Sliding invariants and classification of singular holomorphic foliations in the plane

By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.Comment: 21 pages, 2 figur

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu structures.Comment: New version, with a completely new section which clarifies the relationship between singular foliations and Nambu structures. The size of the paper has doubled from 10 to 20 page

Formes normales de singularités de feuilletages et invariants de glissement

Les objets étudiés dans cette thèse sont les germes de feuilletages holomorphes singuliers dans le plan. Elle est divisée en deux parties: La première partie est consacrée à donner des formes normales formelles de feuilletages topologiquement quasi-homogènes dans des conditions génériques. Toute forme normale est donnée comme la somme de trois termes: un terme générique quasi-homogène initial, un terme hamiltonien et un terme radial. Nous montrons également que le nombre de coefficients libres dans les termes hamiltoniens sont conformes à la dimension de l'espace des modules de Mattei de déploiement. Dans la deuxième partie, par l'introduction d'un nouvel invariant appelé l'ensemble des glissements, nous donnons une classification stricte complète de la classe des germes de feuilletages holomorphes non dicritiques dont les indices de Camacho-Sad ne sont pas rationnels. Par ailleurs, nous allons montrer que, dans cette classe, le nouvel invariant est de détermination finie. Par conséquent, nous obtenons la détermination finie de la classe des feuilletages non dicritiques isoholonomiques et de feuilletages absolument dicritiques qui ont les même applications de Dulac.The objects studied in this thesis are the germs of singular holomorphic foliations in the plan. It is divided into two part: The first part is devoted to the construction of formal normal forms of topologically quasi-homogeneous foliations under generic conditions. Any such normal form is given as the sum of three terms: an initial generic quasi-homogeneous term, a hamiltonian term and a radial term. We also show that the number of free coefficients in the hamiltonian terms are consistent with the dimension of Mattei's moduli space of unfoldings. In the second part, by introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomy non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved

Beam division multiple access for millimeter wave massive MIMO: Hybrid zero-forcing beamforming with user selection

Massive multiple-input multiple-output (MIMO) systems are considered a promising solution to minimize multiuser interference (MUI) based on simple precoding techniques with a massive antenna array at a base station (BS). This paper presents a novel approach of beam division multiple access (BDMA) which BS transmit signals to multiusers at the same time via different beams based on hybrid beamforming and user-beam schedule. With the selection of users whose steering vectors are orthogonal to each other, interference between users is significantly improved. While, the efficiency spectrum of proposed scheme reaches to the performance of fully digital solutions, the multiuser interference is considerably reduced

QUANTUM TELEPORTATION OF ENTANGLED STATES VIA GENERALIZED PHOTON-ADDED PAIR COHERENT STATE

In this paper, we study the quantum teleportation of an unknown atomic state based on the two-photon Jaynes-Cummings model, consisting of an effective two-level atom with a two-mode field in the generalized photon-added pair coherent state (GPAPCS). By applying the detecting method, we use a scheme that includes two two-level atoms and a cavity field to teleport the unknown atomic state from a sender to a receiver. The results show that the number of photons added to the field and the intensity of the initial field influence the average fidelity and success probability of the teleportation process. The time-evolution dependence of the average fidelity is also considered and compared for the field in the pair coherent state and in the GPAPCS

Application of generalized photon-added pair coherent state to quantum teleportation via atom-field entangled channel

In this paper, we introduce a field-atom entangled state that represents the entanglement over time between a two-level atom and a field in a generalized photon-added pair coherent state in the Jaynes-Cummings model. This entangled state is applied for quantum teleportation of an unknown atomic state from a sender to a receiver that is geographically distant. Accordingly, we use an average fidelity criterion to quantify quantum teleportation processing. The results have shown that the process of teleportation is influenced by parameters such as initial field strength, the amplitude of a state to be teleported, and the number of added photons to two modes of the field. In addition, we also compare the success of teleportation in two cases of the field: the pair coherent state and the generalized photon-added pair coherent state

NONCLASSICAL PROPERTIES OF TWO-MODE DISPLACED FOCK STATE

In this paper, we investigate a variety of nonclassical properties of a two-mode displaced Fock state (TMDFS). Nonclassical properties such as sum squeezing, difference squeezing, antibunching effect, and the Cauchy-Schwarz inequality will be discussed analytically for the TMDFS. The results show that by changing the Fock state parameters between modes and the displacement parameter, the TMDFS exhibits enhanced nonclassical behaviors and makes a valuable contribution to quantum optics

CÁC TÍNH CHẤT ĐỘNG LƯỢNG TỬ CỦA TRẠNG THÁI KẾT HỢP BỘ BA THÊM PHOTON TRONG MÔ HÌNH JAYNES-CUMMINGS HAI MODE

In this paper, we studied the quantum dynamic properties via the two-mode Jaynes-Cummings (JC) model where an effective two-level atom interacts with a field in a photon-added trio coherent state (PATCS). The quantum dynamic properties of the atom and the field were investigated via the photon distribution function, the second-order correlation function of the same and different modes, and the probability of finding the atom in the excited state in the cavity. The results show the dynamic properties of the atom and the non-classical ones of the field in the PATCS. In addition to the factors that affect these dynamic properties, such as the original field strength, the role of adding photons to the modes of the field in the PATCS is important when compared with the field in the original state as the trio coherent state.Trong bài báo này, chúng tôi nghiên cứu các tính chất động lượng tử thông qua mô hình Jaynes-Cummings (JC) hai mode, trong đó một nguyên tử hai mức hiệu dụng tương tác với trường ở trạng thái kết hợp bộ ba thêm photon (PATCS). Các tính chất động lượng tử của nguyên tử và trường ở PATCS đã được khảo sát thông qua các hàm phân bố photon, các hàm tương quan bậc hai cùng mode và khác mode và hàm xác suất tìm nguyên tử ở trạng thái kích thích trong hốc cộng hưởng. Các kết quả khảo sát đã chỉ ra các tính chất động của nguyên tử và tính phi cổ điển của trường ở PATCS. Ngoài các yếu tố ảnh hưởng đến các tính chất động trên như cường độ trường ban đầu thì vai trò của việc thêm photon vào các mode của trường ở PATCS là rất quan trọng khi so sánh với trường ở trạng thái gốc ban đầu là trạng thái kết hợp bộ ba