共查询到20条相似文献,搜索用时 14 毫秒
1.
Marilyn Breen 《Periodica Mathematica Hungarica》2009,59(1):99-107
Fix k, d, 1 ≤ k ≤ d + 1. Let $
\mathcal{F}
$
\mathcal{F}
be a nonempty, finite family of closed sets in ℝ
d
, and let L be a (d − k + 1)-dimensional flat in ℝ
d
. The following results hold for the set T ≡ ∪{F: F in $
\mathcal{F}
$
\mathcal{F}
}.
Assume that, for every k (not necessarily distinct) members F
1, …, F
k
of $
\mathcal{F}
$
\mathcal{F}
,∪{F
i
: 1 ≤ i ≤ k} is starshaped and the corresponding kernel contains a translate of L. Then T is starshaped, and its kernel also contains a translate of L. 相似文献
2.
Let k be non–negative integer. The unoriented bordism classes, which can be represented
as [RP(ξ
k
)] where ξ
k
is a k–plane bundle, form an ideal of the unoriented bordism ring MO*. A group
of generators of this ideal expressed by a base of MO* and a necessary and sufficient condition for a
bordism class to belong to this ideal are given.
This work is supported by HNSF (Grant No: 103144) and NNSF of China (10371029) 相似文献
3.
Hanno Lefmann 《Discrete and Computational Geometry》2008,40(3):401-413
We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1]
d
, such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ
k,d
(n), the supremum of this minimum volume over all distributions of n points in [0,1]
d
, we show that c
k,d
⋅(log n)1/(d−k+1)/n
k/(d−k+1)≤Δ
k,d
(n)≤c
k,d
′/n
k/d
for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ
k,d
(n)≤c
k,d
″/n
k/d+(k−1)/(2d(d−1)), where c
k,d
,c
k,d
′,c
k,d
″>0 are constants.
A preliminary version of this paper appeared in COCOON ’05. 相似文献
4.
We consider the differential operators Ψ
k
, defined by Ψ1(y) =y and Ψ
k+1(y)=yΨ
k
y+d/dz(Ψ
k
(y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ
k
F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z
2+β
z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ
k
(F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ
k
(f
′/f) =f
(k)/f, we deduce in particular that iff andf
(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f
′/f :f ∈F} is normal.
The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999,
and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank
Günter Frank for helpful discussions. 相似文献
5.
Riccardo Benedetti Francois Loeser Jean Jacques Risler 《Discrete and Computational Geometry》1991,6(1):191-209
For every polynomial mapf=(f
1,…,f
k): ℝ
n
→ℝ
k
, we consider the number of connected components of its zero set,B(Z
f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off
i), and thek-tuple (Δ1,...,Δ4), Δ
k
being the Newton polyhedron off
i respectively. Our aim is to boundB(Z
f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ
d
(n)=d(2d−1)
n−1. Considered as a polynomial ind, μ
d
(n) has leading coefficient equal to 2
n−1. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ
d
(n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n
k−1
dn, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument,
Smith's theory, and information about the sum of Betti numbers of complex complete intersections. 相似文献
6.
Given two sets
A, B í \Bbb Fqd{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d
, the set of d dimensional vectors over the finite field
\Bbb Fq{\Bbb F}_q
with q elements, we show that the sumset
A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\}
contains a geometric progression of length k of the form vΛ
j
, where j = 0,…, k − 1, with a nonzero vector
v ? \Bbb Fqd{\bf v} \in {\Bbb F}_q^d
and a nonsingular d × d matrix Λ whenever
# A # B 3 20 q2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k}
. We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic
varieties. 相似文献
7.
A stationary Poisson cylinder process Π
cyl
(d,k) is composed of a stationary Poisson process of k-flats in ℝ
d
that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the
accuracy of normal approximation of the d-volume V
ϱ
(d,k) of the union set of Π
cyl
(d,k) that covers ϱW as the scaling factor ϱ becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance
of V
ϱ
(d,k) that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order
cumulants of V
ϱ
(d,k) under the assumption that the (d − k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated
“Lemma on large deviations” of Statulevičius. 相似文献
8.
Let F\mathcal{F} be a family of compact convex sets in ℝ
d
. We say that F\mathcal{F} has a topological
ρ-transversal of index (m,k) (ρ<m, 0<k≤d−m) if there are, homologically, as many transversal m-planes to F\mathcal{F} as m-planes containing a fixed ρ-plane in ℝ
m+k
. 相似文献
9.
Wojciech Bartoszek 《Israel Journal of Mathematics》1995,90(1-3):115-123
LetK be a compact group of linear operators of thed-dimensional spaceR
d
andG
K,d
denote the semidirect productK byR
d
. It is shown that if an adapted probability measureμ onG
K,d
is not scattered (i.e. for some compactF we havex
0 ∈ R
d
(gF)>0), then there exists a nonzero vectorx
0 ∈R
d
such thatk
1(x
0)=k
2(x
0) holds for all (k
1,x
1) and (k
2,x
2) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of
thed-dimensional Euclidian space is scattered.
I thank the Foundation for Research Development for financial support. 相似文献
10.
Let M^n be a smooth, compact manifold without boundary, and F0 : M^n→ R^n+1 a smooth immersion which is convex. The one-parameter families F(·, t) : M^n× [0, T) → R^n+1 of hypersurfaces Mt^n= F(·,t)(M^n) satisfy an initial value problem dF/dt (·,t) = -H^k(· ,t)v(· ,t), F(· ,0) = F0(· ), where H is the mean curvature and u(·,t) is the outer unit normal at F(·, t), such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton. 相似文献
11.
Nariankadu D. Shyamalkumar Kasturi Varadarajan 《Discrete and Computational Geometry》2012,47(1):44-63
We consider the problem of fitting a subspace of a specified dimension k to a set P of n points in ℝ
d
. The fit of a subspace F is measured by the L
τ
norm, that is, it is defined as the τ-root of the sum of the τth powers of the Euclidean distances of the points in P from F, for some τ≥1. Our main result is a randomized algorithm that takes as input P, k, and a parameter 0<ε<1; runs in
nd ·2O(\fractk2e log2 \frac ke)nd \cdot2^{O(\frac{\tau k^{2}}{\varepsilon} \log^{2} \frac {k}{\varepsilon})} time, and returns a k-subspace that with probability at least 1/2 has a fit that is at most (1+ε) times that of the optimal k-subspace. 相似文献
12.
M. Abad J. P. Díaz Varela B. F. López Martinolich M. del C. Vannicola M. Zander 《Central European Journal of Mathematics》2006,4(4):547-561
In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L
p,k, and the finite field F(p
k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V(L
p,k) generated by L
p,k into the variety V(F(p
k)) generated by F(p
k) and an interpretation Φ2 of V(F(p
k)) into V(L
p,k) such that Φ2Φ1(B) = B for every B ε V(L
p,k) and Φ1Φ2(R) = R for every R ε V(F(p
k)). 相似文献
13.
Dimitra Kosta 《Proceedings of the Steklov Institute of Mathematics》2009,264(1):102-109
Let X be a complete intersection of two hypersurfaces F
n
and F
k
in ℙ5 of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F
k
is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial. 相似文献
14.
Consider a d-dimensional Brownian motion X = (X
1,…,X
d
) and a function F which belongs locally to the Sobolev space W
1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f
k
(X), X
k
] involving the weak first partial derivatives f
k
of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X
k
] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation
results for forward and backward stochastic integrals.
Received: 16 March 1998 / Revised version: 4 April 1999 相似文献
15.
Let F be a field and let {d 1,…,dk } be a set of independent indeterminates over F. Let A(d 1,…,dk ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d 1,…,dk }. We assume that no d 1 appears twice in A(d 1,…,dk ). We show that if det A(d 1,…,dk ) = 0 then A(d 1,…,dk ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p. 相似文献
16.
Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q
k/2 of the values of d < q
k-1. In this article we shall prove that, for q = p prime and roughly
\frac38{\frac{3}{8}}-th’s of the values of d < q
k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q
k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over
\mathbb Fp{{\mathbb F}_p} of dimension k with minimum distance d < q
k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports. 相似文献
17.
Elizabeth Meckes 《Journal of Theoretical Probability》2012,25(2):333-352
Let X be a d-dimensional random vector and X
θ
its projection onto the span of a set of orthonormal vectors {θ
1,…,θ
k
}. Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from X
θ
to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance,
in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d. The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝ
d
are close to Gaussian, when n and d are large and k=clog (d) for a small constant c. 相似文献
18.
A d-dimensional simplex S is called a k-reptile if it can be tiled without overlaps by simplices S
1,S
2,…,S
k
that are all mutually congruent, and similar to S. For d=2, k-reptile simplices (triangles) exist for many values of k, and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, for d≥3, only one construction of k-reptile simplices is known, the Hill simplices, and it provides only k of the form m
d
, m=2,3,…. 相似文献
19.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
20.
E. J. Cheon 《Designs, Codes and Cryptography》2009,51(1):9-20
In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g
q
(k, d) + 1, k, d]
q
code for sq
k-1 − sq
k-2 − q
s
− q
2 + 1 ≤ d ≤ sq
k-1 − sq
k-2 − q
s
with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n
q
(k, d) = g
q
(k, d) + 1 for (k − 2)q
k-1 − (k − 1)q
k-2 − q
2 + 1 ≤ d ≤ (k − 2)q
k-1 − (k − 1)q
k-2, k ≥ 6, q ≥ 2k − 3; and sq
k-1 − sq
k-2 − q
s
− q + 1 ≤ d ≤ sq
k-1 − sq
k-2 − q
s
, s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1.
This work was partially supported by the Com2MaC-SRC/ERC program of MOST/KOSEF (grant # R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175). 相似文献