Scalar (mathematics)

Summary

A scalar is an element of a field which is used to define a vector space. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector. The ter

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Scalar geometry and masses in Calabi-Yau string models

Daniel Farquet, Claudio Scrucca

We study the geometry of the scalar manifolds emerging in the no-scale sector of Kahler moduli and matter fields in generic Calabi-Yau string compactifications, and describe its implications on scalar masses. We consider both heterotic and orientifold models and compare their characteristics. We start from a general formula for the Kahler potential as a function of the topological compactification data and study the structure of the curvature tensor. We then determine the conditions for the space to be symmetric and show that whenever this is the case the heterotic and the orientifold models give the same scalar manifold. We finally study the structure of scalar masses in this type of geometries, assuming that a generic superpotential triggers spontaneous supersymmetry breaking. We show in particular that their behavior crucially depends on the parameters controlling the departure of the geometry from the coset situation. We first investigate the average sGoldstino mass in the hidden sector and its sign, and study the implications on vacuum metastability and the mass of the lightest scalar. We next examine the soft scalar masses in the visible sector and their flavor structure, and study the possibility of realizing a mild form of sequestering relying on a global symmetry.

Springer2012

Mathematical analysis of plasmonic nanoparticles: the scalar case

Pierre Robert Maurice Millien

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.

2017

Active Contours and Information Theory for Supervised Segmentation on Scalar Images

Valérie Duay, Sara Luti, Gloria Menegaz, Jean-Philippe Thiran

2007