scholarly journals A Closer Look at Lattice Points in Rational Simplices

10.37236/1469 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Matthias Beck

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.

Author(s):  
Richard J. McIntosh

AbstractLet ((x)) =x−⌊x⌋−1/2 be the swatooth function. Ifa, b, cand e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.


Author(s):  
Theodoros Assiotis ◽  
Jonathan P. Keating

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.


Author(s):  
Ansgar Freyer ◽  
Martin Henk

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.


2012 ◽  
Vol 52 (28) ◽  
pp. 333-340 ◽  
Author(s):  
Vagner Cavarzere ◽  
Thiago Vernaschi Vieira da Costa ◽  
Luís Fábio Silveira

Due to rapid and continuous deforestation, recent bird surveys in the Atlantic Forest are following rapid assessment programs to accumulate significant amounts of data during short periods of time. During this study, two surveying methods were used to evaluate which technique rapidly accumulated most species (> 90% of the estimated empirical value) at lowland Atlantic Forests in the state of São Paulo, southeastern Brazil. Birds were counted during the 2008-2010 breeding seasons using 10-minute point counts and 10-species lists. Overall, point counting detected as many species as lists (79 vs. 83, respectively), and 88 points (14.7 h) detected 90% of the estimated species richness. Forty-one lists were insufficient to detect 90% of all species. However, lists accumulated species faster in a shorter time period, probably due to the nature of the point count method in which species detected while moving between points are not considered. Rapid assessment programs in these forests will rapidly detect more species using 10-species lists. Both methods shared 63% of all forest species, but this may be due to spatial and temporal mismatch between samplings of each method.


1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously


2011 ◽  
Vol 2 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Roger D. Applegate ◽  
Robert E. Kissell ◽  
E. Daniel Moss ◽  
Edward L. Warr ◽  
Michael L. Kennedy

Abstract Point count data are used increasingly to provide density estimates of bird species. A favored approach to analyze point count data uses distance sampling theory where model selection and model fit are important considerations. We used uniform and half normal models and assessed model fit using χ2 analysis. We were unsuccessful in fitting models to 635 northern bobwhite Colinus virginianus observations from 85 avian point locations spanning 6 y (P ≤ 0.05). Most observations (74%) occurred in the outermost (>100-m) distance radius. Our results violated the assumptions that all observations at the point are detected. The assumption that birds were assigned to the correct distance interval also was probably violated. We caution managers in implementing avian point counts with distance sampling when estimating northern bobwhite population density. We recommend exploring other approaches such as occupancy-estimation and modeling for estimating detection probabilities.


2020 ◽  
Vol 24 (6) ◽  
pp. 1031-1043
Author(s):  
Darin J. McNeil ◽  
Christina M. Grozinger

Abstract As evidence for global insect population declines continues to amass, several studies have indicated that Orthoptera (grasshoppers, crickets, and katydids) are among the most threatened insect groups. Understanding Orthoptera populations across large spatial extents requires efficient survey protocols, however, many previously established methods are expensive and/or labor-intensive. One survey method widely employed in wildlife biology, the aural point count, may work well for crickets and katydids (suborder: Ensifera) because males produce conspicuous, species-specific mating calls. We conducted repeated point count surveys across an urban-to-rural gradient in central Pennsylvania. Occupancy analyses of ten focal species indicated that, although detection probability rates varied by species from 0.43 to 0.98, detection rates compounded over five visits such that all focal species achieved cumulative > 0.90. Factors associated with site occupancy varied among species with some positively associated with urbanization (e.g., Greater Anglewing, Microcentrum rhombifolium), some negatively associated with urbanization (e.g., Sword-bearing Conehead, Neoconocephalus ensiger), and others exhibiting constant occupancy across a habitat gradient (e.g., Common True Katydid, Pterophylla camellifolia). Our community-level analysis revealed that different species’ habitat associations interacted such that intermediate levels of urbanization (i.e., suburbs) hosted the highest number of species. Implications for insect conservation Ultimately, our analyses clearly support the concept that aural point counts paired with static occupancy modeling can serve as an important tool for monitoring night-singing Orthoptera populations. Applications of point count surveys by both researchers and citizen scientists may improve our understanding Ensifera populations and help in the global conservation of these threatened insects.


Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.


2011 ◽  
Vol 172-174 ◽  
pp. 1119-1127
Author(s):  
Tetsuo Mohri

Continuous Displacement Cluster Variation Method is employed to study binary phase equilibria on the two dimensional square lattice with Lennard-Jones type pair potentials. It is confirmed that the transition temperature decreases significantly as compared with the one obtained by conventional Cluster Variation Method. This is ascribed to the distribution of atomic pairs in a wide range of atomic distance, which enables the system to attain the lower free energy. The spatial distribution of atomic species around a Bravais lattice point is visualized. Although the average position of an atom is centred at the Bravais lattice point, the maximum pair probability is not necessarily attained for the pairs located at the neighboring Bravais lattice points. In addition to the real space information, k-space information are calculated in the present study. Among them, the diffuse intensity spectra due to short range ordering and atomic displacement are discussed.


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