Local Linear Dependence Research Articles

Emergence of local scaling relations in adsorption energies on high-entropy alloys

Alloying has been proposed to circumvent scaling relations between the adsorption energies thus allowing for the complete optimization of multistep reactions. Herein the fidelity of scaling rules on high-entropy alloy (HEA) surfaces is assessed focusing on hydrogen-containing molecules, *AHx for A = C and N (x = 0, 1, 2, 3), A = S (x = 0, 1, 2) and A = O (x = 0, 1). Using an adsorbate- and site-specific deep learning model to rapidly compute the adsorption energies on CoMoFeNiCu HEA surfaces, the energies of *AHx and *A are shown to be linearly correlated if *A and *AHx have identical adsorption site symmetry. However, a local linear dependence emerges between the configuration-averaged adsorption energies irrespective of the site symmetry. Although these correlations represent a weaker form of the scaling relationships, they are sufficient to prohibit the optimization of multistep reactions. The underpinning of this behavior is twofold (1) the nearsightedness principle and (2) the narrow distribution of the adsorption energies around the mean-field value. While the nearsightedness is general for all electronic systems, the second criterion applies in HEAs with relatively strong reactive elements. The present findings strongly suggest that alloys may not generally enable the breaking of scaling relationships.

Local Linear Dependence of Linear Partial Differential Operators

We show that any finite set of linear partial differential operators with continuous coefficients is linearly dependent if and only if it is locally linearly dependent. It follows that the reflexive closure of any finite set of such operators is equal to its linear span. The last statement can be rephrased as a weak nullstellensatz for linear partial differential operators.

Local linear dependence seen through duality II

A vector space S of linear operators between finite-dimensional vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x∈U is annihilated by a non-zero operator in S. By a duality argument, one sees that studying LLD operator spaces amounts to studying vector spaces of matrices with rank less than the number of columns, or, alternatively, vector spaces of non-injective operators.In this article, this insight is used to obtain classification results for LLD spaces of small dimension or large essential range (the essential range being the sum of all the ranges of the operators in S). We show that such classification theorems can be obtained by translating into the context of LLD spaces Atkinson's classification of primitive spaces of bounded rank matrices; we also obtain a new classification theorem for such spaces that covers a range of dimensions for the essential range that is roughly twice as large as that in Atkinson's theorem. In particular, we obtain a classification of all 4-dimensional LLD operator spaces for fields with more than 3 elements (beforehand, such a classification was known only for algebraically closed fields and in the context of primitive spaces of matrices of bounded rank).These results are applied to obtain improved upper bounds for the maximal rank in a minimal LLD operator space.

Local linear dependence seen through duality I

A vector space S of linear operators between vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x∈U is annihilated by a non-zero operator in S. A duality argument bridges the theory of LLD spaces to the one of vector spaces of non-injective operators. This new insight yields a unified approach to rediscover basic LLD theorems and obtain many additional ones thanks to the power of formal matrix computations.In this article, we focus on the minimal rank for a non-zero operator in an LLD space. Among other things, we reprove the Brešar–Šemrl theorem, which states that an n-dimensional LLD operator space always contains a non-zero operator with rank less than n, and we improve the Meshulam–Šemrl theorem that examines the case when no non-zero operator has rank less than n−1.We also tackle the minimal rank problem for a non-zero operator in an n-dimensional operator space that is not algebraically reflexive. A theorem of Meshulam and Šemrl states that, for all fields with large enough cardinality, a non-reflexive operator space with dimension n must contain a non-zero operator with rank at most 2n−2. We show that there are infinitely many integers n for which this bound is optimal for general infinite fields. Moreover, under mild cardinality assumptions, we obtain a complete classification of the non-reflexive n-dimensional operator spaces in which no non-zero operator has rank less than 2n−2. This classification involves a new algebraic structure called left-division-bilinearizable (in abbreviated form: LDB) division algebras, which generalize a situation that is encountered with quaternions and octonions and whose systematic study occupies a large part of the present article.

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.

Local polynomial property and linear independence of refinable distributions

In this paper, local polynomial property, global linear independence, and local linear dependence of the convolution of a B-spline and a refinable distribution supported on a Cantor-like set are studied.

Remarks on the Wronskian and on sums decompositions

We make a number of remarks on some theorems and considerations contained in the recent and valuable monograph [9]. The connection of the Wronskian with the linear dependence, the local linear dependence and the local linear independence is considered. The necessary and/or sufficient conditions in order that a function $ h(x,y) $ has the form $ \sum^n_{i=1} f_i(x)g_i(y) $ , in terms of the Wronski matrix, are discussed. The d'Alembert equation $ hh_{xy} - h_xh_y = 0 $ is investigated, too. We formulate also some open problems.

M-Band Scaling Function with Filter Having Vanishing Moments Two and Minimal Length

In this paper we consider the Hölder continuity, local linearity, linear independence, and interpolation problem of theM-band scaling function, with a filter having vanishing moments two and minimal length. In particular, we find some new properties that are not true whenM=2, such as local linearity, local linear dependence, differentiability at adjointM-adic points, and interpolation problem at integer knots.

On the optimal approximation rates for criss-cross finite element spaces

Let Δ denote the triangulation of the plane obtained by multi-integer translates of the four lines x=0, y=0, x= y and x=− y. By l k, h μ we mean the space of all piecewise polynomials of degree ⩽ k with respect to the scaled triangulation hΔ having continuous partial derivatives of order ⩽μ on R 2 . We show that the approximation properties of l k, h μ are completely governed by those of the space spanned by the translates of all so called box splines contained in l k,h μ . Combining this fact with Fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where μ is the largest degree of smoothness for which these spaces are dense as h tends to zero. Furthermore, we study the question of local linear dependence of the translates of the box splines for the above criss-cross triangulations.

Local Linear Dependence and the Vanishing of the Wronskian

on the interval I. However, it has also long been known that for n functions which are only (n 1)-times differentiable (so that their Wronskian is defined) the sufficiency part of the above statement no longer holds. Peano [12] seems to have been the first to point this out, and Bocher [3] has given an example which shows that even if the functions involved are infinitely differentiable on I, the identical vanishing of the Wronskian is still not sufficient to imply their linear dependence on I. It was then recognized that the functions involved must satisfy other conditions, supplementary to the vanishing of their Wronskian, in order to guarantee their linear Peano [13] and Bocher [5] have given such conditions (for example, Lemma 3 of Sec. 2 of this paper) and Bocher has shown that his conditions include those of Peano. In this paper we look at this problem from a slightly different point of view. Namely, instead of placing linear dependence in the forefront and looking for a condition to supplement the identical vanishing of the Wronskian, rather, we shall put the vanishing of the Wronskian in the forefront and ask for a kind of generalized dependence (necessarily weaker than linear dependence) which is equivalent to it. This point of view has led the author to define a new type of dependencef relation for functions of a real variable which he has called linear dependence. It is shown that local linear dependence and the identical

Local Linear Dependence and the Vanishing of the Wronskian

Local Linear Dependence and the Vanishing of the Wronskian