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High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics. II. Periodic variables

2000
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Journal of Chemical Physics
*

*line͒*, and the FFT on a 64

*orthogonal*grid ͑dotted

*line͒*. ... , seventh-order FD with a 64 ͑cuu͒ initial grid ͑dashed

*line͒*, and the FFT on a 64

*orthogonal*grid ͑dotted

*line͒*. ...

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Gram-Schmidt technique for aberration analysis in telescope mirror testing

1999
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Advanced Telescope Design, Fabrication, and Control
*

After that it is nesessary to make account of scalar multiplication of initial matrix current

doi:10.1117/12.367612
fatcat:ngscgirf6bcxvhpfponzmvkxrq
*line*and all*lines*calculated on previous steps of an*orthogonal*matrix: ( ) U D*k*i*k*i*k*i*k*i*k*i u d u ... FORMATION OF AN*ORTHOGONAL*MATRIX U AND AUXILIARY MATRIX V . 1. The first*line*corresponds to the first*line*of an initial matrix D . ...##
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Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs
[article]

2022
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arXiv
*
pre-print

For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic

arXiv:2203.03663v2
fatcat:xi4otutaivarhokb23qhaw66qa
*approximations*already in ℝ^2. ... )-*approximations*), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time. ... For all*k*∈ [d] and i ∈ [n], if m T*k*,i ∈ M T i exists, then its*center*point is (2k, i n , −1). For all*k*∈ [d], the*center*point of q*k*∈ Q is (2k, 1 2 , −1.6). ...##
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ALORA: Affine Low-Rank Approximations

2018
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Journal of Scientific Computing
*

We show that the optimal affine

doi:10.1007/s10915-018-0885-5
fatcat:3vee4iqatjbajg3s3x3kqwev6q
*approximation*can be obtained by applying an*orthogonal*projection to the matrix before constructing its best*approximation*. ... In this paper we present the concept of affine low-rank*approximation*for an m×n matrix, consisting in fitting its columns into an affine subspace of dimension at most*k*min(m, n). ... The vectors p j and d j denote the projection vectors along and*orthogonal*to u respectively. passing through the gravity*center*of A,ũ = u 1 (Y ). ...##
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Orthogonal least squares algorithm for the approximation of a map and its derivatives with a RBF network

2003
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Signal Processing
*

In this paper, the

doi:10.1016/s0165-1684(02)00397-3
fatcat:soft4fwdyjeydhiqrdekmj2r7i
*Orthogonal*Least Squares (OLS) algorithm for the identification of RBFNs is modified to provide the*approximation*of a non-linear single-input single-output map along with its derivatives ... Several algorithms are known for the*approximation*of a non-linear curve from a sparse data set by means of RBFNs. ... The RBFN can be viewed as a special case of the linear regression model y(*k*) = b + H i=1 w i p i (*k*) + e(*k*), (2) where y(*k*) is the desired*k*-th output sample, e(*k*) is the*approximation*error, and p i ( ...##
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On approximation of a three-dimensional convex body by cylinders

2006
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St. Petersburg Mathematical Journal
*

New results on

doi:10.1090/s1061-0022-06-00906-x
fatcat:ogjt3sev7bdrrg6jbcpvgltjxq
*approximation*of a convex body*K*⊂ R 3 by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained. ... We note that the*orthogonal*projection of*K*⊂ R 3 onto a plane P*orthogonal*to a given*line*l ⊂ R 3 is inscribed into an affine-regular hexagon S. ... In each plane P in question, we mark also the*centers*of the affine-regular hexagons circumscribed about the*orthogonal*projection of the body*K*onto the plane P . ...##
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On complete and incomplete exponential systems
[article]

2020
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arXiv
*
pre-print

We prove that L^2(B_d) does not possess a ϕ-

arXiv:2007.07972v1
fatcat:qqyy6mqugrc3lfdgdvuvpx7r4u
*approximate**orthogonal*basis of exponentials for a wide range of functions ϕ. ... Given a bounded domain Ω, we say that e^2 π i x · a, e^2 π i x · a' are ϕ-*approximately**orthogonal*if |χ_Ω(a-a')| ≤ϕ(|a-a'|), a≠ a' where ϕ: [0, ∞) → [0, ∞) is a bounded measurable function that tends ...*orthogonal*system in L 2 (*K*). ...##
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Highly localized quasiatomic minimal basis orbitals for Mo fromab initiocalculations

2007
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Physical Review B
*

In addition, the transferability of tight-binding parametrizations can be evaluated, for example, the effect of the two-

doi:10.1103/physrevb.76.205119
fatcat:tes72r72jndmtmzs737nxmrr4e
*center**approximation*. ... This is an evidence that the two-*center**approximation*is not adequate. ... The two perpendicular*lines*passing through the*center*of the QUAMBOs indicate the directions of the nearest-neighbor atoms. ...##
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Foundations for the Approximate Synthesis of RCCC Motion Generators
[chapter]

2013
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Mechanisms and Machine Science
*

The

doi:10.1007/978-94-007-7214-4_37
fatcat:ki3z43h4abb7tp5dndazvbxyra
*approximate*synthesis of RCCC linkages for motion generation, a.k.a. rigid-body guidance, is the subject of this paper. ... The dual normality conditions necessary to obtain a feasible least-square*approximation*are established, following which an algorithm for the solution of the problem is proposed. ... The strategy consists in decomposing the increment in two parts, namely, ∆x*k*= ∆v*k*+L*k*∆û*k*(14) withL*k*denoting the*orthogonal*complementL evaluated atx*k*. ...##
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Page 21 of Hewlett-Packard Journal Vol. 38, Issue 5
[page]

1987
*
Hewlett-Packard Journal
*

There are two solution branches depending on whether « = 0 or

*k*# 0.*k*= O means geometrically that the three*centers*m1, m2, and m3 are colinear. ... +(*k*—p )? = (r0+1r3)? (3) These equations are solved by standard elimination tech- niques. ...##
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Ab Initio Calculation of Tight-Binding Parameters

1997
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Materials Research Society Symposium Proceedings
*

The two

doi:10.1557/proc-491-199
fatcat:aouannlbdfdupe7r6kxy26qy6m
*center*expansion is by far the more serious*approximation*. ...*ORTHOGONAL*BASES Here we consider the suitability of*orthogonal*bases by performing Lijwdin transformations 1191, H"' = 0-1/2HO-1/2, t o create (*k*-dependent) Hamiltonian matrix elements, H"', in an*orthogonal*...##
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Construction of fractional spline wavelet bases

1999
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Wavelet Applications in Signal and Image Processing VII
*

We show how they yield continuous-order generalizations of the

doi:10.1117/12.366799
fatcat:dygqz6pv3jb6pntn2mmnstdaxe
*orthogonal*Battle-Lemarie wavelets and of the semi-*orthogonal*B-spline wavelets. ... What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order o f*approximation*( α + 1), while they reproduce the polynomials of degree α . ... In fact, these coefficients are a re-*centered*version of generalized binomials, for it can be shown that r*k*r*k*r = + 2 . (12) Some examples of these fractional*centered*B-splines are shown ...##
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Best approximation and characterization of Hilbert spaces

2019
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Miskolc Mathematical Notes
*

It is well known that for any nonempty closed convex subset C of a Hilbert space, any best

doi:10.18514/mmn.2019.2917
fatcat:vrue7ovbxjcdjks74bsxuebjwm
*approximation*y 2 C of the point x satisfies the inequality kx yk 2 C*k*´ yk 2 Ä kx ´*k*2 for all´2 C . ... In this paper, we first introduce and study a new subset of best*approximations*involving this inequality in general metric spaces. ... A complete family of*centers*of A is a set S such that In a normed linear space .X;*k**k*/, a vector x is said to be Birkhoff*orthogonal*to a vector y (x? ...##
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Page 556 of Physical Review Vol. 86, Issue 4
[page]

1952
*
Physical Review
*

For the lithium crystal

*K*; is the form Kn inon3=*K*(iym+ ina+ ists), where ij, iz, and i; are fixed, mutually*orthogonal*unit vectors, (”:+%2+ms3) is an arbitrary even integer (face-*centered*-cubic reciprocal ... The key to this lack of usefull- ness of the Bloch method*lines*in Eq. (27). Just as we*approximated*Eq. (29) by Eq. (35), we may approxi- mate Eq. (27) by W,(r)SA woXo(r). ...##
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Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits
[article]

2020
*
arXiv
*
pre-print

We demonstrate the importance of the proved results applying polynomial least squares

arXiv:2004.00414v1
fatcat:3ooeqdix3ngfpn5dkxmpznyh5y
*approximation*for the detection of anomalous values in IGS final orbits for GPS and GLONASS satellites. ... We propose a numerically stable method for the construction of discrete*orthogonal*polynomials of high degrees. ... subsets of points of the real*line*-discrete*orthogonal*polynomials. ...
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