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Concept# Linear span

Summary

In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors in S.
For example, two linearly independent vectors span a plane.
The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.
To express that a vector space V is a linear span of a subset S, one commonly uses the following phrases—either: S spans V, S is a spanning set of V, V is spanned/generated by S, or S is a generator or generator set of V.
Definition
Given a vector space V

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This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L-2-norm and the H-1-seminorm. 2022 Elsevier B.V. All rights reserved.

Rainer Beck, Phil Morten Hundt, Maarten Eduard van Reijzen

Quantum state resolved reactivity measurements probe the role of vibrational symmetry on the vibrational activation of the dissociative chemisorption of CH4 on Ni(111). IR-IR double resonance excitation in a molecular beam was used to prepare CH4 in three different vibrational symmetry components, A1, E, and F2, of the 2ν3 antisymmetric stretch overtone vibration as well as in the ν1 + ν3 symmetric plus antisymmetric C–H stretch combination band of F2 symmetry. The quantum state specific dissociation probability S0 (sticking coefficient) was measured for each of the four vibrational states by detecting chemisorbed carbon on Ni(111) as the product of CH4 dissociation by Auger electron spectroscopy. We observe strong mode specificity, where S0 for the most reactive state ν1 + ν3 is an order of magnitude higher than for the least reactive, more energetic 2ν3-E state. Our first principles quantum scattering calculations show that as molecules in the ν1 state approach the surface, the vibrational amplitude becomes localized on the reacting C–H bond, making them very reactive. This behavior results from the weakening of the reacting C–H bond as the molecule approaches the surface, decoupling its motion from the three non-reacting C–H stretches. Similarly, we find that overtone normal mode states with more ν1 character are more reactive: S0(2ν1) > S0(ν1 + ν3) > S0(2ν3). The 2ν3 eigenstates excited in the experiment can be written as linear combinations of these normal mode states. The highly reactive 2ν1 and ν1 + ν3 normal modes, being of A1 and F2 symmetry, can contribute to the 2ν3-A1 and 2ν3-F2 eigenstates, respectively, boosting their reactivity over the E component, which contains no ν1 character due to symmetry.

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A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x(1),...x(r)]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and R-modules. We prove in particular that the Nr-graded R[x(1),...,x(r)]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals. (C) 2016 Elsevier B.V. All rights reserved.