共查询到20条相似文献,搜索用时 15 毫秒
1.
P. V. SemËnov 《Siberian Mathematical Journal》1996,37(2):344-349
2.
Marilyn Breen 《Journal of Geometry》1999,65(1-2):50-53
Let
be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of
have a union which is simply connected and starshaped viak-paths, then
and
is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249 相似文献
3.
Marilyn Breen 《Journal of Geometry》1982,19(1):183-196
Let S be an arbitrary nonempty set in Rd. The following results are true for every k, 0kd: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional -neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional -neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R2, S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S –R2 and int cl SS=, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible. 相似文献
4.
Alejandro Illanes 《Rendiconti del Circolo Matematico di Palermo》2001,50(3):483-498
Adendroid is an arcwise connected hereditarily unicoherent continuum. Ashore set in a dendroidX is a subsetA ofX such that, for each ε>0, there exists a subdendroidB ofX such that the Hausdorff distance fromB toX is less then ε andB∩A=θ.
Answering a question by I. Puga, in this paper we prove that the finite union of pairwise disjoint shore subdendroids of a
dendroidX is a shore set. We also show that the hypothesis that the shore subdendroids are disjoint is necessary. It is still unknown
if the union of two closed disjoint shore subsets of a dendroidX is also shore set. 相似文献
5.
Marilyn Breen 《Journal of Geometry》1989,35(1-2):19-25
Let
and assume that there is a countable collection of lines {L
i
: 1 i} such that (int cl S)
and ((int cl S) S) L
i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336. 相似文献
6.
Marilyn Breen 《Journal of Geometry》1990,37(1-2):48-54
For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let
be a uniformly bounded collection of compact, convex sets inR
d
. For a fixedk, 1kd, dim {MM in
}k if and only if for some > 0, everyf(d, k) members of
contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR
d
. Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H
i
:i1} inR
d
such that clS S {Hi i 1 } and for eachi1, (clS S) H
i
has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336. 相似文献
7.
Marilyn Breen 《Journal of Geometry》1987,28(1):80-85
Let S be a compact set in Rd. Let p be a fixed point of S and let k be a fixed integer, 1 k <d. Then S is starshaped with p ker S if and only if for every k-dimensional flat F through p, F S is starshaped. Moreover, an analogue of this result holds for unions of starshaped sets as well. 相似文献
8.
Douglas S. Bridges 《Mathematical Logic Quarterly》2004,50(3):293-294
It is well known that in Bishop‐style constructive mathematics, the closure of the union of two subsets of ? is ‘not’ the union of their closures. The dual situation, involving the complement of the closure of the union, is investigated constructively, using completeness of the ambient space in order to avoid any application of Markov's Principle. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Marilyn Breen 《Journal of Geometry》1988,32(1-2):1-12
Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9. 相似文献
10.
11.
Marilyn Breen 《Geometriae Dedicata》1991,37(3):287-294
Let S be an open set in R
3 such that: (1) whenever three points x, y, z in S see each other via S, then conv{x, y, z}
S, and (2) every seven points in S see via S a common point. Then S is finitely starlike. The proof uses the topological version of Helly's theorem. 相似文献
12.
《Discrete Mathematics》2001,221(1-3):387-393
A family of sets has the equal union property if and only if there exist two nonempty disjoint subfamilies having the same union. We prove that any n nonempty subsets of an n-element set have the equal union property if the sum of their cardinalities exceeds n(n+1)/2. This bound is tight. Among families in which the sum of the cardinalities equals n(n+1)/2, we characterize those having the equal union property. 相似文献
13.
E. Z. Dyment 《Mathematical Notes》1977,21(2):141-146
It is proved that the set of all natural numbers cannot be represented as the union of a finite number of regressive immune sets. This answers a question of Appel and McLaughlin. Incidentally, we obtain the following two results: 1. If A1, ..., An are regressive immune sets, then there exists a general recursive function f such that Df(0), ..., Df(n), ... is a sequence of pairwise disjoint sets and $$\forall ^x (|D_{f(x)} |) \leqslant n + 1\& D_{f(x)} \cap \overline {A_1 \cup ... \cup A_n } \ne \emptyset )$$ .2. If A1, ..., An are regressive and B is an infinite subset of \(\bigcup\limits_{t = 1}^n {A_1 } \) , then there exists an i that Ai?eB. 相似文献
14.
Marilyn Breen 《Journal of Geometry》1989,35(1-2):14-18
SetS inR
d has propertyK
2 if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c1,c2 (depending onF) such that each point ofF is clearly visible viaS from at least one ci,i = 1,2. The following characterization theorem is established: Let
, d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S
j
} converging toS (relative to the Hausdorff metric) such that each setS
j satisfies propertyK
2. For
, the sufficiency of the condition above still holds, although the necessity fails. 相似文献
15.
Beifang Chen 《Discrete and Computational Geometry》1993,10(1):79-93
The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces,
its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its
combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such
as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial
Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for
polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting
convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator
functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper. 相似文献
16.
M. Zähle 《Geometriae Dedicata》1987,23(2):155-171
For locally finite unions of sets with positive reach in R
d, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of regular sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper. 相似文献
17.
Charles D. Horvath Marc Lassonde 《Proceedings of the American Mathematical Society》1997,125(4):1209-1214
We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.
18.
We discuss the relationship between properties of spaces and their remainders in extensions from the class P fin of all finite unions of locally compact spaces. In particular, we show that a space X ∈ P fin iff the remainder in each (some) compactification of X is in P fin. Then we study the class P fin and the relationship between the remainders of a space from this class in compact extensions and give a generalization of the theorem of Henriksen-Isbell. 相似文献
19.
V. I. Vasyunin 《Journal of Mathematical Sciences》1984,27(1):2448-2450
The traces of functions from H on the union HT of n Carleson subsets of the unit circle form a space of functions on HT for which the first n — 1 divided differences (with respect to the hyperbolic metric) are uniformly bounded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 31–34, 1983. 相似文献
20.
Daniel M. Oberlin 《Israel Journal of Mathematics》2014,202(1):331-342
We prove a generalization of a result of Peres and Schlag on the dimensions of certain exceptional sets of projections and apply it to a geometric problem. 相似文献