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1.
For an l-graph
, the Turán number
is the maximum number of edges in an n-vertex l-graph
containing no copy of
. The limit
is known to exist [8]. The Ramsey–Turán density
is defined similarly to
except that we restrict to only those
with independence number o(n). A result of Erdős and Sós [3] states that
as long as for every edge E of
there is another edge E′of
for which |E∩E′|≥2. Therefore a natural question is whether there exists
for which
.
Another variant
proposed in [3] requires the stronger condition that every set of vertices of
of size at least εn (0<ε<1) has density bounded below by some threshold. By definition,
for every
. However, even
is not known for very many l-graphs
when l>2.
We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we
construct, for each l≥3, infinitely many l-graphs
for which
.
We also prove that the 3-graph
with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies
. The existence of a hypergraph
satisfying
was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based
on different ideas and is simpler than these earlier proofs.
* Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P.
Sloan Research Fellowship.
† Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529. 相似文献
2.
Abstract
By
we denote the set of all propositional formulas. Let
be the set of all clauses. Define
. In Sec. 2 of this paper we prove that for normal modal logics
, the notions of
-expansions and
-expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of
-expansions for I and
-expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of
-expansions for I is the same as that of
-expansions for I.
The project is supported by NSFC 相似文献
3.
John K. McVey 《Archiv der Mathematik》2005,84(6):481-484
When G is a finite nonabelian group, we associate the common-divisor graph (G) with G by letting nontrivial degrees in cd(G) be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set
of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of
and every vertex outside of
is adjacent to some member of
When G has a nonabelian solvable quotient, we show that if (G) is connected and has a diameter of at most 2, then indeed cd(G) has a strongly connective subset.Received: 7 July 2004; revised: 5 October 2004 相似文献
4.
Michael Capalbo 《Combinatorica》2005,25(4):379-391
Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite
family
of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈
satisfy the following properties. The output-set Y has cardinality
times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of
is simple to specify, and each
has fewer than
edges. We then modify
to obtain explicit unique-neighbor concentrators of maximum degree 3.
* Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013. 相似文献
5.
Let H be a finite group,
a field and V a finite dimensional
H-module. We introduce the Fitting submodule FV (H), an
H submodule of V which has properties similar to the generalized Fitting subgroup of a finite group.
Received: 26 August 2005 相似文献
6.
Mowaffaq Hajja 《Results in Mathematics》2006,49(3-4):237-263
An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by
. Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by
. The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in
[1, Chapter IX.B, pp. 294–333].
The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid
, the circumcenter
, and the incenter
does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers
, and
coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers
, and
of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences
, and
implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles.
Several open questions are posed.
Received: June 19, 2006. 相似文献
7.
The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy
of evolution families on the real line. We consider two general classes denoted
and
and we prove that if V,W are Banach function spaces with
and
, then the admissibility of the pair
for an evolution family
implies the uniform dichotomy of
. In addition, we consider a subclass
and we prove that if
, then the admissibility of the pair
implies the uniform exponential dichotomy of the family
. This condition becomes necessary if
. Finally, we present some applications of the main results. 相似文献
8.
Lutz Strüngmann 《Archiv der Mathematik》2006,86(3):193-204
Let R be a unital associative ring and
two classes of left R-modules. In this paper we introduce the notion of a
In analogy to classical cotorsion pairs as defined by Salce [10], a pair
of subclasses
and
is called a
if it is maximal with respect to the classes
and the condition
for all
and
Basic properties of
are stated and several examples in the category of abelian groups are studied.
Received: 17 March 2005 相似文献
9.
Long Miao 《Bulletin of the Brazilian Mathematical Society》2007,38(4):585-594
Let
be a class of groups. A subgroup H of a group G is called
-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ HG belongs to
where HG is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to study some subgroups of Fitting subgroup and generalized Fitting subgroup
-s-supplemented and some new criterions of p-nilpotency of finite groups are obtained.
*This research is supported by the grant of NSFC and TianYuan Fund of Mathematics of China (Grant #10626047). 相似文献
10.
Dietmar Vogt 《Archiv der Mathematik》2006,87(2):163-171
It is shown that for open convex
, d > 1 and a nontrivial polynomial P the space
does not have property
. If P is elliptic or homogeneous, then this holds for every open Ω. For
even
cannot occur and if it occurs for some Ω, then P must be hypoelliptic.
Received: 18 July 2005 相似文献
11.
Let X, Y be Banach spaces. We say that a set
is uniformly p–summing if the series
is uniformly convergent for
whenever (xn) belongs to
. We consider uniformly summing sets of operators defined on a
-space and prove, in case X does not contain a copy of c0, that
is uniformly summing iff
is, where T (φ x) = (T#φ) x for all
and x∈X. We also characterize the sets
with the property that
is uniformly summing viewed in
.
Received: 1 July 2005 相似文献
12.
We show that the hereditary discrepancy of a hypergraph
on n points increases by a factor of at most O(log n) when one adds a new edge to
. 相似文献
13.
Alina Iacob 《Archiv der Mathematik》2005,85(4):335-344
We consider two pairs of complete hereditary cotorsion theories
on the category of left R-modules, such that
We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules
can be computed either using a left
of M and a left
of M or using a right
a right
of N.
Received: 17 December 2004 相似文献
14.
On the Range of the Aluthge Transform 总被引:1,自引:0,他引:1
Let
be the algebra of all bounded linear operators on a complex separable Hilbert space
For an operator
let
be the Aluthge transform of T and we define
for all
where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range
of Δ. We prove that R(Δ) is neither closed nor dense in
However R(Δ) is strongly dense if
is infinite dimensional.
An erratum to this article is available at . 相似文献
15.
We consider a class of elliptic operators
with unbounded coefficients in a smooth exterior domain Ω and we prove that the Cauchy-Neumann problem associated with
admits, for any bounded and continuous initial datum, a unique bounded classical solution. We also provide pointwise gradient
estimates for such a solution.
Received: 5 July 2005; Revised: 20 December 2005 相似文献
16.
Humio Ichimura 《Archiv der Mathematik》2006,87(6):539-545
Let p be an odd prime number and
. Let
be the classical Stickelberger ideal of the group ring
. Iwasawa [6] proved that the index
equals the relative class number
of
. In [2], [4] we defined for each subgroup H of G a Stickelberger ideal
of
, and studied some of its properties. In this note, we prove that when
mod 4 and [G : H] = 2, the index
equals the quotient
.
Received: 13 January 2006 相似文献
17.
A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and
. Let
be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have
This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi.
We also show that equality holds if and only if
consists of all r-element subsets containing a fixed element.
Analogous results are obtained for nonuniform families. 相似文献
18.
Simon M. Goberstein 《Algebra Universalis》2005,53(4):407-432
A partial automorphism of a semigroup S is any isomorphism between its subsemigroups, and the set all partial automorphisms of S with respect to composition is an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be
if their partial automorphism monoids are isomorphic. A class
of semigroups is called
if it contains every semigroup
to some semigroup from
Although the class of all inverse semigroups is not
we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is
It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is
A semigroup is called
if it is isomorphic or antiisomorphic to any semigroup that is
to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are
To Ralph McKenzieReceived April 15, 2004; accepted in final form October 7, 2004. 相似文献
19.
Let L and M be Archimedean vector lattices such that
and
are complex vector lattices. We constructively and intrinsically prove that if
is an order bounded disjointness preserving operator from
into
then the modulus
of
exists in the ordered vector space of all order bounded operators from L into M.
Received February 11, 2005; accepted in final form March 8, 2005. 相似文献
20.
Antonio G. García Miguel A. Hernández-Medina 《Mediterranean Journal of Mathematics》2005,2(3):345-356
Let
be a symmetric operator with compact resolvent defined in a Hilbert space
For any fixed
we consider an entire
function Ka which involves the resolvent of
Associated with Ka we obtain, by duality in
a Hilbert space
of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of
regardless of the anti-linear mapping which has
as its range space. There exists also a sampling formula allowing to recover any function in
from its samples at the sequence of eigenvalues of
This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología. 相似文献