A self-dual map $G$ is said to be \emph{antipodally self-dual} if the dual $G^*$ antipodal embedded in $\mathbb{S}^2$ with respect $G$. In this paper, we investigate necessary and/or sufficient conditions for a antipodally self-dual. particular, present combinatorial characterization terms of certain \emph{involutive labelings}. The latter lead us obtain \emph{strongly involutive} (a notion rel...

in the present study, the effect of two species of amf, glomus mosseae and glomus intraradices, alone and in combination, was evaluated on the growth criteria, chlorophyll content, and root rot disease caused by fusarium solani f. sp. pisi, on chickpea (cicer arietinum l.) under greenhouse conditions. chickpea seeds were sown into pots containing 100 g of amf inoculum (more than 1,000 propagule...

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that g is a xed point free involution. If C is extremal of length a multiple of 24 all the involutions are xed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C...

1 Discrete Fourier analysis In this lecture, we use the dual-sparse approximation theorem from the last lecture to prove some results in discrete Fourier analysis. For simplicity, we restrict ourselves to the setting of G ¿ n 2 , but the theorems hold (when suitably restated) for any finite abelian group G. Fourier analysis over ¿ n 2. We use ¿ 2 {0, 1} to denote the field on two elements. Let ...

In Almost Free Modules, Set-theoretic Methods, Eklof and Mekler [5, p. 455, Problem 12] raised the question about the existence of dual abelian groups G which are not isomorphic to Z ⊕G. Recall that G is a dual group if G ∼= D for some groupD withD = Hom (D,Z). The existence of such groups is not obvious because dual groups are subgroups of cartesian products Z and therefore have very many homo...

We construct dual Lagrangians for G/H models in two space-time dimensions for arbitrary Lie groups G and H ⊂ G. Our approach does not require choosing coordinates on G/H, and allows for a natural generalization to Lie-Poisson duality. For the case where the target metric on G/H is induced from the invariant metric on G, the dual system is a gauged Higgs model, with a nonconstant metric and a co...

The automorphism group of a binary doubly-even self-dual code is always contained in the alternating group. On the other hand, given a permutation group G of degree n there exists a doubly-even self-dual G-invariant code if and only if n is a multiple of 8, every simple self-dual F2G-module occurs with even multiplicity in F n 2 , and G is contained in the alternating group.