On the lattice discrepancy of bodies of rotation with boundary points of curvature zero

This article gives an asymptotic result for the lattice point discrepancy of a large body of rotation in , whose boundary is piecewise smooth and contains points of vanishing Gaussian curvature. Received: 2 November 2006     

The three-dimensional motion of optimalpyramidal bodies

The three-dimensional inertial motion of pyramidal bodies, optimal in their depth of penetration, formed from parts of planes tangential to a circular cone and having a base in the form of a rhombus or a star, consisting of four symmetrical cycles, is investigated using the numerical solution of the Cauchy problem of the complete system of equations of motion of a body. It is assumed that the force action of the medium on the body can be described within the framework of a local model, when the pressure on the body surface can be represented by a two-term formula, quadratic in the velocity, and the friction is constant. It is shown that the stability criterion, obtained previously for the rectilinear motion of a pyramidal body on the assumption that the perturbed motion of the body is planar, also enables one, in the case of an arbitrary specification of the small perturbations of the parameters leading to the tree-dimensional motion of the body, to determine the nature of development of these perturbations. It is shown that if the rectilinear motion of the body is stable, its perturbed three-dimensional motion can be represented in the form of the superposition of plane motions, and when investigating each of them, the analytical solution of the plane problem obtained earlier can be used.     

The construction of extensible polynomial lattice rules with small weighted star discrepancy

In this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance. The construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest. We also show that the construction can be done component by component.

     

Good lattice points,discrepancy, and numerical integration

Summary Methods based on Diophantine approximations lead to a simple and easy construction of rational vectors the multiple of which, reduced modulo 1, form finite sequences of points with certain properties of equipartition over the unit square. Given a function of bounded variation over this square, it is suggested that computing the average of its values at the points of such a sequence can be a practical method of numerical integration. Precise bounds for the error ore obtained. In the general case, these are of the order of the product of the reciprocal and of the logarithm of the number of points; in the case of a function satisfying stated conditions of regularity and periodicity, they are of the order of the logarithm of the number of points divided by an appropriately high power of this number. A slight sharpening of some well-known results on equipartiton is obtained incidentally. A part of this paper was read at a meeting of the Accademia Nazionale di Scienze Lettere e Arti in Modena on April 7th. 1965.     

The determinants of certain matrices arising from the Boolean lattice

Let B be the Boolean lattice on an n-set with B=?B_i the rank decomposition. Let M(n,i) be the incidence matrix between B_i and B_n−i. We obtain a recursive formula for the determinant of the matrix M(n,i).     

On the mean lattice point discrepancy of a convex disc

For a convex planar domain D \cal {D} , with smooth boundary of finite nonzero curvature, we consider the number of lattice points in the linearly dilated domain t D t \cal {D} . In particular the lattice point discrepancy P_D(t) P_{\cal {D}}(t) (number of lattice points minus area), is investigated in mean-square over short intervals. We establish an asymptotic formula for¶¶ ò_{T - L}^{T + L}(P_D(t))²dt \int\limits_{T - \Lambda}^{T + \Lambda}(P_{\cal {D}}(t))^2\textrm{d}t ,¶¶ for any L = L(T) \Lambda = \Lambda(T) growing faster than logT.     

Good lattice rules based on the general weighted star discrepancy

We study the problem of constructing rank- lattice rules which have good bounds on the ``weighted star discrepancy'. Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may be obtained using a component-by-component (CBC) construction of the generating vector. Under appropriate conditions on the weights, these lattice rules satisfy strong tractability bounds on the weighted star discrepancy. Particular classes of weights known as ``order-dependent' and ``finite-order' weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights.

     

On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram determinants Δ_L(X,Y) as defined in (Adv. Math. 142 (1999) 244) and to extend results of Aval et al. (J. Combin. Theory Ser. A, to appear). We recall that M_L denotes the space of all partial derivatives of Δ_L. In this paper, we want to study the space M_i,j^k(X,Y) which is defined as the sum of M_L spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the “shadow” of a given cell (i,j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M_i,j^k(X,Y), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the “shift” operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M_i,j^k(X) consisting of elements of 0 Y-degree.     

The topological structure of maximal lattice free convex bodies: The general case

Given a genericm × n matrixA, the simplicial complexK(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax b}. The main result of this paper is that the topological space associated withK(A) is homeomorphic withR ^m–1 . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by NSF grant SES-9121936 and the program in Discrete Mathematics at Yale University.Partially supported by the Hungarian NSF grant 1909 and the program in Discrete Mathematics at Yale University.     

Join epimorphisms which preserve certain lattice identities

Presented by H. Peter Gumm.     

Constructions of general polynomial lattice rules based on the weighted star discrepancy

In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration.Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy.In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy.     

Parametric controllability of certain systems of rigid bodies

The concept of parametric controllability as applied to systems of rigid bodies is discussed. The topic of discussion is Langrangian systems for which “unfreezing” of the parameters is possible such as, for example, the refinement of a model by taking account of the small variability of the links assumed by rigid bodies in the first approximation. As has been shown, just taking account of a small change in the parameters can ensure the controllability of a mechanism which was not controllable assuming rigidity absolute of the links. Certain sufficient conditions are proposed for parametric controllability in invariant manifolds for objects with cyclic coordinates. A two-link pendulum in the horizontal plane under the action of an internal moment (from the first link to the second) is considered as an example. The effect of its mass inertial parameters on the controllability is investigated. The parametric controllability of such an object in a manifold of zero angular momentum, due to the elastic pliability of the second link or the oscillations of an additional mass on a spring along the second link, is demonstrated. An example of a parametrically controllable planetary mechanism with slippage of discs is also considered.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .