The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p(x) changes from a delta function at the trivial state x=0 to p(x) ~ x^α at small x (with α = −1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for the model considered. Our results assume that the delay time $\tau$ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average < x(t)|x (t-τ) >, where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied. Moreover, we obtain the characteristic correlation time associated to the model. In particular, the validity of the common assumption of statistical independence between the state at time t and that at t-τ is examined. We find that the correlation time diverges at the model's bifurcation line, thus signalling the failure of statistical independence near threshold. We determine the correlation time both by numerical integration of the governing equation, and analytically in the limit of small τ. The correlation time T diverges as T ~ a^-1, where a is the control parameter so that a_{c}=0 is the bifurcation threshold. The small-τ expansion correctly predicts the location of the bifurcation threshold, but there are systematic deviations in the magnitude of the correlation time. / Le diagramme de bifurcation d'une équation différentielle stochastique incluant une échelle de temps retardée est présenté. Nous sommes motivés par des recherches récentes portant sur des modèles de régulation des gènes. Nous débutons avec la forme normale d'une bifurcation de type fourchette auquelle est ajoutée un terme stochastique de manière paramétrique ainsi qu'un terme linéaire incluant le délai. Ce dernier terme introduit une bifurcation de type Hopf où le délai négatif est particulièrement fort dans l'espace des paramètres. Une bifurcation abrupte est trouvée et nous définissons le seuil de bifurcation comme étant le point dans l'espace des paramètres pour lequel la fonction de distribution stationnaire p(x) change d'une fonction delta autour de l'origine à p(x) ~ x^α pour x petit (avec α = -1 exactement au seuil). Nous démontrons que le seuil de bifurcation est modifié par les fluctuations comparé à la limite déterministique par une valeur qui suit une relation linéraire avec l'intensité du bruit. Des expressions analytiques pour les seuils de bifurcation de type fourchette et Hopf du modèle considéré sont présentées. Nos résultats assument que le temps retardé τ est petit comparé aux autres temps charactéristiques du système, une limitation qui n'est pas significative près de la ligne de bifurcation. L'expression pour la bifurcation de type fourchette est déterminée suite à une expansion stochastique de Taylor. La location de cette bifurcation dépend de la moyenne conditionnelle < x(t)|x (t-τ) >, où x(t) est la variable dynamique. Cette probabilité conditionnelle comprend les effets combinés des fluctuations corrélées et du retardement rétroactif. Nous déterminons aussi une expression pour la bifurcation de type Hopf obtenue à l'aide d'une expansion des échelles de temps autour de la solution près du seuil. Nous obtenons une équation de Fokker-Planck associée à la dynamique des amplitudes des oscillations et nous utilisons celle-ci pour déterminer la location du seuil. Contrairement au cas sans délai, nous démontrons que la location des lignes de bifurcation est modifiée comparé à la limite déterministique et donc que l'intéraction entre les corrélations des fluctuations et le délai affectent la stabilité des solutions du modèle étudié. De plus, nous obtenons le temps de corrélation charactéristique associé au modèle. En particulier, la validité de l'hypothèse d'indépendance statistique entre l'état au temps t ainsi qu'au temps t-τ est examinée. Nous trouvons que le temps de corrélation diverge à la bifurcation, signalant l'infirmation de l'indépendence statistique au seuil de bifurcation. Nous déterminons le temps de corrélation par une intégration directe de l'équation gouvernant le modèle ainsi qu'analytiquement dans la limite où le délai est petit. Le temps de corrélation T diverge suivant une relation du type T ~ a^-1, où a est le paramètre de contrôle et où a_{c}=0 est la location de la bifurcation. L'expansion dans la limite où le délai est petit prédit correctement la location du seuil de bifurcation. Néanmoins, des déviations systématiques dans la magnitude du temps de corrélation sont observées entre les deux résultats.

The study of string theory has recently opened the way to many ground-breaking ideas in theoretical physics. An aspect that had been neglected until recently concerns the role played by time in this theory. It is an important subject because of its possible connections with the field of cosmology. In the first part of this thesis we study S(pacelike)D-branes which are objects arising naturally in string theory when Dirichlet boundary conditions are imposed on the time direction. SD-brane physics is inherently time-dependent. We set up the problem of coupling the most relevant open-string tachyonic mode to massless closed-string modes in the bulk, with back-reaction and Ramond-Ramond fields included. We find solutions (numerically) that are asymptotically flat in the future but plagued with a singularity in the past. The second part of the thesis is concerned with the study of important aspects related to the proposed duality between quantum gravity in de Sitter space and a Euclidean eonformal field theory: the dS/CFT correspondence. First, we study solutions of Einstein gravity coupled to a positive cosmological constant and matter which are asymptotically de Sitter and homogeneous. These solutions are 'tall', meaning that the perturbed universe lives through enough eonformal time for an entire spherical Cauchy surface to enter any observer's past light cone. Our main focus is on the implications of tall universes for the correspondence. Particular attention is given to the associated renormalization group flows, leading to a more general de Sitter c-theorem. We also discuss the eonformal diagrams for various classes of homogeneous flows. Then, we consider the evolution of massive scalar fields in (asymptotically) de Sitter spacetimes of arbitrary dimension. Through the dS/CFT correspondence, our analysis points to the existence of new non-local dualities for the eonformal field theory.

We use a quantum master equation approach to study the transport properties of a triple quantum dot ring. Unlike double quantum dots and triple quantum dot chains, this geometry gives two transport paths with a relative phase sensitive to magnetic flux via the Aharonov-Bohm effect. This gives rise to a coherent population trapping effect and what is known as a "dark state". Unlike other master equation techniques valid only in the high bias voltage limit, our treatment reproduces such results as well as giving an analytic zero-bias conductance formula. As well as providing a more robust signature of this "dark state" physics, our model further predicts a negative differential resistance in connection with high bias rectification already predicted. / Nous utilisons une approche d´equation quantique maîtresse pour étudier les propriétés de transport des points quantiques triples en forme d'anneau. Contrairement aux points quantiques doubles et triples en forme de chaînes, cette géométrie offre deux chemins pour le transport avec une phase quantique relative qui est sensible au flux magnétique en raison de l'effet Aharonov-Bohm. Ceci méne à un effet de piégeage de population cohérent et cela est connu sous le nom d'un "état sombre". Contrairement à d'autres techniques d'équation maîtresse qui sont seulement valides dans la limite d'un potentiel électrique élevé, notre méthode reproduit les résultats de ces derniers en plus de donner une expression analytique pour la conductance différentielle de zéro potentiel électrique. En plus de donner une optique plus robuste de la physique "d´etats sombres", notre modèle prédit une résistance différentielle négative qui est reliée au phénomène déjà prédit de rectification à potentiel élevé.

Placing observational limits on cosmic strings would provide important confirmation of or constraints on early universe models. Cosmic strings imprint the cosmic microwave background (CMB) with a distinct position space signature, leaving line discontinuities in the temperature maps due to a combination of gravitational lensing and the Doppler effect. To improve theoretical observational constraints, I wrote sky map simulations with and without cosmic strings, edge detection and counting algorithms, and programs to differentiate statistically between the ambient edges due to the inflationary background and the string signals. Our application of position space algorithms, specifically the Canny edge detection algorithm, was highly successful and allowed us to establish improved limits, by more than an order of magnitude, on the contribution of cosmic strings to the total fluctuation spectrum from simulated data. We extended our analysis of the Canny algorithm to distinguish between abelian cosmic strings and cosmic superstrings through the presence of three-string junctions in cosmic superstring maps. To this end, I wrote the first simulations of maps with junctions and found a disparity in the density of edges in maps of string networks with and without junctions. This work resulted in a statistic to differentiate between different cosmic string models including string theory models and models with different numbers of cosmic strings. I also modeled the position space polarization signal induced by cosmic string wakes. Due to the gravity of a moving string, matter falls into overdense two dimensional sheets of matter, wakes, behind the string. As photons propagate through the wakes, they encounter this overdensity of ionized matter which can polarize them. In our work we placed the first limits on the CMB polarization signature of cosmic strings including their signature non-linear structures, wakes, and found they would produce an observable signal when used in combination with an algorithm such as the Canny algorithm. The cosmic string wakes lead to a gradient in the intensity of polarization in the shape of the wake, greater at earlier positions of the string. The overdense wakes produced by strings provide more opportunity for the 21 cm hydrogen spin flip as well, so we found that cosmic strings wakes can be observed as three dimensional wedges in maps of 21 cm radiation absorption. Our paper is among the first to study the cosmic string signature in 21 cm and the first to examine the influence of wakes on the 21 cm signal. / La mise en place de limites d'observation sur les cordes cosmiques permettraient de confirmer, ou a tout le moins de contraindre de facon importante, les presents modeles de l'Univers primordial. Les cordes cosmiques impregnent le fond diffus cosmologique (ou CMB) d'une signature distincte dans l'espace, sous forme de discontinuites lineiques dans les cartes de temperatures. Ces dernieres sont dues a une combinaison des effets de lentilles gravitationnelles et de l'effet Doppler. Afin d'ameliorer les contraintes observationnelles theoriques, j'ai ecris des simulations de cartes du ciel avec et sans cordes cosmiques, des algorithmes de detection et de comptages des lignes de discontinuite, ainsi que des programmes permettant de distinguer de facon statistique les lignes de discontinuite produites naturellement dans le contexte de l'inflation cosmique de celles dues a la presence de cordes cosmiques. L'application d'algorithmes dans le domaine spatial (par opposition au domaine des harmoniques spheriques), en particulier l'application de l'algorithme de detection de discontinuites lineiques Canny, fut tres reussie et nous a permis d'etablir des limites ameliorees de plus d'un ordre de grandeur sur la contribution des cordes cosmiques a l'ensemble du spectre de fluctuations pour les donnees simulees. Nous avons etendu notre analyse utilisant l'algorithme Canny pour nous permettre de distinguer les cordes cosmiques abeliennes des supercordes cosmiques en analysant la presence de jonctions de trois cordes dans les cartes de supercordes cosmiques. A cette fin, j'ai ecris les premieres simulations de cartes incorporant ces jonctions et ai constate une disparite dans la densite de lignes de discontinuite detectee dans les cartes de reseaux de cordes avec et sans jonctions. Ce travail m'a permis d'elaborer un modele statistique permettant de differencier les differents modeles de cordes cosmiques, y compris les modeles provenant de la theorie des cordes et les modeles avec differents nombres de cordes cosmiques. J'ai aussi modelise, dans le domaine spatial, le signal de polarisation induit par les sillages de cordes cosmiques. En raison de la geometrie specifique de son champs gravitationnel, une corde cosmique en mouvement cause l'accretion de matiere dans des regions de sur densite bidimentionnelles, ou sillages, derriere elle. Lorsque des photons se propagent a travers ces sillages, ils peuvent etre polarises par leur rencontre avec la region de surdensite de matiere ionisee qui s'ytrouve. Par notre travail, nous avons impose les premieres limites sur la signature possible laissee dans le champs depolarisation du CMB, en incluant les effets de leur structures non-lineaires, c'est-a-dire de leurs sillons, et trouve que ces dernieres produiraient un signal detectable lorsqu'utilisees de concert avec un algorithme tel quel'algorithme Canny. Les sillages d'une corde cosmique donnent naissance a un gradient d'intensite dans le spectre de polarisation present dans la structure des sillages, la polarisation etant plus importante aux points les plus recules dans l'histoire du deplacement de la corde. Les zones de surdensite produites par les sillages des cordes donnent egalement lieu a une gamme d'opportunites concernant la raie d'emission a 21 cm de l'hydrogene. Ainsi, nous avons decouvert que les sillages de cordes pourraient etre observe sous la forme de prismes triangulaires tridimensionnels dans les cartes d'absorption de la raie a 21 cm. Notre article est parmis les premiers a etudier la signature des cordes cosmiques dans le spectre d'absorption de la raie a 21 cm, et est le premier a examiner l'influence possible des sillages sur ce dernier.

Self-assembled quantum dots have attracted recent interest because of their broad potential applications to quantum information processing and fundamental research. While such structures present many obstacles to standard electronic characterization, recent experimental and theoretical work has demonstrated the use of a capacitively coupled AFM cantilever as a sensitive probe of quantum dot charge dynamics. The work in this thesis extends previous theoretical results to explore the case of an AFM tip coupled to a double quantum dot. Compared to single dot results, the DQD system involves significant new physical phenomena. In addition to fluctuations in the total charge, the double dot system can respond to the cantilever through changes in the distribution of a single electron. In particular, new features arise due to coherent interdot tunneling. After presenting the theoretical approach to this problem, we will discuss the results in the context of current experimental efforts using InAs dots. These effects should be accessible in a wide variety of experimental regimes. / Grâce à plusieurs applications prometteuses---en particulier dans le champ de l'informatique quantique---les points quantiques (PQ) auto-assemblées ont récemment merité beaucoup d'attention. Tandis que tels structures presentent des obstacles aux méthodes traditionnelles de characterization, des nouveaux avancements théoriques et expérimentals démontrent l'utilization d'un oscillateur d'une microscope à force atomique (AFM) pour characterizer la dynamique des electrons dans l'échantillon. Cette technique profite du fait que la force électrostatique entre l'AFM et le PQ dépend de la charge, qui change selon la position de l'oscillateur. Le résultat est une force variable qui produit des changements dans la fréquence résonante et la dissipation méchanique de l'oscillateur. Dans cette thèse, une exploration theorique considère le cas où l'oscillateur interagit avec deux points quantiques à la fois. En plus de la charge totale du système, il faut maintenant considérer sa distribution entre les deux points quantiques. Cette distribution, qui peut changer par l'entremise des effets tunnels et cohérents et incohérents, répond à la motion de l'oscillateur, produisant des effets dynamics qui devraient être mesurables dans une large étendue des régimes expérimentals.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008. / Source: Dissertation Abstracts International, Volume: 69-05, Section: B, page: 3070. Adviser: Robert Leigh. Includes bibliographical references (leaves 202-207) Available on microfilm from Pro Quest Information and Learning.

Thèse (Ph.D.)--Université de Sherbrooke (Canada), 2007. / Titre de l'écran-titre (visionné le 1 février 2007). In ProQuest dissertations and theses. Publié aussi en version papier.

Thesis (Ph.D.)--Indiana University, Dept. of Physics, 2009. / Title from PDF t.p. (viewed on Jul 12, 2010). Source: Dissertation Abstracts International, Volume: 70-10, Section: B, page: 6316. Adviser: Mike Berger.

This thesis explores the evolution of different types of black holes, and the ways in which black hole dynamics can be used to answer questions about other physical systems. We first investigate the differences in observable gravitational effects between a four-dimensional Randall-Sundrum (RS) braneworld universe compared to a universe without the extra dimension, by considering a black hole solution to the braneworld model that is localized on the brane. When the brane has a negative cosmological constant, then for a certain range of parameters for the black hole, the intersection of the black hole with the brane approximates a Banados-Teitelboim-Zanelli (BTZ) black hole on the brane with corrections that fall off exponentially outside the horizon. We compute the quasinormal modes of the braneworld black hole, and compare them to the known quasinormal modes of the three-dimensional BTZ black hole. We find that there are two distinct regions for the braneworld black hole solutions that are reflected in the dependence of the quasinormal modes on the black hole mass. The imaginary parts of the quasinormal modes display phenomenological similarities to the quasinormal modes of the three-dimensional BTZ black hole, indicating that nonlinear gravitational effects may not be enough to distinguish between a lower-dimensional theory and a theory derived from a higher-dimensional braneworld. Secondly, we consider the evolution of non-extremal black holes in N=4, d=2 supergravity, and investigate how such black holes might evolve over time if perturbed away from extremality. We study this problem in the probe limit by finding tunneling amplitudes for a Dirac field in a single-centered background, which gives the decay rates for the emission of charged probe black holes from the central black hole. We find that there is no minimum to the potential for the probe particles at a finite distance from the central black hole, so any probes that are emitted escape to infinity. If the central black hole is BPS in the extremal limit, then the potential is flat and so there is no barrier to the emission of probes. If the central black hole is non-BPS in the extremal limit, then there is a barrier to emission and we compute the decay rate, which depends both on the charge of the central black hole and the charges of the emitted black holes. Finally, we consider the possibility that an extremal black hole, the end-point of the evolution of a non-extremal black hole through evaporation, may itself split into a multi-centered black hole solution through quantum tunneling, via a gravitational instanton analogous to the instanton for the symmetric double well in elementary quantum mechanics. We find a gravitational instanton that connects two vacuum states: one state corresponding to a single-centered extremal Reissner-Nordstrom (ERN) black hole configuration, and another state corresponding to a multi-centered ERN configuration. We evaluate the Euclidean action for this instanton and find that the amplitude for the tunneling process is equal to half the difference in entropy between the initial and final configurations.

In this thesis we aim to characterize the self-dual strings of six-dimensional superconformal field theories (SCFTs) and develop a variety of techniques to compute their elliptic genera, which provide information about their spectra. All known N = (1, 0) supersymmetric SCFTs in 6d can be obtained by compactifying F-theory on an elliptic Calabi-Yau threefold with singular, noncompact base B. Resolving the singularity in B and wrapping D3 branes on the exceptional curves leads to strings on the tensor branch of the SCFT. The strings become tensionless at the superconformal fixed point and appear to play an important role in the dynamics of the SCFT. For a number of SCFTs we identify two-dimensional N = (0, 4) quiver gauge theories describing bound states of self-dual strings and obtain their elliptic genera by localization. We achieve this for 6d SCFTs describing M5 or D5 branes at a singularity, as well as n M5 branes probing an M9 plane (corresponding to n small E8 instantons). More generally, we relate the elliptic genera of strings of 6d SCFTs to the counting of supersymmetric (BPS) particles that arise upon compactification to 5d. This enables us to compute elliptic genera by topological string techniques. Following this approach we obtain information about the strings associated to SCFTs with a single tensor multiplet and minimal gauge group. Additionally, for SCFTs whose strings arise from M2 branes suspended between M5 or M9 branes we find a quantum mechanical picture in which M5 branes or M9 planes are represented by domain wall operators or states; in this context, elliptic genera are expressed in terms of suitable correlators which we compute. Finally, we find a non-perturbative completion of the topological string partition function that can be employed to compute five-sphere partition functions of 5d SCFTs and superconformal indices of 6d SCFTs. In the latter case, we argue that the superconformal index can be written in terms of elliptic genera of the self-dual strings, providing further evidence of the important role these strings play in the dynamics of 6d SCFTs. / Physics