Numerické řešení matematických modelů popisujících chování materiálů s jemnou strukturou (kompozitní materiály, jemně perforované materiály, atp.) obvykle vyžaduje velký výpočetní výkon. Proto se při numerickém modelování původní materiál nahrazuje ekvivalentním materiálem homogenním. V této práci je k nalezení homogenizovaného materiálu použita dvojškálová konvergence založena na tzv. rozvinovacím operátoru (anglicky unfolding operator). Tento operátor poprvé použil J. Casado-Díaz. V disertační práci je operátor definován jiným způsobem, než jak uvádí původní autor. To dovoluje pro něj dokázat některé nové vlastnosti. Analogicky je definován operátor pro funkce definované na perforovaných oblastech a jsou dokázány jeho vlastnosti. Na závěr je rozvinovací operátor použit k nalezení homogenizovaného řešení speciální skupiny diferenciálních problémů s integrální okrajovou podmínkou. Odvozené homogenizované řešení je ilustrováno na numerických experimentech.

In this thesis, I deal with the measurement of the transverse momentum spectrum of the top quark produced in top-antitop pairs at the LHC at center of mass energy 7 TeV. The analysis is carried out within the ATLAS collaboration. In the single lepton decay channel, I have performed studies on the simulation which were necessary to obtain the final spectrum from real data. I describe basic event selection rules to reduce background events. I study the efficiency of top-antitop pairs reconstruction. I study the unfolding of the measured spectrum which corrects for effects caused by imperfect resolutions. At the end, I show the measured top quark transverse momentum spectrum obtained from my analysis.

<div> The folding and unfolding processes of poly-protein has been tremendously studied recently. The poly-protein dynamics under an external force can play an important role in addressing the issue of the mechanics of muscle tissue. In this research, we use a single-molecule technique: magnetic tweezers to observe the dynamics of 8-mer poly-protein L under different loads applied and then in different Tris-buffered salines. Our result shows that more protein domains unfold as the force load becomes larger. At 6, 7 and 8 pN loads, the poly-protein is most likely to stay in state 1, 3 and 6 with 1, 3 and 6 domains unfolded, respectively according to the probability distribution. This can be well explained by our constructed free energy-related model. The fit results give protein L parameters of persistence length of 0.4 nm, contour length of 18.8 nm and the unfolding energy of 6.5 kT, all in reasonable ranges based on previously reported literature.</div><div> Besides, we also find the dependency of transition rate on force load and salt. The poly-protein has lower transition rate at high force than at low force due to the free energy tilting effect since high force extremely decreases the possibility of protein unfolding that results in a huge drop in the total number of folding and unfolding events. This inverse proportion effect can also be seen in different TRIS-buffered salines (TRIS-150mM NaCl, TRIS-1M NaCl, and TRIS-1M KCl,). We explore the effect of salt concentration, when the concentration of NaCl is increased, the transition rate increases while the probability distribution remains almost the same, indicating the protein unfolding barrier is lowered without altering the overall energy landscape. We attribute this to, first, the charge shielding effect that more interactions between ions and water molecules occur, causing fewer water molecules available to interact with the charged part of protein than before, and, second, more direct interactions of ions with protein that might affect the electrostatic-related transition rate. Considering the effect of salt type, the two 1M alkali metal-chloride salines are compared. We conclude that ions with larger size have less effect on transition rate because ions with smaller size (Na+) can create stronger bonds with water that increase the interference on the protein interaction with water and can easier penetrate into protein to directly interact with the protein.</div>

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This thesis contains several distinct studies aimed at better understanding fracturing in compressional fault-cored folds. At outcrops of growth strata in the Oliana anticline in the Spanish Pyrenees, the relationship of two joint sets may reflect changing mechanical properties (i.e. via diagenesis) during the folding process. Using a Schmidt hammer, I assess the rigidity contrast between the individual units and suggest that late-stage, throughgoing joints formed in strata with conditions similar to those of the present day and that early, bed-contained joints formed when the rigidity contrast between beds was significantly greater than the present day contrast. Modeling algorithms that are used for fracture prediction assume plane strain to construct, model and restore fault-cored folds. Using mechanical models that allow heterogeneous transport in three dimensions, I explore the distribution and magnitude of out-of-plane transport in plunging fault-cored anticlines and provide guidelines of where plane strain should and should not be applied. I show that out-of-plane transport is significant in the simplest non-cylindrical folds, and suggest that complex non-cylindrical structures should not be modeled using plane strain. I mapped five bed-orthogonal fracture sets associated with folding and faulting events at Sant Corneli anticline, a non-cylindrical, fault related anticline in the Spanish Pyrenees. Fold axis perpendicular, calcite healed joint sets associated with similarly oriented normal faulting both pre-date, and are cross cut by calcite healed, N-NW striking joints. Later bed strike oblique joint sets are distinguished by the presence of iron oxide mineralization that probably occurred during Paleocene-Oligocene time. This study directly links fold-related fracturing to fold evolution because fracture sets can be dated relative to the structural evolution of the anticline. I use three-dimensional restorations of Sant Corneli anticline in the Spanish Pyrenees to test the fracture prediction capability of a fully three-dimensional finite element geomechanical restoration algorithm. Reconstruction of the three-dimensional architecture of the syn-tectonic strata provides a template for incrementally unfolding the anticline. Strains predicted by the restorations are compared to the fracture sets that formed over the corresponding time intervals, which are consistent with the observed fracture patterns at Sant Corneli anticline.

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