Dissertations / Theses on the topic 'Q-Analogs'

Ce travail est consacré à l'étude des q-analogues de nombre réels. La q-déformation d'un nombre rationnel utilisée est une fraction rationnelle à coefficients entiers qui a été introduite par Sophie Morier-Genoud et Valentin Ovsienko en 2019. Il s'agit dans un premier temps de préciser les propriétés algébriques et de donner des interprétations combinatoires des q-rationnels. On manipule les différentes notions liées aux nombres rationnels: les fractions continues, les matrices de PSL(2,Z), les continuants d'Euler, les triangulations de polygones et le graphe de Farey, et leurs versions q-déformées. Les traces des q-matrices de PSL(2,Z) obtenues sont étudiées et interprétées dans le modèle combinatoire de triangulations d'anneaux. Dans un second temps, nous nous intéressons à la q-déformation des nombres irrationnels et plus particulièrement des irrationnels quadratiques. Nous obtenons une formule explicite permettant d'écrire les q-irrationnels quadratiques. On s'intéresse aux rayons de convergence des séries de Laurent obtenues à partir des q-déformations des nombres réels. Enfin, nous introduisons un second paramètre pour obtenir des (q,t)-déformations de nombres. Ces dernières sont étudiées sous un aspect combinatoire pour affiner les interprétations dans les modèles déjà présentes mais également dans les graphes en serpents
This work is devoted to the study of q-analogs of real numbers. The q-deformation of a rational number that we use is a rational funtion with integer coefficientswhich was introduced by Sophie Morier-Genoud and Valentin Ovsienko in 2019. The first step is to elaborate algebraic properties and to give combinatorial interpretations of the q-rationals. We use different notions linked to rational numbers: continued fractions, PSL(2,Z) matrices, Euler continuants, polygon's triangulations and the Farey graph, and their q-deformed versions.The traces of the q-matrices of PSL(2,Z) that we obtained are studied and interpreted in the combinatorial model of triangulation of annulus. In a second stage, we focus on the q-deformations of irrational real numbers, and more precisely on quadratic irrational real numbers. We obtain an explicit formula to describe q-deformed quadratic irrationals. We give estimate for the radii of convergence of the Laurent series obtained from the q-deformations of real numbers. Finally, we introduce a second parameter to obtain (q, t)-deformations of the rationals. The latter is studied in its combinatorial aspect, in the models already described but also in terms of snake graphs

Nous presentons un resume rapide de la theorie de cristallisation pour les modules de dimension finie de l'algebre enveloppante quantifiee u q(g). Nous construisons d'operateurs cristallins sur les modules de verma de u q(g), ayant des proprietes de compatibilite avec la forme de shapovalov et avec la tensorisation a gauche par des modules de dimension finie, generalisant celles des operateurs cristallins sur les modules de dimension finie. En particulier, nous introduisons la classe des semi-cristaux, incluant celle des cristaux, et admettant un produit tensoriel. Les restrictions sur cette construction impliquent qu'elle est probablement unique. Nous demontrons que la generalisation directe aux modules de verma de la construction des bases cristallines de kashiwara est impossible. Il est possible toutefois, de definir des fonctions i, fonctions caracterisant un semi-cristal en tant qu'ensemble, pour ces modules, de maniere compatible avec la tensorisation a gauche par des modules de dimension finie, et les inclusions entre modules de verma correspondant a l'ordre de bruhat faible. Ces fonctions permettent l'introduction de nouveaux q-analogues multiples de la multiplicite, les caracteres cristallins. Les resultats pointent vers l'existence d'une formule de weyl pour ces caracteres. Nous considerons ensuite la construction algebrique par r. K. Brylinsky des q-analogues de lusztig. Ces analogues ne sont connus qu'essentiellement pour les poids dominants d'un module simple de dimension finie. Nous presentons une methode de calcul de ces analogues pour n'importe quel poids se basant sur la l'existence d'une base p. B. W. Special du o-dual d'un module de verma, ainsi que sur la combinatoire du modele de chemins de littelmann. Nous l'appliquons pour calculer les polynomes de brylinsky pour g sl 3. Nous conjecturons que les coefficients de ces polynomes sont invariants sous l'action du groupe de weyl. Nous calculons leur terme principal lorsque g est simplement lacee, ce qui implique dans ce cas la validite de la conjecture pour les u (g) modules simples de dimension finie dont les espaces poids sont de dimension 1.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 67).
The main result of this work is a q-analogue relationship between nilpotent transformations and spanning trees. For example, nilpotent endomorphisms on an n-dimensional vector space over Fq is a q-analogue of rooted spanning trees of the complete graph Kn. This relationship is based on two similar bijective proofs to calculate the number of spanning trees and nilpotent transformations, respectively. We also discuss more details about this bijection in the cases of complete graphs, complete bipartite graphs, and cycles. It gives some refinements of the q-analogue relationship. As a corollary, we find the total number of nilpotent transformations with some restrictions on Jordan block sizes.
by Jingbin Yin.
Ph.D.

In this thesis we investigate the rational and Riccati type special solutions for particular parameter values of a q-discrete analogue of the third Painlevé equation, with rational surface A(1) 5 and affine Weyl group (A2 + A1)(1). The general solutions of this equation are highly transcendental in nature. We work closely with an associated system of discrete linear equations, which we refer to as ‘the linear problem.’We demonstrate that the linear problem can be solved both in terms of q-Gamma functions and series expansions for different parameter values of our discrete Painlevé equation. By developing a Schlesinger transformation, which transforms a series expansion of the linear problem for one parameter value to another series expansion of the linear problem for another parameter value, we are able to develop determinantal representations of the rational and Riccati type special solutions. These determinantal forms appear different to those discovered previously by Kajiwara. This technique has only been used to develop the determinantal forms of two other continuous and discrete Painlevé equations and hence the results presented here further indicate its potential.

Le modèle du tas de sable est un modèle de diffusion discret et isotrope introduit par les physiciens Bak, Tang et Wiesenfeld comme illustration de la criticalité auto-organisée. Pour tout graphe, souvent supposé fini, Dhar a formalisé de nombreuses propriétés simplifiant son analyse. Cette thèse propose des études de ce modèle sur la grille bidimensionnelle usuelle et certains de ses sous-graphes également infinis que sont les bandes bi-infinies de hauteur finie. Des approximations du comportement de la pile de sable peuvent se rapprocher de certains modèles de bootstrap percolation avec un support de stabilisation rectangulaire. Les lois sur son demi-périmètre peuvent se décrire à l’aide de statistiques sur les permutations. Un sous-produit de ce travail fait apparaître une différence de deux séries génératrices comptant des permutations selon deux statistiques mahoniennes classiques dont est extrait un polynôme à coefficients entiers et surtout positifs. La suite de cette thèse revisite dans le cadre de ces graphes infinis, des structures jusque-là bien définies uniquement dans le cas des graphes finis, notamment la récurrence. Dans le modèle sur une bande de hauteur finie H, l’existence donnée par Járai et Lyons d’automates finis reconnaissant les configurations récurrentes lues colonne par colonne est étendu par une construction explicite d’automates avec un nombre moindre d’états, se rapprochant de la conjecture de Gamlin. Dans une seconde approche, l’étude se concentre sur les configurations sur la grille entière qui sont périodiques dans les deux directions. Le puits, un sommet du graphe garantissant la terminaison de la stabilisation, est placé à l’infini dans une direction de pente rationnelle. Ceci permet à la fois de préserver la bipériodicité et de proposer une forme affaiblie du critère de Dhar caractérisant ainsi par un algorithme effectif les configurations récurrentes. Ces configurations récurrentes bipériodiques sont des candidates naturelles pour être les éléments de sous-groupes finis de l’éventuel groupe du tas de sable sur la grille. Des éléments de construction de cette loi de groupe donnent expérimentalement quelques sous-groupes finis
The sandpile model is a discrete model for diffusion of grains on a graph introduced by physicists Bak, Tang and Wiesenfeld as an illustration for self-organised criticality. For any finite graph, Dhar identified many of its numerous structures which simplify its analysis. This thesis focus on the usual square lattice and its subgraphs which are strips of height H, both notions of infinite graphs. Approximations on the behaviour of the stabilisation of a large stack of grains at the origin of the square lattice lead to some random distribution of grains, which stabilisation is connected to some models of bootstrap percolation where modified vertices by this stabilisation forms a rectangle. The laws of the half-perimeter of this rectangle are described by statistics on permutations. As a byproduct, the difference between the generating functions over some permutations of two classical mahonian statistics on permutations appears to mainly be a polynomial with coefficients which are integers and especially positive. Then, this thesis visits in the case of the studied infinite graphs some well-defined structures on finite graphs, in particular the recurrence. In the model on an horizontal strip of height H, we extend the existence of finite automata recognizing recurrent configurations read column by column presented by Járai and Lyons to new automata with significantly less states and these numbers are closer to a conjecture due to Gamlin. An implementation leads to explicit automata for heights 3 and 4 while up to now only the case 2 was obtained by hand. In a second approach, we consider the configurations on the twodimensional square lattice which are periodic in two directions. We suggest to place the sink ensuring that the stabilisation ends at infinity in a direction of rational slope which allows to preserve biperiodicity and a weaker form of Dhar criterion for recurrent configurations. Hence we obtain an effective algorithm defining recurrent configurations among the biperiodic and stable configurations. These biperiodic and recurrent configurations are natural candidates for being the elements of finite subgroups of the hypothetical group on configurations of the sandpile model on the square lattice. We discuss some notions allowing the definition of the law of such a group and experimentally provide some finite subgroups

We prove the positivity of Lusztig's q-analogue of weight multiplicity in a purely combinatorial way for rank 2 Lie algebras. Each summand in the polynomial can be interpreted as a linear combination of positive roots. We prove that all negative coefficients are cancelled in the polynomial. Further, the analysis of the root systems allows us to state formulae for every coefficient in Lusztig's q-analogue for rank 2 Lie algebras.
Ph. D.

A perda da função das células beta acelera a deterioração do controle metabólico em pessoas com diabetes tipo 2. Além da lipo- e da glicotoxicidade, os AGEs parecem contribuir para esse processo, promovendo a apoptose das ilhotas pancreáticas. Em outros tecidos, os AGEs interagem com seu receptor específico (RAGE), produzindo espécies reativas de oxigênio (ROS) e ativando o NF-kB. Para investigar o efeito temporal dos AGEs sobre a apoptose de ilhotas, bem como o potencial de compostos antioxidantes para diminuir danos causados pelos AGEs, ilhotas pancreáticas de ratos foram tratadas durante 24, 48, 72, 96 e 120 h com AGEs gerados a partir de co-incubação de albumina de soro bovino (BSA) com Dgliceraldeído (GAD, 5 mg/mL) ou tampão fostato (controle). A apoptose foi avaliada pela quantificação do DNA fragmentado (ELISA), atividade de caspase 3 e detecção da permeabilidade da membrana mitocondrial (MitoProbe JC-1). O estresse oxidativo foi avaliado pela detecção de espécies de oxigênio (Image-iT LIVE Green) e a atividade da NADPH oxidase foi mensurada pelo método de quimioluminescência da lucigenina. A expressão dos genes Bax, Bcl2 e Nfkb1 foi avaliada por reação em cadeia da polimerase quantitativa após transcrição reversa (RT-qPCR). Em um dos tempos em que foi detectado o aumento da apoptose, o efeito de dois compostos antioxidantes foi avaliado: benfotiamina (350 M), uma vitamina B1 lipossolúvel, e Mito Q (1 M), um derivado da ubiquinona com alvo seletivo para a mitocôndria. Em 24 e 48 h, os AGES promoveram um aumento do índice de apoptose em relação ao controle, concomitantemente com o aumento na expresssão do gene Bcl2 (gene anti-apoptótico) e uma redução na expressão do gene Nfkb1. Em contraste, após 72, 96 h e 120 h, os AGEs promoveram um aumento do índice de apoptose em comparação com a condição de controle, concomitantemente com uma diminuição na expressão do gene Bcl2 e um aumento na expressão do gene Nfkb1. Em 24 h, os AGEs promoveram uma diminuição do conteúdo de ROS nas ilhotas, enquanto que nos tempos de 48 e 72 h, os AGEs promoveram um efeito oposto. A benfotiamina e o Mito Q foram capazes de diminuir o índice de apoptose e o estresse oxidativo de ilhotas expostas aos AGEs por 72 h. Em conclusão, os AGEs exerceram um duplo efeito em cultura de ilhotas pancreáticas, sendo de proteção contra a apoptose após exposição curta, mas pró-apoptótica após exposição prolongada. O Mito Q e e a benfotiamina merecem ser adicionalmente estudados como drogas com o potencial de oferecer proteção às ilhotas pancreáticas em condições de hiperglicemia crônica
Loss of beta cell function hastens the deterioration of metabolic control in people with type 2 diabetes. Besides lipo- and glucotoxicity, AGEs seem to contribute to this process by promoting islet apoptosis. In other tissues, AGEs interact with their specific receptors (RAGE) and elicit reactive oxygen species (ROS) generation and NF-kB activation. In order to investigate the temporal effect of AGEs on islet apoptosis as well as the potential of antioxidant compounds to decrease islet damage caused by AGEs, rat pancreatic islets were treated for 24, 48, 72, 96 and 120 h with either AGEs generated from co-incubation of bovine serum albumin (BSA) with D-glyceraldehyde (GAD, 5 mg/mL) or phosphate-buffered saline (PBS, control). Apoptosis was evaluated by quantification of DNA fragmentation (ELISA), caspase-3 enzyme activity and detection of mitochondrial permeability transition (MitoProbe JC-1). Oxidative stress was evaluated by oxygen species detection (Image-iT LIVE Green) and the activity of NADPH oxidase was measured by the lucigenin-enhanced chemiluminescence method. The expression of the genes Bax, Bcl2 and Nfkb1 was evaluated by reverse transcription real-time quantitative polymerase chain reaction (RT-qPCR). In one of the time points at which increased apoptosis was detected, the effect of two antioxidant compounds was evaluated: benfotiamine (350 M), a liposoluble vitamin B1, and Mito Q (1 M), a derivative of ubiquinone targeted to mitochondria. In 24 and 48 h, AGEs elicited a significant decrease in the apoptosis rate in comparison to the control condition concomitantly with a significant increase in the RNA expression of the antiapoptotic gene Bcl2 and a significant decrease in the Nfkb1 RNA expression. In contrast, after 72 and 96 h, AGEs promoted a significant increase in the apoptosis rate in comparison to the control condition concomitantly with a significant decrease in Bcl2 RNA expression and a significant increase in Nfkb1 RNA expression. In 24 h, AGEs elicited a significant decrease in the islet content of ROS while after 48 and 72 h, AGEs promoted an opposite effect. Benfotiamine and Mito Q were able to decrease the apoptosis rate and the ROS content in islets exposed to AGEs for 72 h. In conclusion, AGEs exerted a dual effect in cultured pancreatic islets, being protective against apoptosis after short exposition but proapoptotic after prolonged exposition. Mito Q and benfotiamine deserve further evaluation as drugs that could offer islet protection in conditions of chronic hyperglycemia

This thesis is divided into the following three parts. A categorical reconstruction of crystals and quantum groups at q = 0. The quantum co-ordinate algebra A_q(𝔤) associated to a KacMoody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite dimensional irreducible U_q(𝔤) modules. In Part I we investigate whether an analogous result is true when q = 0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over ℤ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at v = ∞. A Tannakian Reconstruction Theorem for IndBanach Spaces. Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In Part II we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and Galois descent. A large portion of Part II is dedicated to such examples. On analytic analogues of quantum groups. In Part III we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid's double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a p-adic Drinfel'd-Kohno Theorem, which should relate this work to Furusho's p-adic Drinfel'd associators. Finally, we adapt these constructions to working over Archimedean fields.

Due to the demand for low-cost, small-form factor and large-scale integration of system-on-chip wireless transceivers, the image-reject, zero-IF and low-IF receiver architectures have become the main topologies used in mainstream wireless communication systems. Consequently, signal sources with quadrature phase outputs [quadrature oscillators (QOs)] are therefore essential, and their phase noise, driving capability, tuning range, oscillation frequency, and power consumption have a major impact on the overall receiver performance. Additionally, it is required that the QO synthesize precise I/Q waveforms across the signal bandwidth over process, voltage, and temperature variations for adequate image-rejection and signal modulation/demodulation. While the use of symmetrical layout and large inter-digitated devices minimize both systematic and random mismatches, this solution alone may not succeed in achieving the stringent performance requirements dictated by modern wireless standards particularly as the technology scales into the sub-100nm regime, necessitating both phase and gain calibration of the mismatched I/Q channels post-fabrication. Given the necessity for precise RF quadrature signal synthesis, the goal of this work is to investigate low-power low-phase-noise quadrature oscillator (QVCO) topologies with an integrated phase calibration feature. The first part of this work focuses on the analysis and modeling of cross-coupled LC QVCOs. The analysis focuses on understanding the oscillator basic performance characteristics, design trade-offs, phase-noise performance, effect of including phase shift in the coupling paths, and on examining the quadrature accuracy in presence of process variations. New design parameters and circuit insight are developed and a generalized first order linear model and a one-port model are proposed. Particularly, we introduce the concept of an effective core and coupling transconductances to explain various oscillator properties. Additionally, a new incremental circuit element — the quadrature resistance — is introduced to evaluate the effect of coupling on the open-loop quality factor and hence on the oscillator phase noise performance. Mechanisms affecting the mode selectivity are identified and modeled. A qualitative and quantitative study of the effect of mismatch on the phase imbalance and amplitude error is presented. Particularly, closed-form intuitive expressions of the phase imbalance and amplitude error are derived and verified via circuit simulation. Based on our understanding of the various mechanisms affecting the quadrature accuracy, the second part of this work introduces a very efficient quadrature phase calibration technique based on the disconnected-source parallel-coupled LC QVCO topology. The phase-tunable LC QVCO (PT-QVCO) achieves an ultra-wide I/Q phase tuning range without affecting the relative amplitude error or consuming additional power or chip area. Additionally, in restoring the phase balance, it is observed that the proposed method restores the phase noise performance to its optimal value which presents a potential advantage over classical calibration techniques. Time domain measurements performed on a 5 GHz prototype show that I/Q signals with phase error up to ~±30°, beyond which the VCO cores are unlocked, can be driven to perfect quadrature phase. The PT-QVCO can be tuned from 3.87-4.45 GHz at the negative mode and 4.4-5.4 GHz at the positive mode, a total of ~1.5 GHz. The fabricated circuit including pad structures occupies an area of 1.1x0.7 mm² and drains 18mW (excluding buffer circuits) from a 1.8 V supply voltage. The third part of this work introduces a new low-power, low-phase-noise super harmonic injection-coupled LC QVCO (IC-QVCO) topology. Analysis of the waveform accuracy reveals an inverse dependence of the quadrature error on the tank quality factor thus allowing circuit optimization for both low phase noise and precise quadrature synthesis. Additionally, a tunable tail filter (TTF) is incorporated to calibrate the residual quadrature imbalance in presence of a 3-σ variation in the device parameters. An X-band IC-QVCO prototype with a TTF implemented in a 0.18μm RF CMOS process, achieves a measured phase noise figure-of-merit ranging from 177.3 to 182.6 dBc/Hz along the 9.0 to 9.6 GHz frequency tuning range while dissipating only 9mW from the 1.8V supply. The TTF reduces both the 1/f² and 1/f³ phase noise and calibrates the residual phase error within ±11° post-fabrication without affecting the relative amplitude error or the phase noise performance. The circuit performance compares favorably with recently published work. In the fourth part of this work, we explore the implementation of LC QVCOs as potential I/Q sources at millimeter-wave (MMW) frequencies. Among the several design challenges that emerge as the oscillator frequency is scaled into the MMW band, precise quadrature synthesis and adequate frequency tuning range are among the hardest to achieve. After describing the limitation of using an MOS varactor and a digitally controlled switch capacitor array for frequency tuning, we propose an alternative frequency tuning technique based on the fundamental operation of LC QVCOs. The off-resonance operation, which is defined by the coupling network, suggests varying the coupling current to achieve frequency tuning. In essence, by modifying the bias current of the coupling transistors (G_Mc-tuning), a wide and linear frequency tuning range can be achieved. Extensive simulation results of a 60 GHz prototype, implemented in a 90 nm commercial RF CMOS process, demonstrates a 5 GHz of frequency tuning range (57.5 GHz → 62.5 GHz), a tuning sensitivity of 1GHz/mA, and a 4dB improvement in the phase noise compared to a varactor solution. Finally, the Appendix includes recent research work on the analysis and design of gm-boosted common-gate low-noise amplifiers (CG-LNAs). While this topic seems to diverge from the main theme of the dissertation, we believe that the comprehensive analysis and the originality of the circuit design introduced in this work are worth acknowledging.
Ph.D.
While resting in bed due to illness, the Dutch scientist Christiaan Huygens keenly observed that the pendulums of two clocks hanging on the wall moved synchronously when the clocks were hung close to each other. He concluded that these two oscillatory systems were forced to move in unison by virtue of mechanical coupling through the wall. In essence, each pendulum injected mechanical vibrations into the wall that was strong enough to lock the adjacent pendulum into synchronous motion. Injection locking of oscillatory systems plays a critical role in communication systems ranging from frequency division, to generating clocks (oscillators) with finer phase separation, to the synthesis of orthogonal (quadrature) clocks. All communication systems have the same basic form. Firstly, there will some type of an information or data source which can be a keyboard or a microphone in a smartphone. The source is connected to a receiver by some sort of a channel. In wireless systems, the channel is the air medium. Moreover, to comply with the FCC and 3GPP requirements, data can only be transmitted wirelessly within a predefined set of frequencies and with stringent emission requirements to avoid interference with other wireless systems. These frequencies are generated by high fidelity clock sources, also known as oscillators. Consider a group of people sharing the same room and hence the same channel want to share information. Without regulating the “loudness” of each communicating ensemble, the quality of communication can be severely impaired. Moreover, it is to be expected that information can be shared more efficiently if each pair is allocated non-overlapping timeslots – speak when others are quiet. Called time orthogonality, all wireless systems require precise orthogonal (quadrature) clock sources to improve the communication efficiency. The precision of quadrature clocks is determined by the amplitude and phase accuracy. This dissertation takes a deep dive into the analysis and implementation of high accuracy quadrature (I/Q) clock sources using the concept of injection locking. These I/Q clocks or oscillators, also known as quadrature voltage controlled oscillators (QVCOs), have gained enormous popularity in the last decade. The first part of this work focuses on the analysis and modeling of QVCOs. The analysis focuses on understanding the oscillator basic performance characteristics, and on examining the quadrature accuracy in presence of process variations. New design parameters and circuit insight are developed and a generalized first order linear model and a one-port model are proposed. A qualitative and quantitative study of the effect of mismatch on the phase imbalance and amplitude error is presented. Particularly, closed-form intuitive expressions of the phase imbalance and amplitude error are derived and verified via circuit simulation. Based on our understanding of the various mechanisms affecting the quadrature accuracy, the second part of this work introduces a very efficient quadrature phase calibration technique based The phase-tunable QVCO (PT-QVCO) achieves an ultra-wide I/Q phase tuning range without affecting the oscillator other performance metrics. The proposed topology was successfully verified in silicon using a 5GHz prototype. The third part of this work introduces a new low-power, low-phase-noise injection coupled QVCO (IC-QVCO) topology. An X-band IC-QVCO prototype was successfully verified in a 0.18m RF CMOS process. In the fourth part of this work, we explore the implementation of QVCOs as potential I/Q sources at millimeter-wave (MMW) frequencies. Among the several design challenges that emerge as the oscillator frequency is scaled into the MMW band, precise quadrature synthesis and adequate frequency tuning range are among the hardest to achieve. After describing the limitation of using an conventional frequency tuning techniques, we propose an alternative approach based on the fundamental operation of QVCOs that outperforms existing solutions.

Au cours des deux dernières décennies, des travaux actifs ont été menés pour étendre des résultats classiques liés au groupe symétrique à d'autres groupes plus généraux. Cette thèse a pour objectif d’étendre aux produits en couronne les résultats concernant le tableau de différence d’Euler. Elle est divisée en cinq chapitres. Le tableau de différence d’Euler lié à la suite {n!} conduit naturellement à la formule du nombre de dérangements. Nous étudions dans les deux premiers chapitres, le tableau de différence d’Euler associé à la suite {rnn!} et la généralisation du problème de dérangements. Pour les coefficients de ce dernier tableau, nous donnons des interprétations combinatoires en termes de k-successions sur les produits en couronne. Clarke et al. ont introduit un q-analogue du tableau de différence d’Euler sur le groupe symétrique. Dans le troisième chapitre, nous étendons leurs résultats sur les produits en couronne. En généralisant leur bijection, nous montrons que « (fix, exc, fmaj) » et « (fix, exc, fmaf) » sont équidistribués sur les produits en couronne où «fmaf» est une nouvelle statistique mahonienne. D’autre part, Foata et Han ont récemment construit deux transformations. Nous prouvons dans le quatrième chapitre que ses bijections fournissent une factorisation de la bijection de Clarke et al.. Dans le cinquième chapitre nous donnons une extension de la seconde transformation fondamentale de Foata sur les mots r-colorés. Nous prouvons l’équidistribution sur les produits en couronne de « (fmaj , des*) » et « (finv , col) » où « col » est la somme des couleurs et « des* » une nouvelle statistique
In the last two decades, much effort has been made to extend various enumerative results on symmetric groups to other more general groups. The main objective of this thesis is to extend to wreath products the results that concern the Euler's difference table. It is divided into five chapters. Euler's difference table associated to the sequence {n!} leads naturally to the counting formula for the derangements. In the first two chapters, we study Euler's difference table associated to the sequence {rnn!} and the generalized derangement problem. For the coefficients appearing in the later table, we give the combinatorial interpretations in terms of k-successions on wreath products. Clarke et al. introduced a q-analogue of Euler's difference table on symmetric group. In the third chapter, we extend their results to wreath products. By generalizing their bijection, we prove the equidistribution of the triple statistics “(fix, exc, fmaj)” and “(fix, exc, fmaf)” on wreath products, where “fmaf” is a new mahonian statistic on wreath products. On the other hand, Foata and Han have recently constructed two new transformations. We prove in fourth chapter that their two bijections provide a factorization of Clarke et al.'s bijection. In the fifth chapter we give an extension of Foata’s second fundamental transformation on r-colored words. We show that the bistatistics “(fmaj , des*)” and “(finv , col)” are equidistributed on wreath products, where “col” is the sum of color and “des*” a new statistic

Modelos de Relações Quantitativas entre Estrutura Química e Atividade Biológica (QSAR), previamente gerados no grupo para uma série de brometos de N-benzil-N,N-dimetil-2-(4-X-benzamido)etan-1-amônio substituídos (compostos estruturalmente análogos à procaína), mostraram uma contribuição positiva da lipofilicidade (expressa pelo parâmetro lipofílico, logPapp) para a atividade de bloqueio da transmissão neuromuscular observada para estes. Como os compostos desta série são sais de amônio quaternários, sendo alguns altamente hidrofílicos, as determinações dos seus valores de logPapp pelo método shake-flask acarretam dificuldades experimentais, levando a altas incertezas. Para contornar estas dificuldades, em outro trabalho iniciado no grupo, sugeriu-se, como estratégia metodológica, avaliar a aplicação do conceito do par iônico, usando, para tanto, um contraíon grande e hidrofóbico. Assim, observou-se, para um único composto desta série, o aumento da sua lipofilicidade, por possível formação de par iônico, tornando-o menos hidrofílico. Nesta dissertação, visando ampliar os estudos envolvendo a aplicação desta estratégia metodológica e seus efeitos sobre a lipofilicidade, outros compostos foram selecionados para que uma faixa mais ampla de valores de LogPapp pudesse ser avaliada. Para tanto, os valores de logPapp de cinco compostos desta série foram determinados pelo método shake-flask, usando o sistema n-octanol/água, avaliando-se, ainda, os efeitos da força iônica na partição. Para estes, a faixa de valores de LogPapp determinados variou de -1,73 a 0,99. Para avaliar a aplicação do conceito do par iônico como estratégia, os valores de LogPapp destes compostos foram determinados na presença de diferentes concentrações do contraíon hidrofóbico n-octil sulfato de sódio (OcS). Verificou-se aumento da lipofilicidade com aumento das concentrações de OcS. Obteve-se, também, os correspondentes valores de LogPapp extrapolados para concentração zero de OcS (LogPapp0), que concordaram, dentro do erro experimental, com os valores determinados diretamente na ausência de OcS. Estes resultados indicam que o conceito do par iônico pode ser aplicado como estratégia metodológica, para melhorar as condições experimentais e diminuir as incertezas nas determinações de valores de LogPapp para esta série de brometos de amônio quaternários hidrofílicos. Finalmente, foram realizados experimentos complementares para verificar a formação de par iônico: (i) determinações dos valores de logP da benzamida (composto sem carga permanente) na presença de OcS; (ii) análise por RMN de um dos compostos desta série, na presença de OcS. Ambos indicaram que, somente para os brometos de amônio quaternários estudados, o aumento da lipofilicidade observado ocorre por possível formação de par iônico. E, ainda, foi avaliada a aplicação do conceito do par iônico e seus efeitos sobre a lipofilicidade do QX-222 e do brometo de neostigmina (compostos altamente hidrofílicos), revelando limitações na sua aplicação como estratégia metodológica.
Quantitative Structure-Activity Relationships (QSAR) models previously generated in our group for a set of N-benzyl-N,N-dimethyl-2-(4-X-benzamido)ethan-1-ammonium bromides (procaine structural analogs) showed a positive contribution of the lipophilic term (expressed by the partition coefficient, logPapp) to their neuromuscular blockage activity. However, since some of them are hydrophilic quaternary ammonium bromides, their logPapp measurements were usually associated with experimental shortcomings, leading to low accuracy logPapp values. In order to make these experiments more feasible, the application of the ion-pair concept was evaluated as a methodological strategy in another previous work, initially to one compound of this set. For this purpose, logPapp measurements have been done in the presence of a large and hydrophobic counter ion, leading to an increase of its lipophilicity, due to possible ion-pair formation. In the present work, in order to further investigate this strategy and its effects on the lipophilicity, we include more compounds, allowing the assessment of a wider range of logPapp values. Thus, logPapp values of five compounds were obtained by the shake-flask method, using n-octanol/water, with and without ionic strength adjustment. For them, logPapp range values were determined from -1.73 to 0.99. To evaluate the application of the ion-pair concept as strategy, logPapp values of these compounds were determined in the presence of eleven sodium octyl sulfate (OcS) concentrations. It was observed that the lipophilicity of these compounds, increased with the addition of OcS. Moreover, we obtained the corresponding logPapp values extrapolated to zero OcS concentration (logPapp0), which agreed, within the experimental error, with the corresponding values determined in absence of OcS. These results suggest that the ion-pair concept can be applied as a strategy to improve the experimental conditions, increasing the accuracy of logPapp measurements for this set of hydrophilic quaternary ammonium bromides. Finally, supplementary experiments were done to verify the ion-pair formation: (i) measurement of logP values of benzamide (not a permanent ion) in presence of OcS; (ii) NMR spectroscopy analysis of one compound of this set, also in presence of OcS. Both indicated that only for the studied quaternary ammonium bromides the lipophilicity increase is due to possible ion-pair formation. Additionally, the application of the ion-pair concept and its effects on the lipophilicity was evaluated for QX-222 and for neostigmine bromide (highly hydrophilic compounds), which revealed limitations in its application as a methodological strateg

Au cours des deux dernières décennies, des travaux actifs ont été menés pour étendre des résultats classiques liés au groupe symétrique à d'autres groupes plus généraux. Cette thèse a pour objectif d'étendre aux produits en couronne les résultats concernant le tableau de différence d'Euler. Elle est divisée en cinq chapitres. Le tableau de différence d'Euler lié à la suite {n!} conduit naturellement à la formule du nombre de dérangements. Nous étudions dans les deux premiers chapitres, le tableau de différence d'Euler associé à la suite {rnn!} et la généralisation du problème de dérangements. Pour les coefficients de ce dernier tableau, nous donnons des interprétations combinatoires en termes de k-successions sur les produits en couronne. Clarke et al. ont introduit un q-analogue du tableau de différence d'Euler sur le groupe symétrique. Dans le troisième chapitre, nous étendons leurs résultats sur les produits en couronne. En généralisant leur bijection, nous montrons que " (fix, exc, fmaj) " et " (fix, exc, fmaf) " sont équidistribués sur les produits en couronne où "fmaf" est une nouvelle statistique mahonienne. D'autre part, Foata et Han ont récemment construit deux transformations. Nous prouvons dans le quatrième chapitre que ses bijections fournissent une factorisation de la bijection de Clarke et al.. Dans le cinquième chapitre nous donnons une extension de la seconde transformation fondamentale de Foata sur les mots r-colorés. Nous prouvons l'équidistribution sur les produits en couronne de " (fmaj , des*) " et " (finv , col) " où " col " est la somme des couleurs et " des* " une nouvelle statistique.

Gyroscopes are inertial sensors that measure the rate or angle of rotation. One of the most promising technologies for reaching a high-performance MEMS gyroscope has been development of the micro-hemispherical shell resonator. (μHSR) This thesis presents the electronic control and read-out interface that has been developed to turn the μHSR into a fully functional micro-hemispherical resonating gyroscope (μHRG) capable of measuring the rate of rotation. First, the μHSR was characterized, which both enabled the design of the interface and led to new insights into the linearity and feed-through characteristics of the μHSR. Then a detailed analysis of the rate mode interface including calculations and simulations was performed. This interface was then implemented on custom printed circuit boards for both the analog front-end and analog back-end, along with a custom on-board vacuum chamber and chassis to house the μHSR and interface electronics. Finally the performance of the rate mode gyroscope interface was characterized, showing a linear scale factor of 8.57 mv/deg/s, an angle random walk (ARW) of 34 deg/sqrt(hr) and a bias instability of 330 deg/hr.

Nowadays, small autonomous underwater robots are strongly preferred for remote exploration of unknown and unstructured environments. Such robots allow the exploration and monitoring of underwater environments where a long term underwater presence is required to cover a large area. Furthermore, reducing the robot size, embedding electrical board inside and reducing cost are some of the challenges designers of autonomous underwater robots are facing. As a key device for reliable operation-decision process of autonomous underwater robots, a relatively fast and cost effective controller based on Fuzzy logic and proportional-integral-derivative method is proposed in this thesis. It efficiently models nonlinear system behaviors largely present in robot operation and for which mathematical models are difficult to obtain. To evaluate its response, the fault finding test approach was applied and the response of each task of the robot depicted under different operating conditions. The robot performance while combining all control programs and including sensors was also investigated while the number of program codes and inputs were increased.

Honors thesis (B.A.)-Ithaca College Dept. of Mathematics, 2008.
Title from abstract page. "April 15, 2008." includes abstract Includes bibliographical references (leaf 33). Also available in print form in the Ithaca College Archives.

A thesis submitted in fulfilment of the requirements for the degree of Master of Science, 2018
In this dissertation, we start by giving basic definitions in graph theory and some classes of graphs which are relevant in this work. We then introduce the chromatic polynomial of a graph which is the main focus of this dissertation. In addition, we discuss some properties of a graph, and the chromatic polynomials which are relevant to this work. In particular, explicit expression of chromatic polynomials of certain classes of graphs are presented. Certain graph operations and the chromatic polynomials of the resultant graphs are discussed. Finally, we introduce the 2-vertex join of a graph and we study its chromatic polynomial, in particular we give the explicit expression of the flow polynomial of the 2-vertex join of a path Pn.
XL2019