In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged rule. D. Prawitz calls such improper as opposed to proper ones. Improper more complicated than ones and difficult understand. 2022, we provided a sequent based solely on application of rules. present paper, basis our from classify

The probabilistic interpretation of Laplace transforms is used to help describe the Transform $L(s)$ improper random variables. In particular, busy periods in queueing models are examined. value $L(0)$ explained certain special cases.

There is currently a strong need to understand noncollinear magnetic phases where generalized chiral interactions can play prominent role. Here, the authors give clear and accessible discussion of improper how they arise from proper isotropic or interactions. For example, energy landscape magnets with degeneracy be mistakenly interpreted in terms large Dzyaloshinskii-Moriya that fact originate ...