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O jednom přibližném řešení marginálního problému
Janžura, Martin
With the aid of the Maximum Entropy principle, a solution to the marginal problem is obtained in a form of parametric exponential (Gibbs-Markov) distribution. The unknown parameters can be calculated by an optimization procedure that agrees with the maximum likelihood estimate but it is numerically hardly feasible for highly dimensional systems. A numerically easily feasible solution can be obtained by the algebraic Möbius formula. The formula, unfortunately, involves terms that are not directly available but can be approximated. And the main aim of the present paper consists in this approximation.
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Výnosová křivka neumožňující arbitráž
Dobiáš, Vladimír ; Kodera, Jan (advisor) ; Pelikán, Jan (referee) ; O Sullivan, Conall (referee)
We address the issue of market incompleteness in the time dimension. Specifically, we focus on interest rate markets and the yield curve extraction. The lack of information about interest rates manifest itself in a non-invertible linear system. The usual approach to circumvent this problem is by applying various curve fitting methods - both parametric and non-parametric. We argue in favor of a novel method relying on information theory, which reformulates the ill-posed linear algebra problem into a well-posed optimization problem, where the linear pricing equations are used as constraints. Local cross entropy is used to determine the optimal solution among the admissible solutions, while all the input prices reflected in constraints are perfectly matched. Large-scale optimization package called AMPL is used extensively throughout this work to obtain the optimal solution as well as to demonstrate the implementation details.
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Fyzika z teorie pravděpodobnosti
Gottvald, Aleš
Following basic ideas of information physics, probability theory features as the inner symmetries of physical laws. Consequently, we conjecture that many fundamental physical facts are already hidden in the unique logical structure of probability theory and need not be postulated. A link with statistical thermodynamics emerges via the exponential (MaxEnt) mapping between probability and entropy, whose scaling symmetry also makes a natural bridge to fractal physics and projective geometries. To facilitate links with many other symmetries and physical areas, the exponential mapping between Lie groups and Lie algebras is suggested as a generalization of the MaxEnt relationship. We point out that the natural space of probability theory is an intrinsically 6-dimensional manifold with two fundamental governing equations imposed, which gives a novel straightforward rationale for the emergence of the 4+6=10-dimensional hyperspace, particularly important in modern particle physics.
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Generating random data from biometric samples
Sachová, Romana ; Říha, Zdeněk (advisor) ; Vondruška, Pavel (referee)
Title: Generating random data from biometric samples Author: Bc. Romana Sachová Department: Department of Algebra Supervisor: Ing. Mgr. Zdeněk Říha, Ph.D. Supervisor's e-mail address: zriha@fi.muni.cz Abstract: This thesis aims to achieve the generation of random data from the bio- metric samples. Studying the biometric characteristics, randomness and generation of random data suitable for cryptography as well the variability of fingerprint, iris, face and human voice. In the practical part has been tested the variability of 200 prints from the same finger, using three factors: 1) The coordinates of fingerprints cores. Due to the repeatability of coordinates the obtained entropy was low. 2) Fingerprint area approximation. It was able to verify the diversity of all areas. The maximum available entropy remains around 15 bits. 3) Ridge lines distortion. From the core to the top of the fingerprint has been taken boxes containing part of the ridge line. For all boxes was calculated the average phase angle of the gradient which represents the change of intensity in the box. Vector of phase angles describes the ridge line distortion. Maximum estimated entropy of this vectors was estimated at 71,586 bits. Keywords: biometry, randomness, entropy 1
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Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations
Papež, Jan ; Strakoš, Zdeněk (advisor) ; Ramage, Alison (referee) ; Vejchodský, Tomáš (referee)
Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe- matics Abstract: Solution of algebraic problems is an inseparable and usually the most time-consuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...
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Algebraic and Kripke semantics of substructural logics
Arazim, Pavel ; Bílková, Marta (advisor) ; Běhounek, Libor (referee)
This thesis is about the distributive full Lambek calculus, i.e., intuicionistic logic without the structural rules of exchange, contraction and weakening and particularly about the two semantics of this logic, one of which is algebraic, the other one is a Kripke semantic. The two semantics are treated in separate chapters and some results about them are shown, for example the disjunction property is proven by amalgamation of Kripke models. The core of this thesis is nevertheless the relation of these two semantics, since it is interesting to study what do they have in common and how can they actually differ, both being a semantics of the same logic. We show how to translate frames to algebras and algebras to frames, and, moreover, we extend such translation to morphisms, thus constructing two functors between the two categories. Key words:distributive FL logic, distributive full Lambek calculus, structural rules, distributive residuated lattice, Kripke frames, frame morphisms, category, functor 2
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States on algebras
Štěpánová, Martina
States on algebras Abstract: States are defined as special cases of a mapping into a set of real numbers. In the thesis, we intro- duce states on ordered Abelian groups, many valued algebras (MV-algebras), generalized many valued algebras (GMV-algebras) and commutative dually residuated lattice ordered monoids (commutative DRl-monoids). We describe some properties of above-mentioned algebras and present a connection among them. For example, GMV-algebras (an algebraic counterpart of the non-commutative infinite valued propositional logic) are a non-commutative generalization of MV-algebras (an algebraic analogy of the Łukasiewicz infinite valued propositional logic) and we can obtain MV-algebras as special cases of DRl-monoids. Existence theorems for states, con- ditions for the uniqueness of states and formulas for the ranges of values of states are introduced here.
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Víceúrovňové hierarchické předpodmínění (úvod do problematiky)
Blaheta, Radim ; Byczanski, Petr
The paper describes hierarchical decompositions and AMLI preconditioners, analysis of hierarchical decomposition methods through CBS constant in 2D and 3D, discusses robustness with respect to anisotropy and element shape and introduces fully algebraic AMLI with aggregation or agglomeration. Finally, the paper discusses recent results concerning algebraic theory of AMLI for nonconforming FEś.
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