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The Tutte polynomial of matroid perspectives
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Recall (see Definition 4.72) that the lattice of flatsL(M) of a matroid M is the set of flats of M ordered by inclusion. This partially ordered set is known to be a geometric lattice, the Möbius function of which can be used to calculate the Tutte polynomial of M (see Section 4.7). We refer the reader to Definition 4.141 or [1182] for the definition of the Möbius function, and to Equation (9.9) or [247] for the relation with the Tutte polynomial in matroids. This property of the Tutte polynomial generalizes to matroid perspectives [752], as stated below.
The Electromagnetic Phenomena as Incitants
Published in William J. Rea, Kalpana D. Patel, Air Pollution and the Electromagnetic Phenomena as Incitants, 2018
William J. Rea, Kalpana D. Patel
This should have stopped all biochemists dead in their tracks. Semiconduction only occurs in materials having a very orderly molecular structure, such as crystals, in which electrons can move easily from one electron cloud to another around one nucleus to cloud very close to one another. The atoms must be aligned in a precise geometric lattice rather than the random jumbled display that is found in most liquids and solids. Some of the crystalline materials have spaces in their lattice structures where other atoms can fit. The atoms of these “other impurities” may have more or fewer electrons than the atoms of the lattice material. Forces in this lattice structure hold the same number of electrons in place around each atom. The extra electrons of the impurities then become free to move across the crystalline structure at the speed of light, especially when very small currents are placed across this molecular arrangement. This is where grounding and light photons from the photoelectric effect affect liquid crystalline structures in cells. The number of electrons in the “impurity atom” also determines which structure will act as a positive or a negative (PN) part of the semiconductor. Semiconduction is not explained using the math of adenosine triposphate (ATP) hydrolysis or modern biochemistry.
Alkaline Liquid Electrolyte for Water Electrolysis
Published in Lei Zhang, Hongbin Zhao, David P. Wilkinson, Xueliang Sun, Jiujun Zhang, Electrochemical Water Electrolysis, 2020
Xuefeng Guo, Shanyong Chen, Yu Zhang, Mingjiang Xie, Jian Chen
Ito et al. reported that nitrogen and sulfur co-doping led to the high catalytic activity of a nanoporous graphene in the HER at a low operating potential, comparable to that of the best Pt-free HER catalyst, two-dimensional 2D MoS2.45 The reported N, S-graphene was prepared by the CVD method using pyridine and thiophene as precursors. The interplay between the chemical dopants and geometric lattice defects of the nanoporous graphene plays a fundamental role in the superiority of the HER catalysis. And the decisive reason for the excellent HER activity of the prepared catalysts is that the co-doping of N and S species reduce the ∆GH* of the reaction.
Poisson degenerate central moments related to degenerate Dowling and degenerate r-Dowling polynomials
Published in Applied Mathematics in Science and Engineering, 2022
Taekyun Kim, Dae San Kim, Hye Kyung Kim
A finite lattice L is geometric if it is a finite semimodular lattice which is also atomic. Dowling constructed an important finite geometric lattice out of a finite set of n elements and a finite group G of order m, called Dowling lattice of rank n over a finite group of order m. If L is the Dowling lattice of rank n over a finite group G of order m, then the Whitney numbers of the first kind and the Whitney numbers of the second kind are respectively denoted by and . The Whitney numbers and satisfy the following Stirling number-like relations: For , Dowling polynomials are given by