LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:
There are two crosst-intersecting Hamming spheresA0,?0 with centerX such that |A| ≤ |A0| and|?| ≤ |?0| hold.
Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.
Using these claims, the following conjecture of Frankl is proven:
Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereKln is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
Ifn + t = 2k then |A| |? ≤ (Kkn)2 holds.
The extremal configurations are also determined. 相似文献
Let be an n-uniform hypergraph on 2n vertices. Suppose that and holds for all F1,F2,F3 ∈ . We prove that the size of is at most .
The second author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027 相似文献
Let n and r be positive integers. Suppose that a family
satisfies F1∩···∩Fr ≠∅ for all F1, . . .,Fr ∈
and
. We prove that there exists ε=ε(r) >0 such that
holds for 1/2≤w≤1/2+ε if r≥13. 相似文献
We prove that every finite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive. 相似文献
Nanoparticles of various elements such as Si, Al, and Zr were formed by plasma-induced cathodic discharge electrolysis in
molten chloride electrolyte under a 1 atm Ar atmosphere. Al and Si nanoparticles with 100 nm diameters were obtained from
an LiCl-KCl-CsCl melt at 300°C. Zr nanoparticles with diameters less than 50 nm were obtained from an LiCl-KCl at 450°C. Then
with a newly designed and constructed “rotating disk anode type electrolytic cell”, Ti nanoparticles with diameters less than
20 nm were obtained. Finally, to find more appropriate condition for obtaining finer and more uniform nanoparticles, the effects
of the pulse conditions of the applied current and the rotating velocity of a disk anode on size and morphology among the
obtained nanoparticles were investigated by choosing Ni nanoparticle formation as an example. The results showed that quick
removal of the formed fine nanoparticles from the melt surface, where the discharge column is standing, is the most important
factor to obtain smaller and more uniform nanoparticles. 相似文献
La1-xAxCoO3-δ (A = Sr, Ba) nanoparticles used as an inert anode in molten salts were characterized using phonon vibrations, and their compositions and morphologies were investigated. These nanoparticles were used for nanostructure fabrication of an inert anode to reduce oxide ion transportation. The singularity structure changes with increasing Sr ion content in La1-xSrxCoO3-δ nanoparticles showed a transient of spin state change from a low-spin state to intermediate- and/or high-spin states. The valencies of Co ion in La1-xSrxCoO3-δ were 3.2 and 3.3 for La0.8Sr0.2CoO3-δ and La0.6Sr0.4CoO3-δ, respectively, suggesting that oxygen defects were introduced by Sr ion doping in La1-xSrxCoO3-δ nanoparticles. In contrast, the valencies of Co ion in La1-xBaxCoO3-δ were 3.1 and 3.0 for La0.5Ba0.5CoO3-δ and La0.4Ba0.6CoO3-δ, respectively, suggesting that oxygen defects were introduced slightly by Ba ion doping in La1-xBaxCoO3-δ nanoparticles. The isotropic phonon vibrations of La1-xAxCoO3-δ nanoparticles were estimated using high-temperature synchrotron radiation X-ray diffraction measurements. Crystal anisotropy measurements of phonon vibrations indicated that the oxide ions diffused preferentially along the (a, b) plane in the La1-xSrxCoO3-δ crystal lattice and toward the c-axis direction in the La1-xBaxCoO3-δ crystal lattice. These results suggest that the oxide ion transportation was curtailed using layered nanoparticles to fabricate an inert anode.
The electronic structure and the magnetic properties of the ferromagnetic organic polymer poly (m-phenylcarbene ) was studied by application of the unrestricted Hartree-Fock (UHF) crystal orbital (CO) method. In comparison with the restricted Hartree-Fock (RHF) result, it was revealed that the ferromagnetic state is more stable than the non-magnetic state. According to a detailed energy analysis, the stability originates from both the triplet spin configuration at the carbene centre and the delocalized π spins in an antiferromagnetic fashion over the phenyl ring. 相似文献