Tiling space and slabs with acute tetrahedra

We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than 74.21°, and another which tiles a slab in space.     

Fibre bundles over subspaces of partition space

Acta Mathematica Hungarica -     

A census of genus-two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karábaš, Mali?ký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.     

Study of multiscale global optimization based on parameter space partition

Inverse problems in geophysics are usually described as data misfit minimization problems, which are difficult to solve because of various mathematical features, such as multi-parameters, nonlinearity and ill-posedness. Local optimization based on function gradient can not guarantee to find out globally optimal solutions, unless a starting point is sufficiently close to the solution. Some global optimization methods based on stochastic searching mechanisms converge in the limit to a globally optimal solution with probability 1. However, finding the global optimum of a complex function is still a great challenge and practically impossible for some problems so far. This work develops a multiscale deterministic global optimization method which divides definition space into sub-domains. Each of these sub-domains contains the same local optimal solution. Local optimization methods and attraction field searching algorithms are combined to determine the attraction basin near the local solution at different function smoothness scales. With Multiscale Parameter Space Partition method, all attraction fields are to be determined after finite steps of parameter space partition, which can prevent redundant searching near the known local solutions. Numerical examples demonstrate the efficiency, global searching ability and stability of this method.     

Calculus of generalized hyperbolic tetrahedra

We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.     

Continuous flattening of truncated tetrahedra

Each Platonic polyhedron P can be folded using a continuous folding process into a face of P so that the resulting shape is flat and multilayered, while two of the faces are rigid during the motion. In previous works, explicit formulas of continuous functions for such motions were given and the same result as above was shown to hold for any tetrahedron. In this paper, we show that a truncated regular tetrahedron can be folded continuously via explicit continuous folding mappings into a flat (folded) state, such that two of the hexagonal faces are rigid. Furthermore, given any general tetrahedron P and any truncated tetrahedron Q of P, we show that if Q contains the largest inscribed sphere of P and satisfies some condition, then Q can be folded continuously into a flat folded state such that two of the hexagonal faces of Q are rigid during the motion.     

On the length of the tail of a vector space partition

A vector space partition of a finite dimensional vector space V=V(n,q) of dimension n over a finite field with q elements, is a collection of subspaces U₁,U₂,…,U_t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of consists of the subspaces of least dimension d₁ in , and the length n₁ of the tail is the number of subspaces in the tail. Let d₂ denote the second least dimension in .Two cases are considered: the integer q^d₂−d₁ does not divide respective divides n₁. In the first case it is proved that if 2d₁>d₂ then n₁≥q^d₁+1 and if 2d₁≤d₂ then either n₁=(q^d₂−1)/(q^d₁−1) or n₁>2q^d₂−d₁. These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d₁ respectively d₂.In case q^d₂−d₁ divides n₁ it is shown that if d₂<2d₁ then n₁≥q^d₂−q^d₁+q^d₂−d₁ and if 2d₁≤d₂ then n₁≥q^d₂. The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail.     

Blow-up criteria for 3D Boussinesq equations in the multiplier space

In this paper, we consider the three-dimensional Boussinesq equations with the incompressibility condition. We obtain some regularity conditions for the three-dimensional Boussinesq equations in the multiplier space.     

3D domain decomposition problems,consistent in time and space

During the past years tremendous developments have been achieved in the field of domain decomposition problems as well as in the field of time integration schemes. In the present work actual developments in both fields are merged; the result of this is an extremely stable numerical method in space and in time. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Morrey-Campanato空间中三维 Navier—Stokes方程的正则性准则

利用能量不等式和一些临界空间中的不等式,在Morrey—Campanato空间获得了两个只涉及水平速度场的正则性准则,改进了一些已有的结果．     

Ill-posedness of the Navier-Stokes equations in a critical space in 3D

We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed in in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in can produce solutions arbitrarily large in after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in at the origin.     

Diffusion processes with identical bridges

It is shown that diffusions with identical bridges compose classes of stochastic processes for which the methods of calculation of distributions of functionals are identical as well. Some particular classes of such processes are considered. Bibliography: 8 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 29–42.This research was supported in part by the Russian Foundation for Basic Research, grants 02-01-00265, 00-15-96019, and 99-01-04027 (joint with DFG).Translated by V. N. Sudakov.     

Reconstruction of tetrahedra from sets of edge lengths

Mathematical Notes - The problemof reconstructing tetrahedra fromgiven sets of edge lengths is studied. This is a special case of the problem of determining, up to isometry, the position of a...     

Separation of partition inequalities with terminals

Given a graph with n nodes each of them having labels equal either to 1 or 2 (a node with label 2 is called a terminal), we consider the (1,2)-survivable network design problem and more precisely, the separation problem for the partition inequalities. We show that this separation problem reduces to a sequence of submodular flow problems. Based on an algorithm developed by Fujishige and Zhang the problem is reduced to a sequence of O(n⁴) minimum cut problems.