A modified k-deck of a graph G is obtained by removing k edges of G in all possible ways, and adding k (not necessarily new) edges in all possible ways. Krasikov and Roditty asked if it was possible to construct the usual k-edge deck of a graph from its modified k-deck. Earlier I solved this problem for the case when k = 1. In this paper, the problem is completely solved for arbitrary k. The pr...

A modified k-deck of a graph is obtained by removing k edges in all possible ways and adding k (not necessarily new) edges in all possible ways. Krasikov and Roditty used these decks to give an independent proof of Müller’s result on the edge reconstructibility of graphs. They asked if a k-edge deck could be constructed from its modified k-deck. In this paper, we solve the problem when k = 1. W...

Carlos Simpson CNRS, Université de Nice-Sophia Antipolis Much interest has recently focused on the problem of comparing different definitions of n-categories. 1 Leinster has made a useful compendium of 10 definitions [8]; May has proposed another definition destined among other things to make comparison easier [9], and he has also led the creation of an umbrella research group including compari...

We classify all pairs (G, V ) with G a closed subgroup in a classical group G with natural module V over C, such that G and G have the same composition factors on V ⊗k for a fixed k ∈ {2, 3, 4}. In particular, we prove Larsen’s conjecture stating that for dim(V ) > 6 and k = 4 there are no such G aside from those containing the derived subgroup of G. We also find all the examples where this fai...

Algorithms for classification and taxonomy based on criteria such as information entropy and its production are proposed. As an example, the feasibility of replacing a given anaesthetic by similar ones in the composition of a complex drug is studied. Some local anaesthetics currently in use are classified using characteristic chemical properties of different portions of their molecules. Many cl...

Following Braverman–Finkelberg–Feigin–Rybnikov (arXiv: 1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W -algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang–Mills theory with surface operators. Our new observation is that the C-fixed point set of a handsaw quiver variety is...

The multiplicities a λ,µ of simple modules L µ in the composition series of Kac modules V λ for the Lie superalgebra gl(m/n) were described by Serganova, leading to her solution of the character problem for gl(m/n). In Serganova's algorithm all µ with nonzero a λ,µ are determined for a given λ; this algorithm turns out to be rather complicated. In this Letter a simple rule is conjectured to fin...

A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. The Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that all triangle-free graphs G with connectivity two such that diam(G) = 2 or diam(G) = diam(G) = 3 are reconstructibl...

In this paper we de2ne 2ve families of simple graphs {fi,f2,f^,t^ and F=5) such that the reconstruction conjecture is true if reconstruction is proved for either the families $[,$2 & % or the families F=3;F=4 and F=5. These families are quite restrictive in that each has diameter two or three. Then we prove that families F=1;F=2;F=3;F=4 and F=5 are recognizable by proving that graphs of diamete...