Formal orthogonal polynomials revisited. Applications

From some results concerning the formal orthogonal polynomials, already proved in [5], we develop new properties of generalized adjacent polynomials which correspond to a change in the weight function. A new structure of the singular blocks is given. These results have a direct application to Lanczos methods, theG and -algorithms.     

Markov-Bernstein inequalities for generalized Gegenbauer weight

The Markov-Bernstein inequalities for generalized Gegenbauer weight are studied. A special basis of the vector space Pn of real polynomials in one variable of degree at most equal to n is proposed. It is produced by quasi-orthogonal polynomials with respect to this generalized Gegenbauer measure. Thanks to this basis the problem to find the Markov-Bernstein constant is separated in two eigenvalue problems. The first has a classical form and we are able to give lower and upper bounds of the Markov-Bernstein constant by using the Newton method and the classical qd algorithm applied to a sequence of orthogonal polynomials. The second is a generalized eigenvalue problem with a five diagonal matrix and a tridiagonal matrix. A lower bound is obtained by using the Newton method applied to the six term recurrence relation produced by the expansion of the characteristic determinant. The asymptotic behavior of an upper bound is studied. Finally, the asymptotic behavior of the Markov-Bernstein constant is O(n²) in both cases.     

Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight

The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999) and Draux (Numer Algor 24:31–58, 2000).      

Raman spectral imaging of single cancer cells: probing the impact of sample fixation methods

Raman spectroscopy has proven its potential for the analysis of cell constituents and processes. However, sample preparation methods compatible with clinical practice must be implemented for collection of accurate spectral information. This study aims at assessing, using micro-Raman imaging, the effects of some routinely used fixation methods such as formalin-fixation, formalin-fixation/air drying, cytocentrifugation, and air drying on intracellular spectral information. Data were compared with those acquired from single living cells. In parallel to these spectral information, cell morphological modifications that accompany sample preparation were compared. Spectral images of isolated cells were first analyzed in an unsupervised way using hierarchical cluster analysis (HCA), which allowed delimitation of the cellular compartments. The resulting nuclei cluster centers were compared and revealed at the molecular level that fixation induced changes in spectral information assigned to nucleic acids and proteins. In a second approach, a supervised fitting procedure using model spectra of DNA, RNA, and proteins, chemically extracted from living cells, revealed very small modifications at the level of the localization and quantification of these macromolecules. Finally, HCA and principal components analysis (PCA) performed on individual spectra randomly selected from the nuclear regions showed that formalin-fixation and cytocentrifugation are sample preparation methods that have little impact on the biochemical information as compared to living conditions. Any step involving cell air drying seems to accentuate the spectral deviations from the other preparation methods. It is therefore important in a future context of spectral cytology to take into account these variations.     

Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities

Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L ² for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

Raman imaging of single living cells: probing effects of non-cytotoxic doses of an anti-cancer drug

Identifying cell response to a chemotherapy drug treatment, in particular at the single cell level, is an important issue in patient management. This study aims at evaluating the effect of gemcitabine on single living cells using micro-Raman imaging. We used as a model the non-small lung cancer cell line, Calu-1, exposed to cytostatic doses (1 nM to 1 μM for 24 h and 48 h) of gemcitabine, an antitumor drug currently used in the treatment of lung cancer. Following drug treatment as a function of doses and incubation times, the Raman maps of single living cells were acquired. Cell biomolecules (DNA, RNA, and proteins) were chemically extracted and their spectral signatures used to evaluate their respective distribution in the cellular spectral information of control and treated cells. The quantification of these distributions reveals a significant effect of 100 nM gemcitabine at 48 h incubation (concomitant decrease of nucleic acids and increase of proteins). PCA analyses performed both on nuclear and extracted biomolecules spectra show a time-dependent effect of the drug. These promising results reveal that effects of subtoxic doses can be monitored at the single cell level highlighting the importance of such studies for clinical applications.     

Conglomerate crystallization in organic compounds II. The conformation,configuration, and spontaneous resolution of 1-phenyl-2-cyano-cyclopropane

The crystal structure of 1 -phenyi-2-cyano-cyclopropane was investigated to probe the stereochemical effects of placing different groups at the 2-position of the phenylcyclopropane moiety. The substance crystallized in space group P2₁2₁2₁ with cell constants a=16.921(3), b=7.699(2), and c=6.251(2) Å; V=814.33 ų and D (calc; z=4) =1.168 g·cm^–3. Final R(F)=0.04, using unit weights. The phenyl substituent is almost exactly in the bi-secting conformation with respect to the C-C-C angle at the point of attachment to cyclopropane. The average value of the C-C distances in the three-membered ring is 1.501 Å, which is on the low side of the ranges cited in previous studies. When we apply the additivity principle to our measured geometrical data and calculate the asymmetry parameters, the value of _cns [0.012(3)] is a little low, but within experimental error it is in accord with the value (0.017) given by Allen. However, our value of _cns (0.006(3)] is substantially lower than Allen's, which is surprising, since the bisecting phenyl ring can exert its maximum effect on the asymmetry of cyclopropane. The substance undergoes spontaneous resolution into a conglomerate of chiral crystals. Unfortunately, in the absence of anomalous scatterers, the absolute configuration could not be determined; nonetheless, the space group in which the compound crystallizes is P2₁2₁2₁, which leaves no ambiguity to the fact that the crystals are a conglomerate.     

Studies of chemical fixation effects in human cell lines using Raman microspectroscopy

The in vitro study of cellular species using Raman spectroscopy has proven a powerful non-invasive modality for the analysis of cell constituents and processes. This work uses micro-Raman spectroscopy to study the chemical fixation mechanism in three human cell lines (normal skin, normal bronchial epithelium, and lung adenocarcinoma) employing fixatives that preferentially preserve proteins (formalin), and nucleic acids (Carnoy’s fixative and methanol–acetic acid). Spectral differences between the mean live cell spectra and fixed cell spectra together with principal components analysis (PCA), and clustering techniques were used to analyse and interpret the spectral changes. The results indicate that fixation in formalin produces spectral content that is closest to that in the live cell and by extension, best preserves the cellular integrity. Nucleic acid degradation, protein denaturation, and lipid leaching were observed with all fixatives and for all cell lines, but to varying degrees. The results presented here suggest that the mechanism of fixation for short fixation times is complex and dependent on both the cell line and fixative employed. Moreover, important spectral changes occur with all fixatives that have consequences for the interpretation of biochemical processes within fixed cells. The study further demonstrates the potential of vibrational spectroscopy in the characterization of complex biochemical processes in cells at a molecular level.     

Generalized qd algorithm for block band matrices

The generalized qd algorithm for block band matrices is an extension of the block qd algorithm applied to a block tridiagonal matrix. This algorithm is applied to a positive definite symmetric block band matrix. The result concerning the behavior of the eigenvalues of the first and the last diagonal block of the matrix containing the entries q ^(k) which was obtained in the tridiagonal case is still valid for positive definite symmetric block band matrices. The eigenvalues of the first block constitute strictly increasing sequences and those of the last block constitute strictly decreasing sequences. The theorem of convergence, given in Draux and Sadik (Appl Numer Math 60:1300?C1308, 2010), also remains valid in this more general case.     

Rectangular matrix Padé approximants and square matrix orthogonal polynomials

In this paper we study Padé-type and Padé approximants for rectangular matrix formal power series, as well as the formal orthogonal polynomials which are a consequence of the definition of these matrix Padé approximants. Recurrence relations are given along a diagonal or two adjacent diagonals of the table of orthogonal polynomials and their adjacent ones. A matrix qd-algorithm is deduced from these relations. Recurrence relations are also proved for the associated polynomials. Finally a short presentation of right matrix Padé approximants gives a link between the degrees of orthogonal polynomials in right and left matrix Padé approximants in order to show that the latter are identical. This revised version was published online in June 2006 with corrections to the Cover Date.