A generalization of the katona theorem for crosst-intersecting families

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:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. 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:

1. 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, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

The Maximum Size of 3-Wise Intersecting and 3-Wise Union Families

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     

Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family satisfies F₁∩···∩F_r ≠∅ for all F₁, . . .,F_r ∈ and . We prove that there exists ε=ε(r) >0 such that holds for 1/2≤w≤1/2+ε if r≥13.     

Every Graph Is an Integral Distance Graph in the Plane

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.     

Plasma-induced cathodic discharge electrolysis to form various metal/alloy nanoparticles

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.     

Lattice-dynamics-based characterization of La1-xAxCoO3-δ (A = Sr,Ba) nanoparticles for an inert anode in molten salts

La_1-xA_xCoO_3-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 La_1-xSr_xCoO_3-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 La_1-xSr_xCoO_3-δ were 3.2 and 3.3 for La_0.8Sr_0.2CoO_3-δ and La_0.6Sr_0.4CoO_3-δ, respectively, suggesting that oxygen defects were introduced by Sr ion doping in La_1-xSr_xCoO_3-δ nanoparticles. In contrast, the valencies of Co ion in La_1-xBa_xCoO_3-δ were 3.1 and 3.0 for La_0.5Ba_0.5CoO_3-δ and La_0.4Ba_0.6CoO_3-δ, respectively, suggesting that oxygen defects were introduced slightly by Ba ion doping in La_1-xBa_xCoO_3-δ nanoparticles. The isotropic phonon vibrations of La_1-xA_xCoO_3-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 La_1-xSr_xCoO_3-δ crystal lattice and toward the c-axis direction in the La_1-xBa_xCoO_3-δ crystal lattice. These results suggest that the oxide ion transportation was curtailed using layered nanoparticles to fabricate an inert anode.

     

Extending the Erdős–Ko–Rado theorem

Let be a k‐uniform hypergraph on n vertices. Suppose that holds for all . We prove that the size of is at most if satisfies and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs