Lattice Points

Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo’s pseudorandom numbers. Clearly, the generation of appropriate high-quality quasirandom sequences is crucial to the success of quasi-Monte Carlo methods. The Halton sequence is one of the standard (along with (t,s)(t,s)-sequences and lattice points) low-discrepancy sequences, and one of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this phenomenon is to use a randomized (scrambled) version of the Halton sequence. An alternative approach to this is to find an optimal Halton sequence within a family of scrambled sequences. This paper presents a new algorithm for finding an optimal Halton sequence within a linear scrambling space. This optimal sequence is numerically tested and shown empirically to be far superior to the original. In addition, based on analysis and insight into the correlations between dimensions of the Halton sequence, we illustrate why our algorithm is efficient for breaking these correlations. An overview of various algorithms for constructing various optimal Halton sequences is also given.

We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of -trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical ba...

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditio...

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.

Two new methods for computing with hypergeometric distributions on lattice points are presented. One uses Fourier analysis, and the other uses Gröbner bases in the Weyl algebra. Both are very general and apply to log-linear models that are graphical or non-graphical.

El máximo común divisor entre un número primo p y cada uno de los enteros positivos menores que p es igual a 1 y, como el máximo común divisor se relaciona con la función
parte entera según una fórmula explícita dada por el matemático brasileño M. Polezzi (1997), entonces se halló una interesante proposición que relaciona los números primos, la
función parte entera, los números cuadrados y los números triangulares. Esa proposición sirve como un nuevo test para probar la primalidad de un número.

In Multiple-Input Multiple-Output (MIMO) systems, Maximum-Likelihood (ML) decoding is equivalent to £nding the closest lattice point in an N dimensional complex space. In (1), we have proposed several quasi- maximum likelihood relaxation models for decoding in MIMO systems based on semi-de£nite programming. In this paper, we propose randomization algorithms that £nd a near-optimum solution of the decoding problem by exploring the solution of the corresponding semi-de£nite relaxations.

If $P\subset \R^d$ is a rational polytope, then $i_P(n):=#(nP\cap \Z^d)$ is a quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period of $i_P(n)$ must divide $\LL(P)= \min \{n \in \Z_{> 0} \colon nP \text{is an integral polytope}\}$. Few examples are known where the period is not exactly $\LL(P)$. We show that for any $\LL$, there is a 2-dimensional triangle $P$ such that $\LL(P)=\LL$ but such that the period of $i_P(n)$ is 1, that is, $i_P(n)$ is a polynomial in $n$. We also characterize all polygons $P$ such that $i_P(n)$ is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a rank 1 quadratic form being an extreme ray of the corresponding L-type domain. An n-dimensional lattice determines two normal partitions of the n space Rn into polytopes. These are the Voronoi partition and the Delaunay parti- tion. These partitions are dual, i.e. a k-dimensional face of one partition is orthogonal to an (n k)-dimensional face of the other partition. Besides, a vertex of one partition is the center of a polytope of the other partition. The Voronoi partition consists of Voronoi polytopes with its centers in lattice points. Moreover, any polytope of the Voronoi partition is obtained by a translation of the Voronoi polytope with the center in the origin (=the zero lattice point). Call this polytope the Voronoi polytope. It consists of those points of Rn that are at least as closed to 0 as to any other lattice point. The Delaunay partition consists of Delaunay polytopes which are, in gen- eral, not congruent. The set of all Delaunay polytopes having 0 as a vertex is called the star of Delaunay polytopes. Each Delaunay polytope is the con- vex hull of all lattice points lying on an empty sphere. This sphere is called empty, since no lattice point is an interior point of the sphere. The Voronoi polytope and the Delaunay polytopes of the star are tightly related to minimal vectors of cosets 2L in L. A coset Q is called simple if it contains, up to sign, only one minimal vector.

A description of crystal lattices in terms of automata is presented. The words of a language represented by an automata are mapped to points in R 2 and R 3 defining lattice points and their connections. These automata descriptions of crystal lattices reveal subtle properties that are difficult to see in other descriptions. A few applications of these automata are

Let X be a compact region of area n in the plane. We show that if X is a strictly convex region, or a region bounded by an irreducible algebraic curve, then X can be translated to a position where it covers exactly n lattice points. If X is a polygon, or a convex region, then it can be rotated and translated so that it covers exactly n lattice points.

A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1,1) and (1, −1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0,0) and finish at (2n,0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p1 and p2, for the number of Dyck paths starting at p1, ending at p2, and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.

In this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.