On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold where is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and a discrete subgroup of acting on by left translations. For this purpose, we shall first show that if is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric then the restriction of to the center of is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group . If is one such group, we prove that no timelike geodesic in can be translated by an element of By the way, we rediscover that the Heisenberg Lie group admits a flat left-invariant Lorentz metric if and only if

     

On the number of zero-patterns of a sequence of polynomials

Let be a sequence of polynomials of degree in variables over a field . The zero-pattern of at is the set of those ( ) for which . Let denote the number of zero-patterns of as ranges over . We prove that for and

for . For , these bounds are optimal within a factor of . The bound () improves the bound proved by J. Heintz (1983) using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than () follow from results of J. Milnor (1964), H.E.  Warren (1968), and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary ``linear algebra bound'.

Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the ``branching program' model in the theory of computing, asserts that over any field , most graphs with vertices have projective dimension (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon, gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

     

Tubular Neighbourhoods of 2-Flats in 4-Manifolds of Nonpositive Curvature

We show that every orientable ²-vector bundle over the 2-torus arises as a tubular neighbourhood of a 2-flat in a closed 4-manifold of nonpositive sectional curvature and rank one.     

The Model Companion of Width-Two Orders

The class of width-two orders has a model companion. The model companion is complete, decidable, non-finitely axiomatizable, and has continuum many countable models. Generalizations of some results in (Pouzet, M., 1978, J. Combin. Theory Ser. B 25) are presented in the width-n case, for n2.     

Tensor Products in Categories of Topological Spaces

In this paper symmetric monoidal closed structures on coreflective subcategories of the category of (Hausdorff) topological spaces are studied. We describe all such structures on the category of (Hausdorff) pseudoradial spaces and some of its subcategories and give an example of a coreflective subcategory of the category of Hausdorff topological spaces admitting a proper class of symmetric monoidal closed structures.     

方程(dx/dt)=x~n sum from i=0 to (n-1) a_i(t)x~2的闭解的个数及存在的条件

本文给出方程n＝3时分别正好存在1个闭解，3个闭解，以及至少存在2个闭解的充分条件，并研究了这些闭解的稳定性．当n＝4且ai（t）（i＝0，1，2，3）为t的P次多项式时，文[1]曾猜想其时闭解重数的上界为max｛4，p＋3｝．本文举例指出，即使p＝3，闭解重数的上界也可以大于7．这说明该猜想不成立     

二次自催化反应系统的极限环

本文用广义旋转向量场的方法,证明了系统(1)在B≤A~2时,全平面无极限环;从而与文[1]的结论合在一起,完整地解决系统(1)极限环的存在性问题。     

NUMERICAL SIMULATION OF BENARD CONVECTION IN A CLOSED RECTANGLE (Ⅱ)——THREE-DIMENSIONAL CASE

In the paper, numerical simulation is performed for Benard convection in a closed three-dimensional rectangle with non-slippery bound. Numerical results show that when Rayleigh number Ra<3.6×104, Benard convection is steady, and when Ra≥3.6×104 it is unsteady and irregular. The cross sections and correlation coefficients of various fields are studied, and it is found that the external correlation scales of flow decrease as Ra increases when Ra≥ 7.5×104. Moreover, statistical analyses show that the Taylor inner scales (λv,λθv and so on) also decrease as Ra increases, and that the changing rates of λv and λθv with height are very different to each other in the vicinity of upper and lower bounds. Furthermore, statistical analyses show that the computed Nusselt number Nu is close to the lower limit of many experimental values, and in the transient region of flow pattern d1gNu/d1gRa is relatively greater than that in other region. In addition, other statistical quantities of the fields of temperat     

F-retracts,L-retracts and WAE(n)

The notion of a weakly absolute extensor in the dimension n for the class of bicompacts (WAE(n)) is intruduced in [5]. In this paper, the notions of F-retraction and L-retraction are introduced and using these notions, some properties of WAE(n) are given. Mathematics subject classification numbers, 1980/85. Primary 54C55; 54C99; Secondary 54B1T, 54C15, 54C20, 54C50.