This thesis covers different problems that arise due to crystal and pseudospin anisotropy present in 3d and 5d transition metal oxides. We demonstrate that the methods of computational quantum chemistry can be fruitfully used for quantitative studies of such problems. In Chapter 2, Chapter 3, and Chapter 7 we show that it is possible to reliably calculate local multiplet splittings fully ab initio, and therefore help to assign peaks in experimental spectra to corresponding electronic states. In a situation of large number of peaks due to low local symmetry such assignment using semi-empirical methods can be very tedious and non-unique. Moreover, in Chapter 4 we present a computational scheme for calculating intensities as observed in the resonant inelastic X-ray scattering and X-ray absorption experiments. In our scheme highly-excited core-hole states are calculated explicitly taking into account corresponding orbital relaxation and electron polarization. Computed Cu L-edge spectra for the Li2CuO2 compound reproduce all features present in experiment. Unbiased ab initio calculations allow us to unravel a delicate interplay between the distortion of the local ligand cage around the transition metal ions and the anisotropic electrostatic interactions due to second and farther coordination shells. As shown in Chapter 5 and Chapter 6 this interplay can lead to the counter intuitive multiplet structure, single-ion anisotropy, and magnetic g factors. The effect is quite general and may occur in compounds with large difference between charges of metal ions that form anisotropic environment around the transition metal, like Ir 4+ in plane versus Sr 2+ out of plane in the case of Sr2IrO4. An important aspect of the presented study is the mapping of the quantum chemistry results onto simpler physical models, namely extended Heisenberg model, providing an ab initio parametrization. In Chapter 5 we employ the effective Hamiltonian technique for extracting parameters of the anisotropic Heisenberg model with single-ion anisotropy in the case of quenched orbital moment and second-order spin-orbit coupling. Calculated strong easy-axis anisotropy of the same order of magnitude as the symmetric exchange is consistent with experimentally-observer all-in/all-out magnetic order. In Chapter 6 we introduce new flavour of the mapping procedure applicable to systems with first-order spin-orbit coupling, such as 5d 5 iridates based on analysis of the wavefunction and interaction with magnetic field. In Chapter 6 and Chapter 7 we use this new procedure to obtain parameters of the pseudospin anisotropic Heisenberg model. We find large antisymmetric exchange leading to the canted antiferromagnetic state in Sr2IrO4 and nearly ideal one-dimensional Heisenberg behaviour of the CaIrO3, both agree very well with experimental findings.

Für die theoretische Beschreibung verschiedener Substanzen, so z.B. für diverse Kuprate die Anwendungen als Hochtemperatur-Supraleiter finden, spielt das t-J-Modell eine wichtige Rolle. In vielen Fällen kann man Abweichungen der Verbindungen vom idealen translationsinvarianten Festkörper vernachlässigen, für bestimmte Eigenschaften ist jedoch der Einfluß von Störstellen,z.B. Dotieratomen, bedeutsam. Um solche Verunreinigungen einzubeziehen, behandelt die vorliegende Arbeit das t-J-Modell mit einer zusätzlichen on-site-Energie mit über die Gitterpläte zufallsverteilten Werten. Um für dieses Modell die Einteilchen-Greensfunktion zu bestimmen, wird ein Verfahren entwickelt, welches auf der Projektionstechnik basiert und die Einbeziehung des Unordnungsterms ermöglicht. Die notwendige Mittelung über die möglichen Unordnungskonfigurationen erfolgt näherungsweise durch Faktorisierung und ist verwandt mit der sogenannten average T-matrix approximation, wird hier jedoch auf ein stark korreliertes System erweitert. Zur Illustration wird der Grundzustand von La2−xSrxCuO4 und Nd2−xCexCuO4 bei einem zusätzlichen Ladungsträger über Halbfüllung untersucht. Wie Bandstrukturrechnungen zeigen, ruft die Dotierung der elektronendotierten Substanz gerade einen solchen Zufallsterm hervor. Dies wurde in der bisherigen Literatur meist vernachlässigt. Bei der Übertragung der Bandstrukturergebnisse in die Modellparameter des t-t′-t′′-J-Modells zeigt sich, daß der Einfluß der Dotieratome bei La2−xSrxCuO4 um etwa eine Größenordnung geringer ist als in Nd2−xCexCuO4 . Als wichtige Ursache hierfür wird der Einfluß der Apex-Sauerstoffatome angesehen, die im Fall von La2−xSrxCuO4 die Seltenerd- Dotieratome gegenüber der Kupferoxidebene abschirmen. Für das mit diesen Parametern belegte Modell wird anschließend die Einteilchen- Greensfunktion berechnet, die Ausgangspunkt der Berechnung verschiedener Observablen ist. Die für die elektronendotierte Substanz auftretende lokale Mode gibt zu dem Vorschlag Anlaß, daß die unterschiedliche Stabilität der antiferromagnetischen Phase für die beiden betrachteten Substanzen nicht nur auf die Art der Ladungsträger zurückzuführen ist, sondern auch auf die Struktur der Elementarzelle. / The t-J-Modell can be applied to several classes of materials, e.g. high-Tc cuprate superconductors. Often translational invariance can be assumed, but sometimes it is necessary to take into account the effects of the doping atoms at randomly distributed sites. Therefore a t-J-Modell with an additional randomly distributed on-site energy is investigated. To calculate the one-particle Green’s function considering this term of disorder, a method is developed which bases on projection technique. The average over the possible configurations of the dopand atoms is approximated by factorization and is similar to the so-called average T-matrix approximation. Here it is extended to a model with strong correlations. In order to illustrate the methode the single-particle ground state of La2−xSrxCuO4 and Nd2−xCexCuO4 is analyzed. Band-structure calculations exhibit that for the electron-doped case the doping atoms (in first approximation) induce indeed a term of disordered on-site energies. The transformation of the values of this energies at the copper sites into the parameters in the t − t′ − t′′ − J-model shows that the influence of doping in La2−xSrxCuO4 is by about an order of magnitude smaller than in Nd2−xCexCuO4 . The existence of apex oxygen atoms between the rare-earth plane and the copper-oxygen plane in La2−xSrxCuO4 is one important reason for that effect. The single-particle Greens function for the t-t′-t′′-J-model with these parameters is calculated. A local mode appears in the electron-doped case, which suggests that the differences of the stability of the antiferromagnetic phases in both compounds are not only due to the type of charge carriers but also due to the structure of the unit cell.

The IEEE 802.11a protocol is part of the IEEE 802 protocols for implementing WLAN Wi- Fi computer communications in various frequencies. These protocols find applications worldwide, covering a wide range of devices like mobile phones, computers, laptops, household appliances, etc. Since wireless communication is being used, data that is transmitted is susceptible to noise. As a means to recover from noise, the data transmitted is encoded using convolutional encoding and correspondingly decoded on the receiver side. The decoder used is the Viterbi decoder, in the PHY layer of the protocol. This thesis investigates soft-decision Viterbi decoder implementations that meet the requirements of the IEEE 802.11a protocol. It aims to implement a rate-flexible design as a coarse grain re-configurable architecture using the SiLago framework. SiLago is a modular approach towards ASIC design. Components are designed as hardened blocks, which means they are synthesised and pre-verified. Each block is also abuttable like LEGO blocks, which allows users to connect compatible blocks and make designs specific to their requirements, while getting performance similar to that of traditional ASICs. This approach significantly reduces the design costs, as verification is a one-time task. The thesis discusses the strongly connected trellis Viterbi decoding algorithm and proposes a design for a soft decision Viterbi decoder. The proposed design meets the throughput requirements of the communication protocol and it can be reconfigured to work for 45 different code rates, with programmable soft decision width and parallelism. The algorithm used is compared against MATLAB for its BER performance. Results from RTL simulations, advantages and disadvantages of the proposed design are discussed. Recommendations for future improvements are also made. / IEEE 802.11a-protokollet är en del av IEEE 802-protokollen för att implementera WLAN Wi-Fi-datorkommunikation i olika frekvenser. Dessa protokoll används i applikationer över hela världen som täcker ett brett spektrum av produkter som mobiltelefoner, datorer, bärbara datorer, hushållsapparater etc. Eftersom trådlös kommunikation används är data som överförs känslig för brus. Som ett medel för att återhämta sig från brus kodas överförd data med hjälp av faltningskodning och avkodas på motsvarande sätt på mottagarsidan. Den avkodare som används är Viterbi-avkodaren, i PHY-skiktet i protokollet. Denna avhandling undersöker mjuka beslut Viterbi avkodarimplementeringar som uppfyller kraven i IEEE 802.11a protokollet. Det syftar till att implementera en hastighetsflexibel design som en grovkornig konfigurerbar arkitektur som använder SiLago ramverket. SiLago är ett modulärt synsätt på ASIC design. Komponenterna är utformade som härda block, vilket innebär att de är syntetiserade och förverifierade. Varje block kan också kopplas ihop, som LEGO block, vilket gör det möjligt för användare att ansluta kompatibla block och göra designer som är specifika för deras krav, samtidigt som de får prestanda som liknar traditionella ASICs. Detta tillvägagångssätt minskar designkostnaderna avsevärt, eftersom verifiering är en engångsuppgift. Avhandlingen diskuterar den starkt kopplade trellis Viterbi-avkodningsalgoritmen och föreslår en design för en mjuk Viterbi-avkodare. Den föreslagna designen uppfyller kommunikationsprotokollets genomströmningskrav och den kan konfigureras om för att fungera för 45 olika kodhastigheter, med programmerbar mjuk beslutsbredd och parallellitet. Algoritmen som används jämförs mot MATLAB för dess BER-prestanda. Resultat från RTL-simuleringar, fördelar och nackdelar med den föreslagna designen diskuteras. Rekommendationer för framtida förbättringar görs också.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

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In this work, we advance the understanding of the fundamental limits of computation for binary polynomial optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is β-acyclic. We note that the β-acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on α-acyclic instances.Our algorithm can also be applied to any general BPO problem that contains β-cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance.

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