562 research outputs found
Author(s): Ezra Brown and Nicholas Loehr Source
No full textJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Why is PSL(2, 7) = GL(3, 2)? Mathematical Association of America Ezra Brown and Nicholas Loehr 1. INTRODUCTION. The groups of invertible matrices over finite fields are among the first groups we meet in a beginning course in modern algebra. Eventually, we find out about simple groups and that the unique simple group of order 168 has two representations as a group of matrices. And this is where we learn that the group of 2x2 unimodular matrices over a seven-element field, with / and -/ identified, is isomorphic to the group of invertible 3x3 matrices over a 2-element field. In short, it is a fact that PSL(2, 7) = GL(3, 2). Many of us are surprised by this fact: why should a group of 2 x 2 matrices with mod-7 integer entries be isomorphic to a group of 3 x 3 binary matrices? There are a number of proofs of this remarkable theorem. Dickson [1, p. 303] gives a proof based on his general theorem giving uniform sets of generators and relations for the family of groups SL(2, q), where q is any prime power. One checks that the relations appearing in Dickson's presentation of PSL(2, 7) are satisfied by certain generators of GL(3, 2), implying that these groups have the same presentations and are therefore isomorphic. Dummit and Foote [2, show that every simple group of order 168 is necessarily isomorphic to the automorphism group Aut(.F) of the Fano plane T. They then show that Aut(^) = GL(3, 2) and that PSL(2, 7) is a simple group of order 168; the isomorphism theorem follows. Rotman gives the result as an exercise [5, Exercise 9.26, p. 281]. A hint is to begin with a simple group G of order 168 and use the seven conjugates of a Sylow 2-subgroup P of G to construct a sevenpoint projective plane; the proof is similar to Dummit and Foote's proof. Jeurissen [4] proves the result by showing that both PSL(2, 7) and GL(3, 2) are subgroups of index 2 of the automorphism group of a Coxeter graph. Elkies The aim of this paper is to give a proof that PSL(2, 7) = GL(3, 2) that is elementary in the sense that it uses neither simplicity, nor projective geometry, nor block designs. We will not prove the fact that any two simple groups of order 168 are isomorphic, nor will we use this fact in our proof. What makes our proof work is that: (a) we can identify GL(3, 2) with the set of invertible F2-linear transformations on the finite field with eight elements; (b) 7 = 23 -1; (c) the nonzero squares mod 7 are precisely the powers of 2 mod 7; (d) squaring mod 2 is additive (the Freshman's Dream); and (e) the mapping k h+ -i/k mod 7 translates to a bit-switch mod 2 -which is linear. We begin by giving functional descriptions for both groups, determining their sizes
Interspecific competition between Diadegma semiclausum Hellen and Diadegma mollipla (Holmgren), parasitoids of the diamondback moth, Plutella xylostella (L), feeding on a new host plant
No full textInterspecific competition between an introduced parasitoid species aimed at controlling a herbivorous pest species and a native parasitoid parasitising the same host may influence the success of classical biological control programmes. In Kenya, interspecific competition between an introduced and a local parasitoid on two diamondback moth populations (DBM, Plutella xylostella) was investigated on two different host plants. We tested simultaneous and delayed competition of the local parasitoid Diadegma mollipla Holmgren and its exotic congenus D. semiclausum Hellen on a newly aquired DBM host plant (snowpea) in the laboratory. Under simultaneous competition, D. mollipla produced more progeny than D. semiclausum on snowpea. A head start of D. Mollipla, of four and eight hours before its congenus was introduced, resulted in a similar number of progeny of both species. In delayed competition (time intervals of 24h, 48h and 72h), progeny production was similar for both parasitoids when the time interval was 24h, irrespective of which species parasitized first. More progeny was produced by the species which attacked first, when the time interval was greater than 24h, although it was only significant at 72 h. Competitive abilites of both parasitoids on the new host plant differed largely between laboratory and semi-field conditions. The influence of two host plants (snowpea and cabbage) on competition was studied in the greenhouse with different host and parasitoid densities. Parasitism levels of D. semiclausum were significantly higher than those of D. mollipla, regardless of host plant, host and parasitoid densities, but progeny production of D. mollipla on snowpea was still slightly higher than on cabbage. As compared to the confinement of parasitoids and larvae to small containers, D. mollipla parasitized very few larvae in the cages. Competitive ability of the two parasitoid species tested was influenced both by the density of the searching females and by parameters related to either the host plant and/or the herbivorous hosts.the German Research Foundation (DFG
Does a specialist parasitoid adapt to its host on a new host plant?
No full textThe host plant expansion of a diamondback moth, Plutella xylostella (L.) (DBM) strain to snowpea (Pisum sativum L.) raised the question whether a specialist parasitoid Diadegma semiclausum (DS) could be conditioned to locate and parasitize its host on the new host plant. In a specialist parasitoid a behavioural change towards a plant outside the normal host plant range of its host due to developmental experience is not expected. The responsive behaviour, parasitism rates and fitness of three subsequent DS generations were investigated on the snowpea-strain of DBM. After three generations of DS on the pea 62.5% of females chose an DBM-infested pea plant over DBM infested cabbage. Only 16.4% of cabbage-reared DS was attracted to infested pea. Rearing of the parasitoid in host larvae on peas significantly increased the number of larvae parasitized on this host plant in the first generation; however, there was no further increase in generations 2 and 3. Larval mortality was similar for all parasitoid/DBM combinations on both host plants, but significantly higher mortality occurred in parasitoid pupae from peas. Development time of the parasitoid was slightly prolonged on the pea strain of DBM. The number of females produced by parasitoids reared on the pea strain of DBM was significantly reduced as compared to D. semiclausum reared on the cabbage strain on both host strains. Results show that DS has the potential to change its responsive behaviour in order to locate its host on a new host plant. According to the current view, a specialist parasitoid is not expected to change its reaction to a plant outside the normal host plant range of its host. Within 3 generations, responsive behaviour towards snowpea could be increased. However, fitness trade-offs, especially an extreme shift in sex ratio to males reduced reproductive success
Parasitism of Plutella xylostella L. feeding on a new host plant
No full textThe diamondback moth Plutella xylostella L. (Lepidoptera: Plutellidae) is known to be an oligophagous pest on crucifers. Recently, a population of diamondback moths was found to infest sugar snap and snowpea in the Rift Valley in Kenya, causing heavy damage. The effect of resident parasitoids on diamondback moths in snowpea (Pisum sativum L.) was studied in the field. In addition, parasitism of diamondback moths by the newly introduced parasitoid Diadegma semiclausum (Hellen) was evaluated. Snowpea provided an enemy-free space for diamondback moths. Local parasitoids attacked diamondback moths only sporadically and in very low numbers (mean of 0.25 individuals/ 20 plants during the first study period) on diamondback moths on pea compared with a mean of 15.4 individuals/20 plants on kale (Brassica oleracea acephala L.) However, diamondback moth density was higher on kale (mean of 61.1 larvae/20 plants) than on snowpea (mean of 11.4 larvae/20 plants). Differential diamondback moth infestation and level of parasitism might have been affected by a fungal infection (Ascochyta) of the pea plants. In both crops, the most abundant parasitoid was Oomyzus sokolowskii Kurdjumov. After the release of D. semiclausum, the number of diamondback moths in kale decreased drastically to < 2 individuals/20 plants; however, the parasitoid had little effect on diamondback moths on snowpea. Percentage parasitism on snowpea increased from 2.3 to 4%, whereas on kale, it increased from 25.6 to 75.7%. A host plant expansion could be of future advantage for the diamondback moth to avoid higher enemy pressure caused by D. semiclausum
Can low release numbers lead to establishment and spread of an exotic parasitoid: The case of the diamondback moth parasitoid, Diadegma semiclausum (Hellen), in East Africa
No full textFrom 2001 onwards, Diadegma semiclausum, an exotic parasitoid of the diamondback moth, was introduced and released in Kenya, Tanzania and Uganda. Contrary to common practice where thousands of parasitoids are released, we released very low numbers, 125 females in Kenya, 160 in Uganda and 350 in Tanzania. About 2 years after this single release, the establishment and natural spread of the parasitoids was assessed in all release areas. Two methods were employed: in Kenya, a grid with equidistant points in the four cardinal directions (2-50 km) with the release area in the centre was used and collections were made 27 months after release at the predetermined points. The parasitoid was found up to a distance of 30km from the release site. In Tanzania and Uganda, surveys were made starting from the release area following major roads. At regular intervals, fields were inspected and their position recorded with a Geographic Positioning System (GPS). The results of a field survey conducted 24 months after release indicate that in Tanzania, the parasitoid had spread > 20 km from the release site while in Uganda, the spread was > 30 km. Wherever D. semiclausum was collected, it was the major parasitoid species. Indigenous parasitoids collected were Oomyzus sokolowskii (Hym.: Eulophidae), Diadegma mollipla (Hym.: Ichneumonidae) and Apanteles sp. (Hym.: Braconidae). Overall parasitism and the contribution of the introduced parasitoid to the control of diamondback moth population tended to decrease with increasing distance from the release point. The introduced parasitoid had displaced the indigenous species wherever it was well established. (C) 2007 Elsevier Ltd. All rights reserved
Introduction to Session : Combined Diphtheria and Tetanus Toxoids and Acellular Pertussis, Inactivated Poliovirus, Haemophilus influenzae Type b Conjugate, and Hepatitis B vaccine (Vaxelis)
No full text01-Vaxelis-Loehr-508.pd
Introduction and Terms of Reference: Combined Diphtheria and Tetanus Toxoids and Acellular Pertussis, Inactivated Poliovirus, Haemophilus influenzae Type B Conjugate, and Hepatitis B vaccine (Vaxelis)
No full text01-Vaxelis-Loehr-508.pd
Loehr OHara Finnish soldier data
No full textLife history and morphological measurements of Finnish soldiers who fought in the Winter War (1939-1940). Photographs and individual data gathered from book and internet souces
Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules
No full textHaglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.
A Human Proof for a Generalization of Shalosh B. Ekhad’s 10 n Lattice Paths Theorem
No full textConsider lattice paths in Z 2 taking unit steps north (N) and east (E). Fix positive integers r, s and put an equivalence relation on points of Z 2 by letting v, w be equivalent if v − w = ℓ(r, s) for some ℓ ∈ Z. Call a lattice path valid if whenever it enters a point v with an E-step, then any further points of the path in the equivalence class of v are also entered with an E-step. Loehr and Warrington conjectured that the number of valid paths from (0, 0) n. We prove this conjecture when s = 2. to (nr, ns) is ` r+s r
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