On the cost of computing roots of polynomials

Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods. This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n ³ log₂(n/ε)). This is a worst case analysis and holds for all polynomials without exception. This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim Foundation.     

The behavior of the QR-factorization algorithm with column pivoting

A bound on the performance of QR-factorization with column pivoting is derived and two classes of matrices are constructed for which the bound is sharp or asymptotically sharp.     

A complementary electrochromic system based on Prussian blue and indium hexacyanoferrate

A new complementary electrochromic device (ECD) based on Prussian blue (PB), indium hexacyanoferrate (InHCF), and KCl-saturated poly(2-acrylamido-2-methylpropanesulfonic acid) (K-PAMPS) was proposed and studied in this work. This novel PB-InHCF ECD (PIECD) exhibits blue-to-yellowish electrochromism with a high coloration efficiency of ca. 103 cm²/C at 690 nm. Although the operating voltages for the fully bleached and fully colored states were determined to be 1.2 V and 0 V (InHCF vs. PB), respectively, the major transmittance modulation occurs within a much narrower voltage window (0.9 V↔0.5 V). That is, the PIECD is energetically favorable. Furthermore, it is unnecessary to precolor either electrochromic (EC) electrode during the cell assembly so that the charge balance between two electrochromic films becomes much easier. In addition to the above performance, the compatibility between the K-PAMPS electrolyte and EC electrodes was also demonstrated. In short, this work proposes another promising PB-based ECD and provides a new choice in the EC field. Electronic Publication     

A PL homotopy for finding all the roots of a polynomial

This paper presents a constructive method which gives, for any polynomialF(Z) of the degreen, approximate values of all the roots ofF(Z).. The point of the method is on the use of a piecewise linear function (Z, t) which approximates a homotopyH(Z, t) betweenF(Z) and a polynomialG(Z) of the degreen withn known simple roots. It is shown that the set of solutions to (Z, t) = 0 includesn distinct paths,m of which converges to a root ofF(Z) if and only if the root has the multiplicitym. Starting from givenn roots ofG(Z), a complementary pivot algorithm generates thosen paths.This work was supported by grants from Management Science Development Foundation and Takeda Science Foundation.     

Parallel column liquid chromatography with a single multi-wavelength absorbance detector for enhanced selectivity using chemometric analysis

The selectivity of high performance liquid chromatography (HPLC) separations is increased using a parallel column configuration. In this system, an injected sample is first split between two HPLC columns that provide complementary separations. The effluent from the two columns is recombined prior to detection with a single multiwavelength absorbance detector. Complementary stationary phases are used so that each chemical component produces a detected concentration profile consisting of two peaks. A parallel column configuration, when coupled with multivariate detection, provides increased chemical selectivity relative to a single column configuration with the same multivariate detection. This enhanced selectivity is achieved by doubling the number of peaks in the chromatographic dimension while keeping the run time constant. Unlike traditional single column separation methodology, the parallel column system sacrifices chromatographic resolution while actually increasing the chemical selectivity, thus allowing chemometric data analysis methods to mathematically resolve the multivariate chromatographic data. The parallel column system can be used to reduce analysis times for partially resolved peaks and simplify initial method development as well as provide a more robust methodology if and when subsequent changes in the sample matrix occur (such as when new interferences show up in subsequent samples). Here, a mixture of common aromatic compounds were separated with this system and analyzed using the generalized rank annihilation method (GRAM). Analytes that were significantly overlapped on both stationary phases applied, ZirChrom PBD and CARB phases, when used in traditional single column format, were successfully quantified with a R.S.D.% of typically 2% when the same stationary phases were used in the parallel column format. These results indicate that a parallel column system should substantially improve the chemical selectivity and quantitative precision of the analysis relative to a single-column instrument.     

Single-electron transistors fabricated with sidewall spacer patterning

We have implemented a sidewall spacer patterning method for novel dual-gate single-electron transistor (DGSET) and metal–oxide–semiconductor-based SET (MOSET) based on the uniform SOI wire, using conventional lithography and processing technology. A 30 nm wide silicon quantum wire is defined by a sidewall spacer patterning method, and depletion gates for two tunnel junctions of the DGSET are formed by the doped polycrystalline silicon sidewall. The fabricated DGSET and MOSET show clear single-electron tunneling phenomena at liquid nitrogen temperature and insensitivity of the Coulomb oscillation period to gate bias conditions. On the basis of the phase control capability of the sidewall depletion gates, we have proposed a complementary self-biasing method, which enables the SET/CMOS hybrid multi-valued logic (MVL) to operate perfectly well at high temperature, where the peak-to-valley current ratio of Coulomb oscillation severely decreases. The suggested scheme is evaluated by SPICE simulation with an analytical DGSET model, and it is confirmed that even DGSETs with a large Si island can be utilized efficiently in the multi-valued logic.     

Linear complementarity problems with an invariant number of solutions

This paper studies the class INS of all realn × n matricesM for which the linear complementarity problem (q, M) has exactlyk solutions—k depending only onM—for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. This generalizes the results in Cottle and Stone (1983) which deal with the subclassU in INS wherek equals one.After the first two sections of this paper, which introduce the problem and background material, we move on to examine necessary conditions for a matrixM to be in INS (Section 3) and sufficient conditions under whichM will be in INS (Section 4). Section 5 deals with the possible values whichk may have. Section 6 discusses related results concerning the geometry of linear complementarity problems. Finally, Section 7 deals with some known and new matrix classes which are in INS.     

K11-代数的主同余关系的可补性

研究K11-代数的主同余关系的可补性,给出K11-代数的主同余关系是可补的充要条件。     

Particles and holes in the unitary group method

Based on the definition for complementary Gel'fand states, we proved the simple relationship between the matrix elements of particle states and those of hole states by unitary calculus.