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Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
An inner product is the sum of products between a row vector and a column vector. The multiply operator (*) in Matlab performs an inner product. Thus to calculate a dot product, we only need to multiply the transpose of the first vector by the second vector. The resulting scalar is the dot product of the vectors.
Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The dot product is an operation between two vectors that yields a scalar amount. A·B=‖A‖‖B‖cosθ
Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Moreover, the dot product of a vector with itself is a positive number that is equal to the square of the length of the vector: u⋅u=u2⇔u=u⋅u.
Fusion of Semantic, Visual and Network Information for Detection of Misinformation on Social Media
Published in Cybernetics and Systems, 2022
Nishtha Ahuja, Shailender Kumar
BERT model is based on the transformer architecture. The raw sentences are extracted from the news articles and the word encoding is obtained from them. This transformer architecture is based on the neural machine translation which consists of and encoder-decoder configuration. The most relevant part of the input sequence is learned by performing the self-attention mechanism. Thus, this model is able to capture the long term dependencies in the word sequence. Two blocks exist in the transformers: encoders and decoders. The task of encoder block is to generate the vector form from the given input sequence. On the other hand, the decoder performs the same task in a reverse order that is to generate a sequence from the given vector form. The self-attention mechanism was used in this transformer model which is also called “scalar dot-product attention.” It is used to extract the most relevant and important parts from the input sequences. The attention matrix is calculated by using the following formula:
Topological effects in low-lying electronic states of linear N2H2 + and HBNH+ associated with onset of bending
Published in Molecular Physics, 2018
Anita Das, Rintu Mondal, Debasis Mukhopadhyay
The solution of this equation may be written in an exponentiated line-integral form [7,11] as follows: where S0 and S are the initial and final points located on the contour Γ along which Equation (2.6) is required to be solved. ℘ is the path-ordering operator, the dot stands for a scalar dot product, dS is the differential vector. A(S0) is the initial value of A(S). Another matrix of interest, the topological matrix D(Γ), is identical to the A-matrix and is calculated along the closed contour as [8]
On the kinematics of scalar iso-surfaces in decaying homogeneous, isotropic turbulence
Published in Journal of Turbulence, 2019
Brandon C. Blakeley, Weirong Wang, James J. Riley
To examine Term III in more detail, we note that the strain-rate tensor, , can be decomposed into its principal axes such that [27] where are the eigenvalues of the strain-rate tensor. The eigenvalues are ordered from largest to smallest, so that . Due to incompressibility, the eigenvalues must sum to zero. This formulation guarantees that represents extensive motions, whereas represents compressive motions of the strain-rate tensor. The scalar quantity is the dot product of the scalar normal vector with the eigenvector of , i.e. . Physically, represents the cosine of the angle between the iso-normal vector and the principal axis of the strain-rate tensor. Defined in this way, the effect of the strain-rate on surface area density production can be split into contributions from motions that always act to stretch the fluid (Term III), always act to compress the fluid (Term III), and an intermediate direction that may take on extension or compression to maintain incompressibility (Term III).