Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices

Step by step completion of a left-to-right tiling of a rectangular floor with tiles of a single shape starts from one edge of the floor, considers the possible ways of inserting a tile at the leftmost uncovered square, passes through a sequence of rugged shapes of the front line between covered and uncovered regions of the floor, and finishes with a straight front line at the opposite edge. We count the tilings by mapping the front shapes to nodes in a digraph, then counting closed walks on that digraph with the transfer matrix method. Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3 and modestly wide floors. Equivalent results are shown for the 3-dimensional analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or 1 x 2 x 3 into rectangular containers of small cross sections.Comment: 21 pages, 21 figure

Monomer-dimer tatami tilings of square regions

We prove that the number of monomer-dimer tilings of an $n\times n$ square grid, with $m<n$ monomers in which no four tiles meet at any point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity. In addition, we present a new proof of the result that there are $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings into $n$ classes of size $2^{n-1}$. The sum of these tilings over all monomer counts has the closed form $2^{n-1}(3n-4)+2$ and, curiously, this is equal to the sum of the squares of all parts in all compositions of $n$. We also describe two algorithms and a Gray code ordering for generating the $n2^{n-1}$ tilings with $n$ monomers, which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating tatami mat arrangements of square grids, in: 22nd International Workshop on Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on Tatami tilings at http://alejandroerickson.com/joomla/tatami-blog/collected-resource

Structurally and magnetically complex materials: a magneto-caloric, a multiferroic skyrmion host and a ferrimagnetic semiconductor

The study of structural subtleties and complications in novel materials is often neglected and instead their physical functionality is investigated directly. When this is done however, details in the crystal structure can be missed which have significant consequences for the physical properties of the material. This thesis aims to prove this with the use of investigations of three separate inorganic systems which have very different atomic and magnetic structure, and therefore functionality, from each other. These investigations include the synthesis of polycrystalline and single crystal samples, followed by characterisation with the use of diffraction, magnetisation and transport experiments, which are all introduced in Chapter 2. The most important concepts needed in the analysis of the collected data are explained in Chapter 3. Chapter 4 re-examines the structure of incommensurate composite ferrimagnetic semiconductor Ba1–pCr2Se4–p, finding that a single crystal sample has a modulation vector of q = (0,0,0.24144) partly characterised by crenel occupational modulations, creating Se2− dimers. It also has large disorder in its sixfold channel which is modulated in its basal plane with q ≈ (1/3,1/3,0), although chemical composition is found to vary greatly between samples. The magnetic structure has been determined to have P6/m symmetry with k = (0,0,0) with a total moment of 0.413 μB/Cr3+. There is uniaxial negative thermal expansion along the c axis and a previously undiscovered structural transition at 135 K is observed. It is also determined that Ba1–pCr2Se4–p is a p-type semiconductor with a high resistivity owed to hopping of the charge carriers but with very low thermal conductivity. Chapter 5 aims to ascertain the cause of the phase coexistence in the lacunar spinel GaMo4Se8 below its Jahn-Teller transition. It is found that the strain of the dominant R3m phase is large and so is relieved by a partial distortion to the metastable Imm2 phase, which is determined to likely be insulating with charge-ordering of ions, in contrast to the R3m phase which is likely itinerant and charge-disordered. Single domain crystals are grown, on which magnetisation measurements determine the critical behaviour at the Curie temperature, with critical exponents of β = 0.22(4), γ = 1.19(1) and δ = 6.42(1) not belonging to any one universality class. Magnetoentropic mapping confirms the existence of at least three different non-collinear magnetic phases below the Curie temperature, including a Néel-type skyrmion lattice. Chapter 6 scrutinises the PbFCl-like structure and disorder of magneto-caloric candidate material MnZnSb. In some powder diffraction patterns, there are unusual, asymmetric peak shapes that are fit with a reduction of symmetry to a triclinic cell. In another sample, it is apparent that Rietveld refinements are greatly improved with the inclusion of a partially occupied, Zn containing interstitial site. In variable temperature data, the cell volume changes significantly at the near room temperature magnetic transition, which is identified as a big contributor to the large change in entropy there. It is also theorised that due to differences in diffraction data and results from magneto-caloric calculations, that samples first exist in a metastable triclinic state which then becomes the tetragonal state over time

Surface micro-discharge (SMD)

There is an urgent need for an efficient technology of disinfection and sterilization in view of the alarming dimensions health care-associated infections (HAIs) have reached. Cold atmospheric pressure plasma (CAP) can be utilized for hygienic and medical purposes, in particular in surface decontamination applications, and provides a promising alternative to the conventionally used sterilization techniques. Surface Micro-Discharge (SMD) offers a technology for generating CAP that can be applied to wide areas. In the present work, I investigate the plasma chemistry involved in the inactivation of microorganisms by application of the SMD in order to optimize the antimicrobial effect. For this purpose, different strains of vegetative bacteria and bacterial endospores are exposed to the SMD with experimental parameters such as the gas composition, power input, treatment duration and humidity varied. At the same time, the concentration of ozone produced by the SMD is monitored, and its correlation with the antimicrobial efficacy is investigated. I demonstrate that the bactericidal effect of the SMD on both the Gram-negative Escherichia coli and Gram-positive Enterococcus mundtii is similar and strongly correlates with the ozone concentration. The sporicidal effect on Geobacillus stearothermophilus is crucially affected by the humidity, whereas the ozone concentration appears to have no influence. In addition, I investigate the dynamic behavior of ozone produced by the SMD by varying the geometry and the time interval for the plasma generation and by igniting the plasma in two subsequent phases with different frequencies. Possible explanations for the obtained results are provided. This work fortifies the role of SMD as an efficient sterilization method and discloses diverse possibilities for optimizing the antimicrobial effect.Die Häufigkeit der Krankenhausinfektionen und ihre Auswirkung haben alarmierende Ausmaße angenommen. Angesichts dessen besteht ein akuter Bedarf an einer effizienten Technologie für Desinfektion und Sterilisation. Die Anwendung von kaltem atmosphärischen Plasma (englisch: cold atmospheric plasma, CAP) für die Hygiene und Medizin, insbesondere für die Oberflächendekon-tamination, stellt einen Versuch dar, eine schnelle und effektive Sterilisationstechnik zu schaffen. Die Oberflächenentladung (englisch: Surface Micro-Discharge, SMD) bietet eine Technologie für die Erzeugung und Anwendung von CAP. In der vorliegenden Arbeit analysiere ich die antimikrobiellen Eigenschaften der SMD, um die Inaktivierungsmechnismen besser zu verstehen und den antimikrobiellen Effekt zu optimieren. Hierzu werden unterschiedliche Stämme von vegetativen Bakterien und bakteriellen Endosporen unter unterschiedlichen Bedingungen, inklusive der Gaszusammensetzung, der Eingangsleistung, der Behandlungsdauer und der Luftfeuchtigkeit, dem SMD-Plasma ausgesetzt. Gleichzeitig wird die Konzentration des im SMD-Plasma erzeugten Ozons gemessen, um den Zusammenhang mit dem antimikrobiellen Effekt zu untersuchen. Ich zeige in der vorliegenden Arbeit, dass eine SMD-Behandlung auf die gramnegative Escherichia coli und den grampositiven Enterococcus mundtii eine ähnliche Wirkung erzielt. Zudem hängt die bakterizide Wirkung der SMD stark von der Ozonkonzentration ab. Die Luftfeuchtigkeit spielt eine erhebliche Rolle bei der sporiziden Wirkung der SMD auf Geobacillus stearothermophilus, wobei hier die Ozonkonzentration keine Rolle zu spielen scheint. Darüber hinaus untersuche ich die Dynamik der Konzentration des im SMD-Plasma erzeugten Ozons, indem die Geometrie und das Zeitintervall für die Erzeugung der Entladung variiert werden, und indem die Entladung in zwei aufeinander folgenden Phasen mit verschiedenen Frequenzen gezündet wird. Mögliche Erklärungen für die beobachteten Ergebnisse werden vorgeschlagen. Die vorliegende Arbeit untermauert die Rolle der SMD als eine effiziente Sterilisationstechnik und offenbart unterschiedliche Möglichkeiten für die Optimierung des antimikrobiellen Effekts der SMD. Jedoch konnte im Rahmen dieser Arbeit nicht für alle Beobachtungen eine zufriedenstellende Erklärung geliefert werden. Weiterführende Forschungsarbeiten werden daher nahegelegt

Surface micro-discharge (SMD)

There is an urgent need for an efficient technology of disinfection and sterilization in view of the alarming dimensions health care-associated infections (HAIs) have reached. Cold atmospheric pressure plasma (CAP) can be utilized for hygienic and medical purposes, in particular in surface decontamination applications, and provides a promising alternative to the conventionally used sterilization techniques. Surface Micro-Discharge (SMD) offers a technology for generating CAP that can be applied to wide areas. In the present work, I investigate the plasma chemistry involved in the inactivation of microorganisms by application of the SMD in order to optimize the antimicrobial effect. For this purpose, different strains of vegetative bacteria and bacterial endospores are exposed to the SMD with experimental parameters such as the gas composition, power input, treatment duration and humidity varied. At the same time, the concentration of ozone produced by the SMD is monitored, and its correlation with the antimicrobial efficacy is investigated. I demonstrate that the bactericidal effect of the SMD on both the Gram-negative Escherichia coli and Gram-positive Enterococcus mundtii is similar and strongly correlates with the ozone concentration. The sporicidal effect on Geobacillus stearothermophilus is crucially affected by the humidity, whereas the ozone concentration appears to have no influence. In addition, I investigate the dynamic behavior of ozone produced by the SMD by varying the geometry and the time interval for the plasma generation and by igniting the plasma in two subsequent phases with different frequencies. Possible explanations for the obtained results are provided. This work fortifies the role of SMD as an efficient sterilization method and discloses diverse possibilities for optimizing the antimicrobial effect.Die Häufigkeit der Krankenhausinfektionen und ihre Auswirkung haben alarmierende Ausmaße angenommen. Angesichts dessen besteht ein akuter Bedarf an einer effizienten Technologie für Desinfektion und Sterilisation. Die Anwendung von kaltem atmosphärischen Plasma (englisch: cold atmospheric plasma, CAP) für die Hygiene und Medizin, insbesondere für die Oberflächendekon-tamination, stellt einen Versuch dar, eine schnelle und effektive Sterilisationstechnik zu schaffen. Die Oberflächenentladung (englisch: Surface Micro-Discharge, SMD) bietet eine Technologie für die Erzeugung und Anwendung von CAP. In der vorliegenden Arbeit analysiere ich die antimikrobiellen Eigenschaften der SMD, um die Inaktivierungsmechnismen besser zu verstehen und den antimikrobiellen Effekt zu optimieren. Hierzu werden unterschiedliche Stämme von vegetativen Bakterien und bakteriellen Endosporen unter unterschiedlichen Bedingungen, inklusive der Gaszusammensetzung, der Eingangsleistung, der Behandlungsdauer und der Luftfeuchtigkeit, dem SMD-Plasma ausgesetzt. Gleichzeitig wird die Konzentration des im SMD-Plasma erzeugten Ozons gemessen, um den Zusammenhang mit dem antimikrobiellen Effekt zu untersuchen. Ich zeige in der vorliegenden Arbeit, dass eine SMD-Behandlung auf die gramnegative Escherichia coli und den grampositiven Enterococcus mundtii eine ähnliche Wirkung erzielt. Zudem hängt die bakterizide Wirkung der SMD stark von der Ozonkonzentration ab. Die Luftfeuchtigkeit spielt eine erhebliche Rolle bei der sporiziden Wirkung der SMD auf Geobacillus stearothermophilus, wobei hier die Ozonkonzentration keine Rolle zu spielen scheint. Darüber hinaus untersuche ich die Dynamik der Konzentration des im SMD-Plasma erzeugten Ozons, indem die Geometrie und das Zeitintervall für die Erzeugung der Entladung variiert werden, und indem die Entladung in zwei aufeinander folgenden Phasen mit verschiedenen Frequenzen gezündet wird. Mögliche Erklärungen für die beobachteten Ergebnisse werden vorgeschlagen. Die vorliegende Arbeit untermauert die Rolle der SMD als eine effiziente Sterilisationstechnik und offenbart unterschiedliche Möglichkeiten für die Optimierung des antimikrobiellen Effekts der SMD. Jedoch konnte im Rahmen dieser Arbeit nicht für alle Beobachtungen eine zufriedenstellende Erklärung geliefert werden. Weiterführende Forschungsarbeiten werden daher nahegelegt