Isosceles Triangles Determined by a Planar Point Set

 It is proved that, for any ɛ>0 and n>n ₀(ɛ), every set of n points in the plane has at most triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2.136.) This easily implies the best currently known lower bound, , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi–Cs. Tóth and Tardos. Received: February, 2002 Final version received: September 15, 2002 RID="*" ID="*" Supported by NSF grant CCR-00-86013, PSC-CUNY Research Award 63382-00-32, and OTKA-T-032452 RID="†" ID="†" Supported by OTKA-T-030059 and AKP 2000-78-21     

The Number of Directions Determined by Points in the Three-Dimensional Euclidean Space

   Abstract. Let X be a set of n points in the three-dimensional Euclidean space such that no three points in X are on the same line and there is no plane containing all points in X . An old conjecture states that pairs of points in X determine at least 2n-3 directions. We prove the weaker result that X determines at least 1.75n-2 directions.     

The Number of Directions Determined by Points in the Three-Dimensional Euclidean Space

Abstract. Let X be a set of n points in the three-dimensional Euclidean space such that no three points in X are on the same line and there is no plane containing all points in X . An old conjecture states that pairs of points in X determine at least 2n-3 directions. We prove the weaker result that X determines at least 1.75n-2 directions.     

On the Maximum Number of Translates in a Point Set

Given a finite set P⊆ℝ ^d , called a pattern, t _P (n) denotes the maximum number of translated copies of P determined by n points in ℝ ^d . We give the exact value of t _P (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that t _P (n)=n−m _r (n), where r is the rational affine dimension of P, and m _r (n) is the r -Kruskal–Macaulay function. We note that almost all patterns in ℝ ^d are rational simplices. The function t _P (n) is also determined exactly when | P |≤3 or when P has rational affine dimension one and n is large enough. We establish the equivalence of finding t _P (n) and the maximum number s _R (n) of scaled copies of a suitable pattern R⊆ℝ⁺ determined by n positive reals. As a consequence, we show that s_Ak(n)=n-\varTheta (n^1-1/p(k))s_{A_{k}}(n)=n-\varTheta (n^{1-1/\pi(k)}) , where A _k ={1,2,…,k} is an arithmetic progression of size k, and π(k) is the number of primes less than or equal to k.     

The Number of Congruent Simplices in a Point Set

Abstract. For 1 ≤ k≤ d-1 , let f _k ^(d) (n) be the maximum possible number of k -simplices spanned by a set of n points in R ^d that are congruent to a given k -simplex. We prove that f ₂ ⁽³⁾ (n) = O(n ^5/3 2 ^O(α2(n)) ) , f ₂ ⁽⁴⁾ (n) = O(n^ 2+ ɛ ) , for any ɛ >0 , f ₂ ⁽⁵⁾ (n) = Θ(n ^7/3 ) , and f ₃ ⁽⁴⁾ (n) = O(n^ 20/9+ ɛ ) , for any ɛ >0 . We also derive a recurrence to bound f _k ^(d) (n) for arbitrary values of k and d , and use it to derive the bound f _k ^(d) (n) = O(n^ d/2+ ɛ ) , for any ɛ >0 , for d ≤ 7 and k ≤ d-2 . Following Erdos and Purdy, we conjecture that this bound holds for larger values of d as well, and for k≤ d-2 .     

The Number of Congruent Simplices in a Point Set

   Abstract. For 1 ≤ k≤ d-1 , let f _k ^(d) (n) be the maximum possible number of k -simplices spanned by a set of n points in R ^d that are congruent to a given k -simplex. We prove that f ₂ ⁽³⁾ (n) = O(n ^5/3 2 ^O(α2(n)) ) , f ₂ ⁽⁴⁾ (n) = O(n^ 2+ ɛ ) , for any ɛ >0 , f ₂ ⁽⁵⁾ (n) = Θ(n ^7/3 ) , and f ₃ ⁽⁴⁾ (n) = O(n^ 20/9+ ɛ ) , for any ɛ >0 . We also derive a recurrence to bound f _k ^(d) (n) for arbitrary values of k and d , and use it to derive the bound f _k ^(d) (n) = O(n^ d/2+ ɛ ) , for any ɛ >0 , for d ≤ 7 and k ≤ d-2 . Following Erdos and Purdy, we conjecture that this bound holds for larger values of d as well, and for k≤ d-2 .     

由平面方向决定的新型行列式

设F是一个域.任给F中的一对元素(k1,k2),给出了(k1,k2)-型行列式的定义.我们指出通常的行列式恰是(1,-1)-型.研究了这些(k1,k2)-行列式的性质,指出和通常行列式的相同和不同之处.刻画了一些特殊的(r,-r)-型,(r,0)-型和(r,r)-型行列式的性质.     

Solution techniques for the Large Set Covering Problem

Given a finite set E and a family F={E₁,…,E_m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E_i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.     

关于平面19-点集的空凸分划问题

主要研究了平面上处于一般位置的19-点集,根据其凸包边数的不同,分别讨论了其所含空凸多边形的个数,得出G(19)≤5.在此基础上,对平面上处于一般位置的n-点集得出G(n)≤[11n/42],从而改进了G(n)的上界.     

匹配数确定的树的序集

源于理论化学，对顶点数相同的树据其匹配的大小可为树定义一个凝序．文献[1]～[8]研究了这一凝序．本文进一步给出此凝序下的四个全序子集．不同于以前的研究，本文讨论的是边较为密集的情形，而以前确定的序集多属二度顶点数较多的情形．     

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.     

Asymptotics of a Solution to the Quasiliniear Neumann Problem in a Neighborhood of a Conical Point

The existence of a small solution to the quasilinear Neumann problem is established in a H\"older class with separated asymptotics. The lower-order asymptotic terms are constructed. The homogeneity exponents of these terms turn out to be dependent on the value of the solution at a conical point. Bibliography: 33 titles.     

奇点处分岔解支的数目问题*

本文在孤立奇点的假设下证明了带参数非线性方程组解的孤立性,同时指出在分岔点处必有而且仅有有限条解支分岔出来．这一结论是分岔问题数值方法的一个理论基础．     

Endofunctors of Set Determined by Their Object Map

Functors uniquely determined by their object map are called DVO functors. In this paper, we continue the research into DVO endofunctors of the category Set of all sets and maps as started by V. Trnková, R. El Bashir and A. Barkhudaryan. Two families of connected DVO functors containing an isomorphic copy of the identity functor Id are described.     

二向受力不等的平面薄膜自由振动问题解

本文中求解了双向受力不等的矩形、圆形、椭圆形平面薄膜的自振频率与振型,还给出了任意外形边界的平面薄膜的近似解.矩形薄膜,先经过坐标变换将方程变换成常见的薄膜振动方程,因此很容易求得解.圆形薄膜.先将坐标作与上述同样的变换,再把它变换成椭圆坐标,将方程化为马丢(Mathieu)方程,这样利用马丢函数的性质,不难求得其解.椭圆形薄膜解法与圆形薄膜相似.文末还给出了例题.     

On Helicoidal Surfaces in a Conformally Flat 3-Space

In this paper, we discuss the problem of finding explicit parametrizations for the helicoidal surfaces in a conformally flat 3-space \(\mathbb {E}^3_F\) with prescribed extrinsic curvature or mean curvature given by smooth functions. Also, we give examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in \(\mathbb {E}^3_F\).     

关于三维Minkowski空间中直线汇的一些注记

本文研究了三维Minkowski空间中直线汇的一些性质,特别是关于类时线汇的性质．讨论了线汇基本元素的存在性,并证明了关于三维Minkowski空间中类时线汇的配分参数的一个结果,推广了苏步青1927年的—个成果．     

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.