is approximately greater than 15 MeV/G (E is approximately greater than 165 keV at L is approximately 3) are surprisingly well described (typically within a factor of approximately less than 4) by the quasilinear theory of radial diffusion. This holds even for the case of an individual model storm characterized by substorm-associated impulses in the convection electric field, provided that the actual spectrum of the electric field is incorporated in the quasilinear theory. Correction of the quasilinear diffusion coefficient D(sub LL)(sup ql) for drift-resonance broadening (so as to define D(sub LL)(sup ql)) reduced the typical discrepancy with the diffusion coefficients D(sub LL)(sup sim) deduced from guiding-center simulations of representative-particle trajectories to a factor of approximately 3. The typical discrepancy was reduced to a factor of approximately 1.4 by averaging D(sub LL)(sup sim), D(sub LL)(sup ql), and D(sub LL)(sup rb) over an ensemble of model storms characterized by different (but statistically equivalent) sets of substorm-onset times

Dielectric Constant and Charging Energy in Array of Touching Nanocrystals

We calculate the effective macroscopic dielectric constant $\varepsilon_a$ of a periodic array of spherical nanocrystals (NCs) with dielectric constant $\varepsilon$ immersed in the medium with dielectric constant $\varepsilon_m \ll \varepsilon$. For an array of NCs with the diameter $d$ and the distance $D$ between their centers, which are separated by the small distance $s=D-d \ll d$ or touch each other by small facets with radius $\rho\ll d$ what is equivalent to $s < 0$, $|s| \ll d$ we derive two analytical asymptotics of the function $\varepsilon_a(s)$ in the limit $\varepsilon/\varepsilon_m \gg 1$. Using the scaling hypothesis we interpolate between them near $s=0$ to obtain new approximated function $\varepsilon_a(s)$ for $\varepsilon/\varepsilon_m \gg 1$. It agrees with existing numerical calculations for $\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of $\varepsilon_a(s)$ near $s = 0$. We also show that in this case the charging energy $E_c$ of a single NC in an array of touching NCs has a non-trivial relationship to $\varepsilon_a$, namely $E_c = \alpha e^2/\varepsilon_a d$, where $\alpha$ varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for $\varepsilon_a(s)$ can be used instead of mean field Maxwell-Garnett and Bruggeman approximations to describe percolation like transitions near $s=0$ for other material characteristics of NC arrays, such as conductivity

Phase-Sensitive Impurity Effects in Vortex Core of Moderately Clean Chiral Superconductors

We study impurity effects in vortex core of two-dimensional moderately clean su perconductors within the quasiclassical theory. The impurity scattering rate \G amma(E) of the Andreev bound states in vortex core with +1 vorticity of p-wav e superconductors with {\mib d}=\hat{\mib z}(p_x+\iu p_y) is suppre ssed, compared to the normal state scattering rate $\Gamma_{\rm n}$ in the energ y region \Gamma_{\rm n}^3/E_\delta^2\ll E\ll E_\delta\equiv |\delta_0|\Delta_\i nfty with scattering phase shift $\delta_0$ $(|\delta_0|\ll 1)$ and the pair-po tential in bulk $\Delta_\infty$. Further we find that $\Gamma(E)/\Gamma_{\rm n}$ for p-wave superconductors with {\mib d}=\hat{\mib z}(p_x-\iu p_y) is at most {\cal O}(E/\Delta_\i nfty). These results are in marked contrast to the even-parity case (s,d-wave), where $\Gamma(E)/\Gamma_{\rm n}$ is known to be proportional to \ln(\Delta_\i nfty/E) . Parity- and chirality-dependences of impurity effects are attributed to the Andr eev reflections involved in the impurity-induced scattering between bound states . Implications for the flux flow conductivity is also discussed. Novel enhanceme nt of flux flow conductivity is expected to occur at $T\ll E_\delta$ for {\mib d}=\hat{\mib z}(p_x+\iu p_y) and at $T\ll \Delta_\infty$ for {\mib d}=\hat{\mib z}(p_x-\iu p_y).Comment: 9 pages, No figures, To appear in JPSJ Vol. 69, No. 10 (2000

Kang-Redner Anomaly in Cluster-Cluster Aggregation

The large time, small mass, asymptotic behavior of the average mass distribution \pb is studied in a $d$-dimensional system of diffusing aggregating particles for $1\leq d \leq 2$. By means of both a renormalization group computation as well as a direct re-summation of leading terms in the small reaction-rate expansion of the average mass distribution, it is shown that \pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}} for $m \ll t^{d/2}$, where $e_{KR}=\epsilon +O(\epsilon ^2)$ and $\epsilon =2-d$. In two dimensions, it is shown that \pb \sim \frac{\ln(m) \ln(t)}{t^2} for $m \ll t/ \ln(t)$. Numerical simulations in two dimensions supporting the analytical results are also presented.Comment: 11 pages, 6 figures, Revtex

Fixed-N Superconductivity: The Crossover from the Bulk to the Few-Electron Limit

We present a truly canonical theory of superconductivity in ultrasmall metallic grains by variationally optimizing fixed-N projected BCS wave-functions, which yields the first full description of the entire crossover from the bulk BCS regime (mean level spacing $d \ll$ bulk gap $\tilde\Delta$) to the ``fluctuation-dominated'' few-electron regime ($d\gg\tilde\Delta$). A wave-function analysis shows in detail how the BCS limit is recovered for $d\ll \tilde \Delta$, and how for $d \gg \tilde \Delta$ pairing correlations become delocalized in energy space. An earlier grand-canonical prediction for an observable parity effect in the spectral gaps is found to survive the fixed-N projection.Comment: 4 pages, 3 figures, RevTeX, V2: minor charges to mach final printed versio