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The problem of detecting deep space objects with certain probabilities was investigated. A mathematical model was then developed from given problem specifications that deals with the trade-off of various parameters involved in the detection problem. A software package that allows the user to input data interactively was written to implement the model. The completed program as well as an analysis of the tested results are included. / M.S.

This work is concerned with the characterization of certain classes of stochastic processes via duality formulae. In particular we consider reciprocal processes with jumps, a subject up to now neglected in the literature. In the first part we introduce a new formulation of a characterization of processes with independent increments. This characterization is based on a duality formula satisfied by processes with infinitely divisible increments, in particular Lévy processes, which is well known in Malliavin calculus. We obtain two new methods to prove this duality formula, which are not based on the chaos decomposition of the space of square-integrable function- als. One of these methods uses a formula of partial integration that characterizes infinitely divisible random vectors. In this context, our characterization is a generalization of Stein’s lemma for Gaussian random variables and Chen’s lemma for Poisson random variables. The generality of our approach permits us to derive a characterization of infinitely divisible random measures. The second part of this work focuses on the study of the reciprocal classes of Markov processes with and without jumps and their characterization. We start with a resume of already existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. Thus we are able to connect the results of characterizations via duality formulae with the theory of stochastic mechanics by our interpretation, and to stochastic optimal control theory by the mathematical approach. As an application we are able to prove an invariance property of the reciprocal class of a Brownian diffusion under time reversal. In the context of pure jump processes we derive the following new results. We describe the reciprocal classes of Markov counting processes, also called unit jump processes, and obtain a characterization of the associated reciprocal class via a duality formula. This formula contains as key terms a stochastic derivative, a compensated stochastic integral and an invariant of the reciprocal class. Moreover we present an interpretation of the characterization of a reciprocal class in the context of stochastic optimal control of unit jump processes. As a further application we show that the reciprocal class of a Markov counting process has an invariance property under time reversal. Some of these results are extendable to the setting of pure jump processes, that is, we admit different jump-sizes. In particular, we show that the reciprocal classes of Markov jump processes can be compared using reciprocal invariants. A characterization of the reciprocal class of compound Poisson processes via a duality formula is possible under the assumption that the jump-sizes of the process are incommensurable. / Diese Arbeit befasst sich mit der Charakterisierung von Klassen stochastischer Prozesse durch Dualitätsformeln. Es wird insbesondere der in der Literatur bisher unbehandelte Fall reziproker Klassen stochastischer Prozesse mit Sprungen untersucht. Im ersten Teil stellen wir eine neue Formulierung einer Charakterisierung von Prozessen mit unabhängigen Zuwächsen vor. Diese basiert auf der aus dem Malliavinkalkül bekannten Dualitätsformel für Prozesse mit unendlich oft teilbaren Zuwächsen. Wir präsentieren zusätzlich zwei neue Beweismethoden dieser Dualitätsformel, die nicht auf der Chaoszerlegung des Raumes quadratintegrabler Funktionale beruhen. Eine dieser Methoden basiert auf einer partiellen Integrationsformel fur unendlich oft teilbare Zufallsvektoren. In diesem Rahmen ist unsere Charakterisierung eine Verallgemeinerung des Lemma fur Gaußsche Zufallsvariablen von Stein und des Lemma fur Zufallsvariablen mit Poissonverteilung von Chen. Die Allgemeinheit dieser Methode erlaubt uns durch einen ähnlichen Zugang die Charakterisierung unendlich oft teilbarer Zufallsmaße. Im zweiten Teil der Arbeit konzentrieren wir uns auf die Charakterisierung reziproker Klassen ausgewählter Markovprozesse durch Dualitätsformeln. Wir beginnen mit einer Zusammenfassung bereits existierender Ergebnisse zu den reziproken Klassen Brownscher Bewegungen mit Drift. Es ist uns möglich die Charakterisierung solcher reziproken Klassen durch eine Dualitätsformel physikalisch umzudeuten in eine Newtonsche Gleichung. Damit gelingt uns ein Brückenschlag zwischen derartigen Charakterisierungsergebnissen und der Theorie stochastischer Mechanik durch den Interpretationsansatz, sowie der Theorie stochastischer optimaler Steuerung durch den mathematischen Ansatz. Unter Verwendung der Charakterisierung reziproker Klassen durch Dualitätsformeln beweisen wir weiterhin eine Invarianzeigenschaft der reziproken Klasse Browscher Bewegungen mit Drift unter Zeitumkehrung. Es gelingt uns weiterhin neue Resultate im Rahmen reiner Sprungprozesse zu beweisen. Wir beschreiben reziproke Klassen Markovscher Zählprozesse, d.h. Sprungprozesse mit Sprunghöhe eins, und erhalten eine Charakterisierung der reziproken Klasse vermöge einer Dualitätsformel. Diese beinhaltet als Schlüsselterme eine stochastische Ableitung nach den Sprungzeiten, ein kompensiertes stochastisches Integral und eine Invariante der reziproken Klasse. Wir präsentieren außerdem eine Interpretation der Charakterisierung einer reziproken Klasse im Rahmen der stochastischen Steuerungstheorie. Als weitere Anwendung beweisen wir eine Invarianzeigenschaft der reziproken Klasse Markovscher Zählprozesse unter Zeitumkehrung. Einige dieser Ergebnisse werden fur reine Sprungprozesse mit unterschiedlichen Sprunghöhen verallgemeinert. Insbesondere zeigen wir, dass die reziproken Klassen Markovscher Sprungprozesse vermöge reziproker Invarianten unterschieden werden können. Eine Charakterisierung der reziproken Klasse zusammengesetzter Poissonprozesse durch eine Dualitätsformel gelingt unter der Annahme inkommensurabler Sprunghöhen.

Automated map generalization is a necessary technique for the construction of multi-scale vector map databases that are crucial components in spatial data infrastructure of cities, provinces, and countries. Nevertheless, this is still a dream because many algorithms for map feature generalization are not parameter-free and therefore need human’s interference. One of the major reasons is that map generalization is a process of spatial similarity transformation in multi-scale map spaces; however, no theory can be found to support such kind of transformation. This thesis focuses on the theory of spatial similarity relations in multi-scale map spaces, aiming at proposing the approaches and models that can be used to automate some relevant algorithms in map generalization. After a systematic review of existing achievements including the definitions and features of similarity in various communities, a classification system of spatial similarity relations, and the calculation models of similarity relations in the communities of psychology, computer science, music, and geography, as well as a number of raster-based approaches for calculating similarity degrees between images, the thesis achieves the following innovative contributions. First, the fundamental issues of spatial similarity relations are explored, i.e. (1) a classification system is proposed that classifies the objects processed by map generalization algorithms into ten categories; (2) the Set Theory-based definitions of similarity, spatial similarity, and spatial similarity relation in multi-scale map spaces are given; (3) mathematical language-based descriptions of the features of spatial similarity relations in multi-scale map spaces are addressed; (4) the factors that affect human’s judgments of spatial similarity relations are proposed, and their weights are also obtained by psychological experiments; and (5) a classification system for spatial similarity relations in multi-scale map spaces is proposed. Second, the models that can calculate spatial similarity degrees for the ten types of objects in multi-scale map spaces are proposed, and their validity is tested by psychological experiments. If a map (or an individual object, or an object group) and its generalized counterpart are given, the models can be used to calculate the spatial similarity degrees between them. Third, the proposed models are used to solve problems in map generalization: (1) ten formulae are constructed that can calculate spatial similarity degrees by map scale changes in map generalization; (2) an approach based on spatial similarity degree is proposed that can determine when to terminate a map generalization system or an algorithm when it is executed to generalize objects on maps, which may fully automate some relevant algorithms and therefore improve the efficiency of map generalization; and (3) an approach is proposed to calculate the distance tolerance of the Douglas-Peucker Algorithm so that the Douglas-Peucker Algorithm may become fully automatic. Nevertheless, the theory and the approaches proposed in this study possess two limitations and needs further exploration. • More experiments should be done to improve the accuracy and adaptability of the proposed models and formulae. The new experiments should select more typical maps and map objects as samples, and find more subjects with different cultural backgrounds. • Whether it is feasible to integrate the ten models/formulae for calculating spatial similarity degrees into an identical model/formula needs further investigation. In addition, it is important to find out the other algorithms, like the Douglas-Peucker Algorithm, that are not parameter-free and closely related to spatial similarity relation, and explore the approaches to calculating the parameters used in these algorithms with the help of the models and formulae proposed in this thesis.

Mathematics has been used to describe phenomena and problems in many re- search fields for centuries. The basic elements used in the description are formu- lae which express information symbolically. However, searching for mathematical knowledge in digital form using available tools is still cumbersome. We address this issue by presenting the mathematical search engine EgoMath, based on a full text searching, which can search for mathematical formulae and text. We perform an eval- uation over a large collection of documents showing that our solution is usable. Our approach can be used with huge document collections by applying one specialised technique. In order to provide a valuable evaluation of the quality, we built an al- ternative mathematical search engine using the feature extraction technique proposed by Ma et al. We propose important improvements to this solution achieving interest- ing results. We perform the first ever cross-evaluation of mathematical search engines based on different algorithms. A comprehensive survey of existing techniques avail- able, presented in this thesis, completes the picture of mathematical searching.

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El texto completo de este trabajo no está disponible en el Repositorio Académico UPC por restricciones de la casa editorial donde ha sido publicado. / This article proposes a mechanical development of a dry cell in order to obtain HHO through water electrolysis. Calculations and technical specifications of the materials used for implementation are supported by mathematical, physical and chemical formulas and theories (Faraday´s Law, electrolysis process and mechanical design). The importance of mechanical design is focused on achieving efficient use of the energy provided to the cell that allows the H2 and O2 molecules to be separated without overheating the cell, evaporating the water, loss of current due to the geometry of the electrodes (Foucault Current). Moreover, choosing materials for proper implementation and physical robustness is mandatory. In addition, the mechanical design is not justified in different articles. Nevertheless, the mechanical design of the cell and the efficiency in the production of HHO are related. Therefore, the mechanical design and the calculations were performed, as well as the construction of the dry cell to obtain HHO. The results of the implementation and production were placed and compared with what theoretically the dry cell should produce from the law of Faraday. Finally, the volumetric flow of HHO obtained was 2.70 L per minute. It means a production efficiency of 98.68%. It is higher than the majority of the dry cells. / Revisión por pares

No / Rectangular concrete-filled steel tube (CFST) may be subjected to local bearing forces transmitted from brace members while being used as a chord of a truss, and thus development of finite element analysis (FEA) and simple design calculation method for rectangular CFSTs under local bearing forces are very important to ensure the safety and reliable design of such a truss with rectangular CFST chords in engineering practices. A three-dimensional FEA model was developed using ABAQUS software package to predict the performance of thin-walled rectangular CFST under local bearing forces. The preciseness of the predicted results was evaluated by comparison with experimental results reported in the available literature. The comparison and analysis show that the predicted failure pattern, load versus deformation curves and bearing capacity of rectangular CFST under local bearing forces obtained from FEA modelling were generally in good agreement with the experimental observations. After the validation, the FEA model was adopted for the mechanism analysis of typical rectangular CFSTs under local bearing forces. Finally, based on the parametric analysis, simple design equations were proposed to be used to calculate the bearing capacity of rectangular CFST under local bearing forces. / National Natural Science Foundation of China (51421064) and the Natural Science Foundation of Liaoning Province (2013020125). The financial support is gratefully acknowledged.

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