|
|
The tree property and the continuum function
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Cummings, James (referee) ; Brooke-Taylor, Andrew (referee)
The continuum function is a function which maps every infinite cardinal κ to 2κ. We say that a regular uncountable cardinal κ has the tree property if every κ-tree has a cofinal branch, or equivalently if there are no κ-Aronszajn trees. We say that a regular uncountable cardinal κ has the weak tree property if there are no special κ-Aronszajn trees. It is known that the tree property, and the weak tree property, have the following non-trivial effect on the continuum function: (∗) If the (weak) tree property holds at κ++, then 2κ ≥ κ++. In this thesis we show several results which suggest that (∗) is the only restriction which the tree property and the weak tree property put on the continuum function in addition to the usual restrictions provable in ZFC (monotonicity and the fact that the cofinality of 2κ must be greater than κ; let us denote these conditions by (∗∗)). First we show that the tree property at ℵ2n for every 1 ≤ n < ω, and the weak tree property at ℵn for 2 ≤ n < ω, does not restrict the continuum function below ℵω more than is required by (∗), i.e. every behaviour of the continuum function below ℵω which satisfies the conditions (∗) and (∗∗) is realisable in some generic extension. We use infinitely many weakly compact cardinals (for the tree property) and infinitely many Mahlo...
|
|
| |
|
|
The tree property and the continuum function
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Cummings, James (referee) ; Brooke-Taylor, Andrew (referee)
The continuum function is a function which maps every infinite cardinal κ to 2κ. We say that a regular uncountable cardinal κ has the tree property if every κ-tree has a cofinal branch, or equivalently if there are no κ-Aronszajn trees. We say that a regular uncountable cardinal κ has the weak tree property if there are no special κ-Aronszajn trees. It is known that the tree property, and the weak tree property, have the following non-trivial effect on the continuum function: (∗) If the (weak) tree property holds at κ++, then 2κ ≥ κ++. In this thesis we show several results which suggest that (∗) is the only restriction which the tree property and the weak tree property put on the continuum function in addition to the usual restrictions provable in ZFC (monotonicity and the fact that the cofinality of 2κ must be greater than κ; let us denote these conditions by (∗∗)). First we show that the tree property at ℵ2n for every 1 ≤ n < ω, and the weak tree property at ℵn for 2 ≤ n < ω, does not restrict the continuum function below ℵω more than is required by (∗), i.e. every behaviour of the continuum function below ℵω which satisfies the conditions (∗) and (∗∗) is realisable in some generic extension. We use infinitely many weakly compact cardinals (for the tree property) and infinitely many Mahlo...
|
|
| |
|
|
(100) substrate processing optimization for fabrication of smooth boron doped epitaxial diamond layer by PE CVD
Mortet, Vincent ; Fekete, Ladislav ; Ashcheulov, Petr ; Taylor, Andrew ; Hubík, Pavel ; Trémouilles, D. ; Bedel-Pereira, E.
Boron doped diamond layers were grown in an SEKI AX5010 microwave plasma enhanced chemical vapour deposition system. Effect of surface preparation, i.e. polishing and O2/H2 plasma etching on epitaxial growth on type Ib (100) HPHT synthetic diamonds were investigated. Using optimized substrate preparation, smooth (RRMS ~ 1 nm) boron doped diamond layers with metallic conduction and free of un-epitaxial crystallites were grown with a relatively high growth rate of 3.7 μm/h. Diamond were characterized by optical microscopy, optical profilometry, atomic force microscopy and Hall effect.
|
|
| |
|
|
Nanodiamond seeding of rough substrates
Vlčková, M. ; Stefanovič, M. ; Petrák, V. ; Fendrych, František ; Taylor, Andrew ; Fekete, Ladislav ; Nesládek, M.
Efficient growth of nanocrystalline diamond (NCD) requires nucleation enhancement before chemical vapour deposition step. Nanodiamond (ND) seeding is a commonly used technique that yields high nucleation densities. This technique is well established for conventional planar substrates with low surface roughness. However, many engineering application requires NCD grow on rough and/or non-planar substrates. In this work, we investigate quality of nanodiamond seeding on silicon substrates of high surface roughness (RMS roughness <1 mm). Seeded substrates and deposited diamond films were analysed by atomic force microscopy (AFM), scanning electron microscopy (SEM) and Raman spectroscopy. We discuss influence of nanodiamond particles in seeding solution and seeding technique on nucleation density and quality of deposited NCD film.
|
guest ::
