Separable reduction theorems by the method of elementary submodels

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets "to be dense, nowhere dense, meager, residual or porous" and in properties of functions "to be continuous, semicontinuous or Fr\'echet differentiable". Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as "be continuous on a dense subset", "be Fr\'echet differentiable on a residual subset", etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well.Comment: 27 page

Metric Construction, Stopping Times and Path Coupling

In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling with stopping times, using a metric which allows us to restrict analysis to standard one-step path coupling. This approach provides insight for the design of non-standard metrics giving improvements in the analysis of specific problems. We give illustrative applications to hypergraph independent sets and SAT instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general stopping times theorem (section 2.2), and additonal remarks in section

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We give a new method for analysing the mixing time of a Markov chain using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4 and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19 page

Segmentation of articular cartilage and early osteoarthritis based on the fuzzy soft thresholding approach driven by modified evolutionary ABC optimization and local statistical aggregation

Articular cartilage assessment, with the aim of the cartilage loss identification, is a crucial task for the clinical practice of orthopedics. Conventional software (SW) instruments allow for just a visualization of the knee structure, without post processing, offering objective cartilage modeling. In this paper, we propose the multiregional segmentation method, having ambitions to bring a mathematical model reflecting the physiological cartilage morphological structure and spots, corresponding with the early cartilage loss, which is poorly recognizable by the naked eye from magnetic resonance imaging (MRI). The proposed segmentation model is composed from two pixel's classification parts. Firstly, the image histogram is decomposed by using a sequence of the triangular fuzzy membership functions, when their localization is driven by the modified artificial bee colony (ABC) optimization algorithm, utilizing a random sequence of considered solutions based on the real cartilage features. In the second part of the segmentation model, the original pixel's membership in a respective segmentation class may be modified by using the local statistical aggregation, taking into account the spatial relationships regarding adjacent pixels. By this way, the image noise and artefacts, which are commonly presented in the MR images, may be identified and eliminated. This fact makes the model robust and sensitive with regards to distorting signals. We analyzed the proposed model on the 2D spatial MR image records. We show different MR clinical cases for the articular cartilage segmentation, with identification of the cartilage loss. In the final part of the analysis, we compared our model performance against the selected conventional methods in application on the MR image records being corrupted by additive image noise.Web of Science117art. no. 86

Minimization of Quantum Circuits using Quantum Operator Forms

In this paper we present a method for minimizing reversible quantum circuits using the Quantum Operator Form (QOF); a new representation of quantum circuit and of quantum-realized reversible circuits based on the CNOT, CV and CV$^\dagger$ quantum gates. The proposed form is a quantum extension to the well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the usage of different quantum gates. Therefore QOF permits minimization of quantum circuits by using properties of different gates than only the multi-control Toffoli gates. We introduce a set of minimization rules and a pseudo-algorithm that can be used to design circuits with the CNOT, CV and CV$^\dagger$ quantum gates. We show how the QOF can be used to minimize reversible quantum circuits and how the rules allow to obtain exact realizations using the above mentioned quantum gates.Comment: 11 pages, 14 figures, Proceedings of the ULSI Workshop 2012 (@ISMVL 2012

Torque vectoring differential utilization in the design of the current automobiles

Cílem této bakalářské práce je vytvoření analýzy konstrukčního řešení „Torque Vectoring“ diferenciálů a hnacího ústrojí vozidel, které využívají tento typ diferenciálu. Začátek práce je věnován vysvětlení, co to je diferenciál a jak funguje a dále pak rozdělení diferenciálů. Další část se zabývá aktivními diferenciály, jejich principem funkce a konstrukčním řešením. Konec práce obsahuje konstrukční řešení hnacího ústrojí některých modelů vozidel využívajících aktivní diferenciály. Práce má charakter rešerše a neobsahuje žádné výpočty ani vlastní konstrukční návrhy.The object of this bachelor thesis is to create analysis of the structural design ”Torque Vectoring” differentials and powertrain vehicles which use this type of differential. The project outset is devoted to an explanation of what the differential is and how it works, and then the distribution of differentials. The next section deals with active differentials, their principle function and design solutions. The end of thesis includes design solutions powertrain some models of vehicles using active differentials. The thesis has the character of search and contains no calculations or custom designs.

Numerical Integration in FITkit Platform

Tato práce pojednává o uplatnění numerické metody řešení diferenciálních rovnic pomocí Taylorovy řady. Tento postup byl implementován pomocí specializovaných elementárních numerických procesorů v hradlovém poli FPGA obsaženém ve FITkitu. Bylo tedy nutné vyvinout prostředí, které by dokázalo přijaté hodnoty interpretovat a zobrazit do grafu. Dále bylo pro dobré pochopení postupu výpočtu bylo nutné vyvinout aplikace, které dokázaly prvky zobrazit v podobě, která je vhodná pro neznalého pozorovatele. Je zde také experimentováno s různými úpravami tohoto postupu, aby byla zvýšena rychlost výpočtu.This work discusses the application of numerical methods for solving differential equations using the Taylor series. This procedure was implemented using elementary numerical specialized processors in FPGA contained in FITkit. It was necessary to develop an environment that would be able to interpret the received values and display the graph. It was for a good understanding of the calculation procedure was necessary to develop applications that are able to view the elements in the form appropriate for the uninformed observer. It also experimented with various modifications of this procedure in order to increase the speed of calculation.

Gravitational waves and electroweak baryogenesis in a global study of the extended scalar singlet model

We perform a global fit of the extended scalar singlet model with a fermionic dark matter (DM) candidate. Using the most up-to-date results from the $\mathit{Planck}$ measured DM relic density, direct detection limits from the XENON1T (2018) experiment, electroweak precision observables and Higgs searches at colliders, we constrain the 7-dimensional model parameter space. We also find regions in the model parameter space where a successful electroweak baryogenesis (EWBG) can be viable. This allows us to compute the gravitational wave (GW) signals arising from the phase transition, and discuss the potential discovery prospects of the model at current and future GW experiments. Our global fit places a strong upper $\mathit{and}$ lower limit on the second scalar mass, the fermion DM mass and the scalar-fermion DM coupling. In agreement with previous studies, we find that our model can simultaneously yield a strong first-order phase transition and saturate the observed DM abundance. More importantly, the GW spectra of viable points can often be within reach of future GW experiments such as LISA, DECIGO and BBO.Comment: 42 pages, 10 figures and 2 tables; v2: updated references, submitted to JHEP; v3: corrected typos and updated references, matches version published in JHE