Lp extremal polynomials. Results and perspectives

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.

On Thom Polynomials for A4(−) via Schur Functions

2000 Mathematics Subject Classification: 05E05, 14N10, 57R45.

Some Coefficient Estimates for Polynomials on the Unit Interval

2000 Mathematics Subject Classification: 26C05, 26C10, 30A12, 30D15, 42A05, 42C05.

A Unified Group-Theoretic Method on Improper Partial Semi-Bilateral Generating Functions

2000 Mathematics Subject Classification: 33A65, 33C20.

Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams

An experimental setup to measure the three-dimensional phase-intensity distribution of an infrared laser beam in the focal region has been presented. It is based on the knife-edge method to perform a tomographic reconstruction and on a transport of intensity equation-based numerical method to obtain the propagating wavefront. This experimental approach allows us to characterize a focalized laser beam when the use of image or interferometer arrangements is not possible. Thus, we have recovered intensity and phase of an aberrated beam dominated by astigmatism. The phase evolution is fully consistent with that of the beam intensity along the optical axis. Moreover, this method is based on an expansion on both the irradiance and the phase information in a series of Zernike polynomials. We have described guidelines to choose a proper set of these polynomials depending on the experimental conditions and showed that, by abiding these criteria, numerical errors can be reduced.

The generating function of the generalized Fibonacci sequence

Using tools of the theory of orthogonal polynomials we obtain the generating function of the generalized Fibonacci sequence established by Petronilho for a sequence of real or complex numbers {Qn} defined by Q0 = 0, Q1 = 1, Qm = ajQm−1 + bjQm−2, m ≡ j (mod k), where k ≥ 3 is a fixed integer, and a0, a1, . . . , ak−1, b0, b1, . . . , bk−1 are 2k given real or complex numbers, with bj #0 for 0 ≤ j ≤ k−1. For this sequence some convergence proprieties are obtained.

Idealähtöistä koulualgebraa : IDEAA-opetusmallin kehittäminen algebran opetukseen peruskoulun 7. luokalla

This research is based on the problems in secondary school algebra I have noticed in my own work as a teacher of mathematics. Algebra does not touch the pupil, it remains knowledge that is not used or tested. Furthermore the performance level in algebra is quite low. This study presents a model for 7th grade algebra instruction in order to make algebra more natural and useful to students. I refer to the instruction model as the Idea-based Algebra (IDEAA). The basic ideas of this IDEAA model are 1) to combine children's own informal mathematics with scientific mathematics ("math math") and 2) to structure algebra content as a "map of big ideas", not as a traditional sequence of powers, polynomials, equations, and word problems. This research project is a kind of design process or design research. As such, this project has three, intertwined goals: research, design and pedagogical practice. I also assume three roles. As a researcher, I want to learn about learning and school algebra, its problems and possibilities. As a designer, I use research in the intervention to develop a shared artefact, the instruction model. In addition, I want to improve the practice through intervention and research. A design research like this is quite challenging. Its goals and means are intertwined and change in the research process. Theory emerges from the inquiry; it is not given a priori. The aim to improve instruction is normative, as one should take into account what "good" means in school algebra. An important part of my study is to work out these paradigmatic questions. The result of the study is threefold. The main result is the instruction model designed in the study. The second result is the theory that is developed of the teaching, learning and algebra. The third result is knowledge of the design process. The instruction model (IDEAA) is connected to four main features of good algebra education: 1) the situationality of learning, 2) learning as knowledge building, in which natural language and intuitive thinking work as "intermediaries", 3) the emergence and diversity of algebra, and 4) the development of high performance skills at any stage of instruction.

Microprocessor-based polynomial generator

A new method of generating polynomials using microprocessors is proposed. The polynomial is generated as a 16-bit digital word. The algorithm for generating a variety of basic 'building block' functions and its implementation is discussed. A technique for generating a generalized polynomial based on the proposed algorithm is indicated. The performance of the proposed generator is evaluated using a commercially available microprocessor kit.

Microprocessor-based polynomial generator

A new method of generating polynomials using microprocessors is proposed. The polynomial is generated as a 16-bit digital word. The algorithm for generating a variety of basic 'building block' functions and its implementation is discussed. A technique for generating a generalized polynomial based on the proposed algorithm is indicated. The performance of the proposed generator is evaluated using a commercially available microprocessor kit.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

The Dπ form factors from analyticity and unitarity

We study the shape parameters of the Dπ scalar and vector form factors using as input dispersion relations and unitarity for the moments of suitable heavy-light correlators evaluated with Operator Product Expansions, including O(α 2 s) terms in perturbative QCD. For the scalar form factor, a low energy theorem and phase information on the unitarity cut are implemented to further constrain the shape parameters. We finally determine points on the real axis and isolate regions in the complex energy plane where zeros of the form factors are excluded.

Delay optimized zero-forcing channel shortening algorithm for wireless communication

In Orthogonal Frequency Division Multiplexing and Discrete Multitone transceivers, a guard interval called Cyclic Prefix (CP) is inserted to avoid inter-symbol interference. The length of the CP is usually greater than the impulse response of the channel resulting in a loss of useful data carriers. In order to avoid long CP, a time domain equalizer is used to shorten the channel. In this paper, we propose a method to include a delay in the zero-forcing equalizer and obtain an optimal value of the delay, based on the location of zeros of the channel. The performance of the algorithms is studied using numerical simulations.

The Ginibre Ensemble and Gaussian Analytic Functions

We show that as n changes, the characteristic polynomial of the n x n random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Polya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This suggests another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

On the curvature ODE associated to the Ricci flow

In the vector space of algebraic curvature operators we study the reaction ODE which is associated to the evolution equation of the Riemann curvature operator along the Ricci flow. More precisely, we give a partial classification of the zeros of this ODE up to suitable normalization and analyze the stability of a special class of zeros of the same. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces where is an Einstein (locally) symmetric space of compact type and not a spherical space form when .

O wielomianach cyklotomicznych

Wydział Matematyki i Informatyki