China
Africa
Explore chapters and articles related to this topic
Introduction to Nanosensors
Published in Vinod Kumar Khanna, Nanosensors, 2021
Two functions, f and g, are said to be orthogonal if their inner product is zero for f ≠ g. The inner product (or dot product, or scalar product) is an operation on two vectors that produces a scalar. For a weak perturbation, q’(t) = A cos(2πft) with f = f0 + Δf, f0 = (1/2π)√(k/μ*) and |Δf| << f0. In a first-order approximation, the frequency shift is given by Δf=−f02kA∫01/f0Ftsq′cosm2πf0tdt=−f0kA2Ftsq′
Neurons, Neural Networks, and Linear Discriminants
Published in Stephen Marsland, Machine Learning, 2014
To see this, think about the matrix notation we used in the implementation, but consider just one input vector x. The neuron fires if x·wT ≥ 0 (where w is the row of W that connects the inputs to one particular neuron; they are the same for the OR example, since there is only one neuron, and wT denotes the transpose of w and is used to make both of the vectors into column vectors). The a · b notation describes the inner or scalar product between two vectors. It is computed by multiplying each element of the first vector by the matching element of the second and adding them all together. As you might remember from high school, a · b = ||a||||b|| cos θ, where θ is the angle between a and b and ||a|| is the length of the vector a. So the inner product computes a function of the angle between the two vectors, scaled by their lengths. It can be computed in NumPy using the np.inner() function.
Hilbert Spaces
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A positive Hermitian sesquilinear form on X $ \mathcal{X} $ whose associated seminorm (11.1) is a norm is called an inner product. A vector space equipped with an inner product is called an inner product space. An inner product space that is complete with respect to the induced norm is called a Hilbert space.
Darboux theory of integrability for polynomial vector fields on 𝕊n
Published in Dynamical Systems, 2018
Let be a C1 map. A hypersurface Ω = {(x1,… ,xn + 1) is said to be regular if the gradient ∇G of G is not equal to zero on . Of course, if Ω is regular, then it is smooth. We say that Ω is algebraic if G is an irreducible polynomial. If the degree of the polynomial G is d, then we say that Ω is algebraic of degreed. A polynomial vector fieldon the regular hypersurface Ω (or simply a polynomial vector field on Ω) is a polynomial vector field in satisfying where the dot denotes the inner product of two vectors in . If the polynomial vector field in has degree m, then we say that the vector field on Ω is of degree m.
Research on compressive sensing of strong earthquake signals for earthquake early warning
Published in Geomatics, Natural Hazards and Risk, 2021
Jiening Xia, Yuanxiang Li, Yuxiu Cheng, Juan Li, Shasha Tian
As mentioned, the step 1 in OMP is to calculate the inner product of the residual and each atom (column vector) in the measurement matrix. The geometric interpretation of vectors’ inner products is the product of the projection of one vector on another vector, that is, the product in the same direction. From the value of the inner product, the degree of proximity of the two vectors in the direction can be seen.