Waveform Detection of Conspecifics in the Weakly Electric Fish Gymnotus Carapo

Weakly electric fish use electric fields for communication and location of objects. Electroreceptors that are located around the mouth and along the length of the body are used in order to "decode" the electric organ discharge (EOD). The knollenorgan in Mormyriformes aids in distinguishing between different EODs. Gymnotiformes, however, have no such electroreceptors. How then are Gyrnnotiformes distinguishing between conspecific EODs? In this study scan sampling was investigated to determine whether Gymnotus carapo uses this mechanism to differentiate between distinct EODs. After determining whether Gymnotus carapo was discriminating between neighbor and stranger EODs, these same EODs were played to the test fish either jittered (the EOD of the test fish and that of the playback could not coincide) or non-jittered (the two EODs could coincide). The results show that the test fish was not discriminating between neighbor and stranger EODs. Thus, conclusions about the use of scan sampling by Gymnotus carapo to distinguish between EODs cannot be made.

A Method for Testing Object Localization as a Function of Eod Amplitude in the Weakly Electric Fish Brachyhypopomus Pinnicaudatus

I would like to thank Dr. Philip Stoddard for his patience and guidance throughout the past four years. He has not only taught me about behavior and electricity, but he has also taught me how to think scientifically. Vielka Salazar for making herself available to answer my questions and to help me with my projects. Montserrat Alfaro for providing me with support under times of frustration. Fabian A. Pal, who has often made himself available when I needed help to finish my projects, for being supportive, and for believing in me and my abilities. Most importantly, I would like to thank my parents who have shown tremendous support and patience during the past years. I would also like to thank the Honors Committee, specially Dr. Richards for taking the time to review my thesis and helping me modify it. Finally, I would like to thank the MARC program for providing me with financial assistance and the opportunity to perform this project.

The spectra of linear fractional composition operators on weighted Dirichlet spaces

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context.

Rota’s universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In articular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

Plasticity varies with boldness in a weakly-electric fish

The expression of animal personality is indicated by patterns of consistency in  individual behaviour. Often, the differences exhibited between individuals are consistent across situations. However, between some situations, this can be biased by variable levels of individual plasticity. The interaction between individual plasticity and animal personality can be illustrated by examining situation-sensitive personality traits such as boldness (i.e. risk-taking and exploration tendency). For the weakly electric fish Gnathonemus petersii, light condition is a major factor influencing behaviour. Adapted to navigate in low-light conditions, this species chooses to be more active in dark environments where risk from visual predators is lower. However, G. petersii also exhibit individual differences in their degree of behavioural change from light to dark. The present study, therefore, aims to  examine if an increase of motivation to explore in the safety of the dark, not only affects mean levels of boldness, but also the variation between individuals, as a result of differences  in individual plasticity.  Results: Boldness was consistent between a novel-object and a novel-environment situation in bright light. However, no consistency in boldness was noted between a bright (risky) and a  dark (safe) novel environment. Furthermore, there was a negative association between boldness and the degree of change across novel environments, with shier individuals  exhibiting greater behavioural plasticity.  Conclusions: This study highlights that individual plasticity can vary with personality. In  addition, the effect of light suggests that variation in boldness is situation specific. Finally,  there appears to be a trade-off between personality and individual plasticity with shy but  plastic individuals minimizing costs when perceiving risk and bold but stable individuals  consistently maximizing rewards, which can be maladaptive.

Spectrally isometric elementary operators

We present criteria for unital elementary operators (of small length) on unital semisimple Banach algebras to be spectral isometries. The surjective ones among them turn out to be algebra automorphisms.

More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator x↦∑ⁿ_i=1a_ixb_i on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete characterisation when A acts irreducibly on a Banach space of dimension greater than three.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Elementary operators-still not elementary?

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Non-Gaussian distribution of collective operators in quantum spin chains

We numerically analyse the behavior of the full distribution of collective observables in quantum spin chains. While most of previous studies of quantum critical phenomena are limited to the first moments, here we demonstrate how quantum fluctuations at criticality lead to highly non-Gaussian distributions. Interestingly, we show that the distributions for different system sizes collapse on thesame curve after scaling for a wide range of transitions: first and second order quantum transitions and transitions of the Berezinskii–Kosterlitz–Thouless type. We propose and analyse the feasibility of an experimental reconstruction of the distribution using light–matter interfaces for atoms in optical lattices or in optical resonators.

Orbits of semigroups of truncated convolution operators

We prove that a semigroup generated by finitely many truncated convolution operators on $L_p[0, 1]$ with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Pico Cell Integration to Operator’s Incident Management Process

Mobile network coverage is traditionally provided by outdoor macro base stations, which have a long range and serve several of customers. Due to modern passive houses and tightening construction legislation, mobile network service is deteriorated in many indoor locations. Typically, solutions for indoor coverage problem are expensive and demand actions from the mobile operator. Due to these, superior solutions are constantly researched. The solution presented in this thesis is based on Small Cell technology. Small Cells are low power access nodes designed to provide voice and data services.. This thesis concentrates on a specific Small Cell solution, which is called a Pico Cell. The problem regarding Pico Cells and Small Cells in general is that they are a new technological solution for the mobile operator, and the possible problem sources and incidents are not properly mapped. The purpose of this thesis is to figure out the possible problems in the Pico Cell deployment and how they could be solved within the operator’s incident management process. The research in the thesis is carried out with a literature research and a case study. The possible problems are investigated through lab testing. Pico Cell automated deployment process was tested in the lab environment and its proper functionality is confirmed. The related network elements were also tested and examined, and the emerged problems are resolvable. Operators existing incident management process can be used for Pico Cell troubleshooting with minor updates. Certain pre-requirements have to be met before Pico Cell deployment can be considered. The main contribution of this thesis is the Pico Cell integrated incident management process. The presented solution works in theory and solves the problems found during the lab testing. The limitations in the customer service level were solved by adding the necessary tools and by designing a working question pattern. Process structures for automated network discovery and pico specific radio parameter planning were also added for the mobile network management layer..