The gauge including magnetically induced current method

An overview of applications of the recently developed gauge including magnetically induced current method (GIMIC) is presented. The GIMIC method is used to obtain magnetically induced current densities in molecules. It provides detailed information about electron delocalization, aromatic character, and current pathways in molecules. The method has been employed in aromaticity studies on hydrocarbons, complex multi-ring organic nanorings, M?bius twisted molecules, inorganic and all-metal molecular rings and open-shell species. Recent studies on hydrogen-bonded molecules indicate that GIMIC can also be used to estimate hydrogen-bond strengths without fragmentation of the system. Preliminary results are presented on the applicability of GIMIC for investigating current transport in molecules attached to clusters simulating molecular conductivity measurements. Advantages and limitations of the GIMIC method are reviewed and discussed.     

Universal radius of injectivity for locally quasiconformal mappings

Ifn>2 and iff is a locally quasiconformal mapping from the ballB ⁿ= {x∈R ⁿ:⋎x⋎<1} intoR ⁿ ∪ {∞} thenf is injective inB ⁿ (r)={x∈R ⁿ:⋎x⋎ <r} wherer>0 depends only onn and the maximal dilatation off. Supported in part by the Samuel Neaman Fund, Special Year in Complex Analysis, Technion, I.I.T., Haifa, Israel, 1975/76.     

Some remarks on an existence problem for degenerate elliptic systems

The system which yields Beltrami's system if , is considered. Under a condition for the coefficients a non-existence theorem is proved.

     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

An optimization approach for communal home meal delivery service: A case study

This paper is the first to discuss the communal home meal delivery problem. The problem can be modelled as a multiple travelling salesman problem with time windows, that is closely related to the well-studied vehicle routing problem with time windows. Experimental results are reported for a real-life case study from Central Finland over several alternative scenarios using the SPIDER commercial solver. The comparison with current practice reveals that a significant savings potential can be obtained using off-the-shelf optimization tools. As such, the potential for supporting real-life communal routing problems can be considered to be important for VRP practitioners.