Le codage de reseau (CR) est une nouvelle technique reposant, sur la realisation par les noeuds du reseau, des fonctions de codage et de decodage des donnees afin d’ameliorerle debit et reduire les retards. En utilisant des algorithmes algebriques, le codage consiste àcombiner ensemble les paquets transmis et le decodage consiste à restaurer ces paquets. Cette operation permet de reduire le nombre total de transmissions de paquets pour echanger les donnees, mais requiere des traitements additionnels au niveau des noeuds. Le codage de reseau peut etre applique au niveau de differentes couches ISO.Toutefois dans ce travail, sa mise en noeuvre est effectuee au niveau de la couche reseau. Dans ce travail de thèse, nous presentons des techniques de codage de reseau s’appuyantsur de nouveaux protocoles permettant d’optimiser l’utilisation de la bande passante,D’ameliorer la qualite de service et de reduire l’impact de la perte de paquets dans les reseaux a pertes. Plusieurs defis ont ete releves notamment concernant les fonctions de codage/decodage et tous les mecanismes connexes utilises pour livrer les paquets echanges entre les noeuds. Des questions comme le cycle de vie des paquets dans le reseau, lacardinalite des messages codes, le nombre total d’octets transmis et la duree du temps de maintien des paquets ont ete adressees analytiquement, en s’appuyant sur des theoremes, qui ont ete ensuite confirmes par des simulations. Dans les reseaux a pertes, les methodes utilisees pour etudier precisement le comportement du reseau conduisent a la proposition de nouveaux mecanismes pour surmonter cette perte et reduire la charge.Dans la premiere partie de la these, un etat de l’art des techniques de codage de reseauxest presente a partir des travaux de Alshwede et al. Les differentes techniques sont detaillees mettant l’accent sur les codages lineaires et binaires. Ces techniques sont decrites en s’appuyant sur differents scenarios pour aider a comprendre les avantages etles inconvenients de chacune d’elles. Dans la deuxieme partie, un nouveau protocole base sur la correlation des adresses (ACNC) est presente, et deux approches utilisant ce protocole sont introduites ; l’approche centralisee ou le decodage se fait aux noeuds d’extremites et l’approche distribueeou chaque noeud dans le reseau participe au decodage. Le decodage centralise est elabore en presentant d’abord ses modeles de decision et le detail du decodage aux noeuds d’extremites. La cardinalite des messages codes recus et les exigences de mise en mémoire tampon au niveau des noeuds d’extremites sont etudiees et les notions d’age et de maturite sont introduites. On montre que le decodage distribue permet de reduire la charge sur les noeuds d’extremite ainsi que la memoire tampon au niveau des noeuds intermediaires. La perte et le recouvrement avec les techniques de codage de reseau sont examines pour les deux approches proposees. Pour l’approche centralisee, deux mecanismes pour limiter l’impact de la perte sont presentes. A cet effet, le concept de fermetures et le concept dessous-ensembles couvrants sont introduits. Les recouvrements optimaux afin de trouver l’ensemble optimal de paquets a retransmettre dans le but de decoder tous les paquets reçus sont definis. Pour le decodage distribue, un nouveau mecanisme de fiabilite saut a saut est propose tirant profit du codage de reseau et permettant de recuperer les paquets perdus sans la mise en oeuvre d’un mecanisme d’acquittement. / Network coding (NC) is a new technique in which transmitted data is encoded and decoded by the nodes of the network in order to enhance throughput and reduce delays. Using algebraic algorithms, encoding at nodes accumulates various packets in one message and decoding restores these packets. NC requires fewer transmissions to transmit all the data but more processing at the nodes. NC can be applied at any of the ISO layers. However, the focus is mainly on the network layer level. In this work, we introduce novelties to the NC paradigm with the intent of building easy to implement NC protocols in order to improve bandwidth usage, enhance QoS and reduce the impact of losing packets in lossy networks. Several challenges are raised by this thesis concerning details in the coding and decoding processes and all the related mechanisms used to deliver packets between end nodes. Notably, questions like the life cycle of packets in coding environment, cardinality of coded messages, number of bytes overhead transmissions and buffering time duration are inspected, analytically counted, supported by many theorems and then verified through simulations. By studying the packet loss problem, new theorems describing the behavior of the network in that case have been proposed and novel mechanisms to overcome this loss have been provided. In the first part of the thesis, an overview of NC is conducted since triggered by the work of Alshwede et al. NC techniques are then detailed with the focus on linear and binary NC. These techniques are elaborated and embellished with examples extracted from different scenarios to further help understanding the advantages and disadvantages of each of these techniques. In the second part, a new address correlated NC (ACNC) protocol is presented and two approaches using ACNC protocol are introduced, the centralized approach where decoding is conducted at end nodes and the distributed decoding approach where each node in the network participates in the decoding process. Centralized decoding is elaborated by first presenting its decision models and the detailed decoding procedure at end nodes. Moreover, the cardinality of received coded messages and the buffering requirements at end nodes are investigated and the concepts of aging and maturity are introduced. The distributed decoding approach is presented as a solution to reduce the overhead on end nodes by distributing the decoding process and buffering requirements to intermediate nodes. Loss and recovery in NC are examined for both centralized and distributed approaches. For the centralized decoding approach, two mechanisms to limit the impact of loss are presented. To this effect, the concept of closures and covering sets are introduced and the covering set discovery is conducted on undecodable messages to find the optimized set of packets to request from the sender in order to decode all received packets. For the distributed decoding, a new hop-to-hop reliability mechanism is proposed that takes advantage of the NC itself and depicts loss without the need of an acknowledgement mechanism.

No description available.

The objective of this thesis is to reach theoretical understanding of the unusual relationship between charge-ordering and superconductivity in correlated-electron systems. The competition between these broken symmetries and magnetism in the cuprate high temperature superconductors has been extensively discussed, but exists also in many other correlated-electron superconductors, including quasi-two-dimensional organic charge-transfer solids. It has been suggested that the same attractive interaction is responsible for both charge-order and superconductivity. We propose that the specific interaction is the tendency in correlated-electron systems to form spin-singlet bonds, which is strongly enhanced at the commensurate carrier density p of ½ a charge carrier per site, characteristic of all superconducting charge-transfer solids. To probe superconductivity driven by electron correlations, a necessary condition is that electron-electron interactions enhance superconducting pair-pair correlations, relative to the non-interacting limit. We have performed state of the art numerical calculations on the two-dimensional Hubbard model on different triangular lattices, as well as other lattices corresponding to K-BEDT-TTF based organic charge transfer solids, for the complete range of carrier densities per site p (0 ≤ p ≤ 1). We have shown that pair-pair correlation for each cluster is enhanced by electron-electron interaction only for p ≃ 0.5, far away from the density range thought to be important for superconductivity. Although initial focus is on charge-transfer solids, the results of the research will impact the field of correlated electrons as a whole. We believe our calculations will provide fundamental and fresh insight to the theory of superconductivity in strongly correlated systems.

Transition metal oxides and chalcogenides are an ideal platform for demonstrating and investigating many interesting electronic phases of matter. These phases emerge as a result of collective many body interactions among the electrons. The omnipresent electron, depending on its interaction with other electrons and with the underlying lattice, can generate diverse phases of matter with exotic physical properties. The ultimate objective of Materials Science is to provide a complete microscopic understanding of these myriad electronic phases of matter. A proper understanding of the collective quant-tum behaviour of electrons in different system can also help in designing and tuning new electronic phases of matter that may have strong impact in the field of microelectronics, well beyond that predicted by Moore s law. Strong electron correlation effects produce a wide spectrum of ground state prop-retires like superconductivity, Metal Insulator Transition (MIT), charge-orbital ordering and many more. Similarly, different spin interactions among electrons, essentially due to various kinds of exchange coupling, give rise to varying magnetic ground state prop-retires like ferromagnetism, anti-ferromagnetism, spin glass, among others. The main objective of this thesis is to understand and rationalize diverse electronic and magnetic phases of matter in some selected strongly correlated systems. In chapter 1 we have provided an overview of various electronic and magnetic phases of matter which are relevant and necessary for understanding the chapters that follow. The first part of this chapter describes the fundamental concepts of the so called Metal Insulator Transition (MIT). A small section is dedicated to the subtle interactions among electrons and lattice that actually drive a system from a highly conducting metallic state to a strongly resistive insulating state. The second part of this chapter offers a compilation of different magnetic ground states which are discussed in detail in the last two chapters. In Chapter 2, we have explained various methodologies and experimental tech-antiques that have been used in the work reported in this thesis. In Chapter 3, we have provided a detailed understanding of the MIT in different polymorphic forms of Vanadium dioxide (VO2). Although VO2 exhibits a number of polymorphic forms, only the rutile/monoclinic VO2 phase has been studied extensively compared to other polymorphic forms. This phase shows a well-established MIT across ∼340 K, which has been extensively investigated in order to understand the relative importance of many body electron correlation effects arising primarily from on-site Coulomb interactions within the Vanadium 3d manifold, and single electron effects flounced by the dimerization of Vanadium atoms. Unlike the rutile phase of VO2, little is known about the MIT appearing across 212 K in the metastable B-phase of VO2. This phase shows dimerization of only half of the Vanadium atoms in the insulating state, in contrast to rutile/monoclinic VO2, which show complete dimerization. There is a long standing debate about the origin of the MIT in the rutile/monoclinic phase, that contrasts the role of the many-body Hubbard U term, with single particle effects of the dimerization. In light of this debate, the MIT in the B-phase offers a unique opportunity to understand and address the competition between many body and single particle effects, that has been unresolved over several decades. In this chapter we have investigated different polymorphs of VO2 to understand the underlying electronic structure and the nature of the MIT in these polymorphic forms. The MIT in VO2 B phase is very broad in nature. X-ray photoemission and optical conductivity data indicate that in case of VO2 B phase both correlation effects and dimerization is necessary to drive the MIT. We have also established that the correlation effects are more prominent for VO2 B phase compared to rutile/monoclinic phase. In Chapter 4, we have discussed the electronic structure of LaTiO3 (LTO)-SrTiO3 (STO) system. At the interface between polar LTO and non-polar (STO) oxides, an unique two dimensional electron gas (2DEG) like state appears, that exhibits a phenomenal range of unexpected transport, magnetic, and electronic properties. Thus, this interface stands as a prospective candidate for not only fundamental scientific investigation, but also application in technological and ultimately commercial frontiers. In this chapter, using variable energy Hard X-ray photoemission spectroscopy (HAXPES), we have experimentally investigated the layer resolved evolution of electronic structure across the interface in LTO-STO system. HAXPES results suggest that the interface is more coherent in nature and the coherent to incoherent feature ratio changes significantly as we probe deeper into the layer In chapter 5, we have investigated the electronic structure of the chemically exfoliated trigonal phase of MoS2. This elusive trigonal phase exists only as small patches on chemically exfoliated MoS2, and is believed to control functioning of MoS2 based devices. Its electronic structure is little understood, with total absence of any spec-troscopic data, and contradictory claims from theoretical investigations. We have ad-dressed this issue experimentally by studying the electronic structure of few layered chemically exfoliated MoS2 systems using spatially resolved X-ray photoemission spec-otoscopy and micro Raman spectroscopy in conjunction with electronic structure calculations. We have established that the ground state of this unique trigonal phase is actually a small gap (∼90 meV) semiconductor. This is in contrast with most of the claims in existing literature. In chapter 6, we have re-examined and revaluated the electronic structure of the late 3d transition metal monoxides (NiO, FeO, and CoO) using a combination of HAX-PES and state-of-the-art theoretical calculations. We have observed a strong evolution in the valence band spectra as a function of excitation energy. Theoretical results show that a combined GW+LDA+DMFT scheme is essential for explaining the observed experimental findings. Additionally, variable temperature HAXPES measurement In chapter 8, we have differentiated the surface and the bulk electronic structure in Sr2FeMoO6 and also have provided a new route to increase the Curie temperature of this material. Sr2FeMoO6 is well known for its high Curie temperature (Tc ∼410 K), half-metallic ferromagnetism, and a spectacularly large tunnelling magnetoresistance. The surface electronic structure of Sr2FeMoO6 is believed to be different from the bulk; leading to a Spin-Valve type Magnetoresistance. We have carried out variable energy HAXPES on Sr2FeMoO6 to probe electronic structure as a function of surface depth. Our experimental results indicate that surface is more Mo6+ rich. We have also demonstrated what we believe is the first direct experimental evidence of hard ferro-magnetism in the surface layer using X Ray Magnetic Circular Dichroism (XMCD) with dual detection mode. In the second part of this chapter we have designed a new route to increase the Curie temperature and have been successfully able to achieve a Curie temperature as high as 515 K.

High-quality platelet single crystals of Tl₂Ba₂CuO₆±δ (Tl-2201) have been grown using a novel time-varying encapsulation scheme, minimizing the thallium oxide loss that has plagued other attempts and reducing cation substitution. This encapsulation scheme allows the melt to be decanted from the crystals, a step previously impossible, and the remaining cation substitution is homogenized via a high-temperature anneal. Oxygen annealing schemes were developed to produce sharp superconducting transitions from 5 to 85 K without damaging the crystals. The crystals' high homogeneity and high degree of crystalline perfection are further evidenced by narrow rocking curves; the crystals are comparable to YSZ-grown YBa₂Cu₃O₆₊δ by both metrics. Electron probe microanalysis (EPMA) ascertained the crystals' composition to be Tl₁.₉₂₀₍₂₎Ba₁.₉₆₍₂₎Cu₁.₀₈₀₍₂₎O₆₊δ; X-ray diffraction found the composition of a Tc = 75 K crystal to be Tl₁.₉₁₄₍₁₄₎Ba₂Cu₁.₀₈₆₍₁₄₎O₆.₀₇₍₅₎, in excellent agreement. X-ray refinement of the crystal structure found the crystals orthorhombic at most dopings, and their structure to be in general agreement with previous powder data. Cation-substituted Tl-2201 can be orthorhombic, orthorhombic crystals can be prepared, and these superconduct, all new results. X-ray diffraction also found evidence of an as yet unidentified commensurate superlattice modulation. The Tl-2201 crystals' electronic structure were studied by X-ray absorption and emission spectroscopies (XAS/XES). The Zhang-Rice singlet band gains less intensity on overdoping than expected, suggesting a breakdown of the Zhang-Rice singlet approximation, and one thallium oxide band does not disperse as expected. The spectra correspond very closely with LDA band structure calculations, and do not exhibit the upper Hubbard bands arising from strong correlations seen in other cuprates. The spectra are noteworthy for their unprecedented (in the high-Tc cuprates) simplicity. The startling degree to which the electronic structure can be explained bodes well for future research in the cuprates. The overdoped cuprates, and Tl-2201 in particular, may offer a unique opportunity for understanding in an otherwise highly confusing family of materials. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In a many-body fermionic system, the suppression of continuous transitions to absolute zero can result in a low temperature quantum fluid which deviates strongly from typical metallic behaviour; unconventional superconductivity can be induced by the strange metal region surrounding the zero-temperature phase transition. In this thesis we focus on three systems which demonstrate a highly tunable phase transition, with the aim of pushing them toward the border of a zero-temperature phase transition, and potentially superconductivity. CeAgSb2 is a uniaxial 4f ferromagnet, where physical pressure or a transverse field may be used to tune the magnetic transition towards T = 0 K. Our investigations, however, did not reveal the presence of superconductivity. It is likely that the field tuned transition does not correspond to a true critical point, whilst the high pressure region may be occupied by an antiferromagnetic phase, with the true critical point at higher pressures. However, other interesting features emerge in the electrical resistivity and AC-susceptibility, along with novel thermodynamic signatures linking the magnetisation to the specific heat. The doping series Lu(1-x)YxFe2Ge2 shows an antiferromagnetic transition which is suppressed to absolute zero at a critical concentration x_c=0.2. YFe2Ge2 displays anomalous low temperature behaviour consistent with the proximity to quantum critical fluctuations, along with a superconducting transition which appears in the electrical resistivity beneath a critical temperature of T_c ~ 1.7 K. Using low temperature DC magnetisation measurements, we show that this is a bulk effect, and that the superconductivity in YFe2Ge2 is of type-II. The thermodynamic and BCS properties of the superconducting phase are analysed in line with the parameters we extract experimentally. The superconducting 3-4-13 stannides (Ca,Sr)3Ir4Sn13 show a high temperature structural transition which may be suppressed by the application of hydrostatic pressure or effective chemical pressure. A superconducting dome is found, which appears to peak near where the structural transition extrapolates to zero temperature. Anomalous exponents are seen in the electrical resistivity over a wide temperature range. We investigate the influence of pressure on the superconducting critical temperature in Ca3Ir4Sn13 and the related compound Co3Ca4Sn13, along with an analysis of the upper critical field and flux-line phenomena in Ca3Ir4Sn13 and Sr3Ir4Sn13.

This thesis consists of six chapters. The first chapter gives an introduction to the field of low-dimensional magnetic and electronic systems and relevant numerical techniques. The recent developments in molecular magnets are highlighted. The numerical techniques are reviewed along with their advantages and disadvantages from the present perspective. Study of entanglement of a system can give a great insight into the system. At the last part of this chapter a general overview is given regarding entanglement, its measures and its significance in studying many-body systems. Chapter 2 deals with the technique that has been developed by us for the full symmetry adaptation of non-relativistic Hamiltonians. It is advantageous both computationally and physically/chemically to exploit both spin and spatial symmetries of a system. It has been a long-standing problem to target a state which has definite total spin and also belongs to a definite irreducible representation of a point group, particularly for non-Abelian point groups. A very general technique is discussed in this chapter which is a hybrid method based on valence-bond basis and the basis of the z-component of the total spin. This technique is not only applicable to a system with arbitrary site spins and belonging to any point group symmetry, it is also quite easy to implement computationally. To demonstrate the power of the method, it is applied to the molecular magnetic system, Cu6Fe8, with cubic symmetry. In chapter 3, the extension of the previous hybrid technique to electronic systems is discussed. The power of the method is illustrated by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and is in the largest non-Abelian point group. All the eigenstates of the model are obtained using our technique. Chapter 4 deals with the thermodynamic properties of an important class of single-chain magnets (SCMs). This class of SCMs has alternate isotropic spin-1/2 units and anisotropic high spin units with the anisotropy axes being non-collinear. Here anisotropy is assumed to be large and negative, as a result, anisotropic units behave like canted spins at low temperatures; but even then simple Ising-type model does not capture the essential physics of the system due to quantum mechanical nature of the isotropic units. A transfer matrix (TM) method is developed to study statistical behavior of this class of SCMs. For the first time, it is also discussed in detail that how weak inter-chain interactions can be treated by a TM method. The finite size effect is also discussed which becomes important for low temperature dynamics. This technique is applied to a real helical chain magnet, which has been studied experimentally. In the fifth chapter a bipartite entanglement entropy of finite systems is studied using exact diagonalization techniques to examine how the entanglement changes in the presence of long-range interactions. The PariserParrPople model with long-range interactions is used for this purpose and corresponding results are com-pared with those for the Hubbard and Heisenberg models with short-range interactions. This study helps understand why the density matrix renormalization group (DMRG) technique is so successful even in the presence of long-range interactions in the PPP model. It is also investigated if the symmetry properties of a state vector have any significance in relation to its entanglement. Finally, an interesting observation is made on the entanglement profiles of different states, across the full energy spectrum, in comparison with the corresponding profile of the density of states. The entanglement can be localized between two noncomplementary parts of a many-body system by performing local measurements on the rest of the system. This localized entanglement (LE) depends on the chosen basis set of measurement (BSM). In this chapter six, an optimality condition for the LE is derived, which would be helpful in finding optimal values of the LE, besides, can also be of use in studying mixed states of a general bipartite system. A canonical way of localizing entanglement is further discussed, where the BSM is not chosen arbitrarily, rather, is fully determined by the properties of a system. The LE obtained in this way, called the localized entanglement by canonical measurement (LECM), is not only easy to calculate practically, it provides a nice way to define the entanglement length. For spin-1/2 systems, the LECM is shown to be optimal in some important cases. At the end of this chapter, some numerical results are presented for j1 −j2 spin model to demonstrate how the LECM behaves.

No description available.

Les systèmes à électrons fortement corrélés sont d’intérêt particulier pour le calcul ab initio, cherchant à modéliser ces systèmes à partir des premiers principes. La théorie de la fonctionnelle de la densité associée à une prise en compte des corrélations locales en DFT+U ou en DFT+DMFT, permet de reproduire qualitativement la physique de ces systèmes. Cependant, ces méthodes font intervenir les paramètres d’interaction effective de Hubbard U et de Hund J. Ces derniers peuvent eux-mêmes être calculés de manière ab initio, notamment avec l’approximation de phase aléatoire contrainte (cRPA), ouvrant la voie au développement de schémas de calcul les plus prédictifs possible. Nous utilisons un schéma DFT+U/cRPA, dont le principe consiste à calculer les paramètres U et J en cRPA et la structure électronique en DFT+U de manière auto-cohérente. Nous appliquons ce schéma aux lanthanides allant du cérium au lutétium (en détaillant le cas du cérium dans ses phases gamma et alpha), et aux dioxydes des actinides allant de l'uranium au curium. Nous effectuons d'abord une étude de l'état fondamental en DFT+U en fonction de U, en détaillant l'influence des états métastables. Nous étudions le rôle de la localisation des orbitales corrélées sur l'interaction effective dans un cas particulier. Nous détaillons ensuite les valeurs de U obtenues en cRPA en fonction de celles utilisées pour le calcul DFT+U. Nous nous intéressons plus particulièrement aux effets des processus d’écrantage sur les valeurs de U obtenues. Nous montrons que les limitations du schéma observées (multiplicité des valeurs auto-cohérentes de U obtenues et/ou incompatibilité de ces valeurs avec les spectres de photoémission expérimentaux) sont causées par certains processus d’écrantage spécifiques. Cela suggère d'améliorer la description de ces processus d'écrantage en modifiant le modèle. / Strongly correlated electron systems are particularly interesting for ab initio calculus, which aims to model these systems from first principles. Density functional theory, improved by taking into account local correlations within DFT+U or DFT+DMFT, enables to reproduce qualitatively the physics of these systems. Nonetheless, these methods require the Hubbard and Hund effective interaction parameters U and J. These can be computed from first principles as well, within the constrained random phase approximation (cRPA), paving the way for numerical schemes as predictive as possible. Here, we use a DFT+U/cRPA scheme, which aims to compute self-consistently the U and J parameters with cRPA and the DFT+U electronic structure. We apply this scheme to lanthanides from cerium to lutetium (insisting on cerium and its gamma and alpha phases), and dioxides of actinides from uranium to curium. We study the DFT+U ground state in function of U, giving more details about the influence of metastable states. We study as well the influence of the localization of correlated orbitals on the effective interaction in a particular case. We then detail the values of U obtained with cRPA, in function of those used for the DFT+U calculation. We study more particularly the effects of screening processes on the obtained values of U. We show that the limitations of the scheme (multiplicity of self-consistent values of U and/or their incompatibility with experimental photoemission spectra) are caused by specific screening processes. This suggests to improve the description of these screening processes by modifying the model.

Non-inferiority tests are well developed for randomized parallel group trials where the control and experimental groups are independent. However, these tests may not be appropriate for assessing non-inferiority in correlated one-to-many matched data. We propose a new statistical test that extends Farrington-Manning’s (FM) test to the case where many (≥1) control subjects are matched to each experimental subject. We conducted a Monte Carlo simulation study to compare the size and power of the proposed test with tests developed for clustered one-to-one matched pair data and tests based on generalized estimating equations (GEE). For various correlation patterns, the sizes of tests developed for clustered matched pair data and GEE-based tests are inflated when applied to the case where many control subjects are matched to each experimental subject. The size of the proposed test, on the other hand, is close to the nominal level for a variety of correlation patterns. There is a debate in the literature regarding whether or not statistical tests appropriate for independent samples can be used to assess the statistical significance of treatment effects in propensity-score matched studies. We used Monte Carlo simulations to examine the effect on assessing non-inferiority via risk difference when a method for independent samples (i.e. FM test) is used versus when a method for correlated matched samples is used in propensity-score one-to-many matched studies. If propensity-score matched samples are well-matched on baseline covariates and contain almost all of the experimental treated subjects, a method for correlated matched samples is preferable with respect to power and Type I error than a method for independent samples. Sometimes there are more experimental subjects to choose from for matching than control subjects. We conducted a Monte Carlo simulation study to compare the size and power of the previously mentioned tests when many (≥1) experimental subjects are matched to each control subject. In this case, the Nam-Kwon test for clustered data performs the best in controlling the type I error rate for a variety of correlation patterns. Therefore, the appropriate non-inferiority test to use for correlated matched data depends, in part, on the sample size allocation of subjects.