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1.
The main notion dealt with in this article is
where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from
investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.
Some of the main results of the paper are:
• (1) If there is a maximal family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal
family of size λ of pairwise almost disjoint subsets of κ + each of size κ.
• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the
factors; and a similar characterization for weak products.
• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all
cardinalities of partitions of 1 of that BA.
• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras.
Received: 9 October 1997 / Published online: 21 March 2001 相似文献
2.
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 A 1, ..., A n are regressive immune sets, then there exists a general recursive function f such that D f(0), ..., D f(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 A 1, ..., A n are regressive and B is an infinite subset of \(\bigcup\limits_{t = 1}^n {A_1 } \) , then there exists an i that A i? eB. 相似文献
3.
We focus on
% MathType!End!2!1!( A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless r-maximal set. We answer a long-standing question by showing that there are infinitely many pairwise non-isomorphic filters
of this type.
Research partially supported NSF Grants DMS-96-3465 (Cholak) and DMS-95-00983 (Nies). The authors would like to thank Mike
Stob for his help and interest. 相似文献
4.
For any odd integer n 3 and prime power q, it is known that PG(n–1, q 2) can be partitioned into pairwise disjoint subgeometries isomorphic to PG(n–1, q) by taking point orbits under an appropriate subgroup of a Singer cycle of PG(n–1, q 2). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes. 相似文献
5.
An n× m sonar sequence is a subset of the n× m grid with exactly one point in each column, such that the
vectors determined by them are all distinct. We show that for fixed n the maximal m for which a sonar sequence exists satisfies n– Cn
11/20< m< n+4 n
2/3 for all n and m> n+ c log n log log n for infinitely many n.Another problem concerns the maximal number D of points that can be selected from the n× m grid so that all the
vectors have slopes. We prove n
1/2Dn
4/5
Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341. 相似文献
6.
Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ 3. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family
of pairwise disjoint compact convex sets.
The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's
research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported
in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220. 相似文献
7.
Let G be a compact group with dual object . A decomposition of into pairwise disjoint finite sets provides a way of defining partial sums of Fourier series on G. We employ a theorem of Olevskii to present a condition on such a decomposition which, if satisfied, guarantees that the corresponding Lebesgue constants are unbounded. We apply this to certain summation methods on tori and compact connected semisimple Lie groups. 相似文献
8.
Given a pair of finite disjoint sets A and B in R
n, a fundamental problem with many important applications isto efficiently determine a hyperplane H( w,) which separates these sets when they are separable, or nearly separates themwhen they are not. We seek a hyperplane which minimizes a natural errormeasure in the latter case, and so will surgically separate the sets. Whenthe sets are separable in a strong sense, we show that the problem is aconvex program with a unique solution, which has been investigated byothers. Using the KKT conditions, we improve on an existing algorithm. Whenthe sets are not separable, the problem is nonconvex, generally with properlocal solutions, and we solve an equivalent problem by Branch and Bound.Numerical results are presented. 相似文献
9.
For an ordered set P let P
P
denote the set of all isotone self-maps on P, that is, all maps f from P to P such that xy implies f( x) f( y), and let Aut ( P) the set of all automorphisms on P, that is, all bijective isotone self-maps in P
P
. We establish an inequality relating ¦ P
P
¦ and ¦ Aut( P)¦ in terms of the irreducibles of P. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship. 相似文献
10.
We prove that there exist finitely generated algebras, which are pseudosimple but not simple. This problem goes back to Henkin, Monk, Tarski [71]. In fact, for any limit ordinal i, there exists a pseudosimple algebra, which has no proper subalgebra and whose congruence lattice is
i+1. (Here i denotes ordinal power).Presented by George Grätzer.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810. 相似文献
11.
We describe the set
of values of the parameter for which there exists a Hilbert space H and n partial reflections A
1,...,A
n
(self-adjoint operators such that A
k
3
=A k or, which is the same, self-adjoint operators whose spectra belong to the set {-1,0,1}) whose sum is equal to the scalar operator I
H
. 相似文献
12.
By the query-time complexity of a relativized algorithm we mean the total length of oracle queries made; the query-space complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles.A notable unresolved question in relativized complexity theory is the separation of NP A co NP A from P
A under random oracles A. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity.If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages L A and {0, 1} *L A under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept L A. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1–) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets. 相似文献
13.
A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism
of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure
is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism
of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between
partially ordered sets A and B is a mapping f : A → B satisfying ). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected
components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X
5-dense locally bounded partially ordered sets.
Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017. 相似文献
14.
Let A be a family of n pairwise disjoint compact convex sets in R
d. Let
. We show that the directed lines in R
d, d 3, can be partitioned into
sets such that any two directed lines in the same set which intersect any AA generate the same ordering on A. The directed lines in R
2 can be partitioned into 12 n such sets. This bounds the number of geometric permutations on A by 1/2
d
for d3 and by 6 n for d=2. 相似文献
15.
Summary We shall in this paper consider the problem of determination a row or column scaling of a matrix A, which minimizes the condition number of A. This problem was studied by several authors. For the cases of the maximum norm and of the sum norm the scale problem was completely solved by Bauer [1] and Sluis [5]. The condition of A subordinate to the pair of euclidean norms is the ratio /, where and are the maximal and minimal eigenvalue of ( A
H
A) 1/2 respectively. The euclidean case was considered by Forsythe and Strauss [3]. Shapiro [6] proposed some approaches to a numerical solution in this case. The main result of this paper is the presentation of necessary and sufficient conditions for optimal scaling in terms of maximizing and minimizing vectors. A uniqueness proof for the solution is offered provided some normality assumption is satisfied. 相似文献
16.
We continue to study upper sets ${\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}}We continue to study upper sets [(A)\tilde]={(x,r) ? A×R+ :$y ? A\{x}, r=|x-y|}{\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}} equipped by hyperbolic metric. We define analogous of quasiconvexity, simply connectedness and nearlipschitz functions. We
give a new definition of quasisymmetry as nearlipschitz characteristic on [(A)\tilde]{\widetilde{A}}. In the final part in terms of upper sets we give the following extension property of A ì R2{A\subset R^2}. For 0 £ e £ d{0\le\varepsilon\le \delta}, each (1+e){(1+\varepsilon)}-bilipschitz map f : A → R
2 has an extension to a (1+Ce){(1+C\varepsilon)}-bilipschitz map F : R
2 → R
2. 相似文献
17.
AbstractGiven a topological space X = ( X, T ), we show in the Zermelo-Fraenkel set theory ZF that: Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T4 and the countable axiom of choice holds true. We also show:It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite. It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets. 相似文献
18.
We study the exact rate of convergence of frequencies of digits of “normal” points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of ? d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in ? d—namely, sets with full Hausdorff dimension—for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm. 相似文献
19.
A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear.
We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph with n vertices and m> k
4
n edges contains k+1 pairwise disjoint edges. We also prove that, given a set of points V and a set of axis-parallel rectangles in the plane, then either there are k+1 rectangles such that no point of V belongs to more than one of them, or we can find an at most 2·10 5
k
8 element subset of V meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem
on partially ordered sets.
The research by János Pach was supported by Hungarian National Foundation for Scientific Research Grant OTKA-4269 and NSF
Grant CCR-91-22103. 相似文献
20.
Given a set A in R
2 and a collection S of plane sets, we say that a line L separates A from S if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane.We prove that, for any collection F of n disjoint disks in R
2, there is a line L that separates a disk in F from a subcollection of F with at least ( n–7)/4 disks. We produce configurations H
n
and G
n
, with n and 2 n disks, respectively, such that no pair of disks in H
n
can be simultaneously separated from any set with more than one disk of H
n
, and no disk in G
n
can be separated from any subset of G
n
with more than n disks.We also present a set J
m
with 3 m line segments in R
2, such that no segment in J
m
can be separated from a subset of J
m
with more than m+1 elements. This disproves a conjecture by N. Alon et al. Finally we show that if F is a set of n disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a line L that separates one of the segments from at least n/3+1 elements of F. 相似文献
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