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1.
Two of the pillars of combinatorics are the notion of choosing an arbitrary
subset of a set with n elements
(which can be done in 2n ways),
and the notion of choosing
a k-element subset of a set with
n elements (which can be done in
ways). In this article
I sketch the beginnings of a theory that would import these notions into the category
of hedral sets in the sense of Morelli and the category of polyhedral sets in the sense of
Schanuel. Both of these theories can be viewed as extensions of the theory of finite sets and
mappings between finite sets, with the concept of cardinality being replaced by the more
general notion of Euler measure (sometimes called combinatorial Euler characteristic). I
prove a functoriality theorem (Theorem 1) for subset-selection in the context of polyhedral
sets, which provides quasi-combinatorial interpretations of assertions such as
.
Furthermore, the operation of forming a power set can be viewed as a special
case of the operation of forming the set of all mappings from one set to another; I conclude
the article with a polyhedral analogue of the set of all mappings between two finite sets,
and a restrictive but suggestive result (Theorem 2) that offers a hint of what a general
theory of exponentiation in the polyhedral category might look like. (Other glimpses into
the theory may be found in [11].)Dedicated to the Memory of Gian-Carlo Rota 相似文献
2.
Christian Richter 《Journal of Geometry》2006,84(1-2):117-132
Let
be a group of affine transformations of the Euclidean plane
. Two topological discs D,
are called congruent by dissection with respect to
if D can be dissected into a finite number of subdiscs that can be rearranged by maps from
to a dissection of E.
Our main result says in particular that
admits congruence by dissection of any circular disc C with any square S if and only if
contains a contractive map and all orbits
,
, are dense in
. In this case any two discs D and E are congruent by dissection with respect to
and every disc D is congruent by dissection with n copies of D for every n ≥ 2.
Moreover, we give estimates on minimal numbers of pieces that are needed to realize congruences by dissection.
Dedicated to Irmtraud Stephani on the occasion of her 70th birthday 相似文献
3.
F. Móricz 《Archiv der Mathematik》2003,81(1):82-89
We extend the concept of and basic results on statistical convergence from
ordinary (single) sequences to multiple sequences of (real or complex) numbers.
As an application to Fourier analysis, we obtain the following Theorem 3: (i)
If $f \in L(\textrm{log}^{+} L)^{d-1}(\mathbb{T}^d)$, where $\mathbb{T}^d := [-\pi, \pi)^{d}$
is the d-dimensional torus, then the Fourier
series of f is statistically convergent to $f({\bf t})$
at almost every ${\bf t} \in \mathbb{T}^d$; (ii) If $f \in C(\mathbb{T}^d)$, then the
Fourier series of f
is statistically convergent to $f ({\bf t})$ uniformly on $\mathbb{T}^d$.
Received: 5 November 2001 相似文献
4.
5.
Melvin Hausner 《Combinatorica》1985,5(3):215-225
Ifμ is a positive measure, andA
2, ...,A
n
are measurable sets, the sequencesS
0, ...,S
n
andP
[0], ...,P
[n] are related by the inclusion-exclusion equalities. Inequalities among theS
i
are based on the obviousP
[k]≧0. Letting
=the average average measure of the intersection ofk of the setsA
i
, it is shown that (−1)
k
Δ
k
M
i
≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS
0=1,
whenS
1≧N−1, and
for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN,
for all sequencesM
0, ...,M
n
of sufficiently large length if and only if
for 0<t<1. 相似文献
6.
David J. Grynkiewicz 《Combinatorica》2006,26(4):445-453
An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct so that they can be considered
as sets. If S is a sequence of m+n−1 elements from a finite abelian group G of order m and exponent k, and if
is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence
of S such that
. This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and wi = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that
if S has an n-set partition A=A1, . . .,An such that |wiAi| = |Ai| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A′1, . . .,A′n of S such that
and
for all i, where wiAi={wiai |ai∈Ai}. 相似文献
7.
8.
Ahmet Yıldız Cengizhan Murathan Kadri Arslan Rıdvan Ezentaş 《Monatshefte für Mathematik》2007,151(3):247-256
Let
be (2n + 1)-dimensional Sasakian space form of constant ϕ-sectional curvature (c) and M
n
be an
n
-dimensional C-totally real, minimal submanifold of
. We prove that if M
n
is pseudo-parallel and
, then M
n
is totally geodesic. 相似文献
9.
We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained. 相似文献
10.
We consider cyclic d-polytopes
P that are realizable with
vertices on the moment curve $M_d:t\longrightarrow (t,t^2,\ldots,t^d)$
of order $d\geq 3$. A hyperplane H bisects a
j-face of P
if H meets its relative interior. For $\ell\geq
1$, we investigate the maximum number of vertices that P
can have so that for some $\ell$ hyperplanes, each j-face of
P is bisected by one of the
hyperplanes. For $\ell > 1$, the problem translates to the existence
of certain codes, or equivalently, certain paths on the cube
$\{0,1\}^\ell$. 相似文献
11.
We prove that the size of the largest face of a 4-critical planar graph with 4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows
.All three authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada. 相似文献
12.
We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P}
_{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28
intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such
conclaves (see [6]) $\mathcal{P}_{8}$ of planes
in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence
with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to
the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit
$\operatorname{orb}(2\gamma)$ (see [4]). The fact
that, under the action of $\operatorname{GL}(5,2),$ the stabilizer
groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have
the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a
regulus-free partial spread $\mathcal{S}_{8}$ in
$\operatorname{PG}(4,2)$ we also give a construction of a conclave of
planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$ 相似文献
13.
For a finite lattice L, let $ \trianglelefteq_L $ denote
the reflexive and transitive closure of the join-dependency relation on L,
defined on the set J(L) of all join-irreducible elements of L.
We characterize the relations of the form $ \trianglelefteq_L $, as follows:
Theorem. Let $ \trianglelefteq $ be a quasi-ordering on
a finite set P. Then the following conditions are
equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is
isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L,
let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L
is the congruence lattice of L. It is well-known that the
inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive
lattice D, let us define the join-
excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice
D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$
, for any finite distributive lattice D;
the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite
lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota 相似文献
14.
Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ([? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component Ga contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component Ga contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group. 相似文献
15.
The cd-index is a
polynomial which encodes the flag f-vector of a convex polytope. For
polytopes U and
V, we determine explicit
recurrences for computing the cd-index of the free join
and the cd-index of the
Cartesian product U x
V. As an application of these
recurrences, we prove the inequality
involving the
cd-indices of three
polytopes. 相似文献
16.
Let K be an arbitrary fieldof characteristic 0, and $\mathbf{A}^n$ then-dimensional affine spaceover K.A well-known cancellation problem asks, given twoalgebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphiccylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether$V_1$ and $V_2$ themselves are isomorphic.In this paper, we focus on a related problem: given twovarieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$)cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are$V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$?We call this stable equivalence problem.We show that the answer is positivefor any two curves $V_1, V_2 \subseteq \mathbf{A}^2$.For an arbitrary $n \ge 2$, we consider a special, arguably themost important, case of both problems, where one of the varieties isa hyperplane. We show that a positive solution of the stableequivalence problem in this case implies a positive solution ofthe cancellation problem. 相似文献
17.
Vladislav Kargin 《Probability Theory and Related Fields》2007,139(3-4):397-413
Let X
i
denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X
i
it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX
i
= 1, then is between and c
2
n for certain constant c
1 and c
2. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X
i
, and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables. 相似文献
18.
Marilyn Breen 《Archiv der Mathematik》2005,84(3):282-288
Let k and d be fixed integers, 0kd, and let
be a collection of sets in
If every countable subfamily of
has a starshaped intersection, then
is (nonempty and) starshaped as well. Moreover, if every countable subfamily of
has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of
is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004 相似文献
19.
Cvetan Jardas Josip Peari Nikola Sarapa 《Rendiconti del Circolo Matematico di Palermo》1998,47(3):481-492
In this paper we study the problem of convergence in the weak and the vague topology of the sequence
where μ and ν are probability measures on locally compact commutative semigroupS andA
n
=[a
ij
(n)
(i, j, n ɛ N) are double stochastic matrices satisfying some additional conditions. Our results generalize the results in [9]. Theorem
1 also holds if we make some changes on the topological assumptions onS, i.e. if we suppose thatS is a polish space. 相似文献
20.
Astrid Baumann 《Aequationes Mathematicae》2003,65(3):201-235
Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where
$\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or
$x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.
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