Brain-computer interfaces (BCIs) are very convenient tools to assess locked-in (LIS) and completely locked-in state (CLIS) patients' hidden states of consciousness. For the time being, there is no ground-truth data in respect to these states for above-mentioned patients. This lack of gold standard makes this problem particularly challenging. In addition to consciousness assessment, BCIs also provide them with a communication device that does not require the presence of motor responses, which they are lacking. Communication plays an important role in the patients' quality of life and prognosis. Significant progress have been made to provide them with EEG-based BCIs in particular. Nonetheless, the majority of existing studies directly dive into the communication part without assessing if the patient is even conscious. Additionally, the few studies that do essentially use evoked brain potentials, mostly the P300, that necessitates the patient's voluntary and active participation to be elicited. Patients are easily fatigued, and would consequently be less successful during the main communication task. Furthermore, when the consciousness states are determined using resting state data, only one or two features were used. In this thesis, different sets of EEG features are used to assess the consciousness level of CLIS patients using resting-state data. This is done as a preliminary step that needed to be succeeded in order to engage to the next step, communication with the patient. In other words, the 'conversation' is initiated only if the patient is sufficiently conscious. This variety of EEG features is utilised to increase the probability of correctly estimating the patients' consciousness states. Indeed, each of them captures a particular signal attribute, and combining them would allow the collection of different hidden characteristics that could have not been obtained from a single feature. Furthermore, the proposed method should allow to determine if communication shall be initiated at a specific time with the patient. The EEG features used are frequency-based, complexity related and connectivity metrics. Besides, instead of analysing results from individual channels or specific brain regions, the global activity of the brain is assessed. The estimated consciousness levels are then obtained by applying two different soft-clustering analysis methods, namely Fuzzy c-means (FCM) and Gaussian Mixture Models (GMM), to the individual features and ensembling their results using their average or their product. The proposed approach is first applied to EEG data recorded from patients with unresponsive wakefulness syndrome (UWS) and minimally conscious state (MCS) (patients with disorders of consciousness (DoC)) to evaluate its performance. It is subsequently applied to data from one CLIS patient that is unique in its kind because it contains a time frame during which the experimenters affirmed that he was conscious. Finally, it is used to estimate the levels of consciousness of nine other CLIS patients. The obtained results revealed that the presented approach was able to take into account the variations of the different features and deduce a unique output taking into consideration the individual features contributions. Some of them performed better than others, which is not surprising since each person is different. It was also able to draw very accurate estimations of the level of consciousness under specific conditions. The approach presented in this thesis provides an additional tool for diagnosis to the medical staff. Furthermore, when implemented online, it would enable to determine the optimal time to engage in communication with CLIS patients. Moreover, it could possibly be used to predict patients' cognitive decline and/or death.

This thesis demonstrates the standard for the design of an experimental model to be used for numerical validation purposes. It is proposed that numerical models may be assessed more accurately and directly by validation with a completely described experimental model, consisting of accurate descriptions of the operating conditions, fluid properties, and experimental uncertainties. This idea is demonstrated using an experimental model of a swirling jet at three Reynolds numbers (Re = 550, 2560, and 3650), with vortex breakdown existing in the higher two Reynolds number cases. Measurements of the swirling jet were obtained at two locations upstream of the jet exit with the intent to provide the flow profiles to the numerical model and four downstream locations used to assess the accuracy of the model. Numerical simulations using the laminar model and k-e, k-w, and k-e-v^2-f turbulence models were used for turbulence closure. Detached Eddy Simulation (DES) and Reynolds-stress model results were also obtained to demonstrate unsteady numerical solutions. The results of the experimental and numerical models are compared to understand the influence on validation using a completely described experimental model.

We study the properties of thinning and Markov chain thinning of renewal processes. Among others, we investigate whether some special renewal processes can be obtained through Markov chain thinning.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

This thesis presents methods for detecting consciousness in patients with complete locked-in syndrome (CLIS). CLIS patients are unable to speak and have lost all muscle movement. Externally, the internal brain activity of such patients cannot be easily perceived, but CLIS patients are considered to be still conscious and cognitively active. Detecting the current state of consciousness of CLIS patients is non-trivial, and it is difficult to ascertain whether CLIS patients are conscious or not. Thus, it is vital to develop alternative ways to re-establish communication with these patients during periods of awareness, and a possible platform is through brain–computer interface (BCI). Since consciousness is required to use BCI correctly, this study proposes a modus operandi to analyze not only in intracranial electrocorticography (ECoG) signals with greater signal-to-noise ratio (SNR) and higher signal amplitude, but also in non-invasive electroencephalography (EEG) signals. By applying three different time-domain analysis approaches sample entropy, permutation entropy, and Poincaré plot as feature extraction to prevent disease-related reductions of brainwave frequency bands in CLIS patients, and cross-validated to improve the probability of correctly detecting the conscious states of CLIS patients. Due to the lack a of 'ground truth' that could be used as teaching input to correct the outcomes, k-Means and DBSCAN these unsupervised learning methods were used to reveal the presence of different levels of consciousness for individual participation in the experiment first in locked-in state (LIS) patients with ALSFRS-R score of 0. The results of these different methods converge on the specific periods of consciousness of CLIS/LIS patients, coinciding with the period during which CLIS/LIS patients recorded communication with an experimenter. To determine methodological feasibility, the methods were also applied to patients with disorders of consciousness (DOC). The results indicate that the use of sample entropy might be helpful to detect awareness not only in CLIS/LIS patients but also in minimally conscious state (MCS)/unresponsive wakefulness syndrome (UWS) patients, and showed good resolution for both ECoG signals up to 24 hours a day and EEG signals focused on one or two hours at the time of the experiment. This thesis focus on consistent results across multiple channels to avoid compensatory effects of brain injury. Unlike most techniques designed to help clinicians diagnose and understand patients' long-term disease progression or distinguish between different disease types on the clinical scales of consciousness. The aim of this investigation is to develop a reliable brain-computer interface-based communication aid eventually to provide family members with a method for short-term communication with CLIS patients in daily life, and at the same time, this will keep patients' brains active to increase patients' willingness to live and improve their quality of life (QOL).

A frequency response tracer technique was used to study the hydraulic properties of a laboratory flow through reactor with variations in reactor size, flow rate and applied mixing. At any one set of conditions the reactor was studied over a range of input sine wave frequencies. Theoretical models consisting of in-series networks of completely mixed segments, plug flow segments, and dead space allowances were developed to approximate the experimental findings. / Thesis / Master of Engineering (ME)

The thesis focuses on an algebraic description of S-boxes by the special type of quadratic equations, defined as biaffine equations. Biaffine equations satisfying S-boxes of higher order may not even exist. However, the special type of S-boxes en- ables to find such equations also for S-boxes of higher order. The S-box in the block cipher Rijndael, composed of the inverse function and the affine transformation, is an example of such special type of S-boxes. The thesis proves that a number of biaffine equations satisfying an S-box of this type does not depend on the affine function. The thesis also proves that for every S-box of order n formed by the in- verse function there exist at least 3n − 1 biaffine equations satisfying this S-box. 1

- ABSTRACT - Noncommutative Choquet theory Let S be a linear subspace of a commutative C∗ -algebra C(X) that se- parates points of C(X) and contains identity. Then the closure of the Choquet boundary of the function system S is the Šilov boundary relati- ve to S. In the case of a noncommutative unital C∗ -algebra A, consider S a self-adjoint linear subspace of A that contains identity and generates A. Let us call S operator system. Then the noncommutative formulation of the stated assertion is that the intersection of all boundary representa- tions for S is the Šilov ideal for S. To that end it is sufficient to show that S has sufficiently many boundary representations. In the present work we make for the proof of that this holds for separable operator system.

Sei G eine p-adische analytische gruppe, welche die direkte Summe einer torsionfreien p-adische analytische gruppe H mit zerfallender halbeinfacher Liealgebra und einer n-dimensionalen abelschen p-adische analytische gruppe Z ist. In Kapitel 3 zeigen wir folgenden Satz: Sei M ein endlich erzeugter Torsionmodul über der Iwasawaalgebra von G, welcher keine nichtrivialen pseudo-null-Untermoduln besitzt. Dann ist q(M), das Bild von M in der Quotientenkategorie Q, genau dann volltreu, wenn M als Modul über der Iwasawaalgebra von Z torsionsfrei ist. Hierbei bezeichne Q den Serre-Quotienten der Kategorie der Moduln über der Iwasawaalgebra von G nach der Serre-Unterkategorie der pseudo-null-Moduln. In Kapitel 4 zeigen wir folgenden Satz: Es bezeichne T die Kategorie, deren Objekte die endlich erzeugten Modulen über der Iwasawaalgebra von G sind, welche auch als Moduln über der Iwasawaalgebra von H endlich erzeugt sind. Seien M, N zwei Objekte von T. Wir nehmen an, dass M, N keine nichttrivialen pseudo-null-Untermoduln besitzen und q(M) in Q volltreu ist. Dann gilt: Ist [M]=[N] in der Grothendieckgruppe von Q, so ist das Bild von N ebenfalls volltreu. In Kapitel 5 zeugen wir folgenden Satz: Sei G eine beliebige p-adische analytische Gruppe, welche keine Element der Ordung p besitzt. Dann sind die Grothendieckgruppen der Algebra stetiger Distributionen und der Algebra beschränkter Distributionen isomorph zu c Kopien des Rings der ganzen Zahlen, wobei c die Anzahl der p-regulären Konjugationsklassen des Quotienten von G nach einer offenen uniformen pro-p-Untergruppe H bezeichnet. / Let G be a compact p-adic analytic group with no element of order p such that it is the direct sum of a torsion free compact p-adic analytic group H whose Lie algebra is split semisimple and an abelian p-adic analytic group Z of dimension n. In chapter 3, we show that if M is a finitely generated torsion module over the Iwasawa algebra of G with no non-zero pseudo-null submodule, then the image q(M) of M via the quotient functor q is completely faithful if and only if M is torsion free over the Iwasawa algebra of Z. Here the quotient functor q is the unique functor from the category of modules over the Iwasawa algebra of G to the quotient category with respect to the Serre subcategory of pseudo-null modules. In chapter 4, we show the following: Let M, N be two finitely generated modules over the Iwasawa algebra of G such that they are objects of the category Q of those finitely generated modules over the Iwasaw algebra of G which are also finitely generated as modules over the Iwasawa algebra of H. Assume that q(M) is completely faithful and [M] =[N] in the Grothendieck group of Q. Then q(N) is also completely faithful. In chapter 6, we show that if G is any compact p-adic analytic group with no element of order p, then the Grothendieck groups of the algebras of continuous distributions and bounded distributions are isomorphic to c copies of the ring of integers where c denotes the number of p-regular conjugacy classes in the quotient group of G with an open normal uniform pro-p subgroup H of G.