The literature on bounded rationality and learning in macroeconomics has often used recursive algorithms to depict the evolution of agents' beliefs over time. In this thesis we assess this practice from an applied perspective, focusing on the use of such algorithms for the computation of forecasts of macroeconomic variables. Our analysis develops around three issues we find to have been previously neglected in the literature: (i) the initialization of the learning algorithms; (ii) the determination and calibration of the learning gains, which are key parameters of the algorithms' specifications; and, (iii) the choice of a representative learning mechanism. In order to approach these issues we establish an estimation framework under which we unify the two main algorithms considered in this literature, namely the least squares and the stochastic gradient algorithms. We then propose an evaluation framework that mimics the real-time process of expectation formation through learning-to-forecast exercises. To analyze the quality of the forecasts associated to the learning approach, we evaluate their forecasting accuracy and resemblance to surveys, these latter taken as proxy for agents' expectations. In spite of taking these two criteria as mutually desirable, it is not clear whether they are compatible with each other: whilst forecasting accuracy represents the goal of optimizing agents, resemblance to surveys is indicative of actual agents behavior. We carry out these exercises using real-time quarterly data on US inflation and output growth covering a broad post-WWII period of time. Our main contribution is to show that a proper assessment of the adaptive learning approach requires going beyond the previous views in the literature about these issues. For the initialization of the learning algorithms we argue that such initial estimates need to be coherent with the ongoing learning process that was already in place at the beginning of our sample of data. We find that the previous initialization methods in the literature are vulnerable to this requirement, and propose a new smoothing-based method that is not prone to this critic. Regarding the learning gains, we distinguish between two possible rationales to its determination: as a choice of agents; or, as a primitive parameter of agents learning-to-forecast behavior. Our results provide strong evidence in favor of the gain as a primitive approach, hence favoring the use of surveys data for their calibration. In the third issue, about the choice of a representative algorithm, we challenge the view that learning should be represented by only one of the above algorithms; on the basis of our two evaluation criteria, our results suggest that using a single algorithm represents a misspecification. That motivate us to propose the use of hybrid forms of the LS and SG algorithms, for which we find favorable evidence as representatives of how agents learn. Finally, our analysis concludes with an optimistic assessment on the plausibility of adaptive learning, though conditioned to an appropriate treatment of the above issues. We hope our results provide some guidance on that respect.

Thesis advisor: Ian Biringer / We are interested in proving the following statement: Given a 3-manifold M with boundary and a homeomorphism of the boundary f : ∂M → ∂M such that there is some power that extends to M, there is some k depending only on the genus g(∂M) and some l < k such that ƒᶩ extends to M. We will prove that the power needed to extend is not uniformly bounded with some examples, we will prove the statement is true if M is boundary incompressible and we will show that the general statement reduces to effectivising some technical results about pure homeomorphisms extending to compression bodies. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Isolating chemically-reactive sites into nanosized compartments is an important mode of control used by Nature to perform chemical transformations with extremely high yields and selectivity. Biological systems are fundamentally organized as bounded and isolated nano- and micro-sized environments featuring distinct localized properties, such as steric crowding, polarity, hydrophobicity, potential for molecular recognition, or pH. Through this compartmentalization, reaction substrates are sequestered away from interfering factors and competing substrates, or are physically prevented from forming alternative products or favoring specific pathways. Inspired by Nature, chemists have explored the rational design and application of various nanocompartments. This work explores three types of nanoconfinement systems capable of catalysis and specific transport: surfactant micelles, block-copolymer micelles, and hollow inorganic nanoparticles. The surfactant micelles are designed as part of a system of self-replicating micelles and are used to show how the chirality of the confinement system effects reaction kinetics. Simple click chemistry between a hydrophilic chiral head and a hydrophobic tail is used to produce an amphiphile under biphasic conditions. Once the product achieves critical micelle concentration, stable micelles can form. These micelles subsequently compartmentalize and pre-concentrate hydrophobic substrates, increasing the reaction rate and resulting in the self-propagation of the micellar structures and their chiralities. The next system explores block-copolymer micelles that are made up of a hydrophobic saturated fluorocarbon block and a hydrophilic block. The amphiphilic copolymers can form aggregates in water and, because of properties unique to the hydrophobic block, this system also increases oxygen solubility in water. Different fluorocarbon monomers are discussed and it was found that the structure of the fluorinated monomer, temperature, and pH effect aggregation behavior and the concentration of dissolved oxygen. Additionally, varying the pH of the system could be used to trigger oxygen release. Similar to the block-copolymer micelles, the hollow inorganic nanoparticles were designed to transport oxygen. Here, hollow silica nanoparticles were designed with a fluorinated interior surface and a hydrophilic exterior. This design allows for highly dispersible nanoparticles in water and facilitates the uptake of saturated fluorocarbon liquids into their cores. Once the saturated fluorocarbon is incorporated, the system can les to increases in oxygen solubility.

El texto completo de este trabajo no está disponible en el Repositorio Académico UPC por restricciones de la casa editorial donde ha sido publicado. / The present investigation focuses on the mathematical concept of convexity, as the main tool for the graphic construction of bounded surfaces explicitly and implicitly described, as well as the construction of unconventional solids using GeoGebra. Two cases are presented in which the importance of the concept of convexity is highlighted, in the first situation the convexity is used in the argument of the surface command together with the curves that delimit it to graph a bounded surface, while in the second situation the convexity is evidenced by expressing the coordinates of the surface in parametric form. On the other hand, the 3D graphic view combined with the GeoGebra AR tool allows one to visualize, manipulate, understand and improve the abstraction of mathematical objects that are built in three-dimensional space in a dynamic and friendly environment. These constructions in three-dimensional space that are complex when sketching them with pencil and paper are easier when linking the mathematical definitions with free software such as GeoGebra. / Revisón por pares

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Simulation methods are used when closed-form solutions do not exist. An interesting simulation method that has been widely used in many scientific fields is the Monte Carlo method. Not only it is a simple technique that enables to estimate the quantity of interest, but it can also provide relevant information about the value to be estimated through its confidence interval. However, the use of classical Monte Carlo method is not a reasonable choice when dealing with rare event probabilities. In fact, very small probabilities require a huge number of simulation runs, and thus, the computational time of the simulation increases significantly. This observation lies behind the main motivation of the present work. In this thesis, we propose efficient importance sampling estimators to evaluate rare events probabilities. In the first part of the thesis, we consider a variety of turbulence regimes, and we study the outage probability of free-space optics communication systems under a generalized pointing error model with both a nonzero boresight component and different horizontal and vertical jitter effects. More specifically, we use an importance sampling approach,based on the exponential twisting technique to offer fast and accurate results. We also show that our approach extends to the multihop scenario. In the second part of the thesis, we are interested in assessing the outage probability achieved by some diversity techniques over generalized fading channels. In many circumstances, this is related to the difficult question of analyzing the statistics of the sum of random variables. More specifically, we propose robust importance sampling schemes that efficiently evaluate the outage probability of diversity receivers over Gamma-Gamma, α − µ, κ − µ, and η − µ fading channels. The proposed estimators satisfy the well-known bounded relative error criterion for both maximum ratio combining and equal gain combining cases. We show the accuracy and the efficiency of our approach compared to naive Monte Carlo via some selected numerical simulations in both case studies. In the last part of this thesis, we propose efficient importance sampling estimators for the left tail of positive Gaussian quadratic forms in both real and complex settings. We show that these estimators possess the bounded relative error property. These estimators are then used to estimate the outage probability of maximum ratio combining diversity receivers over correlated Nakagami-m or correlated Rician fading channels

This thesis is devoted to the study of different generalizations of the classical conception of a function of bounded variation. First, we study the functions of bounded p-variation introduced by Wiener in 1924. We obtain estimates of the total p-variation (1&lt;p&lt;∞) and other related functionals for a periodic function f in Lp([0,1]) in terms of its Lp-modulus of continuity ω(f;δ)p. These estimates are sharp for any rate of decay of ω(f;δ)p. Moreover, the constant coefficients in them depend on parameters in an optimal way. Inspired by these results, we consider the relationship between the Riesz type generalized variation vp,α(f) (1&lt;p&lt;∞, 0≤α≤1-1/p) and the modulus of p-continuity  ω1-1/p(f;δ). These functionals generate scales of spaces that connect the space of functions of bounded p-variation and the Sobolev space Wp1. We prove sharp estimates of vp,α(f) in terms of ω1-1/p(f;δ). In the same direction, we study relations between moduli of p-continuity and q-continuity for 1&lt;p&lt;q&lt;∞. We prove an inequality that estimates ω1-1/p(f;δ) in terms of ω1-1/q(f;δ). The inequality is sharp for any order of decay of ω1-1/q(f;δ). Next, we study another generalization of bounded variation: the so-called bounded Λ-variation, introduced by Waterman in 1972. We investigate relations between the space ΛBV of functions of bounded Λ-variation, and classes of functions defined via integral smoothness properties. In particular, we obtain the necessary and sufficient condition for the embedding of the class Lip(α;p) into ΛBV. This solves a problem of Wang (2009). We consider also functions of two variables. Applying our one-dimensional result, we obtain sharp estimates of the Hardy-Vitali type p-variation of a bivariate function in terms of its mixed modulus of continuity in Lp([0,1]2). Further, we investigate Fubini-type properties of the space Hp(2) of functions of bounded Hardy-Vitali p-variation. This leads us to consider the symmetric mixed norm space Vp[Vp]sym of functions of bounded iterated p-variation. For p&gt;1, we prove that Hp(2) is not embedded into Vp[Vp]sym, and that Vp[Vp]sym is not embedded into Hp(2). In other words, Fubini-type properties completely fail in the class of functions of bounded Hardy-Vitali type p-variation for p&gt;1. / Baksidestext The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis. This thesis is devoted to the investigation of various properties of functions of generalized bounded variation.  In particular, we obtain the following results: sharp relations between spaces of generalized bounded variation and spaces of functions  defined by integral smoothness conditions  (e.g., Sobolev and Besov spaces); optimal properties of certain scales of function spaces of frac- tional smoothness generated by functionals of variational type; sharp embeddings within  the scale of spaces of functions of bounded p-variation; results concerning bivariate functions of bounded p-variation, in particular sharp estimates of total variation in terms of the mixed Lp-modulus of continuity, and Fubini-type properties.

The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.

>Magister Scientiae - MSc / This dissertation is a presentation to generalize boundedness and pseudocompactness in pointfree topology. We rst obtain and introduce a boundedness notion for elements of a frame. This is then further inspiration to introduce a de nition of bounded frame homomorphism whose domain may be any frame E, not just the frame of open sets of the reals.

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