Generalization of real interval matrices to other fields

An interval matrix is a matrix whose entries are intervals in $\R$. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in $\Q$. It is proved that a (real) interval $p \times q$ matrix with the endpoints of all its entries in $\Q$ contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than $\min\{p,q\}$ if and only if it contains a rational matrix with rank smaller than $\min\{p,q\}$; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field $K$ and a matrix $\al$ whose entries are subsets of $K$, a criterion to find the maximal rank of a matrix contained in $\al$ is described.

Simulation-Based Study on the Relationship between the Car Queue Pass Time and the Distance between Cars after the Light Turns Green

For the reason that drivers often act according to the car ahead, it’s possible that there is some relationship between the car queue start time and the interval of the cars after the light turns green. In this case, it may exist a rational interval with which the car queue can pass the crossroad fastest. Based on the collision avoidance model, a simulation program is proposed, in which the acceleration of cars is limited by the real characteristics of car movement. A corresponding experiment is then conducted. The result of the experiment shows that the rational interval is actually nonexistent.