KKT Conditions for Rank-Deficient Nonlinear Least-Square Problems with Rank-Deficient Nonlinear Constraints

In nonlinear least-square problems with nonlinear constraints, the function , where f ₂ is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f ₁(x)=0. This problem is ill-posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill-posed if either the Jacobian of f ₁ or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss–Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.     

Loewner's theorem for kernels having a finite number of negative squares

By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.

     

Pythagoras numbers of fields

A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

     

PLS-原子吸收光谱法同时测定稀土氧化物中的Eu和Cu

用偏最小二乘法（ＰＬＳ）校正了火焰原子吸收分析中Ｅｕ３２４．７５３ｎｍ对Ｃｕ３２４．７５４ｎｍ吸收线重叠干扰,用铜空心阴极灯同时测定了合成样及稀土氧化物中的Ｅｕ和Ｃｕ。     

A Structured Secant Method Based on a New Quasi-Newton Equation for Nonlinear Least Squares Problems

In this paper, a new quasi-Newton equation is applied to the structured secant methods for nonlinear least squares problems. We show that the new equation is better than the original quasi-Newton equation as it provides a more accurate approximation to the second order information. Furthermore, combining the new quasi-Newton equation with a product structure, a new algorithm is established. It is shown that the resulting algorithm is quadratically convergent for the zero-residual case and superlinearly convergent for the nonzero-residual case. In order to compare the new algorithm with some related methods, our preliminary numerical experiments are also reported.     

Combined and Least Squares Empirical Likelihood

In conventional empirical likelihood, there is exactly one structural constraint for every parameter. In some circumstances, additional constraints are imposed to reflect additional and sought-after features of statistical analysis. Such an augmented scheme uses the implicit power of empirical likelihood to produce very natural adaptive statistical methods, free of arbitrary tuning parameter choices, and does have good asymptotic properties. The price to be paid for such good properties is in extra computational difficulty. To overcome the computational difficulty, we propose a least-squares version of the empirical likelihood. The method is illustrated by application to the case of combined empirical likelihood for the mean and the median in one sample location inference.     

Organic soft crystals exhibiting spontaneously reversible mechano-responsive luminescence

Mechano-responsive luminescence, or mechanochromic luminescence (MCL), is a type of luminescence that can be reversibly controlled by the addition of mechanical stimuli. Organic materials exhibiting MCL have been an ongoing area of development since the early 2000s, and the number of reports into such materials has been steadily increasing. While the majority of MCL systems rely on the brittle nature of organic crystalline solids, there is a growing interest in "flexible" organic crystals that exhibit mechanical bending or shape deformation owing to their elasticity/plasticity. Such non-destructive deformed crystals may exhibit a new type of MCL that can be controlled by the magnitude of the force stress. In this review, we describe MCL systems capable of the spontaneous recovery of changes in their luminescent properties in response to the loading/unloading of mechanical stress. We particularly focus on the MCL of flexible crystals based on the density gradient of molecular packing (i.e., elastic and plastic crystals) and an emerging system known as "superelastochromism,” which is based on spontaneously reversible crystal polymorphism. This emerging research area has the potential to play an important role in the promotion of next-generation soft crystals.     

A genetic algorithm for flexible molecular overlay and pharmacophore elucidation

Summary A genetic algorithm (GA) has been developed for the superimposition of sets of flexible molecules. Molecules are represented by a chromosome that encodes angles of rotation about flexible bonds and mappings between hydrogen-bond donor proton, acceptor lone pair and ring centre features in pairs of molecules. The molecule with the smallest number of features in the data set is used as a template, onto which the remaining molecules are fitted with the objective of maximising structural equivalences. The fitness function of the GA is a weighted combination of: (i) the number and the similarity of the features that have been overlaid in this way; (ii) the volume integral of the overlay; and (iii) the van der Waals energy of the molecular conformations defined by the torsion angles encoded in the chromosomes. The algorithm has been applied to a number of pharmacophore elucidation problems, i.e., angiotensin II receptor antagonists, Leu-enkephalin and a hybrid morphine molecule, 5-HT_1D agonists, benzodiazepine receptor ligands, 5-HT₃ antagonists, dopamine D₂ antagonists, dopamine reuptake blockers and FKBP12 ligands. The resulting pharmacophores are generated rapidly and are in good agreement with those derived from alternative means.     

Comparison of various chemometric evaluation approaches for on-line FT-NIR transmission and FT-MIR/ATR spectroscopic data of methyl methacrylate solution polymerization

For the evaluation of vibrational spectroscopic data acquired on-line to a chemical reaction a broad range of different chemometric algorithms is available. The present study reports the comparative results obtained by different chemometric techniques from the data acquired by light-fiber coupled Fourier-transform near infrared (FT-NIR) transmission spectroscopy and Fourier-transform mid-infrared (FT-MIR) spectroscopy in the attenuated total reflection (ATR) mode to monitor the solution polymerization of methyl methacrylate (MMA). We have found that the results obtained by the application of multivariate curve resolution (MCR) methods to the MIR spectral data acquired during the polymerization of MMA are quite comparable to the results derived by partial least-squares calibration. In the case of the NIR data univariate calibration yields somewhat poorer results than multivariate calibration and MCR, but still inside an acceptable range.