We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite–dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that ...

Empirical studies clearly demonstrate the effectiveness of the nested jointree (NJT) representation in probabilistic inference. A NJT is a traditional Markov network (MN) together with a possible local MN nested in each clique. These nested MNs can themselves contain other nested MNs in a recursive manner. However, the NJT representation is not necessarily a faithful representation of a given B...

In this paper, we prove that the twisted Alexander polynomial for the Lawrence-Krammer representation of the braid group B4 is trivial. This gives an answer for the problem whether the twisted Alexander polynomial for given faithful representations is always non-trivial.

Sensory cortices receive inputs not only from thalamus but also from higher-order cortical regions. Here, Zagha et al. (2013) show that motor cortical inputs can switch barrel cortex into a desynchronized state that enables more faithful representation of subtle sensory stimuli.

X.S. Lin’s original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. J. Cha’s fibering obstruction theorem is generalized. The group of a nontrivial virtual knot shown by L. Kauffman to have trivial Jones polynomial is seen also to have a faithful representation that yields a trivial twisted Alexander polynomial.

We prove that the fundamental group of the double of the gure eight knot exterior admits a faithful discrete representation into SO(4; 1; R) for which the image group is separable on its geometrically nite subgroups. 2000 Mathematics Subject Classiication 20H10

We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.

We show that the monoid M2(T) of 2 × 2 tropical matrices is a regular semigroup satisfying the semigroup identity A2B4A2A2B2A2B4A2 =A2B4A2B2A2A2B4A2. Studying reduced identities for subsemigroups of M2(T), and introducing a faithful semigroup representation for the bicyclic monoid by 2 × 2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

It is shown, that the mapping class group of a surface of the genus g ≥ 2 admits a faithful representation into the matrix group GL6g−6(Z). The proof is based on a categorical correspondence between the Riemann surfaces and the AF -algebras.