Weak quasi-randomness for uniform hypergraphs

We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs. Moreover, let K(k) be the complete graph on k vertices and M(k) the line graph of the graph of the k-dimensional hypercube. We will show that the pair of graphs (K(k),M(k)) has the property that if the number of copies of both K(k) and M(k) in another graph G are as expected in the random graph of density d, then G is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to d. (C) 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 1-38, 2012

Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

Hypergraphs with many Kneser colorings

For fixed positive integers r, k and E with 1 <= l < r and an r-uniform hypergraph H, let kappa(H, k, l) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least l elements. Consider the function KC(n, r, k, l) = max(H epsilon Hn) kappa(H, k, l), where the maximum runs over the family H-n of all r-uniform hypergraphs on n vertices. In this paper, we determine the asymptotic behavior of the function KC(n, r, k, l) for every fixed r, k and l and describe the extremal hypergraphs. This variant of a problem of Erdos and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdos-Ko-Rado Theorem (Erdos et al., 1961 [8]) on intersecting systems of sets. (C) 2011 Elsevier Ltd. All rights reserved.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

Diagonal forms, linear algebraic methods and Ramsey-type problems

This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.

As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.

One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.

Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.

Captain Grant K. Chapman in uniform; served in W. W. II

Studio portrait of Captain Grant K. Chapman in uniform; served in W. W. II.

Plasma-assisted self-organized growth of uniform carbon nanocone arrays

The self-organized growth of uniform carbon nanocone arrays using low-temperature non-equilibrium Ar + H 2 + CH 4 plasma-enhanced chemical vapor deposition (PECVD) is studied. The experiment shows that size-, shape-, and position-uniform carbon nanocone arrays can develop even from non-uniformly fragmented discontinuous nickel catalyst films. A three-stage scenario is proposed where the primary nanocones grow on large catalyst particles during the first stage, and the secondary nanocones are formed between the primary ones at the second stage. Finally, plasma-related effects lead to preferential growth of the secondary nanocones and eventually a uniform nanopattern is formed. This does not happen in a CVD process with the same gas feedstock and surface temperature. The proposed three-stage growth scenario is supported by the numerical experiment which generates nanocone arrays very similar to the experimentally synthesized nanopatterns. The self-organization process is explained in terms of re-distribution of surface and volumetric fluxes of plasma-generated species in a developing nanocone array. Our results suggest that plasma-related self-organization effects can significantly reduce the non-uniformity of carbon nanostructure arrays which commonly arises from imperfections in fragmented Ni-based catalyst films.

Inductively coupled plasma-assisted RF magnetron sputtering deposition of highly uniform SiC nanoislanded films

A new deposition technique-inductively coupled plasma-assisted RF magnetron sputtering has been developed to fabricate SiC nanoislanded films. In this system, the plasma production and magnetron sputtering can be controlled independently during the discharge. The deposited SiC nanoislanded films are highly uniform, have excellent stoichiometry, have a typical size of 10-45 nm, and contain small (∼ 6 nm) cubic SiC nanocrystallites embedded in an amorphous SiC matrix.

Generation of uniform plasmas by crossed internal oscillating current sheets : key concepts and experimental verification

The results of comprehensive experimental studies of the operation, stability, and plasma parameters of the low-frequency (0.46 MHz) inductively coupled plasmas sustained by the internal oscillating rf current are reported. The rf plasma is generated by using a custom-designed configuration of the internal rf coil that comprises two perpendicular sets of eight currents in each direction. Various diagnostic tools, such as magnetic probes, optical emission spectroscopy, and an rf-compensated Langmuir probe were used to investigate the electromagnetic, optical, and global properties of the argon plasma in wide ranges of the applied rf power and gas feedstock pressure. It is found that the uniformity of the electromagnetic field inside the plasma reactor is improved as compared to the conventional sources of inductively coupled plasmas with the external flat coil configuration. A reasonable agreement between the experimental data and computed electromagnetic field topography inside the chamber is reported. The Langmuir probe measurements reveal that the spatial profiles of the electron density, the effective electron temperature, plasma potential, and electron energy distribution/probability functions feature a high degree of the radial and axial uniformity and a weak azimuthal dependence, which is consistent with the earlier theoretical predictions. As the input rf power increases, the azimuthal dependence of the global plasma parameters vanishes. The obtained results demonstrate that by introducing the internal oscillated rf currents one can noticeably improve the uniformity of electromagnetic field topography, rf power deposition, and the plasma density in the reactor.

Effective control of ion fluxes over large areas by magnetic fields : from narrow beams to highly uniform fluxes

An effective control of the ion current distribution over large-area (up to 103 cm2) substrates with the magnetic fields of a complex structure by using two additional magnetic coils installed under the substrate exposed to vacuum arc plasmas is demonstrated. When the magnetic field generated by the additional coils is aligned with the direction of the magnetic field generated by the guiding and focusing coils of the vacuum arc source, a narrow ion density distribution with the maximum current density 117 A m-2 is achieved. When one of the additional coils is set to generate the magnetic field of the opposite direction, an area almost uniform over the substrate of 103 cm2 ion current distribution with the mean value of 45 A m-2 is achieved. Our findings suggest that the system with the vacuum arc source and two additional magnetic coils can be effectively used for the effective, high throughput, and highly controllable plasma processing.

Self-assembly of uniform carbon nanotip structures in chemically active inductively coupled plasmas

Self-assembly of carbon nanotip (CNTP) structures on Ni-based catalyst in chemically active inductively coupled plasmas of CH 4 + H 2 + Ar gas mixtures is reported. By varying the process conditions, it appears possible to control the shape, size, and density of CNTPs, content of the nanocrystalline phase in the films, as well as to achieve excellent crystallinity, graphitization, uniformity and vertical alignment of the resulting nanostructures at substrate temperatures 300-500°C and low gas pressures (below 13.2 Pa). This study provides a simple and efficient plasma-enhanced chemical vapor deposition (PECVD) technique for the fabrication of vertically aligned CNTP arrays for electron field emitters.

Calculations of helium separation via uniform pores of stanene-based membranes

The development of low energy cost membranes to separate He from noble gas mixtures is highly desired. In this work, we studied He purification using recently experimentally realized, two-dimensional stanene (2D Sn) and decorated 2D Sn (SnH and SnF) honeycomb lattices by density functional theory calculations. To increase the permeability of noble gases through pristine 2D Sn at room temperature (298 K), two practical strategies (i.e., the application of strain and functionalization) are proposed. With their high concentration of large pores, 2D Sn-based membrane materials demonstrate excellent helium purification and can serve as a superior membrane over traditionally used, porous materials. In addition, the separation performance of these 2D Sn-based membrane materials can be significantly tuned by application of strain to optimize the He purification properties by taking both diffusion and selectivity into account. Our results are the first calculations of He separation in a defect-free honeycomb lattice, highlighting new interesting materials for helium separation for future experimental validation.

Portrait of Moritz Reiss in uniform taken by the Imperial photographer; Vienna Portraits Men

Digital Image

Portrait of Moritz Reiss in uniform taken by the Imperial photographer; Vienna Portraits Men

Digital Image