Lattice Rules Research Articles

Weak decay form factors of heavy pseudoscalar mesons within a light-front constituent quark model

The transition form factors describing the semileptonic decays of heavy pseudoscalar mesons are investigated within a relativistic constituent quark model formulated on the light-front. For the first time the form factors are calculated in the whole accessible kinematical region, adopting meson wave functions derived from various effective q q interactions able to describe the meson mass spectra. It is shown that the decay rates of the B → Dℓ ν ℓ, D → Kℓ ν ℓ and D → πℓ ν ℓ weak decays are mainly governed by the effects of the confinement scale, whereas the B → πℓ ν ℓ decay rate is sensitive to the high-momentum components generated in the meson wave functions by the effective one-gluon-exchange interaction. Our results are consistent with available experimental data and predictions of lattice QCD calculations and QCD sum rules.

Triangular canonical forms for lattice rules of prime-power order

In this paper we develop a theory of t t -cycle D − Z D-Z representations for s s -dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a D D -matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their Z Z -matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. Our main result is to define a unique canonical form for prime-power rules. This ultratriangular form is a triangular form, is easy to recognize, and may be derived in a straightforward manner.

Analysis of volume effects of phase transformation in titanium alloys

Some results of studies have been presented, concerning possibility of computer technology application in the prediction of transformation temperatures, lattice parameters and volume effects of transformations in two-phase titanium alloy heated at various rates. To do so, a methodology has been introduced, based upon a mathematical description of a physical model of a diffusional phase transformation. To estimate the changes of lattice parameters Vegard's rule have been applied that allows the phase lattice parameters to be determined out of its chemical composition. Furthermore, it has been assumed that the external volume effects of transformations are the resultant of changes of the alloy phase composition, the chemical composition of the component phases and their specific volummes. The calculated results have been experimentally verified in high temperature x-ray diffraction studies, dilatometric tests and microanalytical observations. Good agreement of the calculations with the experimental data has been proved.

Constraining spectral functions at finite temperature and chemical potential with exact sum rules in asymptotically free theories.

Within the framework of the operator product expansion and the renormalization group equation, we show that the temperature and chemical potential dependence of the zeroth moment of a spectral function (SF) is completely determined by the one-loop structure in an asymptotically free theory, and in particular in QCD. Logarithmic corrections are found to play an essential role in the derivation. This exact result constrains the shape of SF's, and implies striking effects near phase transitions. Phenomenological parametrizations of the SF, often used in applications such as the analysis of lattice QCD data or QCD sum rule calculations at finite temperature and baryon density, must satisfy these constraints. We also explicitly illustrate in detail the exact sum rule in the Gross-Neveu model.

The determination of canonical forms for lattice quadrature rules

Lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s-dimensional unit hypercube U s = [0, 1) s . For a given lattice rule, say Q L , a set of points L (the integration lattice), regularly spaced in all of R s , is generated by a finite number of rational vectors. The abscissa set for Q L is then P( Q L )= L ∩ U s . It is known that P( Q L ) is a finite Abelian group under addition modulo the integer lattice Z s , and that Q L ( f) may be written in the form of a nonrepetitive multiple sum, Q L(f)= 1 n 1⋯n m ∑ j 1=1 n 1 ⋯ ∑ j m=1 n m f j 1 n 1 z 1+⋯+ j m n m z m , known as a canonical form, in which + denotes addition modulo Z s . In this form, z i ∈ Z s , m is called the rank and n 1, n 2,…, n m are called the invariants of Q L , and n i+1¦n i for i = 1,2,…, m − 1. The rank and invariants are uniquely determined for a given lattice rule. In this paper we provide a construction of a canonical form for a lattice rule Q L , given a generator set for the lattice L. We then show how the rank and invariants of Q L may be determined directly from the generators of the dual lattice L ⊥.

Exact sum rules at finite temperature and chemical potential and their application to QCD

Within the framework of the operator product expansion (OPE) and the renormalization group equation (RGE), we show that the temperature and chemical potential dependence of the zeroth moment of a spectral function (SF) is completely determined by the one-loop structure of an asymptotically free theory. This exact result constrains the shape of SF's, and implies a highly non-trivial functional form for the SF near second order, or weak first order, phase transitions. Phenomenological parameterizations of the SF, often used in applications such as the analysis of lattice QCD data or QCD sum rule calculations at finite temperature and baryon density, must satisfy these constraints.

Computational aspects of Boolean cubature

Boolean methods of interpolation [1,4] have been applied to construct multivariate quadrature rules for periodic functions of Korobov classes which are comparable with lattice rules of numerical integration [6,7]. In particular, we introducedd-variate Boolean trapezoidal rules [3,4] andd-variate Boolean midpoint rules [2,4]. The basic tools for constructing Boolean midpoint rules are Boolean midpoint sums. It is the purpose of this paper to use a modification of these Boolean midpoint sums to compute Boolean trapezoidal rules in an efficient way.

Application of the Smith Normal Form to the Structure of Lattice Rules

Two independent approaches to the theory of the lattice rule have been exploited at length in the literature. One is based on the generator matrix A of the lattice $\Lambda $ whose elements provide the abscissas of Q. The other, based on the t-cycle form $Q(\Lambda )f$ of Sloan and Lyness [Math. Comput., 52 (1989), pp. 81–94], leads to a canonical form for Q. In this paper, a close connection between these approaches is demonstrated. This connection reflects the close relation between the Kronecker decomposition theorem for Abelian groups and the Smith normal form of an integer matrix. It is shown that the invariants of the canonical form of $Q(\Lambda )f$ coincide with the elements of the Smith normal form of $B = (A^T )^{ - 1} $, the reciprocal lattice generator matrix. This fact may be used to provide a straightforward solution to the previously intransigent problem of identifying and removing a repetition in the general t-cycle form.

Modeling water infiltration in unsaturated porous media by interacting lattice gas‐cellular automata

A two‐dimensional lattice gas‐cellular automaton fluid model with long‐range interactions (Appert and Zaleski, 1990) is used to simulate saturated and unsaturated water infiltration in porous media. Water and gas within the porous medium are simulated by applying the dense and the light phase, respectively, of the cellular automaton fluid. Various wetting properties can be modeled when adjusting the corresponding solid‐liquid interactions. The lattice gas rules include a gravity force step to allow buoyancy‐driven flow. The model handles with ease complex geometries of the solid, and an algorithm for generating random porous media is presented. The results of four types of simulation experiments are presented: (1) We verified Poiseuille's law for steady and saturated flow between two parallel plates. (2) We analyzed transient water infiltration between two parallel plates of varying degrees of saturation and various apertures. (3) Philip's infiltration equation was adequately simulated in an unsaturated porous medium. (4) Infiltration into an aggregated medium containing one vertical parallel crack was simulated. Further applications of this lattice gas method for studying unsaturated flow in porous media are discussed.

Ab initio lattice dynamics and charge fluctuations in alkaline-earth oxides.

We present an ab initio calculation of lattice-dynamical properties of the alkaline-earth oxides MgO, CaO, and SrO in the rocksalt structure. We employ the density-functional perturbation theory within the local-density approximation. We show phonon dispersion curves along several symmetry directions in the Brillouin zone. The calculated frequencies agree well with the available experimental data. The investigation of the phonon-induced charge-density fluctuations of MgO and CaO at the L point of the Brillouin zone partially supports the breathing-shell model of lattice dynamics and rules out the charge-transfer model for this class of materials. Moreover, our calculations show that the breathinglike charge-density response is more pronounced for oxygen than for the cations in these compounds.

Integration of nonperiodic functions of two variables by Fibonacci lattice rules

Two-dimensional lattice rules are applied to continuous functions over the unit square which do not have a continuous periodic extension. It is shown that, provided lattice points at vertices and edges are treated appropriately, certain functions (including all bilinear functions) are integrated exactly whenever the lattice contains a (possibly rotated) square unit cell. The Fibonacci lattice with denominators F k for the nodes is then shown to have a square unit cell if and only if k is odd. Numerical experiments for Fibonacci rules and copies of Fibonacci rules confirm that there are significant different between the odd- k and even- k cases.

Medians for weight metrics in the covering graphs of semilattices

We consider the undirected covering graph G of a finite (meet) semilattice X endowed with a lower valuation. More precisely, our main concerns are the lower valuations associated to a weighting of the join-irreducible elements of X and the corresponding minimum path length metrics in G, which are frequently considered in the literature. Some results on the medians for such metrics are obtained, in relation with a lattice majority rule. Especially, these medians are characterized in the case where X is distributive. The unanimity, or Pareto, property is also investigated for such medians.

Lattice Rules by Component Scaling

We introduce a theory of rectangular scaling of integer lattices. This may be used to construct families of lattices. We determine the relation between the Zaremba index $\rho (\Lambda )$ of various members of the same family. It appears that if one member of a family has a high index, some of the other family members of higher order may have extraordinarily high indices. We have applied a technique based on this theory to lists of good lattices available to us. This has enabled us to construct lists of excellent previously unknown lattices of high order in three and four dimensions and of moderate order in five dimensions.

Random Number Generation and Quasi-Monte Carlo Methods.

Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.

Intermediate Rank Lattice Rules for Multidimensional Integration

Every lattice rule in s dimensions may be characterised by an integer m, lying between 1 and s inclusive, called the rank or index of the rule and a set of m positive integers called the invariants. Earlier work has shown that, in a certain precise sense, lattice rules of rank s are better than the commonly used rank-1 rules. Here this earlier work is extended by showing that a similar result holds for certain lattice rules of intermediate rank $m < s$.

Improved Error Bounds for Lattice Rules

We establish results on the worst-case errors that can be achieved by well-chosen lattice rules for standard classes of multivariate periodic functions. These theorems improve or generalize earlier results of this type.

Bounds on the lattice rule criterion 𝑅

Lattice rules are used for the numerical integration of periodic functions over the s-dimensional unit cube. They are normally classified according to their ’rank’; in simple terms, the rank of a lattice rule is the minimum number of sums required to write it down. One criterion for measuring the ’goodness’ of a lattice rule is the quantity R which is the quadrature error for a certain test function. Bounds on R exist for rank-1 and rank-2 lattice rules, but not for lattice rules of higher rank. For 1 ≤ m ≤ s 1 \leq m \leq s , we shall look at certain rank-m rules and obtain bounds on R for them. These rank-m rules have n m r {n^m}r quadrature points, where n and r are relatively prime numbers. In order to obtain these bounds, we make use of a result which shows that R may be considered to be the quadrature error obtained when a modified lattice rule with only r quadrature points is applied to a modified test function. Some numerical results are given.

Lattice rules by component scaling

We introduce a theory of rectangular scaling of integer lattices. This may be used to construct families of lattices. We determine the relation between the Zaremba index ρ ( Λ ) \rho (\Lambda ) of various members of the same family. It appears that if one member of a family has a high index, some of the other family members of higher order may have extraordinarily high indices. We have applied a technique based on this theory to lists of good lattices available to us. This has enabled us to construct lists of excellent previously unknown lattices of high order in three and four dimensions and of moderate order in five dimensions.

The living in a lattice rule language

This paper presents a rule-based query language for an object-oriented database model. The database model supports complex objects, object identity, classes and types, and a class/type hierarchy. The instances are described by ‘object relations’ which are functions from a set of objects to value sets and other object sets. The rule language is based on object-terms which provide access to objects via the class hierarchy. Rules are divided into two classes: object-preserving rules manipulating existing objects (yielding a new ‘view’ on objects available in the object base) and object-generating rules creating new objects with properties derived from existing objects. The derived object sets are included in a class lattice. We give conditions for whether the instances of the ‘rules’ heads are ‘consistent’, i.e. represent object relations where the properties of the derived objects are functionally determined by the objects.

The number of lattice rules having given invariants

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.