On (Sub)stochastic and Transient Weightings of Infinite Strong Digraphs

In the present paper, for a given (possibly, infinite) strongly connected digraph $\cal{D},$ we consider the class $\cal{S}_{<}({\cal D})$ of all truthly substochastic weightings of ${\cal D}$ (here, the word "truthly" means that there exists a vertex whose out-weight is strictly less than $1$). For a finite subdigraph $\cal{F}$ of $\cal{D}$ weighted by $S\in {\cal S}_{<}({\cal D}),$ let $\ell_{max}(\cal{F})$ be the length of its longest directed cycle and $\lambda_{S}(\cal{F})$ be the Perron root (spectral radius) of its weighted adjacency matrix. We prove that the infimum of $\ell_{max}(\cal{F})\bigl(1-\lambda_{S}(\cal{F})\bigr)$ taken over all $\cal{F}$ is positive for every $S\in \cal{S}_{<}({\cal D})$ if and only if $\cal{D}$ admits a finite cycle transversal. The result obtained provides general theorems on the set ${\cal T}({\cal D})$ of transient weightings of ${\cal D}.$ In particular, we present a theorem of alternatives for finite approximations to elements of ${\cal T}({\cal D})$ and simply reprove V. Cyr's criterion for ${\cal T}({\cal D})$ to be empty

Metallic and insulating behaviour of the two-dimensional electron gas on a vicinal surface of Si MOSFETs

The resistance R of the 2DEG on the vicinal Si surface shows an unusual behaviour, which is very different from that in the (100) Si MOSFET where an unconventional metal to insulator transition has been reported. The crossover from the insulator with dR/dT0 occurs at a low resistance of R_{\Box}^c \sim 0.04h/e^2. At the low-temperature transition, which we attribute to the existence of a narrow impurity band at the interface, a distinct hysteresis in the resistance is detected. At higher temperatures, another change in the sign of dR/dT is seen and related to the crossover from the degenerate to non-degenerate 2DEG.Comment: 4 pages, 4 figure