scholarly journals Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

2007 ◽  
Vol 114 (1) ◽  
pp. 97-109 ◽  
Author(s):  
David Forge ◽  
Thomas Zaslavsky
Keyword(s):  
Lattice Point ◽  
Hyperplane Arrangements ◽  
Point Counts ◽  
Shi Arrangement

scholarly journals A Closer Look at Lattice Points in Rational Simplices

10.37236/1469 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Matthias Beck
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Lattice Points ◽  
Point Count ◽  
Point Counts ◽  
Reciprocity Law ◽  
Dilation Factor

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an $n$-dimensional polytope with $n+1$ rational vertices, we use its description as the intersection of $n+1$ halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.

scholarly journals Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

2020 ◽  
pp. 2150019
Author(s):  
Theodoros Assiotis ◽  
Jonathan P. Keating
Keyword(s):  
Explicit Expression ◽  
Lattice Point ◽  
Characteristic Polynomials ◽  
Leading Order ◽  
Unitary Matrices ◽  
Point Count ◽  
Point Counts ◽  
Order Coefficient ◽  
The Matrix ◽  
Random Unitary Matrices

In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand–Tsetlin patterns with constraints.

scholarly journals A lattice point counting generalisation of the Tutte polynomial

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Amanda Cameron ◽  
Alex Fink
Keyword(s):  
Lattice Point ◽  
Tutte Polynomial ◽  
Minkowski Sum ◽  
Point Counting ◽  
Combinatorial Interpretation ◽  
Point Counts ◽  
Standard Simplex ◽  
International Audience

International audience The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

A New Condition of Formation and Stablity of All Crystalline Systems in a Good Agreement with Experimental Data

2015 ◽  
Vol 11 (3) ◽  
pp. 3224-3228
Author(s):  
Tarek El-Ashram
Keyword(s):  
Experimental Data ◽  
Brillouin Zone ◽  
Electron Concentration ◽  
Free Electron ◽  
Lattice Point ◽  
Valence Electron ◽  
Fermi Gas ◽  
Valence Electron Concentration ◽  
Fermi Sphere ◽  
Good Agreement

In this paper we derived a new condition of formation and stability of all crystalline systems and we checked its validity andit is found to be in a good agreement with experimental data. This condition is derived directly from the quantum conditionson the free electron Fermi gas inside the crystal. The new condition relates both the volume of Fermi sphere VF andvolume of Brillouin zone VB by the valence electron concentration VEC as ;𝑽𝑭𝑽𝑩= 𝒏𝑽𝑬𝑪𝟐for all crystalline systems (wheren is the number of atoms per lattice point).

Die Brutvogelfauna eines Nadelwaldes der nördlichen Voralpen nach dem Sturm Lothar

2017 ◽  
Vol 168 (2) ◽  
pp. 59-66
Author(s):  
Pierre Mollet ◽  
René Hardegger ◽  
Res Altwegg ◽  
Pius Korner ◽  
Simon Birrer
Keyword(s):  
Forest Succession ◽  
Coniferous Forest ◽  
Bird Species ◽  
Individual Species ◽  
Mountain Forest ◽  
Breeding Bird ◽  
Point Counts ◽  
Federal Institute ◽  
Central Switzerland ◽  
Bird Fauna

Breeding bird fauna in a coniferous forest in the northern Prealps after storm Lothar In a 70-hectare large coniferous forest located on the northern edge of the Alps in central Switzerland, Canton of Obwalden, at an altitude of 1260 to 1550 metres above sea level, we surveyed the local breeding bird fauna in 2002 and 2013 by means of point counts as well as additional area searches for rare species. In December 1999, hurricane Lothar caused two large windthrow areas and several smaller areas with scattered throws in the survey range. We found a total of 48 breeding bird species, which is a very diverse species composition for a mountain forest. In the eleven years between surveys, a decline in distribution or abundance was recorded for four species, while seven species showed an increase; a further four species showed no change. For the remaining species, the data sets were too small to reliably estimate changes. A comparison with forest structure data provided by the Swiss Federal Institute of Forest, Snow and Landscape Research WSL revealed that for five bird species, the changes in distribution or abundance could be explained at least partially by forest succession. In order to obtain realistic distribution and abundance values in this kind of breeding bird survey, it is essential to collect large enough samples and to consider the detection probability of each individual species using appropriate statistical methods.

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