scholarly journals Shi Threshold Arrangement

10.37236/2046 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Seunghyun Seo
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperplane Arrangement ◽  
Field Method ◽  
Finite Field Method

Richard Stanley suggested the problem of finding the number of regions and the characteristic polynomial of a certain hyperplane arrangement defined by $x_i + x_j=0,1$, which is called the Shi threshold arrangement. We present the answer of the problem, using the finite field method.

scholarly journals Bijections for Faces of the Shi and Catalan Arrangements

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Duncan Levear
Keyword(s):  
Finite Field ◽  
Hyperplane Arrangement ◽  
Field Method ◽  
Binary Trees ◽  
Formula One ◽  
Bijective Proof ◽  
Simple Set ◽  
Shi Arrangement ◽  
Counting Formula ◽  
Finite Field Method

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in ${R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.

scholarly journals On accurate high-order numerical derivatives computations for quantum chemistry purposes

2018 ◽  
Keyword(s):  
Quantum Chemistry ◽  
Finite Field ◽  
Numerical Differentiation ◽  
Field Method ◽  
High Order ◽  
Parameter Choice ◽  
Derivatives Of ◽  
Finite Field Method ◽  
Account Electron

Various molecular parameters in quantum chemistry could be computed as derivatives of energy over different arguments. Unfortunately, it is quite complicated to obtain analytical expression for characteristics that are of interest in the framework of methods that account electron correlation. Especially it relates to the coupled cluster (CC) theory. In such cases, numerical differentiation comes to rescue. This approach, like any other numerical method has empirical parameters and restrictions that require investigation. Current work is called to clarify the details of Finite-Field method usage for high-order derivatives calculation in CC approaches. General approach to the parameter choice and corresponding recommendations about numerical steadiness verification are proposed. As an example of Finite-Field approach implementation characterization of optical properties of fullerene passing process through the aperture of carbon nanotorus is given.

scholarly journals Simulating Electrochemical Systems by Combining the Finite Field Method with a Constant Potential Electrode

2019 ◽  
Vol 123 (19) ◽  
Author(s):  
Thomas Dufils ◽  
Guillaume Jeanmairet ◽  
Benjamin Rotenberg ◽  
Michiel Sprik ◽  
Mathieu Salanne
Keyword(s):  
Finite Field ◽  
Field Method ◽  
Electrochemical Systems ◽  
Potential Electrode ◽  
Constant Potential ◽  
Finite Field Method
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