A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of upsteps u = (1, 1) and down-steps d = (1,−1) which never passes below the x-axis. Let Dn denote the set of Dyck paths of length 2n. A peak is an occurrence of ud (an upstep immediately followed by a downstep) within a Dyck path, while a valley is an occurrence of du. Here, we compute explicit formulas for the genera...

Classical measures of network connectivity are the number of disjoint paths between a pair of nodes, and the size of a minimum cut. For standard graphs, these measures can be computed efficiently using network flow techniques. However, in the Internet on the level of Autonomous Systems (ASs), referred to as AS-level Internet, routing policies impose restrictions on the paths that traffic can ta...

We propose a device in which sheet of graphene is coupled to Weyl semimetal, allowing for the physical access study tunneling from two- three-dimensional massless Dirac fermions. Because reconstructed band structure, we find that this acts as robust valley filter electrons sheet. show that, by appropriate alignment, semimetal draws away current one two valleys, while other pass unimpeded. In co...

We demonstrate by an example that the search for valley-ridge transition (VRT) points used in the commented paper does not always indicate a bifurcation of a reaction path. This was claimed.

Applying a minimum-cost path algorithm to find the path through the bottom of a curvilinear valley yields a biased path through the inside of a corner. DNA molecules, blood vessels, and neurite tracks are examples of string-like (network) structures, whose minimum-cost path is cutting through corners and is less flexible than the underlying centerline. Hence, the path is too short and its shape...

Are the inventor networks of Silicon Valley more densely connected than those of the Boston Route 128 corridor? The evidence remains mostly historical and controversial to date. We develop an analysis of all the patented inventors in both regions since 1975 and find that the networks of Silicon Valley are simultaneously more connected and less robust than those of Boston. Of greatest interest, ...

We report the observation of lifetime-enhanced transport (LET) based on perpendicular valleys in silicon by transport spectroscopy measurements of a two-electron system in a silicon transistor. The LET is manifested as a peculiar current step in the stability diagram due to a forbidden transition between an excited state and any of the lower energy states due to perpendicular valley (and spin) ...

Since new (Power) Transmission Lines (TLs) can have a long-term effect on the makeup of landscape and human living space, it should be expected that route any TL will based objective criteria take into account views public. Geographic Information Science (GIScience) provides powerful tools assist in determination feasible locations for TLs georeferenced data by combining methods from Multi-Crit...