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

Raul Guantes, Stavros C. Farantos
2000 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.  ... 
doi:10.1063/1.1324004 fatcat:4m2ry6odundntpdaraa34ur4qe

Gram-Schmidt technique for aberration analysis in telescope mirror testing

Nadezhda D. Tolstoba, William Roybal
1999 Advanced Telescope Design, Fabrication, and Control  
After that it is nesessary to make account of scalar multiplication of initial matrix current 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 .  ... 
doi:10.1117/12.367612 fatcat:ngscgirf6bcxvhpfponzmvkxrq

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs [article]

Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, André Nusser, Zahra Parsaeian
2022 arXiv   pre-print
For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic 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).  ... 
arXiv:2203.03663v2 fatcat:xi4otutaivarhokb23qhaw66qa

ALORA: Affine Low-Rank Approximations

Alan Ayala, Xavier Claeys, Laura Grigori
2018 Journal of Scientific Computing  
We show that the optimal affine 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 ).  ... 
doi:10.1007/s10915-018-0885-5 fatcat:3vee4iqatjbajg3s3x3kqwev6q

Orthogonal least squares algorithm for the approximation of a map and its derivatives with a RBF network

C. Drioli, D. Rocchesso
2003 Signal Processing  
In this paper, the 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 (  ... 
doi:10.1016/s0165-1684(02)00397-3 fatcat:soft4fwdyjeydhiqrdekmj2r7i

On approximation of a three-dimensional convex body by cylinders

V. V. Makeev
2006 St. Petersburg Mathematical Journal  
New results on 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 .  ... 
doi:10.1090/s1061-0022-06-00906-x fatcat:ogjt3sev7bdrrg6jbcpvgltjxq

On complete and incomplete exponential systems [article]

Alex Iosevich, Azita Mayeli
2020 arXiv   pre-print
We prove that L^2(B_d) does not possess a ϕ-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).  ... 
arXiv:2007.07972v1 fatcat:qqyy6mqugrc3lfdgdvuvpx7r4u

Highly localized quasiatomic minimal basis orbitals for Mo fromab initiocalculations

T.-L. Chan, Y. X. Yao, C. Z. Wang, W. C. Lu, J. Li, X. F. Qian, S. Yip, K. M. Ho
2007 Physical Review B  
In addition, the transferability of tight-binding parametrizations can be evaluated, for example, the effect of the two-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.  ... 
doi:10.1103/physrevb.76.205119 fatcat:tes72r72jndmtmzs737nxmrr4e

Foundations for the Approximate Synthesis of RCCC Motion Generators [chapter]

Jorge Angeles
2013 Mechanisms and Machine Science  
The 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 .  ... 
doi:10.1007/978-94-007-7214-4_37 fatcat:ki3z43h4abb7tp5dndazvbxyra

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.  ... 

Ab Initio Calculation of Tight-Binding Parameters

A. K. Mcmahan, J. E. Klepeis
1997 Materials Research Society Symposium Proceedings  
The two 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  ... 
doi:10.1557/proc-491-199 fatcat:aouannlbdfdupe7r6kxy26qy6m

Construction of fractional spline wavelet bases

Michael A. Unser, Thierry Blu, Michael A. Unser, Akram Aldroubi, Andrew F. Laine
1999 Wavelet Applications in Signal and Image Processing VII  
We show how they yield continuous-order generalizations of the 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  ... 
doi:10.1117/12.366799 fatcat:dygqz6pv3jb6pntn2mmnstdaxe

Best approximation and characterization of Hilbert spaces

Setareh Rajabi
2019 Miskolc Mathematical Notes  
It is well known that for any nonempty closed convex subset C of a Hilbert space, any best 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?  ... 
doi:10.18514/mmn.2019.2917 fatcat:vrue7ovbxjcdjks74bsxuebjwm

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).  ... 

Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits [article]

Sergey P. Tsarev, Alexey A. Kytmanov
2020 arXiv   pre-print
We demonstrate the importance of the proved results applying polynomial least squares 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.  ... 
arXiv:2004.00414v1 fatcat:3ooeqdix3ngfpn5dkxmpznyh5y
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