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Register a new userApplication of arrangement theory to unfolding models
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Arrangement theory plays an essential role in the study of the unfolding model used in many fields. This paper describes how arrangement theory can be usefully employed in solving the problems of counting (i) the number of admissible rankings in an unfolding model and (ii) the number of ranking patterns generated by unfolding models. The paper is mostly expository but also contains some new results such as simple upper and lower bounds for the number of ranking patterns in the unidimensional case.
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We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all braid slices, including those not associated with unfolding models, are in one-to-one correspondence with the chambers of an arrangement. By identifying those which are associated with unfolding models, we find the number of ranking patterns. We also give an upper bound for the number of ranking patterns when the difference by a permutation of objects is ignored.
In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type $D_ell$ explicitly.
In this paper we consider the hyperplane arrangement in $mathbb{R}^n$ whose hyperplanes are ${x_i + x_j = 1mid 1leq i < jleq n}cup {x_i=0,1mid 1leq ileq n}$. We call it the emph{boxed threshold arrangement} since we show that the bounded regions of this arrangement are contained in an $n$-cube and are in one-to-one correspondence with the labeled threshold graphs on $n$ vertices. The problem of counting regions of this arrangement was studied earlier by Joungmin Song. He determined the characteristic polynomial of this arrangement by relating its coefficients to the count of certain graphs. Here, we provide bijective arguments to determine the number of regions. In particular, we construct certain signed partitions of the set ${-n,dots, n}setminus{0}$ and also construct colored threshold graphs on $n$ vertices and show that both these objects are in bijection with the regions of the boxed threshold arrangement. We independently count these objects and provide closed form formula for the number of regions.
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $sigma$ such that every set of $t-1$ consecutive elements in $sigma$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $sigma$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation $Phi$ of $sigma$ is a flower. If $t$ is odd, then $Phi$ is a daisy, but if $t$ is even, then, depending on $M$, it is possible for $Phi$ to be either an anemone or a daisy.
We propose a novel periodicity-free unfolding method of the electronic energy spectra. Our new method solves a serious problem that calculated electronic band structure strongly depends on the choice of the simulation cell, i.e., primitive-cell or supercell. The present method projects the electronic states onto the free-electron states, giving rise to the {it plane-wave unfolded} spectra. Using the method, the energy spectra can be calculated as a completely independent quantity from the choice of the simulation cell. We have examined the unfolded energy spectra in detail for three models and clarified the validity of our method: One-dimensional interacting two chain model, monolayer graphene, and twisted bilayer graphene. Furthermore, we have discussed that our present method is directly related to the experimental ARPES (Angle-Resolved Photo-Emission Spectroscopy) spectra.
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