China
Africa
Explore chapters and articles related to this topic
Elements of Quantum Electronics
Published in Michael Olorunfunmi Kolawole, Electronics, 2020
Before we delve into qubits in state space, it is important to define some notations used in quantum mechanics as compared to classical system. The space of states in classical system is Boolean. Analogous to its Boolean counterpart, the space of states in quantum system is not a mechanical set: it is a vector space (also called Hilbert space). A vector space, say V, over the field of complex numbers ℂ is a non-empty set of elements called vectors. As such, in V, it is defined as the operations of vector addition and multiplication of a vector by a scalar in ℂ. A vector space may have either a finite or an infinite number of dimensions, but in most applications in quantum computation, finite vector spaces are used for completeness; denoted by ℂn. The dimension of a vector space is the number of basis vectors.
Vectorial PWM for Basic Three-Phase Inverters
Published in Dorin O. Neacsu, Switching Power Converters, 2017
where vd, vq, and v0 are also called coordinates. If the vector space has a finite dimension, then all possible bases have the same number of elements. The dimension of a vector space refers to the number of vectors within any base. When applied to three-phase power systems or power converters, the dimension of the vector space is three, which means that any base has three elements. A base is ortho-normalized if all its vectors are unitary and any two different vectors are orthogonal. The mathematical theory of vector spaces also provides tools for making transformations between different bases of the same vector space. These transformations are unique and reversible and can have linear or rotational effects on the vectors.
Fundamentals to Geometric Modeling and Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
In the vector space, vectors are linearly independent if we cannot represent one in terms of the others using the scalar-vector addition operation. The maximum number of linearly independent vectors in the space is defined as the dimension of the vector space. Given two nonparallel vectors u and v, their cross-product determines a vector w which is orthogonal to both u and v, and we have w = u × v. Then we call u, v and w mutually orthogonal. The angle between u and v in the plane containing them is given by |sinθ| = |u| × |v| / (|u||v|).
Transitional points in constructing the preimage concept in linear algebra
Published in International Journal of Mathematical Education in Science and Technology, 2022
Asuman Oktaç, Rita Vázquez Padilla, Osiel Ramírez Sandoval, Diana Villabona Millán
Linear transformations are introduced as functions between vector spaces that satisfy the two linearity properties. Examples are given with emphasis on R2 and R3 as domain and codomain spaces, where linear transformations are defined by an algebraic rule and sometimes accompanied by geometric representations as in the case of rotations, projections and reflections. Some examples on matrix or function spaces are also provided. Matrix representations of linear transformations are studied. Concepts of domain, codomain and image of a linear transformation are defined as well as image of an element under a linear transformation, preimage of an element or subset of the image set and kernel. General results about bases, dimension, kernel and image are established. Composition and inverse of transformations are studied, including conditions for the existence of an inverse transformation. The invertible matrix theorem is explored.
Gait event detection based on inter-joint coordination using only angular information
Published in Advanced Robotics, 2020
Tamon Miyake, Yo Kobayashi, Masakatsu G. Fujie, Shigeki Sugano
Figure 1 shows the trajectory in angular space of the hip, knee, and ankle joints, and the calculation procedure. Blue, red, purple, and green show the regions of the trajectory related to swing up, swing down, loading response, and support, respectively. The positive direction of the hip joint angle was the extension direction, that of the knee joint angle was the bending direction, and that of the ankle joint angle was the dorsiflexion direction. The goniometers were calibrated such that the hip joint angle was zero when the torso and thigh were in line, the knee joint angle was zero when the thigh and shank were in line, and the ankle joint angle was zero when the shank and foot were orthogonal. The calculation procedure consists of two steps. First, the controller derives the planes by extracting parts of the angular data in each phase (block 1 of Figure 1). The four planes are derived from the recorded angular data of one gait cycle. Second, the controller detects the switching points from the support phase to the swing up phase in angular space so as to detect the time points when the swing phase starts (block 2 of Figure 1). If an angular point belongs to a plane, the distance between the plane and the point is close to zero and the angular point transition and plane is parallel. Because the dimension of the plane in the three-dimensional space is two, the coordinates on the plane are expressed with two basis vectors and an initial point on the plane. Therefore, the point Pn on the planes is expressed by where w1n and w2n are the basis vectors, and a1n and a2n are coefficients of the basis vectors in each plane. The initial point Gn of the planes is derived by calculating the average of the extracted angular data for each motion. The suffix n denotes a label of each plane ranging from one to four. The basis vectors are elements of the vector space, which are linearly independent.