Multiplicative renormalization method for orthogonal polynomials

To study the orthogonal polynomials, Asai, Kubo and Kuo recently have developed the multiplicative renormalization method. Motivated by infinite dimensional white noise analysis, it is an alternative to the computational part of the classical Gram-Schmidt process to find the orthogonal polynomials for a given measure. Instead of finding the orthogonal polynomials recursively as described in the Gram-Schmidt process, one analyzes different types of generating functions systematically in order to obtain polynomials after power series expansion. This work also produces the Jacobi-Szego parameters easily and paves the way for the study of one-mode interacting Fock spaces related to these parameters. They have verified the classical measures and their corresponding orthogonal polynomials. In this thesis, we take this to the next level in order to classify measures with certain generating functions which leads to some new measures. We will begin with a description of the mathematical background of the method and we will re-derive all measures with generating functions of the exponential type as did Meixner and Morris in different settings. Next we shall derive all measures with generating functions of the fractional type, which will yield new measures We will also show the relation between consecutive Gegenbauer measures and neigbouring Jacobi polynomials. Finally we will demonstrate the uniqueness of the uniform measure with a certain type of generating functio

Influence of different cytokinins used in in vitro culture on the stoma morphology of pistachio (Pistacia vera L. cv. Siirt)

Three different cytokinins (BA, kinetin and TDZ at 1 mg/l) were tested in order to study their effects on pistachio stoma morphology. Additional BA concentrations of 0.25, 0.625 and 2 mg/l were also tested.The stoma morphology of pistachio leaves in vitro was affected by cytokinins. Stomas obtained from 1 mg/l BA medium were similar to those developed in vivo. TDZ, kinetin and other BA concentrations resulted in abnormal stoma development

Active learning education in Museum

In this study, it was aimed to determine social studies candidates how to practice about make of use museums as a learning environment. It was tried to discuss a case, for this purpose ıt was used case study. The study group consisted of 19 students (12 females, 7 males) studying social studies at a public university. Participants were given museum training lasting 7 weeks and the process started with the visit of the Antalya Archeology Museum. In this study, Museum Evaluation Forms were used as data collection tools. These forms were used at two-stage. The first stage was the Pre-Museum Evaluation Form and the second stage was the Activity Evaluation Forms. Content analysis technique was used in data analysis, and the forms were examined separately and themes and categories were created. In the categories, the statements of some of the participants were given with direct quotations. As a result of the research, the expectations of the students before the museum training are divided into the categories of Learning about the use of the Museum, Historical awareness and Embodiment in the theme of Cognitive Field. In the affective main theme, it was determined that it was divided into the categories of Group Work, Contribution to Professional Knowledge and Being an Effective Citizen. In the Cognitive Field theme, the students' experiences after museum training are divided into the categories of Field Knowledge, Museum Use Learning and Creativity. In the affective Thinking, Contribution to Professional Knowledge, Permanent Learning, and Role-Playing are the categories of the affective categories. Participants stated that to make use of museum to effect on the cognitive area more effective than affective area.

Applicability of multiplicative renormalization method for a certain function

Abstract. We characterize the class of probability measures for which the multiplicative renormalization method can be applied for the function h(x) = 1√ 1−x to obtain orthogonal polynomials. It turns out that this class consists of only uniform probability measures on intervals and probability measures being supported by one or two points. 1. Multiplicative Renormalization Method Let µ be a probability measure on R with finite moments of all orders. Then we can apply the Gram-Schmidt orthogonalization process to the sequence {xn}∞n=0 to get an orthogonal sequence {Pn(x)}∞n=0 in the real Hilbert space L2(R, µ). Here the leading coefficient of Pn is 1 for each n. It is well known that these polynomials satisfy the recursion formula: (x − αn)Pn(x) = Pn+1(x) + ωn−1Pn−1(x), n ≥ 0, (1.1) where by convention ω−1 = 1 and P−1 = 0. The numbers αn, ωn, n ≥ 0, are known as the Jacobi–Szego ̈ parameters of µ. It is natural to ask whether there is a method for deriving {Pn(x)}∞n=0 from µ. A method, called multiplicative renormalization method, has been introduced in [3, 4] to answer this question. This method starts with an analytic function h(x) at 0. Then we define two functions θ(t) = R h(tx) dµ(x), (1.2) θ̃(t, s) = R h(tx)h(sx) dµ(x), (1.3) Theorem 1.1. [3, 4] Let ρ(t) be an analytic function at 0 with ρ(0) = 0 and ρ′(0) 6 = 0. Then the function Θρ(t, s):= θ̃ ρ(t), ρ(s

The characterization of a class of probability measures by multiplicative renormalization

Abstract. We use the multiplicative renormalization method to characterize a class of probability measures on R determined by five parameters. This class of probability measures contains the arcsine and the Wigner semi-circle distributions (the vacuum distributions of the field operators of interacting Fock spaces related to the Anderson model), as well as new nonsymmetric distributions. The corresponding orthogonal polynomials and Jacobi–Szegö parameters are derived from the orthogonal-polynomial generating functions. These orthogonal polynomials can be expressed in terms of the Chebyshev polynomials of the second kind. 1

Numerical Investigation of Turbulent Hydrogen-Methane-Nitrogen Non-Premixed Jet Flame

In this work, the numerical investigation of the two-dimensional axisymmetric turbulent diffusion flame of a composite fuel was performed by using a computational fluid dynamics code to predict flame structure. The composite fuel was an H2/CH4/N2 gas mixture. The amount of H2 and N2 in the fuel mixture varies under constant volumetric fuel flow rate. Fluent, which solves the governing and reaction equations using the finite volume method, was used as the computational fluid dynamics program. The non-premixed model was used for computation of the combustion. The standard k-ε model was used for modeling the turbulent flow. The interaction of the chemistry and turbulence was accounted for by the program with the probability density function model. This model was validated against the experimental data taken from literature. In general, the numerical results of the temperature, velocity, and CO2 concentration distributions were in satisfactory agreement with the experimental results. The numerical results showed that adding H2 to the fuel mixture decreases the flame length and generally increases the maximum temperature of the flame. On the other hand, adding N2 to the mixture decreases both the flame length and maximum flame temperature. The flame length corresponds to the axial position of the peak flame temperature. Keywords: Combustion Modeling; Composite Fuels; Diffusion Flame; H2/CH4/N2 Flame; Flame Length; Emission

Numerical Investigation of the Effect of Different Conical Turbulators on the Performance of a Liquid Fuelled Boiler

The increment in the combustion of fossil fuels for heating and power generation in recent years has led researchers to design more efficient energy conversion systems by increasing the efficiency of the existing systems and by minimizing energy losses. In this study, increasing the efficiency of the liquid fuelled smoke tube boilers used for domestic heating was researched. In this context, turbulators with full conical and frusto-conical geometries placed to smoke tubes of boiler and their effects on flame structure, heat transfer and boiler efficiency were investigated numerically. Calculations were carried out at two dimensional axisymmetric conditions and Fluent was used as the computational fluid dynamics software. In all cases, the standard k-ε model was used for modelling the turbulent flow and the species transport model was used for modelling the combustion. The results obtained by using these turbulators were evaluated for each placement condition of the turbulators. Besides, the temperature and the stream function distribution and the pressure drop in the boiler were investigated according to the type of turbulators. Finally, it was discussed which type of turbulator would be most appropriate at boilers. Keywords: Spray Combustion; Combustion Modelling; Boilers; Conical Turbulator

Effect of plant growth regulators on in vitro shoot multiplication of Amygdalus communis L. cv. Yaltsinki

An efficient protocol was established for in vitro shoot multiplication from apical shoot tips derived from mature trees of almond (Amygdalus communis L.) cultivars, Yaltsinki. Explants were cultured on Murashige and Skoog (1962) (MS) medium containing various concentrations of 6-benzyladenin (BA) and kinetin (kin) for shoot multiplication. Shoot multiplication was best achieved from explant on MS medium containing 30 gl-1 sucrose, 7 gl-1 agar and 1.0 mgl-1 BA. This amount of BA (1.0 mgl-1) gave the best multiple shoot formation response with an average of 16.10 shoots per explant. In addition, shoots were cultured on the media containing 1.0 mgl-1 BA and kin combined with three different auxins (0.25 and 0.5 mgl-1 of IAA, IBA and NAA) separately. It was noted that 1.0 mgl-1 BA and kin combinated with NAA had inhibitory effect on new shoot formation and no shoot formation was induced. However, explants cultivated on medium containing 1.0 mgl-1 BA and 0.5 mgl-1 IAA resulted in 11.25 shoots per explant. The effect of four different sucrose concentrations (20, 30, 40, 50 gl-1) on the multiplication of shoots was also investigated. The best shoot multiplication was obtained in MS media containing 30 gl-1 sucrose with an average of 15.40 shoots per explant