We investigate the impact of boundary bound states on the dynamics of quantum fields, which propagate on star graphs modeling quantum wires. These states are localized in the vertex of the graph and decay exponentially in the bulk. We demonstrate that their contribution is completely fixed by causality (local commutativity), which represents a key point of our investigation. The boundary bound ...

A simple proof of associativity and commutativity of LR-coefficients (or the hive ring) In this paper we propose a simple bijective proof of associativity and commutativity of Littlewood-Richardson coefficient or the hive ring ([13]).

In the Map-Reduce programming model for data parallel computation in a cloud environment, the reducer phase is responsible for computing an output key-value pair, given a sequence of input values associated with a particular key. Because of non-determinism in transmitting key-value pairs over the network, the inputs may not arrive at a reducer in a fixed order. This gives rise to the reducer co...

then we term the elements x and y n-commutative. It is not difficult to verify that »-commutativity and (1—»)-commutativity are equivalent properties of the elements x and y, that ( — ̂ -commutativity implies ordinary commutativity, and that commuting elements are »-commutative. From any concept and property involving the fact that certain elements [or functions of elements] commute, one may der...

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural an...