q=a+ib+jc+kd} its scalar part a is the real number q − i q i − j q j − k q k 4 {\displaystyle {\frac {q-iqi-jqj-kqk}{4}}} . Likewise the scalar part of −qi...

vector field X {\displaystyle X} on the manifold is then defined as the scalar function ∇ ⋅ X {\displaystyle \nabla \cdot X} with the property ( ∇ ⋅ X...

complexity. In typical machine learning algorithms, these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued...

VNM-utility (it is unique up to adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize u...

a branch of mathematical analysis dealing with vector-valued functions. Scalar analysis is a branch of mathematical analysis dealing with values related...

(without skipping any layers), and there is a loss function that computes a scalar loss for the final output, backpropagation can be understood simply by matrix...

as the Calabi flow, as a proposal for finding Kähler metrics of constant scalar curvature.[C82a] More broadly, Calabi introduced the notion of an extremal...

absorbed into the definitions of the covariant tensor derivative ∇h and the scalar curvature R. Pierre Bonnet proved that two quadratic forms satisfying the...

written in collaboration with Marcel Grossmann, in which the single Newtonian scalar gravitational field is replaced by ten fields, which are the components...

metric to a scalar flat metric with constant mean curvature on the boundary", Annals of Mathematics, 1992. "Conformal metrics with prescribed scalar curvature"...

Dacorogna: General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Mathematica, 178, 1997, 1–37. with Irene Fonseca...

geometry and elliptic regularity theory, that is closely related to the scalar curvature operator, which provides an intrinsic curvature for a three-dimensional...

The relation between risk aversion and declining marginal utility for scalar outcomes can be stated formally with Jensen's inequality: risk aversion...

International Congress of Mathematicians (ICM) of 2010 in Hyderabad (on "Scalar curvature, conformal geometry, and the Ricci flow with surgery"), and a...

Surface subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009) Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007)...

or encoding numbers Prime number – Number divisible only by 1 or itself Scalar (mathematics) – Elements of a field, e.g. real numbers, in the context of...

integrable. The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function...

which is both time reversed and spatially inverted. The Ricci curvature scalar and the Ricci curvature tensor are both zero. Non-zero components of the...

the existence of "extremal" Kähler metrics, typically those with constant scalar curvature (see for example cscK metric). Donaldson obtained results in the...

showing that in the case when the solution is vector-valued rather than scalar-valued, it need not be analytic; the example of De Giorgi consists of an...

formulation of quantum mechanics Berestycki, H.; Lions, P.-L. (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal...

Sometimes, an otherwise complicated operation reduces to multiplication by a scalar (a number). Such numbers are called eigenvalues of that operation. Eigenvalues...

formulation of quantum mechanics Berestycki, H.; Lions, P.-L. (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal...

some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality...

}^{(a)},} where all these quantities are now functions of t alone; the scalar quantities, the energy density ε and the pressure of the matter p, are also...

{\textstyle \psi \mapsto D_{f}(\psi )} of f {\displaystyle f} defines a scalar-valued map D f : D ( R ) → C , {\displaystyle D_{f}:{\mathcal {D}}(\mathbb...

vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of w ¯ ∂ ∂ z − z ¯ ∂ ∂ w . {\displaystyle {\bar {w}}{\frac {\partial...

}{\frac {e^{itx}}{x^{2}+1}}\,dx} (which arises in probability theory as a scalar multiple of the characteristic function of the Cauchy distribution) resists...

supremum. Above, for the sake of convenience, the same notation is used for scalar, vector and tensor-valued Lebesgue and Sobolev spaces. The original version...

(collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals...