The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Hardware implementation of pseudo-random number generators based on chaotic maps

We show the usefulness of bifurcation diagrams to implement a pseudo-random number generator (PRNG) based on chaotic maps. We provide details on the selection of the best parameter values to obtain high entropy and positive Lyapunov exponent from the bifurcation diagram of four chaotic maps, namely: Bernoulli shift map, tent, zigzag, and Borujeni maps. The binary sequences obtained from these maps are analyzed to implement a PRNG both in software and in hardware. The software implementation is realized using 32 and 64 bits microprocessor architectures, and with floating point and fixed point computer arithmetic. The hardware implementation is done by using a field-programmable gate array (FPGA) architecture. We developed a serial communication interface between the PRNG on the FPGA and a personal computer to obtain the generated sequences. We validate the randomness of the generated binary sequences with the NIST test suite 800-22-a both in floating point and fixed point arithmetic. At the end, we show that those chaotic maps are suitable to implement a PRNG but according to the hardware resources, the one based on the Bernoulli shift map is better. In addition, another advantage is that the required initial value for the sequences can be within the whole interval \([-1,1]\), including its bounds.     

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.     

The sticking probability,dipole moment,and absolute coverage of CO on Ni(111)

The change in work function of a Ni(111) surface resulting from the adsorption of CO is measured continuously. Work function change is related to absolute coverage through thermal desorption measurements and LEED patterns. From the coverage-time curve, the sticking probability is determined, with an initial sticking probability of 0.91. From the relation between work function change and coverage, the dipole moment of an adsorbed CO molecule is determined to be 0.28 Debye with a negative end of the molecule projecting outward from the surface.     

Quasiparticles in the superconducting state of Bi(2)Sr(2)CaCu(2)O(8+delta)

Recent improvements in momentum resolution lead to qualitatively new angle-resolved photoemission spectroscopy results on the spectra of Bi(2)Sr(2)CaCu(2)O(8+delta) (Bi2212) along the (pi,pi) direction, where there is a node in the superconducting gap. We now see the intrinsic line shape, which indicates the presence of true quasiparticles at all Fermi momenta in the superconducting state, and lack thereof in the normal state. The region of momentum space probed here is relevant for charge transport, motivating a comparison of our results to conductivity measurements by infrared reflectivity.     

Electrocatalytic (bio)platforms for the determination of tetracyclines

Journal of Solid State Electrochemistry - Due to their widespread use in veterinary medicine, the presence of tetracyclines (TCs) and their metabolites in foodstuffs from treated animal represents...     

The finite Littlewood problem in $${{\mathbb {F}}}_p$$

Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\)