Concise row-pruning algorithm to invert a matrix

Presented here, in a vector formulation, is an O(mn2) direct concise algorithm that prunes/identifies the linearly dependent (ld) rows of an arbitrary m X n matrix A and computes its reflexive type minimum norm inverse A(mr)-, which will be the true inverse A-1 if A is nonsingular and the Moore-Penrose inverse A+ if A is full row-rank. The algorithm, without any additional computation, produces the projection operator P = (I - A(mr)- A) that provides a means to compute any of the solutions of the consistent linear equation Ax = b since the general solution may be expressed as x = A(mr)+b + Pz, where z is an arbitrary vector. The rank r of A will also be produced in the process. Some of the salient features of this algorithm are that (i) the algorithm is concise, (ii) the minimum norm least squares solution for consistent/inconsistent equations is readily computable when A is full row-rank (else, a minimum norm solution for consistent equations is obtainable), (iii) the algorithm identifies ld rows, if any, and reduces concerned computation and improves accuracy of the result, (iv) error-bounds for the inverse as well as the solution x for Ax = b are readily computable, (v) error-free computation of the inverse, solution vector, rank, and projection operator and its inherent parallel implementation are straightforward, (vi) it is suitable for vector (pipeline) machines, and (vii) the inverse produced by the algorithm can be used to solve under-/overdetermined linear systems.

Study Of Acoustic Source Localization Algorithm For Planar Arrays

The source localization algorithms in the earlier works, mostly used non-planar arrays. If we consider scenarios like human-computer communication, or human-television communication where the microphones need to be placed on the computer monitor or television front panel, i.e we need to use the planar arrays. The algorithm proposed in 1], is a Linear Closed Form source localization algorithm (LCF algorithm) which is based on Time Difference of Arrivals (TDOAs) that are obtained from the data collected using the microphones. It assumes non-planar arrays. The LCF algorithm is applied to planar arrays in the current work. The relationship between the error in the source location estimate and the perturbation in the TDOAs is derived using first order perturbation analysis and validated using simulations. If the TDOAs are erroneous, both the coefficient matrix and the data matrix used for obtaining source location will be perturbed. So, the Total least squares solution for source localization is proposed in the current work. The sensitivity analysis of the source localization algorithm for planar arrays and non-planar arrays is done by introducing perturbation in the TDOAs and the microphone locations. It is shown that the error in the source location estimate is less when we use planar array instead of the particular non-planar array considered for same perturbation in the TDOAs or microphone location. The location of the reference microphone is proved to be important for getting an accurate source location estimate if we are using the LCF algorithm.

The structure of amorphous platinum disulfide as studied by anomalous x-ray-scattering

Anomalous X-ray scattering (AXS) has been applied to study the structure of amorphous platinum disulfide, Pt1-xS2, prepared by the precipitation process. The local atomic arrangement in amorphous Pt1-xS2 was determined by the least-squares variational method so as to reproduce the experimental differential interference function at the Pt L(III) absorption edge by the AXS method as well as the ordinary interference function by MoK alpha. The structural unit in amorphous Pt1-xS2 is found to be a PtS6 octahedron, similar to that in crystalline PtS2. These octahedra share both their corners and edges, while only edge-sharing linkages occur in crystalline PtS2.

Anomalous X-ray scattering study of local structures in the superionic conducting class (CuI)(0.3)(Cu2O)(0.35)(MoO3)(0.35)

The anomalous X-ray scattering (AXS) method using Cu and Mo K absorption edges has been employed for obtaining the local structural information of superionic conducting glass having the composition (CuI)(0.3)(Cu2O)(0.35)(MoO3)(0.35). The possible atomic arrangements in near-neighbor region of this glass were estimated by coupling the results with the least-squares analysis so as to reproduce two differential intensity profiles for Cu and Mo as well as the ordinary scattering profile. The coordination number of oxygen around Mo is found to be 6.1 at the distance of 0.187 nm. This implies that the MoO6 octahedral unit is a more probable structural entity in the glass rather than MoO4 tetrahedra which has been proposed based on infrared spectroscopy. The pre-peak shoulder observed at about 10 nm(-1) may be attributed to density fluctuation originating from the MoO6 octahedral units connected with the corner sharing linkage, in which the correlation length is about 0.8 nm. The value of the coordination number of I- around Cu+ is estimated as 4.3 at 0.261 nm, suggesting an arrangement similar to that in molten CuI.

Crystal and molecular structure of Boc-Phe-Val-OMe; comparison of the peptide conformation with its dehydro analogue

The crystal structure of the peptide Boc-Phe-Val-OMe determined by X-ray diffraction methods is reported in this paper. The crystals grown from aqueous methanol are orthorhombic, space group P2(1)2(1)2(1), a = 11.843(2), b = 21.493(4), c = 26.676(4)Angstrom and V = 6790 Angstrom(3). Data were collected on a CAD4 diffractometer using MoK2 radiation (lambda = 0.7107 Angstrom) up to Bragg angle theta = 26 degrees. The structure was solved by direct methods and refined by a least-squares procedure to an R value of 6.8% for 3288 observed reflections. There are three crystallographically independent peptide molecules in the asymmetric unit. All the three molecules exhibit extended conformation. The sidechain of the Val(2) residue shows two different conformations. The conformation of the peptide Boc-Phe-Val-OMe is compared with the conformation of Ac-Delta Phe-Val-OH. It is observed that while Boc-Phe-Val-OMe exhibits an extended conformation, Ac-Delta Phe-Val-OH shows a folded conformation. The results of this comparison highlight the conformation constraining property of the Delta Phe residue. Interestingly, even though Boc-Phe-Val-OMe and Ac-Delta Phe-Val-OH are conformationally different, they exhibit similar packing patterns in the solid state. (C) Munksgaard 1995.

Saturated liquid viscosity correlations for alternative refrigerants

This article presents dimensionless equations for the temperature dependence of the saturated liquid viscosity of R32, R123, R124, R125, R134a, R141b, and R152a valid over a temperature range of engineering interest. The correlation has the form Phi(D)(n)=A+BTD where Phi(D) is the dimensionless fluidity (1/eta(D)) and T-D is a dimensionless temperature. n, A, and B are evaluated for each of the above refrigerants based on a least-squares fit to experimental data. This equation is found to provide an improved fit over those existing in the literature up to T-D=0.8.

WLS method for parameter estimation in water distribution networks

The weighted-least-squares method based on the Gauss-Newton minimization technique is used for parameter estimation in water distribution networks. The parameters considered are: element resistances (single and/or group resistances, Hazen-Williams coefficients, pump specifications) and consumptions (for single or multiple loading conditions). The measurements considered are: nodal pressure heads, pipe flows, head loss in pipes, and consumptions/inflows. An important feature of the study is a detailed consideration of the influence of different choice of weights on parameter estimation, for error-free data, noisy data, and noisy data which include bad data. The method is applied to three different networks including a real-life problem.

Structure, Spectra and Redox Behaviour of Copper(11) Complexes of Bis( benzimidazoly1)diamine Ligands

The linear quadridentate ligand N,N'-bis(benzimidazoI-2-ylethyl)ethane-l,2-diamine (L') and its 1 - methylbenzimidazole analogue (L2) and homologues form 1 : 1 complexes with Cu(CIO,),; L' also forms complexes of the types CuL'X, where X = NO,, PF,, Br or CI and CuL'(X)Y where X = CI or Br and Y = CIO, or Br. Deep blue CuL1Br,*2H20 crystallizes in the monoclinic space group C2/c with Z = 4, a = 9.91 9(2), b = 16.626(3), c = 14.1 02(3) le\ and p = 94.39(2)". The structure was solved by Patterson and Fourier difference methods and refined by the least-squares technique to R = 0.064 for 2195 independent reflections with / > 1.50(/). The molecule lies on a two-fold axis symmetrically around Cu". The co-ordination around Cu" is found to be square planar with two amino nitrogens and two benzimidazole nitrogens forming the equatorial plane [CU-N 1.983(3) and 2.037(4) A]. The bromides are at longer distances [3.349(1) A] in axial sites. Ligand field and EPR spectra indicate that one bromide or chloride ion is axially co-ordinated to Cu" in [CuL1l2+. This ion exhibits quasi-reversible redox behaviour. Electrochemical studies of the dihalides in methanol have established the presence of [CuL'X,], [CuL'(X)]+ and [CuL'I2+ in equilibrium. In complexes with 565 [CuL4I2+ [L4 = N,Nbis( benzimidazol-2-ylmethyl)ethane-l,2-diamine] and 555 [CuL3] [L3 = N,N'-bis(1 -methylbenzimidazol- 2-ylmethyl)propane-l,3-diamine] chelate rings, Cull does not seem to lie in the N, square plane, as revealed by their low A values and irreversible electrochemical behaviour. The Cu"-Cu' redox potentials in methanol are in the order [CuL1I2+ < [CuL3I2+ < [CuL4I2+; this illustrates that sixmembered chelate rings are suitable to stabilize Cu", when CU-N 0 interactions are favourable.

Unusual ribose conformation in a ternary copper(II) 2?deoxyribonucleotide complex

The ternary metal deoxyribonucleotide complex [Cu(bzim)(5?-dGMP)(H2O)3](bzim = benzimidazole, 5?-dGMP = 2?-deoxyguanosine 5?-monophosphate) has been prepared and the structure analysed by X-ray diffraction. The compound crystallizes in the space group P1 with a= 7.069(6), b= 13.959(10), c= 14.204(12)Å, ?= 75.12(6), ?= 94.15(6), ?= 97.98(6)° and Z= 2. The structure was solved by the heavy-atom method and refined by full-matrix least-squares procedures on the basis of 2813 observed [I[gt-or-equal] 3?(I)] reflections to final R and R? values of 0.050 and 0.052 respectively. There are two independent molecules in the asymmetric unit and both copper(II) centres have square-pyramidal co-ordination geometry. An unusual feature of the structure is the co-ordination of the metal by N(7) of the base, in the presence of a ?-aromatic amine, bzim. The structure is stabilized by intermolecular base�bzim stacking. The nucleotides of both the molecules have an anti conformation about the glycosyl bond, and a gauche-gauche conformation about the C(4?)�C(5?) bond. A feature of particular interest is the unusual sugar conformation. The base furanose rings of the two nucleotide molecules adopt C(3?)-exo/C(2?)-endo pucker and C(3?)-exo pucker respectively.

Crystal structure of Boc-LAla-Delta Phe-Delta Phe-Delta Phe-Delta Phe-NHMe: a left-handed helical peptide

alpha,beta-Dehydrophenylalanine residues constrain the peptide backbone to beta-bend conformation. A pentapeptide containing four consecutive (Delta Phe) residues has been synthesised and crystallised. The peptide Boc-LAla-Delta Phe-Delta Phe-Delta Phe-Delta Phe-NHMe (C45H46N6O7, MW = 782.86) was crystallised from an acetonitrile/methanol mixture. The crystal belongs to the orthorhombic space group P2(1)2(1)2(1) With a = 19.455(6), b = 20.912(9), c = 11.455(4) Angstrom and Z = 4. The X-ray (MoKalpha, lambda = 0.7107 Angstrom) intensity data were collected using the Rigaku-AFC7 diffractrometer. The crystal structure was determined by direct methods and refined using the least-squares technique, R = 8.41% for 1827 reflections with \F-o\ > 4 sigma\F-o\. The molecule contains the largest stretch of consecutive dehydrophenylalanine residues whose crystal structure has been determined so far. The peptide adopts left-handed 3(10)-helical conformation despite the presence of LAla at the N-terminus. The mean phi, psi values, averaged across the last four residues are 56.8 degrees and 17.5 degrees, respectively. There are four 4-->1 intramolecular hydrogen bonds, characteristic of the 3(10)-helix. In the crystal each molecule interacts with four crystallographically symmetric molecules with one hydrogen bond each.

Role of two consecutive alpha,beta-dehydrophenylalanines in peptide structure: Crystal and molecular structure of Boc-Leu-Delta Phe-Delta Phe-Ala-Phe-NHMe

An N-alpha-protected model pentapeptide containing two consecutive Delta Phe residues, Boc-Leu-Delta Phe-Delta Phe-Ala-Phe-NHMe, has been synthesized by solution methods and fully characterized. H-1-nmr studies provided evidence for the occurrence of a significant population of a conformer having three consecutive, intramolecularly II-bonded beta-bends in solution. The solid state structure has been determined by x-ray diffraction methods. The crystals grown from aqueous methanol are orthorhombic, space group P2(1)2(1)2(1),, a = 11.503(2), b = 16.554(2), c = 22.107(3) Angstrom, V = 4209(1) Angstrom,(3) and Z = 4. The x-ray data were collected on a CAD4 diffractometer using CuKalpha radiation (lambda = 1.5418 Angstrom). The structure was determined using direct methods and refined by full-matrix least-squares procedure. The R factor is 5.3%. The molecule is characterized by a right handed 3(10)-helical conformation ((phi) = -68.2 degrees (psi) = -26.3 degrees), which is made up of two consecutive type III beta-bends and one type I beta-bend. In the solid state the helical molecules are aligned head-to-tail, thus forming long rod like structures. A comparison with other peptide structures containing consecutive Delta Phe residues is also provided. The present study confirms that the -Delta Phe-Delta Phe-sequence can be accommodated in helical structures. (C) 1997 John Wiley & Sons, Inc.

Local environmental structures of Mo in the superionic conducting glass (AgI)(0.6)(Ag2MoO4)(0.4) by anomalous X-ray scattering

The anomalous X-ray scattering (AXS) method using Mo K absorption edges has been employed for obtaining the local structural information of superionic conducting glass having the composition (AgI)(0.6)(Ag2MoO4)(0.4). The possible atomic arrangements in the near-neighbor region of this glass were estimated by coupling the results with the least-squares variational analysis so as to reproduce the differential intensity profile for Mo as well as the ordinary scattering profile. The coordination number of oxygen around Mo is found to be about 4 at the distance of 0.180 mn. This implies that the most probable structural entity in the glass is the MoO4 tetrahedral unit which has been proposed based on infrared spectroscopy. The value of the coordination number of I- around Ag+ is estimated as 4.4 at 0.287 nm, suggesting an arrangement similar to that of crystalline or molten AgI.

Novel subspace methods for blind estimation of multiple CFOs in MIMO-OFDM systems

The problem of estimating multiple Carrier Frequency Offsets (CFOs) in the uplink of MIMO-OFDM systems with Co-Channel (CC) and OFDMA based carrier allocation is considered. The tri-linear data model for generalized, multiuser OFDM system is formulated. Novel blind subspace based estimation of multiple CFOs in the case of arbitrary carrier allocation scheme in OFDMA systems and CC users in OFDM systems based on the Khatri-Rao product is proposed. The method works where the conventional subspace method fails. The performance of the proposed methods is compared with pilot based Least-Squares method.

Electron diffraction structure analysis: structural research with low-quality diffraction data

Electron Diffraction Structure Analysis (EDSA) with data from standard selected-area electron diffraction (SAED) is still the method of choice for structure determination of nano-sized single crystals. The recently determined heavy atom structure α-Ti2Se (Albe & Weirich, 2003) is used as an example to illustrate the developed procedure for structure determination from two-dimensionally SAED data via direct methods and kinematical least-squares refinement. Despite the investigated crystallite had a relatively large effective thickness of about 230 Å as determined from dynamical calculations, the obtained structural model from SAED data was found in good agreement with the result from an earlier single crystal X-ray study (Weirich, Pöttgen & Simon, 1996). Arguments, which support the validity of the used quasi-kinematical approach, are given in the text. The influences of dynamical and secondary scattering on the quality of the data and the structure solution are discussed. Moreover, the usefulness of first-principles calculations for verifying the results from EDSA is demonstrated by two examples, whereas one of the structures was unattainable by conventional X-ray diffraction.

Water quality parameter estimation in steady-state distribution system

The maintenance of chlorine residual is needed at all the points in the distribution system supplied with chlorine as a disinfectant. The propagation and level of chlorine in a distribution system is affected by both bulk and pipe wall reactions. It is well known that the field determination of wall reaction parameter is difficult. The source strength of chlorine to maintain a specified chlorine residual at a target node is also an important parameter. The inverse model presented in the paper determines these water quality parameters, which are associated with different reaction kinetics, either in single or in groups of pipes. The weighted-least-squares method based on the Gauss-Newton minimization technique is used for the estimation of these parameters. The validation and application of the inverse model is illustrated with an example pipe distribution system under steady state. A generalized procedure to handle noisy and bad (abnormal) data is suggested, which can be used to estimate these parameters more accurately. The developed inverse model is useful for water supply agencies to calibrate their water distribution system and to improve their operational strategies to maintain water quality.