Design of Mechanisms Via Constrained Least-Squares Method and Its Variants

1988 ◽  
Vol 110 (4) ◽  
pp. 429-434 ◽  
Author(s):  
Hui Cheng ◽  
K. C. Gupta
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Direct Methods ◽  
Superposition Method ◽  
Linear Superposition ◽  
Normal Equations ◽  
Design Specifications ◽  
Synthesis Of Mechanisms ◽  
Special Case ◽  
Least Squares Techniques

Based on the design equation of mechanism and the least-squares techniques, a rapidly convergent iteration method and simple direct methods for the synthesis of mechanisms are presented. It is proved that the so-called linear superposition method is a special case of the direct methods whose effectiveness depends upon the singular nature of the normal equations of the least-squares method as well as the smallness of the Lagrange multipliers of the compatibility equations for the mechanism. While the significance of the latter has been recognized in the literature, that of the former has not been documented in the literature. By examining the correlation matrix and the condition number for the normal equations, we show that these are near-singular. This property provides a fundamental basis for the direct methods presented in this paper. The sensitivity of solutions to the design specifications and to the precision of floating point computations also is discussed. The theory and associated algorithms can be applied to the synthesis of any planar or spatial mechanism where the use of the least-squares technique is contemplated.

On the estimation of a harmonic component in a time series with stationary dependent residuals

1973 ◽  
Vol 5 (02) ◽  
pp. 217-241 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Harmonic Component ◽  
Asymptotic Distributions ◽  
Finite Variance ◽  
Rigorous Derivation ◽  
Image Position ◽  
Vector Valued ◽  
Special Case

Let observations (X 1, X 2, …, Xn ) be obtained from a time series {Xt } such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu (θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu (θ) = 0 for u > 0, the parameter θ thus being absent.

Crystal structures of optically active naphthidine (4,4′-diamino-1,1′-binaphthyl) and 1,1′-binaphthyl

1983 ◽  
Vol 61 (1) ◽  
pp. 69-71 ◽  
Author(s):  
Richard A. Pauptit ◽  
James Trotter
Keyword(s):  
Least Squares ◽  
Crystal Structures ◽  
Direct Methods ◽  
Optically Active ◽  
Dihedral Angles ◽  
Full Matrix ◽  
Bond Lengths ◽  
Least Squares Techniques

Crystals of optically active naphthidine and of binaphthyl are tetragonal, P41212 (or P43212), a = 7.945(1), 7.164(2), c = 24.264(5), 27.70(1) Å, respectively, Z = 4. The structures were solved by direct methods and refined by full-matrix least-squares techniques to respective R = 0.068 and 0.030 for 548 and 562 reflexions. The dihedral angles between the planes of the naphthalene residues are 87 and 102°, respectively. Bond lengths and angles are normal.

The Crystal Structure of Dimedone

1975 ◽  
Vol 53 (7) ◽  
pp. 1046-1050 ◽  
Author(s):  
Ishwar Singh ◽  
Crispin Calvo
Keyword(s):  
Crystal Structure ◽  
Solid State ◽  
Hydrogen Bonds ◽  
Least Squares ◽  
Direct Methods ◽  
Monoclinic Symmetry ◽  
Enolic Form ◽  
Full Matrix ◽  
Space Group ◽  
Least Squares Techniques

Dimedone, C8H12O2, crystallizes with monoclinic symmetry, a = 10.079(7), b = 6.835(3), c = 12.438(4) Å, β = 110.24(5)°, space group P21/n and Z = 4. The structure of this compound was solved by direct methods and refined by full-matrix least-squares techniques using 1205 unique reflections to a final R of 0.047. In the solid state it exists in the enolic form and these molecules pack in the crystal in systems of infinite chains linked together by hydrogen bonds in the y direction. These results are virtually the same as recently reported by Semmingsen.

scholarly journals Synthesis and Molecular Structure of cis-Tetracarbonyl[N-(diphenylphosphino-kP)-naphthalen-1-yl-P,P-diphenylphosphinous amide-kP]chromium(0)

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Harbi Tomah Al-Masri
Keyword(s):  
Molecular Structure ◽  
Least Squares ◽  
Direct Methods ◽  
Molar Ratio ◽  
Planar Geometry ◽  
Monoclinic Space Group ◽  
Full Matrix ◽  
Space Group ◽  
Distorted Octahedral ◽  
Least Squares Techniques

The reaction of N,N-bis(diphenylphosphanyl)naphthylamine C10H7-1-N(PPh2)2 with (C5H10NH)2Cr(CO)4 (1 : 1 molar ratio) in dichloromethane afforded cis-[Cr(CO)4{C10H7-1-N(PPh2)2}] (1). This complex was crystallized in the monoclinic space group P21/n. The structure was solved by direct methods and refined by full-matrix least squares techniques to an R factor of 0.0313 for 6488 observed reflections. The Cr-metal is coordinated by four terminal CO molecules and a P,P′-bidentate N,N-bis(diphenylphosphanyl)naphthylamine ligand in a distorted octahedral array. The N-atom adopts a planar geometry with the two P-atoms and C-atom attached to it. The four-membered metallacycle ring P2CrN is nearly planar.

Curve-Fitting by the Method of Grouping

1952 ◽  
Vol 5 (2) ◽  
pp. 238
Author(s):  
PG Guest
Keyword(s):  
Least Squares ◽  
Curve Fitting ◽  
Least Squares Method ◽  
Normal Equations

A method of fitting polynomials is described in which the "normal" equations are obtained much more rapidly than the corresponding equations in the least-squares method. Efficiencies are found to be about 90 per cent. The method is illustrated by an example.

X-ray Structural Study of 5-Phenyl-10,11-dihydrodibenzo[b,f]phosphepin-5-oxide

1980 ◽  
Vol 35 (2) ◽  
pp. 133-135 ◽  
Author(s):  
David W. Allen ◽  
Ian W. Nowell ◽  
Philip E. Walker
Keyword(s):  
Least Squares ◽  
Electron Density ◽  
Structural Study ◽  
Title Compound ◽  
Bond Distance ◽  
Direct Methods ◽  
X Ray ◽  
Phenyl Rings ◽  
Density Methods ◽  
Least Squares Techniques

AbstractCrystals of the title compound are triclinic, a = 8.533(5), b = 11.106(6), c = 8.815(5) Å, a = 107.83(6), β = 104.99(6), γ = 81.30(5)°, Z = 2, space group P1̄. The structure was determined by multisolution direct methods and electron density methods. Refinement by least-squares techniques gave a final R = 0.081 for the 1753 independent reflections. The molecule adopts a butterfly-type conformation such that the fused phenyl rings are inclined to each other at an angle of 56.8°. The P-0 bond distance is 1.506(4) Å; the endocyclic angle at phosphorus is 107.2(3)° and the exocyclic angles vary from 106.5 to 111.9(3)°.

The crystal structure of p-diethylaminobenzenediazonium hexafluorophosphate

1985 ◽  
Vol 63 (2) ◽  
pp. 332-335 ◽  
Author(s):  
Richard G. Ball ◽  
Richard MacLeod Elofson
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Triple Bond ◽  
Direct Methods ◽  
Full Matrix ◽  
Space Group ◽  
Bond Lengths ◽  
The Mean ◽  
Least Squares Techniques ◽  
Diazo Group

p-Diethylaminobenzenediazonium hexafluorophosphate, C10H14N3+•PF6−, crystallized in space group [Formula: see text] with a = 12.105(4), b = 12.340(5), c = 10.439(4) Å, α = 96.53(3), β = 104.11(3), γ = 64.44(3)°, and Z = 4. The structure was solved using direct methods and refined with full-matrix least-squares techniques on F, to a final R of 0.054 for 1917 reflections with F2 > 3σ(F2). The mean bond lengths for the diazo group are: N—N 1.096(6); C—N 1.357(7) Å. The geometry of the molecule is discussed in terms of the possible resonance forms and it is shown to be consistent with a form in which the N—N triple bond is essentially intact and the aminobenzene moiety has "quinoidal" character.

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

1999 ◽  
Vol 55 (4) ◽  
pp. 755-760 ◽  
Author(s):  
David A. Langs ◽  
Robert H. Blessing ◽  
Dongyao Guo
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Heavy Atom ◽  
Direct Methods ◽  
Anomalous Dispersion ◽  
Correct Solution ◽  
Phase Ambiguity ◽  
Practical Applications ◽  
The Past

In the past few years, a number of strategies have been outlined to resolve the SAS phase ambiguity given that unique estimates \omega(h, k) of the triple invariants are available. A new least-squares method is described that can in principle resolve the phase ambiguity to determine macromolecular phases provided that \omega(h, k) estimates are unbiased. Limitations of the method in practical applications are discussed. An example is given where the correct solution can be identified by use of the SAS tangent formula in the instance that traditional SAS phasing methods have lead to an incorrect heavy-atom substructure.

Structural studies and redox reactivity of platinum complexes of 14-membered tetraaza macrocyclic ligands

1997 ◽  
Vol 75 (10) ◽  
pp. 1363-1374 ◽  
Author(s):  
S. Chandrasekhar ◽  
W.L. Waltz ◽  
J.W. Quail ◽  
L. Prasad
Keyword(s):  
Cyclic Voltammetry ◽  
Least Squares ◽  
Mass Spectroscopy ◽  
Least Squares Method ◽  
Direct Methods ◽  
Complex Ions ◽  
Orthorhombic Space Group ◽  
Full Matrix ◽  
Space Group ◽  
Electrospray Mass Spectroscopy

Pt(II) monomeric complexes of N-containing macrocycles have been synthesized and characterized in solution by 1H NMR, 13C NMR, and electrospray mass spectroscopy (ESMS) and in the solid state by X-ray crystallography. Crystals of [Pt(cyclam)](ClO4)2 (cyclam = 1,4,8,11-tetraazacyclotetradecane) are orthorhombic, space group P21cn, a = 9.596 (3), b = 14.595 (21), c = 24.8782 (20) Å, Z = 8. The structure was solved by direct methods and was refined by the full-matrix least-squares method to R = 0.073 and Rw = 0.093 for 2976 reflections with I ≥ 2.5σ(I). Crystals of [Pt(N-methylcyclam)]-(ClO4)2•CH3CN (N-methylcyclam = 1,4,8,11 tetramethyl-1,4,8,11-tetraazacyclotetradecane = NMe4cyclam = tmc) are orthorhombic, space group Pmcn, a = 9.574 (8), b = 14.116 (5), c = 17.5456 (15) Å, Z = 4. The structure was solved by direct methods and was refined by the full-matrix least-squares method to R = 0.067 and Rw = 0.091 for 2216 reflections with I ≥ 2.5σ(I). Cyclic voltammetry and EPR spectroscopy have been used to study the redox reactivity of the Pt(II)and Pt(IV) complex ions. The solution chemistry of Pt(IV) complexes based on cyclam and N-methylcyclam was investigated by electrospray mass spectrometry. Electrospray mass spectral evidence has been obtained for the formation of [Pt(III)(cyclam)Cl2]+ and [Pt(III)(N-methylcyclam)Cl2]+ ions. Keywords: Pt macrocycles, crystallography, electrospray mass spectroscopy, cyclic voltammetry, Pt(III) complex ions, tetraaza macrocycles.

The molecular and crystal structure of 4,4-dimethyl-6,6-diphenyl-tetrahydro-2H-pyran-2-one

1989 ◽  
Vol 54 (6) ◽  
pp. 1661-1665
Author(s):  
Jiří Novotný ◽  
Jan Ondráček ◽  
Marián Schwarz ◽  
Bohumil Kratochvíl
Keyword(s):  
Crystal Structure ◽  
Nmr Spectroscopy ◽  
Least Squares ◽  
Heterocyclic Compound ◽  
Least Squares Method ◽  
Lactone Ring ◽  
Direct Methods ◽  
Molecular And Crystal Structure ◽  
Boat Conformation ◽  
C Nmr

The molecular and crystal structure of 4,4-dimethyl-6,6-diphenyl-tetrahydro-2H-pyran-2-one was solved by direct methods and anisotropically refined by the least squares method. The final R-factor value was 0.065 for 2 181 observed reflections (I > 1.96σ (I)) and 272 refined parameters. The symmetry of the structure corresponds to the P21/c space group with lattice parameters a = 11.207(2), b = 8.168(1), c = 16.896(3) Å, β = 90.05(1)°. The unit cell contains four formula units. A lactone structure was found for this geminally substituted heterocyclic compound and was also demonstrated by 1H and 13C NMR spectroscopy. The lactone ring assumes the boat conformation.

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