Tetrads of lines spanning PG(7,2)

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.Comment: Some typos correcte

Tetrads of lines spanning PG(7,2)

Our starting point is a very simple one, namely that of a set L₄ of four mutually skew lines in PG(7,2). Under the natural action of the stabilizer group G(L₄)&lt;GL(8,2) the 255 points of PG(7,2) fall into four orbits ω₁,ω₂,ω₃,ω4, of respective lengths 12,54,108,81. We show that the 135 points ∈ω₂∪ω₄ are the internal points of a hyperbolic quadric H7 determined by L₄, and that the 81-set ω₄ (which is shown to have a sextic equation) is an orbit of a normal subgroup G₈₁≅(Z₃)4 of G(L4). There are 40 subgroups ≅(Z₃)3 of G₈₁, and each such subgroup H&lt;G₈₁ gives rise to a decomposition of ω4 into a triplet {RH,R′H,R′′H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S₃(2) in PG(7,2). This ties in with the recent finding 225-239 --- that each S=S₃(2) in PG(7,2) determines a distinguished Z₃ subgroup of GL(8,2) which generates two sibling copies S′,S′′ of S

Comparing the effects of different approaches to liberalising world grains markets

The success of the current Doha Round of the WTO negotiations on agriculture will require substantial reform in each of the three areas of market access, export subsidies and domestic support. Substantial improvement in market access for agricultural products will be an essential requirement for achieving a successful outcome. However, the extent of improvement in market access resulting from the current negotiations will largely depend on the form and the approach followed to reduce tariffs and expand tariff rate quotas. In this paper different approaches to expanding market access for grains area analysed using a partial equilibrium model. Simulated scenarios include linear reductions in applied tariffs and expansions in tariff rate quotas, which are contrasted with a scenario representing market access proposals of the Cairns Group of countries in the current WTO agricultural negotiations. The effects of these two trade liberalisation scenarios on world prices and trade are analysed and discussed. Results indicate that to achieve a meaningful gain in market access for grains, WTO members must agree to either directly reduce the current applied tariffs or make large percentage reductions to the WTO bound rates, which lead to effective reductions in the current applied rates.Crop Production/Industries, International Relations/Trade,

Swelling and shrinking kinetics of a lamellar gel phase

We investigate the swelling and shrinking of L_beta lamellar gel phases composed of surfactant and fatty alcohol after contact with aqueous poly(ethylene-glycol) solutions. The height change $\Delta h(t)$ is diffusion-like with a swelling coefficient, S: $\Delta h = S \sqrt{t}$. On increasing polymer concentration we observe sequentially slower swelling, absence of swelling, and finally shrinking of the lamellar phase. This behavior is summarized in a non-equilibrium diagram and the composition dependence of S quantitatively described by a generic model. We find a diffusion coefficient, the only free parameter, consistent with previous measurements.Comment: 3 pages, 4 figures to appear in Applied Physics Letter

Nurturing health-related online support groups:exploring the experience of patient moderators

The aim of this study was to examine the views of moderators across a diverse and geographically broad range of online support groups about their moderator experiences and to explore both the personal benefits as well as challenges involved. Thirty-three patient moderators completed an online questionnaire which included a series of open-ended questions. Thematic analysis identified three themes: emergence, empowerment, nurturing. Several moderators declared their own diagnosis and for some, being able to share personal insights motivated them to establish the group and in turn offered validation. They felt empowered by helping others and learned more about the condition through accessing the "communal brain". Some felt the group aided patients' access to health services and their ability to communicate with health professionals while others worried about them becoming over-dependent. Moderators described needing to nurture their group to ensure it offered a safe space for members. Clear rules of engagement, trust, organisation skills, compassion and kindness were considered essential. Patient moderated online support groups can be successfully developed and facilitated and can be empowering for both the group member and moderator alike

Linear sections of GL(4, 2)

For V = V (n; q); a linear section of GL(V ) = GL(n; q) is a vector subspace S of the n 2 -dimensional vector space End(V ) which is contained in GL(V ) [ f0g: We pose the problem, for given (n; q); of classifying the di erent kinds of maximal linear sections of GL(n; q): If S is any linear section of GL(n; q) then dim S n: The case of GL(4; 2) is examined fully. Up to a suitable notion of equiv- alence there are just two classes of 3-dimensional maximal normalized linear sections M3;M0 3 , and three classes M4;M0 4 ;M00 4 of 4-dimensional sections. The subgroups of GL(4; 2) generated by representatives of these ve classes are respectively G3 = A7; G 0 3 = GL(4; 2); G4 = Z15; G 0 4 = Z3 A5; G 00 4 = GL(4; 2): On various occasions use is made of an isomorphism T : A8 ! GL(4; 2): In particular a representative of the class M3 is the image under T of a subset f1; ::: ; 7g of A7 with the property that 1 i j is of order 6 for all i =6 j: The classes M3;M0 3 give rise to two classes of maximal partial spreads of order 9 in PG(7; 2); and the classes M0 4 ;M00 4 yield the two isomorphism classes of proper semi eld planes of order 16