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1.
Suppose thatA ⊂R
n
is a bounded set of diameter 1 and that:f:A →l
2 is a map satisfying the nearisometry condition |x−y|−ɛ≤|fx−fy|≤|x−y|+ɛ withɛ≤1. Then there is an isometryS:A →l
2 such that |Sx−fx|≤c
n √ɛ for allx inA. IfA satisfies a thickness condition and iff:A →R
n
, then there is an isometryS:R
n
→R
n
with |Sx−fx|≤c
nɛ/q, whereq is a thickness parameter. 相似文献
2.
Let P and Q be disjoint point sets with 2r and 2s elements respectively, and M1 and M2 be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M1 and M2 intersect at most |M1|+|M2|−1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r,s} times. This bound is also sharp.
Supported by PAPIIT(UNAM) of México, Proyecto IN110802
Supported by FAI-UASLP and by CONACYT of México, Proyecto 32168-E
Supported by CONACYT of México, Proyecto 37540-A 相似文献
3.
We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4.
The problem originates from Razborov and Vereshchagin’s work on decision tree complexity.
Research supported in part by the Hungarian Research Fund under grant OTKA T-032969. 相似文献
4.
The bubble-sort graph Bn is a bipartite graph. Kikuchi and Araki [Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs. Information Processing Letters, 100, 52- 59 (2006)] have proved that Bn is edge-bipancyclic for n ≥ 5 and Bn-F is bipancyclic when n ≥ 4 and |F | ≤ n-3. In this paper, we improve this result by showing that for any edge set F of Bn with |F | ≤ n-3, every edge of Bn F lies on a cycle of every even length from 6 to n! for n ≥ 5 and every edge of Bn F lies on a cycle of every even length from 8 to n! for n = 4. 相似文献
5.
We characterize the position of a convex bodyK such that it minimizesM(TK)M*(TK) (theMM*-position) in terms of properties of the measures ‖ · ‖
K
d
σ(·) and ‖ · ‖
K
°d
σ(·), answering a question posed by A. Giannopoulos and V. Milman. The techniques used allow us to study other extremal problems
in the context of dual Brunn-Minkowski theory.
Partially supported by MCYT Grant (Spain) BFM2001-1793 and FEDER funds from UE.
Partially supported by MCYT Grant (Spain) BFM2001-1793 and FEDER funds from UE. 相似文献
6.
We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular
bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings Mk exist with |Mk∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k. 相似文献
7.
Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior.
We also show that for non-empty, finite point sets A
1, ..., A
d+1 in ℝ
d
, if the origin is contained in the convex hull of A
i
∪ A
j
for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A
i
|=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version
of Lovász’ colored Helly theorem.
Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays.
Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted
while visiting the Courant Institute of Mathematical Sciences.
Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF. 相似文献
8.
Alexander Koldobsky 《Discrete and Computational Geometry》2012,47(3):538-547
Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ
n
and any origin-symmetric convex body K in ℝ
n
,
m(K) £ \fracnn-1\frac|B2n|\fracn-1n|B2n-1|maxx ? Sn-1 m(K?x^)\operatornameVoln(K)1/n,\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n}, 相似文献
9.
B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R
n
, n ≥ 3:
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