The permanent’s polynomial-time computability over fields of characteristic 3 for k-semi-unitary matrices (i.e. n×n-matrices A such that rank(AA − In) = k) in the case k ≤ 1 and its #3-P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely t...

Abért-Weiss have shown that the Bernoulli shift sΓ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that sΓ × a is weakly equivalent to a. Using random Bernoulli shifts introduced by Abért-Glasner-Virag we generalized this to non-free actions, replacing sΓ with ...

‎in this paper we introduce mixed unitary cayley graph $m_{n}$‎ ‎$(n>1)$ and compute its eigenvalues‎. ‎we also compute the energy of‎ ‎$m_{n}$ for some $n$‎.

We verify in this paper that the linearity and orthogonality structures of a (not necessarily local trivial) Hilbert bundle over a locally compact Hausdorff space Ω determine its unitary structure. In fact, as Hilbert bundles over Ω are exactly Hilbert C0(Ω)-modules, we have a more general set up. A C-linear map θ (not assumed to be bounded) between two Hilbert C∗-modules is said to be “orthogo...

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact non-elementary convergence and lattices in non-compact semi-simple Lie but excludes inner amenable groups. show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions groups this have at most one weakly compact Cartan subalgebra, up to u...

Abért and Weiss have shown that the Bernoulli shift s0 of a countably infinite group 0 is weakly contained in any free measure preserving action a of 0. Proving a conjecture of Ioana, we establish a strong version of this result by showing that s0 × a is weakly equivalent to a. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replaci...

We show that, for a generic measure preserving transformation T, the closed group generated by T is not isomorphic to topological $$L^0(\lambda , {{\mathbb {T}}})$$ of all Lebesgue measurable functions from [0, 1] $${\mathbb {T}}$$ (taken with pointwise multiplication and topology convergence in measure). This result answers question Glasner Weiss. The main step proof consists showing that Koop...

We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary preserves the space, we associate an index integer multiple of 1/p, where p degeneracy. prove additive under composition unitaries. This formalism gives rise to several applications: fracti...