The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.Comment: 18 pages; to appear in Math.

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ? 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios

The Future of Agriculture in Our Community: A Pilot Program to Increase Community Dialogue About Agricultural Sustainability

The Future of Agriculture in Our Community is a program developed to allow Pennsylvania communities to assess and address the needs of local agriculture. This article describes the program in detail and provides results from an evaluation conducted of the pilot program. Findings (n=55) suggest that the program was received very well among participants and seemed to increase community organization skills, knowledge of local agriculture, interest in agriculture and in community life, and intentions to participate in future volunteer efforts. Based on these results, recommendations are offered for those interested in pursuing similar programs

Universality of Leading Relativistic Corrections to Bound State Gyromagnetic Ratios

We discuss the leading relativistic (nonrecoil and recoil) corrections to bound state $g$-factors of particles with arbitrary spin. These corrections are universal for any spin and depend only on the free particle gyromagnetic ratios. We explain the physical reasons behind this universality.Comment: 10 pp., based on talks given at the Gribov-80 Memorial Workshop, ICTP, Trieste, Italy, May 2010 and at PSAS 2010 workshop, Les Houches, June 2010. To be published in the Gribov-80 Proceedings and in Can. J. Phy

Ordering our world: the quest for traces of temporal organization in autobiographical memory

An experiment examined the idea, derived from the Self Memory System model (Conway & Pleydell-Pearce, 2000), that autobiographical events are sometimes tagged in memory with labels reflecting the life era in which an event occurred. The presence of such labels should affect the ease of judgments of the order in which life events occurred. Accordingly, 39 participants judged the order of two autobiographical events. Latency data consistently showed that between-era judgments were faster than within-era judgments, when the eras were defined in terms of either: (a) college versus high school, (b) academic quarter within year, or (c) academic year within school. The accuracy data similarly supported the presence of a between-era judgment effect for the college versus high school dichotomy

Decoupling of the ϵ\epsilon-scalar mass in softly broken supersymmetry

It has been shown recently that the introduction of an unphysical $\epsilon$-scalar mass $\tilde{m}$ is necessary for the proper renormalization of softly broken supersymmetric theories by dimensional reduction (\drbar). In these theories, both the two-loop $\beta$-functions of the scalar masses and their one-loop finite corrections depend on $\tilde{m}^2$. We find, however, that the dependence on $\tilde{m}^2$ can be completely removed by slightly modifying the \drbar renormalization scheme. We also show that previous \drbar calculations of one-loop corrections in supersymmetry which ignored the $\tilde{m}^2$ contribution correspond to using this modified scheme.Comment: 7 pages, LTH-336, NUB-3094-94TH, KEK-TH-40

Filaments in the southern giant lobe of Centaurus A : Constraints on nature and origin from modelling and GMRT observations

Date of acceptance: 22/05/2014We present results from imaging of the radio filaments in the southern giant lobe of CentaurusA using data from Giant Metrewave Radio Telescope observations at 325 and 235 MHz, and outcomes from filament modelling. The observations reveal a rich filamentary structure, largely matching the morphology at 1.4 GHz. We find no clear connection of the filaments to the jet. We seek to constrain the nature and origin of the vertex and vortex filaments associated with the lobe and their role in high-energy particle acceleration. We deduce that these filaments are at most mildly overpressuredwith respect to the global lobe plasma showing no evidence of largescale efficient Fermi I-type particle acceleration, and persist for ~2-3 Myr. We demonstrate that the dwarf galaxy KK 196 (AM 1318-444) cannot account for the features, and that surface plasma instabilities, the internal sausage mode and radiative instabilities are highly unlikely. An internal tearing instability and the kink mode are allowed within the observational and growth time constraints and could develop in parallel on different physical scales. We interpret the origin of the vertex and vortex filaments in terms of weak shocks from transonic magnetohydrodynamical turbulence or from a moderately recent jet activity of the parent AGN, or an interplay of both.Peer reviewe