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5G New Radio Non-Orthogonal Multiple Access
This book provides detailed descriptions of downlink non-orthogonal multiple transmissions and uplink non-orthogonal multiple access (NOMA) from the aspects of majorly used 5G new radio scenarios and system performance. For the downlink, the discussion focuses on the candidate schemes in 3GPP standards which are not only applicable to unicast services but also to broadcast/multicast scenarios.For the uplink, the main target scenario is massive machine-type communications where grant-free transmission can reduce signaling overhead, power consumption of devices and access delays.The design principles of several uplink NOMA schemes are discussed in-depth, together with the analysis of their performances and receiver complexities. Devoted to the basic technologies of NOMA and its theoretical principles, data analysis, basic algorithms, evaluation methodology and simulation results, this book will be an essential read for researchers and students of digital communications, wireless communications engineers and those who are interested in mobile communications in general.
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Symmetry Breaking for Representations of Rank One Orthogonal Groups II
This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup.The study of symmetry breaking operators (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical physics ranging from number theory to differential geometry and quantum mechanics.The first author initiated a program of the general study of symmetry breaking operators.The present book pursues the program by introducing new ideas and techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real rank one, which will serve as models for further research in other settings.In connection to automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad for tempered principal series representations in the case (SO(n + 1, 1), SO(n, 1)).The authors propose a further multiplicity conjecture for nontempered representations.Viewed from differential geometry, this seminal work accomplishes the classification of all conformally covariant operators transforming differential forms on a Riemanniann manifold X to those on a submanifold in the model space (X, Y) = (Sn, Sn-1).Functional equations and explicit formulæ of these operators are also established.This book offers a self-contained and inspiring introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups.This feature will be helpful for active scientists and accessible to graduate students and young researchers in representation theory, automorphic forms, differential geometry, and theoretical physics.
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What is an orthogonal vector?
An orthogonal vector is a vector that is perpendicular to another vector. In other words, two vectors are orthogonal if their dot product is zero. Geometrically, this means that the two vectors form a 90-degree angle with each other. Orthogonal vectors are important in many areas of mathematics and physics, including linear algebra and vector calculus.
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What does the term orthogonal mean?
The term orthogonal refers to two things being perpendicular or at right angles to each other. In mathematics, it often refers to vectors or matrices that are perpendicular to each other. In a broader sense, it can also refer to any two things that are independent or unrelated to each other.
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Are linearly independent vectors always orthogonal?
No, linearly independent vectors are not always orthogonal. Linear independence means that no vector in the set can be written as a linear combination of the others, while orthogonality means that the vectors are perpendicular to each other. It is possible for linearly independent vectors to be orthogonal, but it is not a guarantee. For example, in three-dimensional space, the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) are linearly independent and orthogonal, but the vectors (1, 1, 0) and (0, 1, 1) are linearly independent but not orthogonal.
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When are planes and lines orthogonal?
Planes and lines are orthogonal when the line is perpendicular to the plane. This means that the line forms a 90-degree angle with the plane, creating a right angle. In other words, the direction of the line is perpendicular to the direction of the plane. This relationship is important in geometry and engineering, as it affects the intersection and orientation of different geometric elements.
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Climbing Toubkal : Hiking Jebel Toubkal in Moroccoa??s Atlas Mountains
A guidebook to hiking Jebel Toubkal in Morocco's Atlas Mountains and the multi-day Toubkal circuit trek.The routes are divided into summer and winter ascents with additional acclimatisation options.They are designed for hikers with some experience in summer and winter trekking at altitude. Ascents to the Toubkal summit are detailed for both the popular Ikhibi Sud route and the less-travelled Ikhibi Nord route.The guide also includes access information from Imlil to Toubkal Basecamp, along with three winter and four summer acclimatisation options.The circular Toubkal Circuit trek explores the Toubkal National Park and is described in 6 stages covering 68km (42 miles). Oxford Alpine Club mapping included for each route and trek stage GPX files available to download Detailed information on planning, facilities and transportAll the 4000m peaks includedCovers all beginner winter routes in the Toubkal area
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How do you calculate the orthogonal complement?
To calculate the orthogonal complement of a subspace, you first need to find a basis for the subspace. Then, you can use the Gram-Schmidt process to find an orthonormal basis for the subspace. Finally, the orthogonal complement is the set of all vectors in the vector space that are orthogonal to every vector in the orthonormal basis of the subspace.
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How do I calculate the orthogonal complement?
To calculate the orthogonal complement of a subspace, you first need to find a basis for the subspace. Then, you can use the Gram-Schmidt process to find an orthogonal basis for the subspace. Once you have the orthogonal basis, you can take the orthogonal complement by finding the orthogonal complement of each basis vector and then taking the span of those vectors. This will give you the orthogonal complement of the original subspace.
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What is a proof of two orthogonal?
Two vectors are considered orthogonal if their dot product is equal to zero. This means that the angle between the two vectors is 90 degrees, forming a right angle. Mathematically, if vectors u and v are orthogonal, then u · v = 0. This property can be used to prove that two vectors are orthogonal by calculating their dot product and showing that it equals zero.
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What is a proof for two orthogonal?
Two vectors are orthogonal if their dot product is zero. This can be proven by calculating the dot product of the two vectors and showing that it equals zero. If the dot product is zero, it means that the vectors are perpendicular to each other, which is the definition of orthogonality in Euclidean space.
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