Topic

# Basis (linear algebra)

About: Basis (linear algebra) is a research topic. Over the lifetime, 14069 publications have been published within this topic receiving 278522 citations. The topic is also known as: Hamel basis & algebraic basis.

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TL;DR: The relatively small diffuse function-augmented basis set, 3-21+G, is shown to describe anion geometries and proton affinities adequately as discussed by the authors.

Abstract: The relatively small diffuse function-augmented basis set, 3-21+G, is shown to describe anion geometries and proton affinities adequately. The diffuse sp orbital exponents are recommended for general use to augment larger basis sets.

5,916 citations

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TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.

Abstract: The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h - and p -refinement schemes are presented and a new, more efficient, higher-order concept, k -refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A k -refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.

5,137 citations

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Brown University

^{1}TL;DR: The use of natural symmetries (mirror images) in a well-defined family of patterns (human faces) is discussed within the framework of the Karhunen-Loeve expansion, which results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix.

Abstract: The use of natural symmetries (mirror images) in a well-defined family of patterns (human faces) is discussed within the framework of the Karhunen-Loeve expansion This results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix, without increasing the complexity of the calculation The resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average >

2,686 citations

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TL;DR: This paper shows how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low-dimensional linear space and that this subspace is close to those acquired by the other methods.

Abstract: Previous work has demonstrated that the image variation of many objects (human faces in particular) under variable lighting can be effectively modeled by low-dimensional linear spaces, even when there are multiple light sources and shadowing. Basis images spanning this space are usually obtained in one of three ways: a large set of images of the object under different lighting conditions is acquired, and principal component analysis (PCA) is used to estimate a subspace. Alternatively, synthetic images are rendered from a 3D model (perhaps reconstructed from images) under point sources and, again, PCA is used to estimate a subspace. Finally, images rendered from a 3D model under diffuse lighting based on spherical harmonics are directly used as basis images. In this paper, we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low-dimensional linear space and that this subspace is close to those acquired by the other methods. More specifically, there exist configurations of k point light source directions, with k typically ranging from 5 to 9, such that, by taking k images of an object under these single sources, the resulting subspace is an effective representation for recognition under a wide range of lighting conditions. Since the subspace is generated directly from real images, potentially complex and/or brittle intermediate steps such as 3D reconstruction can be completely avoided; nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse (harmonic) light fields. We validate the use of subspaces constructed in this fashion within the context of face recognition.

2,472 citations

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TL;DR: In this paper, the authors used double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self-consistent field (SCF) energies and correlation energies.

Abstract: The major source of errror in most ab initio calculations of molecular energies is the truncation of the one‐electron basis set. A complete basis set model chemistry is defined to include corrections for basis set truncation errors. This model uses double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self‐consistent‐field (SCF) energies and correlation energies. The small corrections to give the complete basis set SCF energies are then estimated using the l−6 asymptotic convergence of the multicenter angular momentum expansion. The calculated correlation energies of the atoms He, Be, and Ne, and of the hydrides LiH, BH3, CH4, NH3, H2O, and HF, using the double zeta plus polarization basis sets vary from 83.0% to 91.2% of the experimental correlation energies. However, extrapolation of each of the pair energies and pair‐coupling terms to the complete basis set values using the asymptotic convergence of pair natural orbital expansions retrieves from 99.5±0.7% to ...

2,329 citations