Az eszközárazás második alaptétele (The second fundamental theorem of asset pricing)

A dolgozatban röviden bemutatjuk az eszközárazás második alaptételét. A bizonyítás során felhasználjuk a Dalang-Morton-Wilinger tétel bizonyításában használt állításokat. ______ In the article we summarize the results about the second fundamental theorem of asset pricing.

Lattice Boltzmann method for flow and heat transfer in microgeometries

Recent technological developments have made it possible to design various microdevices where fluid flow and heat transfer are involved. For the proper design of such systems, the governing physics needs to be investigated. Due to the difficulty to study complex geometries in micro scales using experimental techniques, computational tools are developed to analyze and simulate flow and heat transfer in microgeometries. However, conventional numerical methods using the Navier-Stokes equations fail to predict some aspects of microflows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. In this study, lattice Boltzmann method (LBM) was used to investigate the flow and heat transfer in micro sized geometries. The LBM depends on the Boltzmann equation which is valid in the whole rarefaction regime that can be observed in micro flows. Results were obtained for isothermal channel flows at Knudsen numbers higher than 0.01 at different pressure ratios. LBM solutions for micro-Couette and micro-Poiseuille flow were found to be in good agreement with the analytical solutions valid in the slip flow regime (0.01 < Kn < 0.1) and direct simulation Monte Carlo solutions that are valid in the transition regime (0.1 < Kn < 10) for pressure distribution and velocity field. The isothermal LBM was further extended to simulate flows including heat transfer. The method was first validated for continuum channel flows with and without constrictions by comparing the thermal LBM results against accurate solutions obtained from analytical equations and finite element method. Finally, the capability of thermal LBM was improved by adding the effect of rarefaction and the method was used to analyze the behavior of gas flow in microchannels. The major finding of this research is that, the newly developed particle-based method described here can be used as an alternative numerical tool in order to study non-continuum effects observed in micro-electro-mechanical-systems (MEMS).

Lattice Boltzmann modeling of fluid flow and solute transport in karst aquifers

A novel modeling approach is applied to karst hydrology. Long-standing problems in karst hydrology and solute transport are addressed using Lattice Boltzmann methods (LBMs). These methods contrast with other modeling approaches that have been applied to karst hydrology. The motivation of this dissertation is to develop new computational models for solving ground water hydraulics and transport problems in karst aquifers, which are widespread around the globe. This research tests the viability of the LBM as a robust alternative numerical technique for solving large-scale hydrological problems. The LB models applied in this research are briefly reviewed and there is a discussion of implementation issues. The dissertation focuses on testing the LB models. The LBM is tested for two different types of inlet boundary conditions for solute transport in finite and effectively semi-infinite domains. The LBM solutions are verified against analytical solutions. Zero-diffusion transport and Taylor dispersion in slits are also simulated and compared against analytical solutions. These results demonstrate the LBM’s flexibility as a solute transport solver. The LBM is applied to simulate solute transport and fluid flow in porous media traversed by larger conduits. A LBM-based macroscopic flow solver (Darcy’s law-based) is linked with an anisotropic dispersion solver. Spatial breakthrough curves in one and two dimensions are fitted against the available analytical solutions. This provides a steady flow model with capabilities routinely found in ground water flow and transport models (e.g., the combination of MODFLOW and MT3D). However the new LBM-based model retains the ability to solve inertial flows that are characteristic of karst aquifer conduits. Transient flows in a confined aquifer are solved using two different LBM approaches. The analogy between Fick’s second law (diffusion equation) and the transient ground water flow equation is used to solve the transient head distribution. An altered-velocity flow solver with source/sink term is applied to simulate a drawdown curve. Hydraulic parameters like transmissivity and storage coefficient are linked with LB parameters. These capabilities complete the LBM’s effective treatment of the types of processes that are simulated by standard ground water models. The LB model is verified against field data for drawdown in a confined aquifer.

Lattice boltzmann modeling of fluid flow to determine the permeability of a karst specimen

A limestone sample was scanned using computed tomography (CT) and the hydraulic conductivity of the 3D reconstructed sample was determined using Lattice- Boltzmann methods (LBM) at varying scales. Due to the shape and size of the original sample, it was challenging to obtain a consistent rectilinear test sample. Through visual inspection however, 91 mm and 76 mm samples were digitally cut from the original. The samples had porosities of 58% and 64% and produced hydraulic conductivity values of K= 13.5 m/s and K=34.5 m/s, respectively. Both of these samples were re-sampled to 1/8 and 1/64 of their original size to produce new virtual samples at lower resolutions of 0.542 mm/lu and 1.084 mm/lu, while still representing the same physical dimensions. The hydraulic conductivity tended to increase slightly as the resolution became coarser. In order to determine an REV, the 91 mm sample was also sub-sampled into blocks that were 1/8 and 1/64 the size of the original. The results were consistent with analytical expectations such as those produced by the Kozeny-Carman equation. A definitive REV size was not reached, however, indicating the need for a larger sample. The methods described here demonstrate the ability of LBM to test rock structures and sizes not normally attainable.

Lattice Boltzmann Modeling of Fluid Flow and Solute Transport in Karst Aquifers

A novel modeling approach is applied to karst hydrology. Long-standing problems in karst hydrology and solute transport are addressed using Lattice Boltzmann methods (LBMs). These methods contrast with other modeling approaches that have been applied to karst hydrology. The motivation of this dissertation is to develop new computational models for solving ground water hydraulics and transport problems in karst aquifers, which are widespread around the globe. This research tests the viability of the LBM as a robust alternative numerical technique for solving large-scale hydrological problems. The LB models applied in this research are briefly reviewed and there is a discussion of implementation issues. The dissertation focuses on testing the LB models. The LBM is tested for two different types of inlet boundary conditions for solute transport in finite and effectively semi-infinite domains. The LBM solutions are verified against analytical solutions. Zero-diffusion transport and Taylor dispersion in slits are also simulated and compared against analytical solutions. These results demonstrate the LBM’s flexibility as a solute transport solver. The LBM is applied to simulate solute transport and fluid flow in porous media traversed by larger conduits. A LBM-based macroscopic flow solver (Darcy’s law-based) is linked with an anisotropic dispersion solver. Spatial breakthrough curves in one and two dimensions are fitted against the available analytical solutions. This provides a steady flow model with capabilities routinely found in ground water flow and transport models (e.g., the combination of MODFLOW and MT3D). However the new LBM-based model retains the ability to solve inertial flows that are characteristic of karst aquifer conduits. Transient flows in a confined aquifer are solved using two different LBM approaches. The analogy between Fick’s second law (diffusion equation) and the transient ground water flow equation is used to solve the transient head distribution. An altered-velocity flow solver with source/sink term is applied to simulate a drawdown curve. Hydraulic parameters like transmissivity and storage coefficient are linked with LB parameters. These capabilities complete the LBM’s effective treatment of the types of processes that are simulated by standard ground water models. The LB model is verified against field data for drawdown in a confined aquifer.

Lattice Boltzmann Method for Flow and Heat Transfer in Microgeometries

Recent technological developments have made it possible to design various microdevices where fluid flow and heat transfer are involved. For the proper design of such systems, the governing physics needs to be investigated. Due to the difficulty to study complex geometries in micro scales using experimental techniques, computational tools are developed to analyze and simulate flow and heat transfer in microgeometries. However, conventional numerical methods using the Navier-Stokes equations fail to predict some aspects of microflows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. In this study, lattice Boltzmann method (LBM) was used to investigate the flow and heat transfer in micro sized geometries. The LBM depends on the Boltzmann equation which is valid in the whole rarefaction regime that can be observed in micro flows. Results were obtained for isothermal channel flows at Knudsen numbers higher than 0.01 at different pressure ratios. LBM solutions for micro-Couette and micro-Poiseuille flow were found to be in good agreement with the analytical solutions valid in the slip flow regime (0.01 < Kn < 0.1) and direct simulation Monte Carlo solutions that are valid in the transition regime (0.1 < Kn < 10) for pressure distribution and velocity field. The isothermal LBM was further extended to simulate flows including heat transfer. The method was first validated for continuum channel flows with and without constrictions by comparing the thermal LBM results against accurate solutions obtained from analytical equations and finite element method. Finally, the capability of thermal LBM was improved by adding the effect of rarefaction and the method was used to analyze the behavior of gas flow in microchannels. The major finding of this research is that, the newly developed particle-based method described here can be used as an alternative numerical tool in order to study non-continuum effects observed in micro-electro-mechanical-systems (MEMS).

O problema da matéria escura e o teorema do virial na teoria de gravidade Eddington-Born-Infeld

Dark matter is a fundamental ingredient of the modern Cosmology. It is necessary in order to explain the process of structures formation in the Universe, rotation curves of galaxies and the mass discrepancy in clusters of galaxies. However, although many efforts, in both aspects, theoretical and experimental, have been made, the nature of dark matter is still unknown and the only convincing evidence for its existence is gravitational. This rises doubts about its existence and, in turn, opens the possibility that the Einstein’s gravity needs to be modified at some scale. We study, in this work, the possibility that the Eddington-Born-Infeld (EBI) modified gravity provides en alternative explanation for the mass discrepancy in clusters of galaxies. For this purpose we derive the modified Einstein field equations and find their solutions to a spherical system of identical and collisionless point particles. Then, we took into account the collisionless relativistic Boltzmann equation and using some approximations and assumptions for weak gravitational field, we derived the generalized virial theorem in the framework of EBI gravity. In order to compare the predictions of EBI gravity with astrophysical observations we estimated the order of magnitude of the geometric mass, showing that it is compatible with present observations. Finally, considering a power law for the density of galaxies in the cluster, we derived expressions for the radial velocity dispersion of the galaxies, which can be used for testing some features of the EBI gravity.

O problema da matéria escura e o teorema do virial na teoria de gravidade Eddington-Born-Infeld

Dark matter is a fundamental ingredient of the modern Cosmology. It is necessary in order to explain the process of structures formation in the Universe, rotation curves of galaxies and the mass discrepancy in clusters of galaxies. However, although many efforts, in both aspects, theoretical and experimental, have been made, the nature of dark matter is still unknown and the only convincing evidence for its existence is gravitational. This rises doubts about its existence and, in turn, opens the possibility that the Einstein’s gravity needs to be modified at some scale. We study, in this work, the possibility that the Eddington-Born-Infeld (EBI) modified gravity provides en alternative explanation for the mass discrepancy in clusters of galaxies. For this purpose we derive the modified Einstein field equations and find their solutions to a spherical system of identical and collisionless point particles. Then, we took into account the collisionless relativistic Boltzmann equation and using some approximations and assumptions for weak gravitational field, we derived the generalized virial theorem in the framework of EBI gravity. In order to compare the predictions of EBI gravity with astrophysical observations we estimated the order of magnitude of the geometric mass, showing that it is compatible with present observations. Finally, considering a power law for the density of galaxies in the cluster, we derived expressions for the radial velocity dispersion of the galaxies, which can be used for testing some features of the EBI gravity.

Reconstrução e reconhecimento de imagens binárias utilizando o algoritmo Máquina de Boltzmann

The objective of this work is to use algorithms known as Boltzmann Machine to rebuild and classify patterns as images. This algorithm has a similar structure to that of an Artificial Neural Network but network nodes have stochastic and probabilistic decisions. This work presents the theoretical framework of the main Artificial Neural Networks, General Boltzmann Machine algorithm and a variation of this algorithm known as Restricted Boltzmann Machine. Computer simulations are performed comparing algorithms Artificial Neural Network Backpropagation with these algorithms Boltzmann General Machine and Machine Restricted Boltzmann. Through computer simulations are analyzed executions times of the different described algorithms and bit hit percentage of trained patterns that are later reconstructed. Finally, they used binary images with and without noise in training Restricted Boltzmann Machine algorithm, these images are reconstructed and classified according to the bit hit percentage in the reconstruction of the images. The Boltzmann machine algorithms were able to classify patterns trained and showed excellent results in the reconstruction of the standards code faster runtime and thus can be used in applications such as image recognition.

Centralizers of normal subgroups and the Z*-theorem

Peer reviewed

Centralizers of normal subgroups and the Z*-theorem

Peer reviewed

Centralizers of normal subgroups and the Z*-theorem

Peer reviewed

Inductive Theorem Proving and Computer Algebra in the Mathweb Software Bus

Reasoning systems have reached a high degree of maturity in the last decade. However, even the most successful systems are usually not general purpose problem solvers but are typically specialised on problems in a certain domain. The MathWeb SOftware Bus (Mathweb-SB) is a system for combining reasoning specialists via a common osftware bus. We described the integration of the lambda-clam systems, a reasoning specialist for proofs by induction, into the MathWeb-SB. Due to this integration, lambda-clam now offers its theorem proving expertise to other systems in the MathWeb-SB. On the other hand, lambda-clam can use the services of any reasoning specialist already integrated. We focus on the latter and describe first experimnents on proving theorems by induction using the computational power of the MAPLE system within lambda-clam.