A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

For positive constants a > b > 0, let P _T (t) denote the lattice point discrepancy of the body tT _a,b , where t is a large real parameter and T = T _a,b is bounded by the surface $$ \partial \tau _{a,b} :\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {(a + b\cos \alpha )\cos \beta } \\ {(a + b\cos \alpha )\sin \beta } \\ {b\sin \alpha } \\ \end{array} } \right), 0 \leqq \alpha ,\beta < 2\pi . $$ In a previous paper [12] it has been proved that $$ P_\tau (t) = \mathcal{F}_{a,b} (t)t^{3/2} + \Delta _\tau (t), $$ where F _a,b (t) is an explicit continuous periodic function, and the remainder satisfies the (“pointwise”) estimate Δ _T (t) ? t ^11/8+? . Here it will be shown that this error term is only ? t ^1+? in mean-square, i.e., that $$ \int\limits_0^T {(\Delta _\tau (t))^2 dt} \ll T^{3 + \varepsilon } $$ for any ? > 0.     

The lattice point discrepancy of a body of revolution: Improving the lower bound by Soundararajan’s method

For a convex body in which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus volume) of a linearly dilated copy is estimated from below. On the basis of a recent method of K. Soundararajan [16], an -bound is obtained that improves upon all earlier results of this kind.Dedicated to the memory of Professor Erich LamprechtThis revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004.     

The lattice discrepancy of bodies bounded by a rotating Lamé’s curv

This article provides an asymptotic result for the lattice point discrepancy of the special three-dimensional body for fixed k > 2 and large t. Authors’ addresses: Ekkehard Kr?tzel, Faculty of Mathematics, University of Vienna, Nordbergstra?e 15, 1090 Wien, ?sterreich; Werner Georg Nowak, Department of Integrative Biology, Institute of Mathematics, Universit?t für Bodenkultur Wien, Gregor Mendel-Stra?e 33, 1180 Wien, ?sterreich     

Description of a Class of Solids in ℝ3

Let be a compact convex body such that for each parallel projection onto any plane no two opposite faces of Q are projected strictly inside the projection of the entire Q. Then Q is either a cone, or a frustum of a trihedral pyramid, or a prism (possibly with nonparallel bases). Bibliography: 1 title.     

Minimal cubature formulas of degree 3 for a torus in ℝ<Superscript>3</Superscript>

We construct minimal cubature formulas of degree 3 for a torus in R³. The cases of a degenerate torus with radius r = 1 and a torus with arbitrary radius r > 1 are considered separately.     

The limit set of trajectory in quasi-homogeneous system in ℝ3

In this article, we study the limit set of trajectory in three-dimensional quasi-homogeneous system. For the trajectory which is close to a singular curve, we show that either it approaches a fixed point or infinity, or it oscillates. Moreover, an oscillating example is given. The behaviour of trajectory which is near a closed orbit is also studied. At the end, we classify the integral manifolds of the system with isolated singularities.     

Qualitative analysis of a singularly perturbed system in ℝ3

We present some qualitative analysis of a singularly perturbed system of ordinary differential equations with two slow variables and one fast variable. The study rests on the method of integral manifolds and its modification in connection with applied problems. The inspection of the system requires studying various types of oscillations. We propose some sufficient conditions for the existence of relaxation oscillations in this system in the case that the slow surface has two folds.     

Clifton-pohl torus and geodesic completeness by a ‘complex’ point of view

We show that a natural complexification and a mild generalization of the idea of completeness guarantee geodesic completeness of Clifton-Pohl torus; we explicitely compute all of its geodesics.     

Observers and a moving object in ℝ3

Doklady Mathematics - Suppose that an object t moves within a given corridor Y in the presence of a groups S of hostile observers S ∉ Y, each having a fixed visibility cone K(S). The problem...     

The existence therems for a class of sublinear elliptic equations in ℝ N

An existence theorem for the solution to the equation

Time Evolution of a Vlasov-Poisson Plasma with Infinite Charge in ℝ3

We study existence and uniqueness of the solution to the Vlasov-Poisson system describing a one-species plasma evolving in ?³, whose particles interact via the Coulomb potential. It is assumed that initially the particles have bounded velocities and are distributed according to a non integrable density.     

The Geometry of Polygons in ℝ5 and Quaternions

We consider the moduli space _r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between _r and a weighted quotient of n-fold products of the quaternionic projective line ¹ by the diagonal PSL(2, )-action. We explore the relation between _r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr_ℂ(2, 4) ⁿ /SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space _r as a quotient of a subspace in the quaternionic Grassmannian Gr_ℍ(2, n) by the action of the group Sp(1) ⁿ . We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on _r .     

The global dynamics of a class of vector fields in ℝ<Superscript>3</Superscript>

In this paper, we find a bridge connecting a class of vector fields in ℝ³ with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ³. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Analogues of the central point theorem for families with d-intersection property in ℝ d

In this paper we consider families of compact convex sets in ? ^d such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.     

The Classification of Equiaffine Indefinite Flat Homogeneous Surfaces in ℝ4

In this paper a complete classification of equiaffine homogeneous surfaces in ⁴ with indefinite flat affine metric is given.