Small Uniformly Resolvable Designs for Block Sizes 3 and 4

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k‐pc and for a given k the number of k‐pcs is denoted r_k. In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r₄ over all permissible v. We obtain a URD({3, 4}; 276) with r₄ = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v) with r₄ = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions.     

Yang—Yang thermodynamics of one-dimensional Bose gases with anisotropic transversal confinement

By combining the thermodynamic Bethe ansatz and local density approximation, we investigate the Yang—Yang thermodynamics of interacting one-dimensional Bose gases with anisotropic transversal confinement. It is shown that with the increase of anisotropic parameter at low temperature, the Bose atoms are distributed over a wider region, while at high temperature the density distribution is not affected obviously. Both the temperature and transversal confinement can strengthen the local pressure of the Bose gases.     

转镜分幅相机中分幅系统放大倍率的校正

理论上分析转镜分幅相机分幅系统放大倍率不一致产生的原因及因素。给出分幅系统放大倍率校正的方法,并以国内普遍使用的FJZ-250型高速转镜分幅相机为例,给出了每一画幅的放大倍率和校正系数。实测结果表明：分幅系统的放大倍率的不一致性与理论计算值差异较大,以校正的数据去分析处理爆轰实验底片,其空间测试精度有较大的提高。     

关于HSOLSSOM（2^nu^1）的谱

本文讨论型为２＾ｎｕ＾１的有对称正交侣的带洞自正交拉丁方（ＨＳＯＬＳＳＯＭ（２＾ｎｕ＾１））的谱。证明当ｎ≤９时,ＨＳＯＬＳＳＯＭ（２＾ｎｕ＾１）存在的充分必要条件是ｕ为偶数且ｎ≥３ｕ／２＋１;当ｎ≥２６３时,若ｕ为偶数且ｎ≥２（ｕ－２）,则ＨＳＯＬＳＳＯＭ（２＾ｎｕ＾１）存在。     

Representation of congruences on regular semigroups with inverse transversals

For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S ^o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component partsI,S ^o and Λ. The structure of images of this type of semigroups is also presented. This work is supported by Natural Science Foundation of Guangdong Province     

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Let K, L ⊂ ℝⁿ be two convex bodies with non–empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t ↦ χ(∂K ∩ (∂L + t)) and t ↦ χ(∂K ∩ (L + t)), t ∈ ℝⁿ, with respect to an n–dimensional Haar measure of ℝⁿ can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let K_r, L_r denote the outer parallel bodies of K, L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂K_r ∩ ∂L_r and ∂K_r ∩ L_r, respectively, is independent of r ≥ 0 if K° ∩ L° ≠ ∅︁ and if ∂K and ∂L intersect almost transversally.As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey .     

The structure of branching in Anosov flows of 3-manifolds

In this article we study the topology of Anosov flows in 3-manifolds. Specifically we consider the lifts to the universal cover of the stable and unstable foliations and analyze the leaf spaces of these foliations. We completely determine the structure of the non Hausdorff points in these leaf spaces. There are many consequences: (1) when the leaf spaces are non Hausdorff, there are closed orbits in the manifold which are freely homotopic, (2) suspension Anosov flows are, up to topological conjugacy, the only Anosov flows without free homotopies between closed orbits, (3) when there are infinitely many stable leaves (in the universal cover) which are non separated from each other, then we produce a torus in the manifold which is transverse to the Anosov flow and therefore is incompressible, (4) we produce non Hausdorff examples in hyperbolic manifolds and derive important properties of the limit sets of the stable/unstable leaves in the universal cover. Received: March 13, 1997     

Arc‐Disjoint Directed and Undirected Cycles in Digraphs

The dicycle transversal number of a digraph D is the minimum size of a dicycle transversal of D, that is a set of vertices of D, whose removal from D makes it acyclic. An arc a of a digraph D with at least one cycle is a transversal arc if a is in every directed cycle of D (making acyclic). In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph such that . It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). In the remaining case (allowing arbitrarily many transversal vertices) the problem is NP‐complete. In this article, we classify the complexity of the arc‐analog of this problem, where we ask for a dicycle B and a cycle C that are arc‐disjoint, but not necessarily vertex‐disjoint. We prove that the problem is polynomially solvable for strong digraphs and for digraphs with a constantly bounded number of transversal arcs (but possibly an unbounded number of transversal vertices). In the remaining case (allowing arbitrarily many transversal arcs) the problem is NP‐complete.