Related papers: Estimates on Lattice Points in the Circle
We have found analytical expressions (polynomials) of the percolation probability for site percolation on a square lattice of size $L \times L$ sites when considering a plane (the crossing probability in a given direction), a cylinder…
By median we mean a scheme that inputs three element of a lattice, and outputs an element that is an average of the three inputs in a certain sense. The medians of a given finite lattice form a new lattice that is usually larger than the…
We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the…
Subject of this letter is the dynamics of a chain obtained performing the continuous limit of a system of links and beads. In particular, the probability distribution of the relative position between two points of the chain averaged over a…
A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks…
A 1/L-expansion for percolation problems is proposed, where L is the lattice finite length. The square lattice with 27 different sizes L = 18, 22 ... 1594 is considered. Certain spanning probabilities were determined by Monte Carlo…
For $n\geq 3$ and $\Gamma$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact…
Lattice QCD should allow a derivation of the $\Delta I=1/2$ rule from first principles, but numerical calculations to date have been plagued by a variety of problems. After a brief review of these problems, we present several new methods…
In this paper, we give bounds on the variance of the number of points of the circular and the Gaussian $\beta$ ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized…
Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of…
We present a novel technique for estimating disk parameters (the centre and the radius) from its 2D image. It is based on the maximal likelihood approach utilising both edge pixels coordinates and the image intensity gradients. We emphasise…
On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of…
New condition is found for the set of points in the plane, for which the locus is a circle. It is proved: the locus of points, such that the sum of the $(2m)$-th powers $S_n^{(2m)}$}of the distances to the vertexes of fixed regular…
Let $\Lambda\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the…
The problem of finding the number of lattice points in a triangle has a classical solution if the lattice is $\mathbf{Z}^2$ and the vertices of the triangle have integer valued coordinates. We consider what happens when we replace the…
We present an $O(n^2\log^4 n)$-time algorithm for computing the center region of a set of $n$ points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes…
In our earlier work, the charge and matter radial distributions of heavy-light mesons were measured on a 16^3 times 24 lattice with a lattice spacing of a = 0.17 fm and a light quark mass about that of the strange quark. Several major…
We present an approximation algorithm for computing the diameter of a point-set in $\Re^d$. The new algorithm is sensitive to the ``hardness'' of computing the diameter of the given input, and for most inputs it is able to compute the exact…
A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent…
A general relation is derived for the action difference between two fixed points and a phase space area bounded by the irreducible component of a heteroclinic tangle. The determination of this area can require accurate calculation of…