The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph Γ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the sp...

This week we will be studying the tropical analogue of Bezout’s theorem, which in classical geometry gives a precise count of the number of intersections between two projective algebraic curves. It turns out, interestingly, that although all the definitions (except that of intersection) are different in tropical geometry, we can also prove a theorem with the same statement. Next time, we will s...

A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of modules over Dedekind rings. A generalized Chinese remainder theorem is derived as a consequence of the above resolution. The GelfandNaimark duality between finit...

The type II1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [P] Section 3.1, for the parameter value k = 1, and depends on a parameter n ≥ 2. Our main results are the explicit calculation of the linear relations of the type II1 unprojection for any value n ≥ 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the c...

We study higher Arthur-Merlin classes de ned via several natural probabilistic operators BP;R and co-R. We investigate the complexity classes they de ne, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bound...

We consider second order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem-Rosen theorem and the GoldblattThomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of S...

We prove a geometrical version of Herbert’s theorem [4] by considering the self-intersection immersions of a self-transverse immersion up to bordism. This generalises Herbert’s theorem to additional cohomology theories and gives a commutative diagram in the homotopy of Thom complexes. The proof is based on Herbert’s and uses Koschorke and Sanderson’s operations [6] and the fact that bordism of ...

Surface diagrams are a new way to specify any smooth closed orientable 4-manifold by an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. The aim of this paper is to announce a uniqueness theorem for these objects (within a fixed homotopy class) that turns out to be similar to the Reide...

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not gene...

In this paper we first defined the partial-union sequence, the partial-intersection sequence, and the partial-difference-union sequence of given sequence of subsets, and then proved the additive theorem of infinite sequences and sub-additive theorem of finite sequences for probability. Further, we defined the monotone class of families of subsets, and discussed the relations between the monoton...