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.     

The lattice point discrepancy of a torus in ℝ3

This article provides an asymptotic formula for the number of integer points in the three-dimensional body $$ \left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b, $$ for fixed a > b > 0 and large t.     

Asymptotic behavior of solutions to the compressible nematic liquid crystal system in ℝ3

In this paper, we study a nematic liquid crystals system in three-dimensional whole space ?³ and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L¹?³ additionally.     

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.     

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 .     

A trajectory in ℝ<Superscript>3</Superscript> concealed from observers

In the problem of tracking an object moving in ?³ by observers, the most concealed trajectory is characterized under the condition that the object is at any time visible to at most two observers.     

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.     

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.     

Discrete and Nondiscrete Isometric Deformations of Surfaces in ℝ3

We prove existence of closed infinitely differentiable surfaces M of each of which can be included in some family F of isometric pairwise noncongruent infinitely differentiable surfaces which is uniformly as close as we want to M. We also prove that F can be more than countable.     

On Borsuk's Problem in ℝ3

Mathematical Notes -     

Mixed finite elements in ℝ3

Summary We present here some new families of non conforming finite elements in ³. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.First, we introduce some notations K is a tetrahedron or a cube, thevolume of which is - K is its boundary - f is a face ofK, thesurface of which is - a is an edge, the length of which is - L ² (K) is the usual Hilbert space of square integrable functions defined onK - H ^m (K) {L ²(K); L ²(K); ||m}, where =(₁, ₂, ₃) is a multi-index; ||=₁+₂+₃ - curlu u, (defined by using the distributional derivative) foru=(u ₁,u ₂,u ₃);u _iL ² (K) - H(curl) {u(L ² (K))³; curlu(L ² (K)) ³} - divu ·u - H(div) {u(L ² (K)) ³; divuL ² (K)} - D ^k u is thek-th differential operator associated tou, which is a (k+1)-multilinear operator acting on ³ - k is an index - _k is the linear space of polynomials, the degree of which is less or equal tok - _k is the group of all permutations of the set {1, 2, ...,k} - c orc will stand for any constant depending possibly on      

Class of Trajectories ℝ3 in Most Remote from Observers

Doklady Mathematics - The set of extremal trajectories is completely described. Their construction is reduced to finding the best routes on a directed graph whose vertices are subsets (boxes) of...     

Nilfactors of ℝ m and configurations in sets of positive upper density in ℝ m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. LetE be a measurable subset of ℝ ^m , with . LetV = {0,v ₁,...,v _k} ⊂ ℝ^m. We show that fort large enough, we can find an isometric copy oftV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property form=k=2.     

The view-obstruction problem for spheres in ℝ5

In the view-obstruction problem, congruent, closed convex bodies centred at the points in ⁿ are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1 and cubes of side 1 these values are known forn=2, 3 and 4. Here we show that in ⁵, this value for the sphere of diameter 1 is .     

The existence of an inverse limit of an inverse system of measure spaces — A purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the modern probability theory. Results from Kolmogorov [11], Bochner [2], Choksi [6], Metivier [15], Bourbaki [4], Mallory and Sion [12] among others have paved the way of the deep understanding of this problem. All the above results, however, call for some topological concepts, or at least the ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sufficient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.     

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

In this paper, the initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating solution with expansions up to any order are given and the convergence rates O(ɛ ^m+1/2) and the optimal convergence rates O(ɛ ^m+1) are obtained respectively. This improves the result of J.G. SHI.