Diffusion of Selenium in Liquid-Phase Epitaxy–Grown Hg<Subscript>0.78</Subscript>Cd<Subscript>0.22</Subscript>Te

We present preliminary results on Se diffusion in liquid-phase epitaxy (LPE)–grown HgCdTe epilayers. The LPE Hg_0.78Cd_0.22Te samples were implanted with Se of 2.0 × 10¹⁴/cm² at 100 keV and annealed at 350–450°C in mercury saturated vapor. Secondary ion mass spectrometry (SIMS) profiles were obtained for each sample. From a Gaussian fit, we find that the Se diffusion coefficient D _Se is about 1–2 orders of magnitude smaller than that of arsenic. The as-implanted Se distribution is taken into account in case of small diffusion length in Gaussian fitting. The D _Se was found to satisfy the Arrhenius relationship .     

Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant–Lambert–Vanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over \mathbbF_p²{\mathbb{F}}_{p^{2}}. We extend their results and demonstrate that they can be applied to the GLV method.     

Basis arrays and successive packing for M-D interleaving

In this paper, we first present a novel concept of 2-D basis interleaving array (also referred to as basis array for short). That is, an m × m interleaved array is said to be a basis array if the shortest distance among all pairs of elements in each of the so-called m-equivalent sets within the m × m array reaches the maximum. It is shown that this maximum is given by ${\lfloor \sqrt{2m} \rfloor}$ and an m × m basis array can be constructed by using a simple cyclic translation method. The previously developed concept of successive packing is then generalized in the sense that it can be applied to any basis array to generate an interleaved array with a larger size. Except that optimality cannot be guaranteed, the concept of basis arrays and successive packing are extended to M-D cases. It is shown that for any M ?? 2, the proposed technique can spread any error burst of block size ${m_{1}^{k} \times m_{2}^{k} \times \cdots \times m_{M}^{k}}$ within an ${ m_{1}^{n} \times m_{2}^{n} \times \cdots \times m_{M}^{n}}$ array (1 ?? k ?? n?1) so effectively that the error burst can be corrected with some simple random error-correcting code (provided the error-correcting code is available). It is shown that important prior results in M-D interleaving such as the t-interleaved array based approach by Blaum et al. and the successive packing approach by Shi and Zhang now become special cases of the framework based on basis arrays and successive packing, proposed in this paper.     

Investigation of Trap States in AlInN/AlN/GaN Heterostructures by Frequency-Dependent Admittance Analysis

We present a systematic study on the admittance characterization of surface trap states in unpassivated and SiN _x -passivated Al_0.83In_0.17N/AlN/GaN heterostructures. C–V and G/ω–V measurements were carried out in the frequency range of 1 kHz to 1 MHz, and an equivalent circuit model was used to analyze the experimental data. A detailed analysis of the frequency-dependent capacitance and conductance data was performed, assuming models in which traps are located at the metal–AlInN surface. The density (D _t) and time constant (τ _t) of the surface trap states have been determined as a function of energy separation from the conduction-band edge (E _c − E _t). The D _st and τ _st values of the surface trap states for the unpassivated samples were found to be D_st @ (4 - 13) ×10¹² D_{\rm{st}} \cong (4 - 13) \times 10^{12}  eV ^{- 1} cm ^{- 2} {\hbox{eV}}^{ - 1} {\hbox{cm}}^{ - 2} and τ _st ≈ 3 μs to 7 μs, respectively. For the passivated sample, D _st decreased to 1.5 ×10¹² 1.5 \times 10^{12}  eV ^{- 1} cm ^{- 2} {\hbox{eV}}^{ - 1} {\hbox{cm}}^{ - 2} and τ _st to 1.8 μs to 2 μs. The density of surface trap states in Al_0.83In_0.17N/AlN/GaN heterostructures decreased by approximately one order of magnitude with SiN _x passivation, indicating that the SiN _x insulator layer between the metal contact and the surface of the Al_0.83In_0.17N layer can passivate surface states.     

VLSI architecture for fast 2D discrete orthonormal wavelet transform

The discrete wavelet transform (DWT) provides a new method for signal/image analysis where high frequency components are studied with finer time resolution and low frequency components with coarser time resolution. It decomposes a signal or an image into localized contributions for multiscale analysis. In this paper, we present a parallel pipelined VLSI array architecture for 2D dyadic separable DWT. The 2D data array is partitioned into non-overlapping groups of rows. All rows in a partition are processed in parallel, and consecutive partitions are pipelined. Moreover, multiple wavelet levels are computed in the same pipeline, and multiple DWT problems can be pipelined also. The whole computation requires a single scan of the image data array. Thus, it is suitable for on-line real-time applications. For anN×N image, anm-level DWT can be computed in time units on a processor costing no more than , whereq is the partition size,p is the length of corresponding 1D DWT filters,C _m andC _a are the costs of a parallel multiplier and a parallel adder respectively, and a time unit is the time for a multiplication and an addition. Forq=N m, the computing time reduces to . When a large number of DWT problems are pipelined, the computing time is about per problem.     

The Influence of Current Direction on the Cu-Ni Cross-Interaction in Cu/Sn/Ni Diffusion Couples

The influence of current direction on the Cu-Ni cross-interaction in the Cu/Sn/Ni joint configuration was investigated in this study. During current stressing, an electric current towards or away from the Ni-side of Cu/Sn/Ni was imposed at 150°C. It was observed that the (Cu,Ni)₆Sn₅ ternary compound was the dominant reaction product at both interfaces, and its growth at the Ni-side strongly depended upon the direction and magnitude of the electron flow. When the electron flow was towards the Ni-side, more Cu was found to be driven to the Ni-side, resulting in an increase in the thickness of (Cu,Ni)₆Sn₅. This is due to the chemical-potential-induced Cu flux (\( J_{\rm{chem}}^{\rm{Cu}} \)) that was enhanced by the electromigration (\( J_{\rm{em}}^{\rm{Cu}} \)). In the case of electron flow away from the Ni-side, the supply of Cu to the Ni-side was retarded due to the fact that \( J_{\rm{em}}^{\rm{Cu}} \) was in the opposite direction to \( J_{\rm{chem}}^{\rm{Cu}} . \) The results of this study revealed that the Ni-side (Cu,Ni)₆Sn₅ thickness remained almost unchanged under current stressing of 10⁴ A/cm² at 150°C, which suggests the inward Cu flux is approximately equal to the outward flux, i.e., \( J_{\rm{chem}}^{\rm{Cu}} \approx J_{\rm{em}}^{\rm{Cu}} . \)     

Phase Equilibria of BaO-R2O3-CuO z Systems (R = Y and Lanthanides) under CO2-free Conditions

For applications ranging from phase equilibria to the processing of second-generation high T _c superconductor-coated-conductors, phase diagrams constructed under carbonate-free conditions are needed. Subsolidus phase equilibria of BaO-R₂O₃-CuO _z (R = Ho) have been investigated at (810°C), 21 kPa (875°C) and 0.1 MPa (850 and 930°C) by applying controlled atmosphere methods to minimize the presence of carbonate and CO₂ and H₂O contamination. Under carbonate-free conditions, most of these phase diagrams are different from those reported in the literature. In this paper, we also review and compare the phase diagrams of ten BaO-R₂O₃-CuO _z systems (R = Nd, Sm, Eu, Gd, Dy, Y, Ho, Er, Tm and Yb) that were previously determined in this laboratory under Among these diagrams, a distinct trend of phase formation and tie-line relationships is observed.     

TD-SCDMA系统中利用中置实现DOA检测的方法

智能天线的一项核心技术是来波方向（DOA）估计。设计物理模型清晰、数学运算复杂度低的DOA估计算法,对于智能天线的工程实现具有重要的意义。本文给出了一种利用TD-SCDMA标准中置序列在基带上实现DOA检测的简单实用的谱搜索方法 MIDOA谱搜索算法。文章首先给出了TD-SCDMA系统8阵元圆阵智能天线系统中置接收机的物理模型,在此基础上导出了与之相对应的数学处理模型,并根据这些数学模型导出了MIDOA谱搜索算法流程图和相应的基带检测MIDOA谱搜索算法。从仿真结果和性能分析可以看出该方法是一种针对TD-SCDMA标准的、简单有效的DOA估计方法;与基于子空间的Music等算法相比,它具有物理模型清晰、算法的稳健度高和可检测的DOA数不受智能天线阵元数的限制等特点。      

An Analysis of the Blockcipher-Based Hash Functions from PGV

Preneel, Govaerts, and Vandewalle (1993) considered the 64 most basic ways to construct a hash function $H{:\;\:}\{0,1\}^{*}\rightarrow \{0,1\}^{n}Preneel, Govaerts, and Vandewalle (1993) considered the 64 most basic ways to construct a hash function H:   {0,1}^*? {0,1}ⁿH{:\;\:}\{0,1\}^{*}\rightarrow \{0,1\}^{n} from a blockcipher E:   {0,1}ⁿ×{0,1}ⁿ? {0,1}ⁿE{:\;\:}\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}^{n}. They regarded 12 of these 64 schemes as secure, though no proofs or formal claims were given. Here we provide a proof-based treatment of the PGV schemes. We show that, in the ideal-cipher model, the 12 schemes considered secure by PGV really are secure: we give tight upper and lower bounds on their collision resistance. Furthermore, by stepping outside of the Merkle–Damg?rd approach to analysis, we show that an additional 8 of the PGV schemes are just as collision resistant (up to a constant). Nonetheless, we are able to differentiate among the 20 collision-resistant schemes by considering their preimage resistance: only the 12 initial schemes enjoy optimal preimage resistance. Our work demonstrates that proving ideal-cipher-model bounds is a feasible and useful step for understanding the security of blockcipher-based hash-function constructions.     

Thermoelectric Oxides: Effect of Doping in Delafossites and Zinc Oxide

The electrical resistivity (ρ) and Seebeck coefficient (S) of the three delafossites CuFe_0.9Cr_0.1O₂, CuCr_0.98Mg_0.02O₂, and CuRh_0.9Mg_0.1O₂ have been measured and their power factor (PF) calculated. These p-type oxides show PF values at 800 K from 1.4 × 10⁻⁴ W K⁻² m⁻¹ to 6.9 × 10⁻⁴ W K⁻² m⁻¹. In contrast to delafossites containing Fe or Cr, for which ρ exhibits a regime, the Rh-based delafossite shows a metallic regime from 5 K to 1000 K. This points toward the role of the transition-metal electronic configuration in the transport properties. Otherwise, similar PF values are obtained in the case of the n-type Al-doped ZnO. For these oxides, the ρ and PF values are minimum and maximum, respectively, for x = 0.01. However, the presence of spinel impurities even for x = 0.01 in Zn_1−x Al _x O or for x = 0.02 in CuCr_1−x Mg _x O₂ calls into question the role of the doping element in the physical properties. This should motivate a deeper insight into the physics of thermoelectric oxides.     

Preparation of Well-Tiled <Emphasis Type="Italic">γ</Emphasis>-Na<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Co<Subscript>2</Subscript>O<Subscript>4</Subscript> by a Novel and Simple Sodium Alginate Gel Method and Its Electrical Properties

In this paper, a novel and simple sodium alginate (SA) gel method was developed to prepare γ-Na _x Co₂O₄. This method involved the chemical gelling of SA in the presence of Co²⁺ ions by cross-linking. After calcining at 700°C to 800°C, single-phase γ-Na _x Co₂O₄ crystals were obtained. The arrangement of about 1 μm to 4 μm flaky particles exhibited a well-tiled structure along the plane direction of the flaky particles. SA not only acted as the control agent for crystal growth, but also provided a Na source for the γ-Na _x Co₂O₄ crystals. The electrical properties of γ-Na _x Co₂O₄ ceramics prepared via ordinary sintering after cold isostatic pressing were investigated. The Seebeck coefficient and power factor of the bulk material were 177 μV K⁻¹ and 4.3 × 10⁻⁴ W m⁻¹ K⁻² at 850 K, respectively.     

Variance and Covariance Computations for 2-D ARMA Processes

An algorithm is presented to compute the variance of the output of a two-dimensional (2-D) stable auto-regressive moving-average (ARMA) process driven by a white noise bi-sequence with unity variance. Actually, the algorithm is dedicated to the evaluation of a complex integral of the form , where and G(z₁,z₂) = B(z₁, z₂) / A(z₁, z₂) is stable (z₁,z₂)-transferfunction. Like other existing methods, the proposed algorithmis based on the partial-fraction decomposition G(z₁,z₂)G(z ₁ ^-1 , z ₂ ^-1 ) = X(z₁, z₁) / A(z₁,z₂)+ X(z ₁ ^-1 , z ₂ ^-1 ) / A(z ₁ ^-1 , z ₂ ^-1 ). However,the general and systematic partial-fraction decomposition schemeof Gorecki and Popek [1] is extended to determine X(z₁,z₂).The key to the extension is that of bilinearly transforming thediscrete (z₁, z₂)-transfer function G(z₁,z₂)into a mixed continuous-discrete (s₁, z₂)-transferfunction . As a result, the partial-fraction decomposition involves only efficient DFT computations for the inversion of a matrix polynomial, and the value of I is finally determined by the residue method with finding the roots of a 1-D polynomial. The algorithm is very easy to implement and it can be extended to the covariance computation for two 2-D ARMA processes.     

Recombination of donor nickel excitons in the solid solutions ZnSe1−y Sy:Ni

We have found that the zero-phonon line of the donor exciton Ni [d ⁷ e] and its vibrational repetitions in the electroabsorption spectrum depend strongly on the composition of the solid solution ZnSe_1−y S_y:Ni (y⩽20%). Using the idea of an intermediate virtual {d ⁸}* state, we interpret this on the basis of the adiabatic potential model as an interaction of the d ⁷ configuration with the nearest-neighbor environment and nonradiative tunneling recombination [d ⁷ e]→(d ⁸)*. Fiz. Tekh. Poluprovodn. 32, 455–457 (April 1998)     

High-Energy-Density All-Organic Dielectric Film via a Continuous Preparation Processing

Dielectric polymers with high power density and breakdown strength (E_b) are indispensably used in electrostatic energy-storing systems and devices. However, the discharged energy density (U_e) of dielectric polymers is severely limited due to the relatively low dielectric constant (K). Although current polymer composites improve K, this approach usually faces challenges in enhancing U_e due to the trade-off relation between K and E_b and difficulties in scalable production of dielectric films. Here, a fully melt-extrudable, meter-scale, and high-U_e ferroelectric polymer-based all organic composite film comprising a poly(p-phenylene terephthalamide)-based fluxible polymer (denoted as f-PPTA) is reported. The polymer composite with only 2 wt% of f-PPTA presents a productivity of 12 m² h⁻¹ and an ultrahigh U_e of 20.7 J cm⁻³, which outperforms other extruded dielectric polymers reported in the literatures. Such enhancements of dielectric and capacitive properties have been comprehensively investigated and attributed to the crystallization behavior modulations and conformation changes induced by f-PPTA. As a demonstration of real applications, the dielectric capacitors established based on the extruded films enable tens of times higher efficacy on powering electronic devices than biaxially oriented polypropylene capacitors, in addition to long-term cyclic stability. This study opens up new avenue for the design and fabrication of high-U_e polymer dielectrics that are totally compatible with industrial production.     

Critical Layer Thickness in Exponentially Graded Heteroepitaxial Layers

Exponentially graded semiconductor layers are of interest for use as buffers in heteroepitaxial devices because of their tapered dislocation density and strain profiles. Here we have calculated the critical layer thickness for the onset of lattice relaxation in exponentially graded In _x Ga_1?x As/GaAs (001) heteroepitaxial layers. Upwardly convex grading with \( x = x_{\infty } \left( {1 - {\rm e}^{ - \gamma /y} } \right) \) was considered, where y is the distance from the GaAs interface, γ is a grading length constant, and x _∞ is the limiting mole fraction of In. For these structures the critical layer thickness was determined by an energy-minimization approach and also by consideration of force balance on grown-in dislocations. The force balance calculations underestimate the critical layer thickness unless one accounts for the fact that the first misfit dislocations are introduced at a finite distance above the interface. The critical layer thickness determined by energy minimization, or by a detailed force balance model, is approximately \( h_{\rm{c}} \approx <Exponentially graded semiconductor layers are of interest for use as buffers in heteroepitaxial devices because of their tapered dislocation density and strain profiles. Here we have calculated the critical layer thickness for the onset of lattice relaxation in exponentially graded In _x Ga_1−x As/GaAs (001) heteroepitaxial layers. Upwardly convex grading with x = x_￥ ( 1 - e ^{- g/y} ) x = x_{\infty } \left( {1 - {\rm e}^{ - \gamma /y} } \right) was considered, where y is the distance from the GaAs interface, γ is a grading length constant, and x _∞ is the limiting mole fraction of In. For these structures the critical layer thickness was determined by an energy-minimization approach and also by consideration of force balance on grown-in dislocations. The force balance calculations underestimate the critical layer thickness unless one accounts for the fact that the first misfit dislocations are introduced at a finite distance above the interface. The critical layer thickness determined by energy minimization, or by a detailed force balance model, is approximately h_c ? < h_{\rm{c}} \approx < Although these results were developed for exponentially graded In _x Ga_1−x As/GaAs (001), they may be generalized to other material systems for application to the design of exponentially graded buffer layers in metamorphic device structures such as modulation-doped field-effect transistors and light-emitting diodes.     

Comparison of Bowing Behaviors Between III–V and II–VI Common-Cation Semiconductor Ternary Alloys

The scope of the present investigation is to make a clear contrast between the bandgap bowing characters of III–V and II–VI compound-semiconductor common-cation ternary alloys. For this aim, both the sp ³ s ^* tight-binding method, with the inclusion of spin–orbit coupling, and the full-potential linear augmented plane-wave technique are used to calculate the partial and total densities of states, the constituent ionicity, and the total electron charge density for the common-cation GaSb _x As_1−x and CdSe _x Te_1−x ternary alloys. The results show that the bowing is sensitive to competition between the anions for trapping/losing electric charges. The lack of this competition would result in complete absence of the bowing, as is the case for common-anion ternary alloys. In the common-cation ternary alloys studied herein, the bowing is found to be proportional to the electronegativity of the anions χ ^anion (i.e., the 6-valency anions are more electronegative than the 5-valency ones, and consequently the former result in stronger intercompetition and yield stronger bowing in the II–VI alloys) and also proportional to the relative mismatch in electronegativity between the competing anions (\Updeltac^anion/c^anion_ave).({\Updelta}\chi^{\rm{anion}}/\chi^{\rm{anion}}_{\rm{ave}}). The excellent agreement between our theoretical results and recent photoluminescence data corroborates our claim.     

A Far IR Study of NbN in the Normal and Superconductive State

A thin film of NbN (thickness t = 300 Å), has been deposited on an MgO and a Si wafer. Both samples have been studied by transmission from 10 or 20 to 120 cm^?1, and have exhibited one maximum of transmission at a given frequency like the classical superconductors as Pb, Sn or Hg in the superconductive state. From the Far IR experimental data, the characteristic temperature θ_c, and the gap frequency (ν_gap (θ) = 2 Δ (θ), Δ(θ) being the energy gap) are immediately obtained (for instance for the NbN / MgO sample, θ_c = 15.5 K; ν_g (5 K) = 39.7 cm^?1), and it is seen that $\frac{{{\text{2}}\vartriangle (4{\text{K)}}}}{{{\text{k}}_{\text{B}} \theta _c }} \approx 3.5$ as expected from the BCS theory for a weak coupling. To fit the data we had to adjust only two additionnal parameters: collision and plasma frequency, ν_c (θ) and ν_p (including all carriers). At θ = 5 K, thebest fit for the NbN / MgO sample is obtained with ν_c = 371 cm^?1 and ν_p = 12,600cm^?1.     

Parallel implementation of neural networks

This paper presents systematic methods, based on graph theoretic approach, for mapping of neural networks onto mesh connected SIMD arrays. The methods are applicable to a large class of multilayer network models, which can be represented in terms of sparse matrix vector operations. The class of computers, that the mappings are suitable for, encompasses most of the experimental and commercial mesh-connected SIMD arrays of processors. There are three methods described in the paper, one for the case of a processor array, which is larger or equal to the network size and two for the partitioned case, i.e. array smaller than the input data size. The methods are illustrated on an example of a multilayer perceptron with back-propagation learning, which consists ofn nuerons ande synaptic connections. For the first method, the processor array is assumed to be of sizeN×N, whereN ² n+e, and the required local memory of processors is limited to only a few registers. The implementation of a single iteration of a recall phase according to the method requires 24(N-1) shifts. For this method we have developed a software tool, which generates a sequence of pseudo instructions, such as elemental data shift and arithmetic operations, that implement a given neural network on a given size processor array. For the two partitioned methods, the processor array is of sizeP×P, whereP ²n+e, and the local memory in the processors is of sizeO(K). The faster of the two methods requiresO(N ³/P ³ K) time for an iteration of the recall or learning phase.This research was supported in part by the National Science Foundation under grant MIP-8714689 and IRI-9145810.Preliminary versions of the results contained in this paper appear in the International Conference on Applications-Specific Array Processors 1990 and the IEEE Workshop on VLSI Signal Processing 1990.     

An Alternative Approach to Improve the Thermoelectric Properties of Half-Heusler Compounds

In this report an alternative approach for optimization of the thermoelectric properties of half-Heusler compounds is presented. The common approaches are partial substitution of elements by elements of nearby groups and substitution with homologs. In this approach we substitute one element by one neighboring element with fewer valence electrons and by one with more electrons. The amounts of the substitutions are chosen such that the amount of deficiency and excess electrons are compensated. In the solid solution TiCo_x(Ni_0.5Fe_0.5)_1-xSb\hbox{TiCo}_{x}(\hbox{Ni}_{0.5}\hbox{Fe}_{0.5})_{1-x}\hbox{Sb}, Co was substituted equally by Fe and Ni. The aim of the substitution was to improve the figure of merit by a reduction of the thermal conductivity accompanied by an unchanged high Seebeck coefficient. The solid solution TiCo_x(Ni_0.5Fe_0.5)_1-xSb\hbox{TiCo}_{x}(\hbox{Ni}_{0.5}\hbox{Fe}_{0.5})_{1-x}\hbox{Sb} was synthesized by arc-melting. The structure of the as-cast samples was analyzed by x-ray diffraction. Rietveld refinements yielded the C1_bC1_b structure type with a small amount of antisite disorder between Co and Sb. The thermoelectric properties of the solid solution were investigated in the temperature range from 2 K to 400 K. A Seebeck coefficient of -260 mV K^-1-260\,\mu\hbox{V\,K}^{-1} at 400 K and a reduction of the thermal conductivity to 3 Wm^-1 K^-13\,\hbox{Wm}^{-1}\,\hbox{K}^{-1} were measured. The figure of merit was enhanced by a factor of about seven to a value of 0.04 at 400 K for TiCo_0.8(Ni_0.1Fe_0.1)Sb\hbox{TiCo}_{0.8}(\hbox{Ni}_{0.1}\hbox{Fe}_{0.1})\hbox{Sb}.     

Symmetry relations in multidimensional Fourier transform pairs

A relation between the types of symmetries that exist in signal and Fourier transform domain representations is derived for continuous as well as discrete domain signals. The symmetry is expressed by a set of parameters, and the relations derived in this paper will help to find the parameters of a symmetry in the signal or transform domain resulting from a given symmetry in the transform or signal domain respectively. A duality among the relations governing the conversion of the parameters of symmetry in the two domains is also brought to light. The application of the relations is illustrated by a number of two-dimensional examples.Notation R the set of real numbers - R ^m R × R × ... × R m-dimensional real vector space - continuous domain real vector - L {¦ – _i , i = 1,2,..., m} - m-dimensional frequency vector - W { – _i ,i=1,2,..., m} - m-dimensional normalized frequency vector - _P {¦ – _i , i=1,2,...,m} - g(ol) g (₁,₂,..., _m ) continuous domain signal - () ( _{1 2},..., _m )=G (j ₁,j ₂,..., j _m ) Fourier transform ofg (ol) - (A,b,,,) parameters ofT- symmetry - N the set of integers - N ^m N × N × ... × N m-dimensional integer vector spacem-dimensional lattice - h(n) h (n ₁,.,n _m ) discrete domain signal - H() Fourier transform ofh (n) - v ₁,v ₂,..., v_m m sample-direction and interval vectors - V (v ₁ v ₂ ...v _m ) sampling basis matrix - [x]^* complex conjugate ofx - detA determinant ofA - X {x¦ – x _i , i=1,2,..., m} - A ^–t [A ^–1] ^t ,t stands for transpose This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-7739 to M. N. S. Swamy and in part by Tennessee Technological University under its Faculty Research support program to P. K. Rajan.