— The words “monodromy” and “isomonodromy” are used in the theory of difference and q-difference equations by Baranovsky-Ginzburg, Jimbo-Sakai, Borodin, Krichever,... although it is not clear that phenomena of branching during analytic continuation are involved there. In order to clarify what is at stake, we survey results obtained during the last few years, mostly by J.-P. Ramis, J. Sauloy and...

Let ZK C be the moment angle complex associated to a simplicial complex K on Œm , together with the natural action of the torus T D U.1/ . Let G T be a (possibly disconnected) closed subgroup and R WD T=G . Let ZŒK be the Stanley– Reisner ring of K and consider ZŒR  WD H .BRIZ/ as a subring of ZŒT  WD H .BTIZ/ . We prove that H G .ZK IZ/ is isomorphic to Tor ZŒR .ZŒK;Z/ as a graded module...

A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as anticipated, it is a part of non-trivial A ∞ structure, involving a chain of pol...

Community structure analysis is a powerful tool for complex networks, which can simplify their functional analysis considerably. Many approaches have recently been proposed to the communities in complex networks, but a method to characterize the node importance to communities is still lacking. In this paper a centrality metric is proposed to measure the importance of network nodes to community ...

We examine the impact of enforcing non-compete covenants (CNC) on the formation and performance of new firms using matched employer-employee data on 30 US states. To identify the impact of CNC, we exploit the inter-state variation in CNC enforcement along with the fact that courts do not enforce such covenants between law firms and departing lawyers in any state. Using a difference-in-differenc...

The statistical non-Hamiltonian theory of fluctuation in the complex systems with a discrete current time is presented. Quasidynamic Liouville equation for the state vector of the complex system serves as a initial point of the discrete analysis. The projection operator in a vector state space of finite dimension allows to reduce Liouville equations to a closed non-Markov kinetic equation for a...

We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. investigate stability linear systems defined by an matrix $A$ and derive conditions for equilibrium points systems. For one-dimensional case where $A =\lambda \in \mathbb {C}$, we find that region, if any is enclosed a boundary curve obtain parametric equation same. Furthermore, there no s...