Laguerre moments and generalized functions

Here we explore the link between the moments of the Laguerre polynomials or Laguerre moments and the generalized functions (as the Dirac delta-function and its derivatives), presenting several interesting relations. A useful application is related to a procedure for calculating mean values in quantum optics that makes use of the so-called quasi-probabilities. One of them, the P-distribution, can be represented by a sum over Laguerre moments when the electromagnetic field is in a photon-number state. Consequently, the P-distribution can be expressed in terms of Dirac delta-function and derivatives. More specifically, we found a direct relation between P-distributions and the Laguerre factorial moments.

An electrode of the second kind for aspirin determination in tablet formulations

This paper describes the construction of an electrode of the second kind, Pt\Hg\Hg-2(Salic)(2)\Graphite, sensitive to salicylate. This electrode responds to the salicylate ion with a sensivity of 58.66 mV/decade over the range 6.0x10(-4) - 1.0x10(-1) mol/l at pH 6.0 and an ionic strength of 0.500 - 3.00 mol/l, adjusted with NaClO4. The electrode is easily constructed, shows a fast response time, is low in cost, has excellent response stability, and has a lifetime greater than 18 months, which is much longer than those reported earlier for other systems. The influence of 10 different carboxylate and inorganic anions on the electrode response showed that there was negligible interference by most of these ions. It was used to determine aspirin in tablets (after hydrolysis of acetylsalicylic acid to salicylate) by means of the standard additions method. The results obtained using this electrode for aspirin determination, in three different samples of antithermic drugs, compared favorably with the results given by the British Pharmacopoeia method.

An electrode of the second kind for benzoate determination in medicinal syrups

This paper describes the construction of an electrode of the second kind, Pt\Hg\Hg-2(Bzt)(2)\graphite, sensitive to benzoate ion (Bzt). The electrode provides a near-Nernstian response (-58.19 mV/decade at 25 degrees C) to benzoate ion in the concentration range 1.11 x 10(-1) to 5.00 x 10(-4) M at pH 7.0 and ionic strength 0.500-3.00M, adjusted with NaClO4. The electrode is easily constructed, shows a fast response time, is low in cost, has excellent response stability, and has a lifetime greater than 24 months. which is much longer than those earlier reported for other systems. A further salient feature of the electrode is that perchlorate does not interfere in its response, which may be advantageous in the analysis of perchloric acid-treated samples. It was applied to the determination of sodium benzoate in some commercial expectorating medicinal syrups using the standard additions method. The results obtained by the proposed procedure compare very favorably with those obtained through the reference method recommended by the United States Pharmacopeia. (C) 1998 Academic Press.

Two prognostic factors in breast cancer: Novel functions of EGFR and beta-catenin

In this study, we demonstrated the novel functions of two important prognostic markers in breast cancer, EGFR and b -catenin in proliferation and/or other transformation phenotype. ^ First we demonstrated that EGFR could be detected in the nucleus in highly proliferating tissues, including primary breast cancer samples and a breast cancer cell line. We found that EGFR contained a strong transactivation domain, complexed with an AT-rich consensus DNA sequence and activated promoters containing this sequence, including cyclin D1 promoter. Therefore, EGFR may function as a transcription factor to activate genes required for highly proliferating activity such as cyclin D1 in breast cancer. ^ In the second part of this study, we identified b -catenin as an important prognostic factor in breast cancer. We found that cyclin D1 was one of the genes regulated by b -catenin in breast cancer cells. The transactivation activity of b -catenin correlated significantly with cyclin D1 expression in both breast cancer cell lines and in breast cancer patient samples, in which high b -catenin activity correlated with poor prognosis of the patients. Moreover, blockage of b -catenin activity significantly inhibited transformation phenotypes in breast cancer cells. Therefore, our results indicate that b -catenin can be involved in breast cancer formation and/or progression and may serve as a target for breast cancer therapy. ^

Differential Expression and Functions of Cortical Myosin IIA and IIB Isotypes during Meiotic Maturation, Fertilization, and Mitosis in Mouse Oocytes and Embryos

To explore the role of nonmuscle myosin II isoforms during mouse gametogenesis, fertilization, and early development, localization and microinjection studies were performed using monospecific antibodies to myosin IIA and IIB isotypes. Each myosin II antibody recognizes a 205-kDa protein in oocytes, but not mature sperm. Myosin IIA and IIB demonstrate differential expression during meiotic maturation and following fertilization: only the IIA isoform detects metaphase spindles or accumulates in the mitotic cleavage furrow. In the unfertilized oocyte, both myosin isoforms are polarized in the cortex directly overlying the metaphase-arrested second meiotic spindle. Cortical polarization is altered after spindle disassembly with Colcemid: the scattered meiotic chromosomes initiate myosin IIA and microfilament assemble in the vicinity of each chromosome mass. During sperm incorporation, both myosin II isotypes concentrate in the second polar body cleavage furrow and the sperm incorporation cone. In functional experiments, the microinjection of myosin IIA antibody disrupts meiotic maturation to metaphase II arrest, probably through depletion of spindle-associated myosin IIA protein and antibody binding to chromosome surfaces. Conversely, the microinjection of myosin IIB antibody blocks microfilament-directed chromosome scattering in Colcemid-treated mature oocytes, suggesting a role in mediating chromosome–cortical actomyosin interactions. Neither myosin II antibody, alone or coinjected, blocks second polar body formation, in vitro fertilization, or cytokinesis. Finally, microinjection of a nonphosphorylatable 20-kDa regulatory myosin light chain specifically blocks sperm incorporation cone disassembly and impedes cell cycle progression, suggesting that interference with myosin II phosphorylation influences fertilization. Thus, conventional myosins break cortical symmetry in oocytes by participating in eccentric meiotic spindle positioning, sperm incorporation cone dynamics, and cytokinesis. Although murine sperm do not express myosin II, different myosin II isotypes may have distinct roles during early embryonic development.

Expression of a second receptor rescues self-specific T cells from thymic deletion and allows activation of autoreactive effector function.

Allelic exclusion at the T-cell receptor alpha chain locus is incomplete resulting in the generation of T cells that express two T-cell receptors. The potential involvement of such T cells in autoimmunity has been suggested [Padovan, E., Casorati, G., Dellabona, P., Meyer, S., Brockhaus, M. & Lanzavecchia, A. (1993) Science 262, 422-424; Heath, W. R. & Miller, J. F. A. P. (1993) J. Exp. Med. 178, 1807-1811]. Here we show that expression of a second T-cell receptor can rescue T cells with autospecific receptors from thymic deletion and allow their exit into the periphery. Dual receptor T cells, created by constitutive expression of two transgenic T-cell receptors on a Rag1-/- background, are tolerant to self by maintaining low levels of autospecific receptor, but selfreactive effector function (killing) can be induced through activation via the second receptor. This opens the possibility that T cells carrying two receptors in the periphery of normal individuals contain putatively autoreactive cells that could engage in autoimmune effector functions after recognition of an unrelated environmental antigen.

Characterization and functional ordering of Slu7p and Prp17p during the second step of pre-mRNA splicing in yeast.

Temperature-sensitive alleles in four genes (slu7-1, prp16-2, prp17-1, and prp18-1) are known to confer a specific block to the second chemical step of pre-mRNA splicing in vivo in the yeast Saccharomyces cerevisiae. Previous studies showed that Prp16p and Prp18p are required solely for the second step in vitro. The RNA-dependent ATPase, Prp16p, functions at a stage in splicing when ATP is required, whereas Prp18p functions at an ATP-independent stage. Here we use immunodepletion to show that the roles of Slu7p and Prp17p are also confined to the second step of splicing. We find that extracts depleted of Prp17p require both Prp17p and ATP for slicing complementation, whereas extracts depleted of Slu7p require only the addition of Slu7p. These different ATP requirements suggest that Prp16p and Prp17p function before Prp18p and Slu7p. Although SLU7 encodes an essential gene product, we find that a null allele of prp17 is temperature-sensitive for growth and has a partial splicing defect in vitro. Finally, high-copy suppression experiments indicate functional interactions between PRP16 and PRP17, PRP16 and SLU7, and SLU7 and PRP18. Taken together, the results suggest that these four factors may function within a multi-component complex that has both an ATP-dependent and an ATP-independent role in the second step of pre-mRNA splicing.

The theory of functions of a complex variable. [Lectures] New York University, Institute for Mathematics and Mechanics.

"First edition 1948. Second printing 1952."

Wage Stabilization Board ; report, Eighty-second Congress, first session, on the disputes functions of the Wage Stabilization Board, 1951, together with the Individual views of Mr. Taft.

At head of title: 82d Cong. 1st sess. Senate. Report no. 1037

Experimental and theoretical characterisation of tunable fibre gratings

This thesis presents details on both theoretical and experimental aspects of UV written fibre gratings. The main body of the thesis deals with the design, fabrication and testing of telecommunication optical fibre grating devices, but also an accurate theoretical analysis of intra-core fibre gratings is presented. Since more than a decade, fibre gratings have been extensively used in the telecommunication field (as filters, dispersion compensators, and add/drop multiplexers for instance). Gratings for telecommunication should conform to very high fabrication standards as the presence of any imperfection raises the noise level in the transmission system compromising its ability of transmitting intelligible sequence of bits to the receiver. Strong side lobes suppression and high and sharp reflection profile are then necessary characteristics. A fundamental part of the theoretical and experimental work reported in this thesis is about apodisation. The physical principle of apodisation is introduced and a number of apodisation techniques, experimental results and numerical optimisation of the shading functions and all the practical parameters involved in the fabrication are detailed. The measurement of chromatic dispersion in fibres and FBGs is detailed and an estimation of its accuracy is given. An overview on the possible methods that can be implemented for the fabrication of tunable fibre gratings is given before detailing a new dispersion compensator device based on the action of a distributed strain onto a linearly chirped FBG. It is shown that tuning of second and third order dispersion of the grating can be obtained by the use of a specially designed multipoint bending rig. Experiments on the recompression of optical pulses travelling long distances are detailed for 10 Gb/s and 40 Gb/s. The characterisation of a new kind of double section LPG fabricated on a metal-clad coated fibre is reported. The fabrication of the device is made easier by directly writing the grating through the metal coating. This device may be used to overcome the recoating problems associated with standard LPGs written in step-index fibre. Also, it can be used as a sensor for simultaneous measurements of temperature and surrounding medium refractive index.

Fractional Integration of the Product of Bessel Functions of the First Kind

Dedicated to 75th birthday of Prof. A.M. Mathai, Mathematical Subject Classification 2010:26A33, 33C10, 33C20, 33C50, 33C60, 26A09

Rational Function Interpolation of Electromagnetic Transfer Functions of High-Speed Interconnect Systems from Discrete Time-Domain and Frequency-Domain Data

We present new methodologies to generate rational function approximations of broadband electromagnetic responses of linear and passive networks of high-speed interconnects, and to construct SPICE-compatible, equivalent circuit representations of the generated rational functions. These new methodologies are driven by the desire to improve the computational efficiency of the rational function fitting process, and to ensure enhanced accuracy of the generated rational function interpolation and its equivalent circuit representation. Toward this goal, we propose two new methodologies for rational function approximation of high-speed interconnect network responses. The first one relies on the use of both time-domain and frequency-domain data, obtained either through measurement or numerical simulation, to generate a rational function representation that extrapolates the input, early-time transient response data to late-time response while at the same time providing a means to both interpolate and extrapolate the used frequency-domain data. The aforementioned hybrid methodology can be considered as a generalization of the frequency-domain rational function fitting utilizing frequency-domain response data only, and the time-domain rational function fitting utilizing transient response data only. In this context, a guideline is proposed for estimating the order of the rational function approximation from transient data. The availability of such an estimate expedites the time-domain rational function fitting process. The second approach relies on the extraction of the delay associated with causal electromagnetic responses of interconnect systems to provide for a more stable rational function process utilizing a lower-order rational function interpolation. A distinctive feature of the proposed methodology is its utilization of scattering parameters. For both methodologies, the approach of fitting the electromagnetic network matrix one element at a time is applied. It is shown that, with regard to the computational cost of the rational function fitting process, such an element-by-element rational function fitting is more advantageous than full matrix fitting for systems with a large number of ports. Despite the disadvantage that different sets of poles are used in the rational function of different elements in the network matrix, such an approach provides for improved accuracy in the fitting of network matrices of systems characterized by both strongly coupled and weakly coupled ports. Finally, in order to provide a means for enforcing passivity in the adopted element-by-element rational function fitting approach, the methodology for passivity enforcement via quadratic programming is modified appropriately for this purpose and demonstrated in the context of element-by-element rational function fitting of the admittance matrix of an electromagnetic multiport.

Methodologies for investigating relationships between concept development and the development of problem solving abilities

This paper is the second in a pair that Lesh, English, and Fennewald will be presenting at ICME TSG 19 on Problem Solving in Mathematics Education. The first paper describes three shortcomings of past research on mathematical problem solving. The first shortcoming can be seen in the fact that knowledge has not accumulated – in fact it has atrophied significantly during the past decade. Unsuccessful theories continue to be recycled and embellished. One reason for this is that researchers generally have failed to develop research tools needed to reliably observe, document, and assess the development of concepts and abilities that they claim to be important. The second shortcoming is that existing theories and research have failed to make it clear how concept development (or the development of basic skills) is related to the development of problem solving abilities – especially when attention is shifted beyond word problems found in school to the kind of problems found outside of school, where the requisite skills and even the questions to be asked might not be known in advance. The third shortcoming has to do with inherent weaknesses in observational studies and teaching experiments – and the assumption that a single grand theory should be able to describe all of the conceptual systems, instructional systems, and assessment systems that strongly molded and shaped by the same theoretical perspectives that are being used to develop them. Therefore, this paper will describe theoretical perspectives and methodological tools that are proving to be effective to combat the preceding kinds or shortcomings. We refer to our theoretical framework as models & modeling perspectives (MMP) on problem solving (Lesh & Doerr, 2003), learning, and teaching. One of the main methodologies of MMP is called multi-tier design studies (MTD).

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.