Weak and Semi Compatible Maps in Probabilistic Metric Space Using Implicit Relation

2000 Mathematics Subject Classification: 54H25, 47H10.

On Differential Inclusions with Unbounded Right-Hand Side

2000 Mathematics Subject Classification: 58C06, 47H10, 34A60.

A Harrod modell strukturális stabilitása (Structural stability of the Harrod model)

In this study it is shown that the nontrivial hyperbolic fixed point of a nonlinear dynamical system, which is formulated by means of the adaptive expectations, corresponds to the unstable equilibrium of Harrod. We prove that this nonlinear dynamical (in the sense of Harrod) model is structurally stable under suitable economic conditions. In the case of structural stability, small changes of the functions (C1-perturbations of the vector field) describing the expected and the true time variation of the capital coefficients do not influence the qualitative properties of the endogenous variables, that is, although the trajectories may slightly change, their structure is the same as that of the unperturbed one, and therefore these models are suitable for long-time predictions. In this situation the critique of Lucas or Engel is not valid. There is no topological conjugacy between the perturbed and unperturbed models; the change of the growth rate between two levels may require different times for the perturbed and unperturbed models.

Szolvenciatőke mint fixpont (Solvency Capital as a Fixpoint)

A tanulmányban a szerző a fixpont-iteráció témájával foglalkozik egy elméleti modellben, a biztosítók szolvenciatőkéjének számolásával kapcsolatban. A téma aktualitását a biztosítók tőkemegfelelésével összefüggő Szolvencia II. európai uniós irányelv előreláthatólag közeljövőben várható bevezetése mutatja. Az eredmények alapján megállapítható, hogy az elméleti modellben a biztosítók szolvenciához kapcsolódó tőkeszükséglete matematikai értelemben vett fixpontként is értelmezhető. Bár a gyakorlati tőkeszükséglet-számítások a tanulmányban bemutatottnál jóval összetettebbek, az elméleti eredmények a szolvenciatőke-modellezés érdekes összefüggéseire világítanak rá. _____ In this study fixed point iteration is analyzed in a theoretical model similar to the practical modelling of solvency capital in insurance companies. Actuality of this subject is shown by the approaching practical implementation of the Solvency II directive in the European Union. The results show that in the theoretical model, solvency capital of insurance companies can be interpreted as a fixed point in a mathematical sense. Although practical solvency capital calculations are more complex than those in the study, theoretical results highlight interesting features of solvency capital modelling.

A Kullback-Leibler relatív entrópia függvény alkalmazása páros összehasonlítás mátrix egy prioritásvektora meghatározására (Applying the Kullback-Leibler relative entropy function for determining priorities for the pairwise comparison matrix)

A dolgozatban a döntéselméletben fontos szerepet játszó páros összehasonlítás mátrix prioritásvektorának meghatározására új megközelítést alkalmazunk. Az A páros összehasonlítás mátrix és a prioritásvektor által definiált B konzisztens mátrix közötti eltérést a Kullback-Leibler relatív entrópia-függvény segítségével mérjük. Ezen eltérés minimalizálása teljesen kitöltött mátrix esetében konvex programozási feladathoz vezet, nem teljesen kitöltött mátrix esetében pedig egy fixpont problémához. Az eltérésfüggvényt minimalizáló prioritásvektor egyben azzal a tulajdonsággal is rendelkezik, hogy az A mátrix elemeinek összege és a B mátrix elemeinek összege közötti különbség éppen az eltérésfüggvény minimumának az n-szerese, ahol n a feladat mérete. Így az eltérésfüggvény minimumának értéke két szempontból is lehet alkalmas az A mátrix inkonzisztenciájának a mérésére. _____ In this paper we apply a new approach for determining a priority vector for the pairwise comparison matrix which plays an important role in Decision Theory. The divergence between the pairwise comparison matrix A and the consistent matrix B defined by the priority vector is measured with the help of the Kullback-Leibler relative entropy function. The minimization of this divergence leads to a convex program in case of a complete matrix, leads to a fixed-point problem in case of an incomplete matrix. The priority vector minimizing the divergence also has the property that the difference of the sums of elements of the matrix A and the matrix B is n times the minimum of the divergence function where n is the dimension of the problem. Thus we developed two reasons for considering the value of the minimum of the divergence as a measure of inconsistency of the matrix A.

Extensões multidimensionais para correntropia e suas aplicações em estimativas robustas

This present work uses a generalized similarity measure called correntropy to develop a new method to estimate a linear relation between variables given their samples. Towards this goal, the concept of correntropy is extended from two variables to any two vectors (even with different dimensions) using a statistical framework. With this multidimensionals extensions of Correntropy the regression problem can be formulated in a different manner by seeking the hyperplane that has maximum probability density with the target data. Experiments show that the new algorithm has a nice fixed point update for the parameters and robust performs in the presence of outlier noise.

Control of pattern formation by time-delay feedback with global and local contributions

We consider the suppression of spatiotemporal chaos in the complex GinzburgLandau equation by a combined global and local time-delay feedback. Feedback terms are implemented as a control scheme, i.e., they are proportional to the difference between the time-delayed state of the system and its current state. We perform a linear stability analysis of uniform oscillations with respect to space-dependent perturbations and compare with numerical simulations. Similarly, for the fixed-point solution that corresponds to amplitude death in the spatially extended system, a linear stability analysis with respect to space-dependent perturbations is performed and complemented by numerical simulations. © 2010 Elsevier B.V. All rights reserved.

Essays on Microeconometrics

My dissertation has three chapters which develop and apply microeconometric tech- niques to empirically relevant problems. All the chapters examines the robustness issues (e.g., measurement error and model misspecification) in the econometric anal- ysis. The first chapter studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treat- ment variable is mismeasured and endogenous. I characterize the sharp identified set for the local average treatment effect under the following two assumptions: (1) the exclusion restriction of an instrument and (2) deterministic monotonicity of the true treatment variable in the instrument. The identification strategy allows for general measurement error. Notably, (i) the measurement error is nonclassical, (ii) it can be endogenous, and (iii) no assumptions are imposed on the marginal distribution of the measurement error, so that I do not need to assume the accuracy of the measure- ment. Based on the partial identification result, I provide a consistent confidence interval for the local average treatment effect with uniformly valid size control. I also show that the identification strategy can incorporate repeated measurements to narrow the identified set, even if the repeated measurements themselves are endoge- nous. Using the the National Longitudinal Study of the High School Class of 1972, I demonstrate that my new methodology can produce nontrivial bounds for the return to college attendance when attendance is mismeasured and endogenous.

The second chapter, which is a part of a coauthored project with Federico Bugni, considers the problem of inference in dynamic discrete choice problems when the structural model is locally misspecified. We consider two popular classes of estimators for dynamic discrete choice models: K-step maximum likelihood estimators (K-ML) and K-step minimum distance estimators (K-MD), where K denotes the number of policy iterations employed in the estimation problem. These estimator classes include popular estimators such as Rust (1987)’s nested fixed point estimator, Hotz and Miller (1993)’s conditional choice probability estimator, Aguirregabiria and Mira (2002)’s nested algorithm estimator, and Pesendorfer and Schmidt-Dengler (2008)’s least squares estimator. We derive and compare the asymptotic distributions of K- ML and K-MD estimators when the model is arbitrarily locally misspecified and we obtain three main results. In the absence of misspecification, Aguirregabiria and Mira (2002) show that all K-ML estimators are asymptotically equivalent regardless of the choice of K. Our first result shows that this finding extends to a locally misspecified model, regardless of the degree of local misspecification. As a second result, we show that an analogous result holds for all K-MD estimators, i.e., all K- MD estimator are asymptotically equivalent regardless of the choice of K. Our third and final result is to compare K-MD and K-ML estimators in terms of asymptotic mean squared error. Under local misspecification, the optimally weighted K-MD estimator depends on the unknown asymptotic bias and is no longer feasible. In turn, feasible K-MD estimators could have an asymptotic mean squared error that is higher or lower than that of the K-ML estimators. To demonstrate the relevance of our asymptotic analysis, we illustrate our findings using in a simulation exercise based on a misspecified version of Rust (1987) bus engine problem.

The last chapter investigates the causal effect of the Omnibus Budget Reconcil- iation Act of 1993, which caused the biggest change to the EITC in its history, on unemployment and labor force participation among single mothers. Unemployment and labor force participation are difficult to define for a few reasons, for example, be- cause of marginally attached workers. Instead of searching for the unique definition for each of these two concepts, this chapter bounds unemployment and labor force participation by observable variables and, as a result, considers various competing definitions of these two concepts simultaneously. This bounding strategy leads to partial identification of the treatment effect. The inference results depend on the construction of the bounds, but they imply positive effect on labor force participa- tion and negligible effect on unemployment. The results imply that the difference- in-difference result based on the BLS definition of unemployment can be misleading

due to misclassification of unemployment.

Moduli Space of (0,2) Conformal Field Theories

In this thesis we study aspects of (0,2) superconformal field theories (SCFTs), which are suitable for compactification of the heterotic string. In the first part, we study a class of (2,2) SCFTs obtained by fibering a Landau-Ginzburg (LG) orbifold CFT over a compact K\"ahler base manifold. While such models are naturally obtained as phases in a gauged linear sigma model (GLSM), our construction is independent of such an embedding. We discuss the general properties of such theories and present a technique to study the massless spectrum of the associated heterotic compactification. We test the validity of our method by applying it to hybrid phases of GLSMs and comparing spectra among the phases. In the second part, we turn to the study of the role of accidental symmetries in two-dimensional (0,2) SCFTs obtained by RG flow from (0,2) LG theories. These accidental symmetries are ubiquitous, and, unlike in the case of (2,2) theories, their identification is key to correctly identifying the IR fixed point and its properties. We develop a number of tools that help to identify such accidental symmetries in the context of (0,2) LG models and provide a conjecture for a toric structure of the SCFT moduli space in a large class of models. In the final part, we study the stability of heterotic compactifications described by (0,2) GLSMs with respect to worldsheet instanton corrections to the space-time superpotential following the work of Beasley and Witten. We show that generic models elude the vanishing theorem proved there, and may not determine supersymmetric heterotic vacua. We then construct a subclass of GLSMs for which a vanishing theorem holds.

Some results on orbifold quotients and related objects

The quotient of a finite-dimensional Euclidean space by a finite linear group inherits different structures from the initial space, e.g. a topology, a metric and a piecewise linear structure. The question when such a quotient is a manifold leads to the study of finite groups generated by reflections and rotations, i.e. by orthogonal transformations whose fixed point subspace has codimension one or two. We classify such groups and thereby complete earlier results by M. A. Mikhaîlova from the 70s and 80s. Moreover, we show that a finite group is generated by reflections and) rotations if and only if the corresponding quotient is a Lipschitz-, or equivalently, a piecewise linear manifold (with boundary). For the proof of this statement we show in addition that each piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge by Thurston and confirms a conjecture by Kwasik and Lee. In the topological category a counterexample to the above mentioned characterization is given by the binary icosahedral group. We show that this is the only counterexample up to products. In particular, we answer the question by Davis of when the underlying space of an orbifold is a topological manifold. As a corollary of our results we generalize a fixed point theorem by Steinberg on unitary reflection groups to finite groups generated by reflections and rotations. As an application thereof we answer a question by Petrunin on quotients of spheres.

Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Análise da estrutura e transporte do zooplâncton no estuário do Rio Caravelas (Bahia-Brasil) através do uso do zooscan

Spatial-temporal dynamics of zooplankton in the Caravelas river estuary (Bahia, Brazil). The survey was conducted in order to describe the zooplankton community of the estuary Caravelas (Bahia, Brazil), to quantify and relate the patterns of horizontal and vertical transport with the type of tide (neap and spring) and tidal phase (flood and ebb). Zooplankton samples were collected with the aid of a suction pump (300L), filtered in plankton nets (300μm) and fixed in saline formalin 4%. Samples were collected at a fixed point (A1), near the mouth of the estuary, with samples taken at neap tides and spring tides during the dry and rainy seasons. Samples were collected for 13 hours, at intervals of 1 hour in 3 depths: surface, middle and bottom. Simultaneous collection of biological, we measured the current velocity, temperature and salinity of the water through CTD. In the laboratory, samples were selected for analysis in estereomicroscope, with 25 groups identified, with Copepoda getting the highest number of species. The 168 samples obtained from temporal samples were subsampled and processed on equipment ZooScan, with the aid of software ZooProcess at the end were generated 458.997 vingnettes. 8 taxa were identified automatically, with 16 classified as a semi-automatic. The group Copepoda, despite the limited taxonomic refinement ZooScan, obtained 2 genera and 1 species identified automatically. Among the seasons dry and wet groups Brachyura (zoea), Chaetognatha, and the Calanoid copepods (others), Temora spp., Oithona spp. and Euterpina acutifrons were those who had higher frequency of occurrence, appearing in more than 70% of the samples. Copepoda group showed the largest percentage of relative abundance in both seasons. There was no seasonal variation of total zooplankton, with an average density of 7826±4219 org.m-3 in the dry season, and 7959±3675 org.m-3 in the rainy season, neither between the types and phases of the tides, but seasonal differences were significant recorded for the main zooplankton groups. Vertical stratification was seen for the major zooplankton groups (Brachyura, Chaetognatha, Calanoida (other), Oithona spp, Temora spp. e Euterpina acutifrons). The scale of this stratification varied with the type (square or tide) and tidal phase (flood or ebb). The instantaneous transport was more influenced by current velocity, with higher values observed in spring tides to the total zooplankton, however, there was a variation of this pattern depending on the zooplankton group. According to the data import and export of total zooplankton, the outflow of organisms of the estuary was higher than the input. The results suggest that the estuary of Caravelas may influence the dynamics of organic matter to the adjacent coast, with possible consequences in National Marine Park of Abrolhos

Selbstgesteuertes Lernen mit Kompetenzrastern. Ein theoretischer Blick auf das Potenzial eines pädagogischen Instruments zum individualisierten Lernen

Kompetenzraster sind pädagogische Instrumente, die zum kompetenzorientierten, individualisierten und selbstgesteuerten Lernen in beruflichen Schulen eingesetzt werden. Sie werden üblicherweise im Rahmen eines pädagogischen Gesamtkonzeptes genutzt, indem die Raster oft ein zentrales Instrument in einem komplexen Gefüge schulischer Lern- und Lehrprozesse sind. Kompetenzraster sind häufig der Fixpunkt, an dem sich andere Instrumente (wie Checklisten und Lernjobs) orientieren und sie definieren die Ausgangs- und Zielpunkte der Lernprozesse. Dabei werden den Schülern üblicherweise Freiheitsgrade eingeräumt, so dass sie (mit-) entscheiden ob, was, wann, wie und woraufhin sie lernen. Die schulische Arbeit mit den Rastern kann als ein Versuch angesehen werden, die Lernenden in den Mittelpunkt pädagogischen Denkens und Handelns zu stellen. Dieser Beitrag hat das Ziel, selbstgesteuertes Lernen aus einer distanzierten, vom einzelnen pragmatischen Modell abstrahierenden und eher theoretischen Perspektive auf das individualisierte Lernen mit Kompetenzrastern zu beziehen. Im Kern wird ein Systematisierungsansatz entwickelt, in dem die komplexen Zusammenhänge des Lernens mit Kompetenzrastern im Kontext von selbstgesteuertem Lernen dargestellt werden. Damit soll ein Beitrag zur Elaboration des Lernens mit Kompetenzrastern in beruflichen Schulen geleistet werden. Konkret wird die folgende Frage fokussiert: Was können Kompetenzraster im Rahmen selbstgesteuerten Lernens leisten? (DIPF/Orig.)

Delays in activity based neural networks

In this paper we study the effect of two distinct discrete delays on the dynamics of a Wilson-Cowan neural network. This activity based model describes the dynamics of synaptically interacting excitatory and inhibitory neuronal populations. We discuss the interpretation of the delays in the language of neurobiology and show how they can contribute to the generation of network rhythms. First we focus on the use of linear stability theory to show how to destabilise a fixed point, leading to the onset of oscillatory behaviour. Next we show for the choice of a Heaviside nonlinearity for the firing rate that such emergent oscillations can be either synchronous or anti-synchronous depending on whether inhibition or excitation dominates the network architecture. To probe the behaviour of smooth (sigmoidal) nonlinear firing rates we use a mixture of numerical bifurcation analysis and direct simulations, and uncover parameter windows that support chaotic behaviour. Finally we comment on the role of delays in the generation of bursting oscillations, and discuss natural extensions of the work in this paper.

Análise da estrutura e transporte do zooplâncton no estuário do Rio Caravelas (Bahia-Brasil) através do uso do zooscan

Spatial-temporal dynamics of zooplankton in the Caravelas river estuary (Bahia, Brazil). The survey was conducted in order to describe the zooplankton community of the estuary Caravelas (Bahia, Brazil), to quantify and relate the patterns of horizontal and vertical transport with the type of tide (neap and spring) and tidal phase (flood and ebb). Zooplankton samples were collected with the aid of a suction pump (300L), filtered in plankton nets (300μm) and fixed in saline formalin 4%. Samples were collected at a fixed point (A1), near the mouth of the estuary, with samples taken at neap tides and spring tides during the dry and rainy seasons. Samples were collected for 13 hours, at intervals of 1 hour in 3 depths: surface, middle and bottom. Simultaneous collection of biological, we measured the current velocity, temperature and salinity of the water through CTD. In the laboratory, samples were selected for analysis in estereomicroscope, with 25 groups identified, with Copepoda getting the highest number of species. The 168 samples obtained from temporal samples were subsampled and processed on equipment ZooScan, with the aid of software ZooProcess at the end were generated 458.997 vingnettes. 8 taxa were identified automatically, with 16 classified as a semi-automatic. The group Copepoda, despite the limited taxonomic refinement ZooScan, obtained 2 genera and 1 species identified automatically. Among the seasons dry and wet groups Brachyura (zoea), Chaetognatha, and the Calanoid copepods (others), Temora spp., Oithona spp. and Euterpina acutifrons were those who had higher frequency of occurrence, appearing in more than 70% of the samples. Copepoda group showed the largest percentage of relative abundance in both seasons. There was no seasonal variation of total zooplankton, with an average density of 7826±4219 org.m-3 in the dry season, and 7959±3675 org.m-3 in the rainy season, neither between the types and phases of the tides, but seasonal differences were significant recorded for the main zooplankton groups. Vertical stratification was seen for the major zooplankton groups (Brachyura, Chaetognatha, Calanoida (other), Oithona spp, Temora spp. e Euterpina acutifrons). The scale of this stratification varied with the type (square or tide) and tidal phase (flood or ebb). The instantaneous transport was more influenced by current velocity, with higher values observed in spring tides to the total zooplankton, however, there was a variation of this pattern depending on the zooplankton group. According to the data import and export of total zooplankton, the outflow of organisms of the estuary was higher than the input. The results suggest that the estuary of Caravelas may influence the dynamics of organic matter to the adjacent coast, with possible consequences in National Marine Park of Abrolhos