Let G = (V,E) be a graph and let f be a function that assigns list sizes to the vertices of G. It is said that G is f -choosable if for every assignment of lists of colors to the vertices of G for which the list sizes agree with f , there exists a proper coloring of G from the lists. The sum choice number is the minimum of the sum of list sizes for f over all choosable functions f for G. The su...

All muscle contractions are dependent on the functioning of motor units. In diseases such as amyotrophic lateral sclerosis (ALS), progressive loss of motor units leads to gradual paralysis. A major difficulty in the search for a treatment for these diseases has been the lack of a reliable measure of disease progression. One possible measure would be an estimate of the number of surviving motor ...

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First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...

In this paper we obtain optimal bounds for the length of the longest arithmetic progression in various kinds of sum-sets. As an application, we derive a sharp estimate for the number of sets A of residues modulo a prime n such that no subsum of A equals x modulo n, where x is a fixed residue modulo n.

In this paper, our objective is maximizing total sum-rate subject to power constraints on total relay transmit power or individual relay powers, for amplify-and-forward single-antenna relay-based wireless communication networks. We derive a closed-form solution for the total power constraint optimization problem and show that the individual relay power constraints optimization problem is a quad...

By disaggregating price indices, it becomes apparent that the real exchange rate consists of the real exchange rate for a single good and a weighted sum of relative prices between goods. When applying a battery of panel unit root tests to this sum and its components, it is found that both the sum and the relative prices are non-stationary. This implies that PPP is invalid even if the LOP holds ...

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the n...

we give a new recursive method to compute the number of cliques and cycles of a graph. this method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. in particular, let $g$ be a graph and let $overline {g}$ be its complement, then given the chromatic polynomial of...