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Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.

Physics --- Physics --- Quantum physics --- Quantum optics --- Statistical physics --- Machine learning --- Elementary particles (Physics) --- Quantum field theory

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Various cosmological observations support not only cosmological inflation in the early universe, which is also known as exponential cosmic expansion, but also that the expansion of the late-time universe is accelerating. To explain this phenomenon, the existence of dark energy is proposed. In addition, according to the rotation curve of galaxies, the existence of dark matter, which does not shine, is also suggested. If primordial gravitational waves are detected in the future, the mechanism for realizing inflation can be revealed. Moreover, there exist two main candidates for dark matter. The first is a new particle, the existence of which is predicted in particle physics. The second is an astrophysical object which is not found by electromagnetic waves. Furthermore, there are two representative approaches to account for the accelerated expansion of the current universe. One is to assume the unknown dark energy in general relativity. The other is to extend the gravity theory to large scales. Investigation of the origins of inflation, dark matter, and dark energy is one of the most fundamental problems in modern physics and cosmology. The purpose of this book is to explore the physics and cosmology of inflation, dark matter, and dark energy.

brans-dicke theory --- dark energy model --- cosmological parameters --- Dark Energy --- statistical analysis --- Baryon Acoustic Oscillation (BAO) --- Supernovae --- cosmological model --- Hubble constant --- Cosmic Microwave Background (CMB) temperature --- n/a --- Dark Energy --- Dark Matter --- memory --- dark matter --- galactic rotation curve --- cosmoligical parameters --- dark energy models --- loop quantum cosmology --- dark energy --- spacetime symmetry --- de Sitter vacuum --- quantum optical systems --- astronomical and space-research instrumentation --- instruments, apparatus, and components common to several branches of physics and astronomy --- normal galaxies, extragalactic objects and systems --- field theory --- comparative planetology --- properties of specific particles --- quantum optics --- fundamental astronomy --- Einstein-Aether theory of gravity --- dosmological parameters --- dark energy models --- cosmology --- particle physics --- cosmo–particle physics --- QCD --- hyper-color --- dark atoms --- composite dark matter --- scalar–tensor gravity --- junction conditions --- null hypersurfaces --- higher dimension gauged super-gravity black hole --- quantum gravity --- quantum tunneling phenomenon --- Hawking radiation --- dynamical Chern–Simons modified gravity --- parametrizations --- cosmological parameters