Mercator projection, loxodromes, and related topics

v českém jazyce- Mercatorova projekce, loxodromy a souvislosti Tato bakalářská práce se zabývá Mercatorovou projekcí sféry do roviny, a to z his- torického i dnešního úhlu pohledu. Pozornost je věnována zejména formálnímu odvození projekce a studiu jejích vlastností. Projekci pak práce zobecňuje i pro obecnou rotační plochu s aplikací pro speciální případ, rotační elipsoid. Dále se zabývá loxodromami na sféře, jejich vlastnostmi a jejich vztahem k Mercatorově projekci. Opět pak tyto poznatky zobecňuje pro obecnou rotační plochu a násled- ně aplikuje na rotační elipsoid. V závěru pak práce zkoumá vztah Mercatorovy projekce se stereografickou projekcí a s exponenciální funkcí v komplexním oboru. 1anglicky- Mercator projection, loxodromes, and related topics This bachelor thesis deals with the Mercator projection from both today's and historical point of view. Special emphasis is placed on the formal derivation of the projection, and on the studying of its properties. The projection for the surface of revolution is also considered. Then the thesis deals with the loxodromes, their properties, and their relation to the Mercator projection. The loxodromes on the surface of revolution are also considered, again. And the thesis ends with some relations between the Mercator projection, the stereographic projection, and the complex exponential function. 1Department of Mathematics EducationKatedra didaktiky matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Mercator projection, loxodromes, and related topics

anglicky- Mercator projection, loxodromes, and related topics This bachelor thesis deals with the Mercator projection from both today's and historical point of view. Special emphasis is placed on the formal derivation of the projection, and on the studying of its properties. The projection for the surface of revolution is also considered. Then the thesis deals with the loxodromes, their properties, and their relation to the Mercator projection. The loxodromes on the surface of revolution are also considered, again. And the thesis ends with some relations between the Mercator projection, the stereographic projection, and the complex exponential function.

Multivariate risk measures in stochastic optimization

The thesis deals with possible generalization of widely used risk measures, Value-at-Risk and Conditio- nal Value-at-Risk, to the multivariate case. First, the theory of p-efficient points, possible generalization of a quantile, is presented. The Prékopa-Vizvári-Badics algorithm for finding p-efficient points in case of random vectors with finite support is presented and a generalization of the algorithm in special case is proposed. Multivariate Value-at-Risk and Multivariate Conditional Value-at-Risk are defined and some of the properties are discussed. A lot-sizing problem for different time horizons is solved.