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"The book is based on the lectures given at the CIME school "Quantum many body systems" held in the summer of 2010. It provides a tutorial introduction to recent advances in the mathematics of interacting systems, written by four leading experts in the field, V. Rivasseau illustrates the applications of constructive Quantum Field Theory to 2D interacting electrons and their relation to quantum gravity, R. Seiringer describes a proof of Bose-Einstein condensation in the Gross-Pitaevski limit and explains the effects of rotating traps and the emergence of lattices of quantized vortices, J.-P. Solovej gives an introduction to the theory of quantum Coulomb systems and to the functional analytic methods used to prove their thermodynamic stability, finally, T. Spencer explains the supersymmetric approach to Anderson localization and its relation to the theory of random matrices. All the lectures are characterized by their mathematical rigor combined with physical insights."

"There have been many recent and important developments based on effective field theory and the renormalization group in atomic, condensed matter, nuclear and high-energy physics. These powerful and versatile methods provide novel approaches to study complex and strongly interacting many-body systems in a controlled manner. The six extensive lectures gathered in this volume combine selected introductory and interdisciplinary presentations focused on recent applications of effective field theory and the renormalization group to many-body problems in such diverse fields as BEC, DFT, extreme matter, Fermi-liquid theory and gauge theories. "

"The dependence on the choice of these projections is studied in detail here and a method to overcome this ambiguity in DFT+U, by self-consistently determining the projections, is introduced. The author shows how nonorthogonal representations for electronic states may be used to construct these projections and, furthermore, how DFT+U may be implemented with a linearly increasing cost with respect to system size. The use of nonorthogonal functions in the context of electronic structure calculations is extensively discussed and clarified, with new interpretations and results, and, on this topic."

"Mekanika kuantum banyak benda menjadi suatu topik khusus dalam fisika dan memiliki banyak aplikasi di bidang fisika partikel, fisika nuklir, maupun fisika zat yang termampatkan. Hal ini tidak terlepas dari sistem yang ditinjau, yakni banyak partikel yang memungkinkan saling berinteraksi. Pada sistem materi termampatkan, hal yang menarik untuk ditinjau adalah interaksi elektron yang terkorelasi kuat. Berdasarkan teknik banyak benda yang diturunkan dari diagram Feynman, kami mengimplementasikan pendekatan ladder diagram untuk menangkap efek interaksi elektron-elektron. Kami menginvestigasi bagaimana efek korelasi ini mempengaruhi sifat yang bergantung suhu pada sistem metal terkorelasi dengan kuat. Sistem yang dipelajari adalah sifat material yang bergantung pada suhu, maka metode ladder diagram perlu diaplikasikan pada domain frekuensi Matsubara untuk mendapatkan self-energy yang self-consistent. Kami melakukan perhitungan kerapatan keadaan density of state dengan menggunakan algoritma ladder diagram untuk sistem metal.

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Quantum mechanics of many body problems has been a special topic in physics and may have many applications in particle physics, nuclear physics, and condensed matter physics. This condition comes because the system which is observed, that many particle with their interaction. In condensed matter physics, the strongly interaction of electrons are fascinating. Based on many body Feynman diagram, we implement the ladder diagram approximation for the electron interaction. We investigate the temperature dependent to capture the physical properties of the system. We formulated the Matsubara self energy then transform to the real frequency use Pad approximant."

"This monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. But it is good to finally have a mathematical confirmation in this sense. This book shed some light on the dynamics of N point masses that move in spaces of non-zero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2-body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics, classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and the theory of polytopes."

"This book highlights the symmetry properties of acoustic fields and describes the gauge invariance approach, which can be used to reveal those properties. Symmetry is the key theoretical framework of metamaterials, as has been demonstrated by the successful fabrication of acoustical metamaterials. The book first provides the necessary theoretical background, which includes the covariant derivative, the vector potential, and invariance in coordinate transformation. This is followed by descriptions of global gauge invariance (isotropy), and of local gauge invariance (anisotropy). Sections on time reversal symmetry, reflection invariance, and invariance of finite amplitude waves round out the coverage. "