The Mahler measure of the Rudin-Shapiro polynomials

Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before

The Matter of Entrepreneurial Learning: A Literature Review

This paper is a comprehensive review of the entrepreneurial learning literature and its engagement with the material aspects of entrepreneurship, as part of the “material turn” in the social sciences. Drawing on actor-network theory, we construct a classificatory scheme and an evaluative matrix to find that this field is dominated by an anthropocentric bias and cognitivist approaches which largely ignore issues of materiality in entrepreneurship. However we also identify some heterogeneous network-based conceptualisations of entrepreneurial learning which could provide the foundations for more materially aware approaches. We conclude by calling for a material turn in entrepreneurial learning and outline some possible avenues for it

Inequalities for Lorentz polynomials

We prove a few interesting inequalities for Lorentz polynomials including Nikolskii-type inequalities. A highlight of the paper is a sharp Markov-type inequality for polynomials of degree at most n with real coefficients and with derivative not vanishing in the open unit disk. The result may be compared with Erdos's classical Markov-type inequality (1940) for polynomials of degree at most n having only real zeros outside the interval (-1,1)

Wave propagation in steady stratified one-dimensional cylindrical waveguides

Aims. This paper studies the propagation of longitudinal magnetic tube waves in a stratified isothermal flux tube with an internal equilibrium background flow. Methods. The governing differential equation is solved by means of Laplace transforms and temporal and spatial solutions are developed, with boundary conditions given by various footpoint drivers, namely a monochromatic source, a delta function pulse, and a sinusoidal pulse. The effect of the background flow is to introduce an increase in amplitude of the wave perturbation and changes in phase shift when compared with the corresponding static case. Results. Results are presented and applied to conditions in the solar atmosphere. When the source is driven continuously, the forced atmospheric oscillations are shown to have large percentage differences when compared to the corresponding static case. For the free atmospheric oscillations, percentage increases in amplitude merely a few percent are found and vary greatly in height but are practically unaltered in time. Phase shifts up to a radian are introduced and weakly depend on both height and time. Conclusions. The results presented in this paper may have interesting observational consequences, especially when using the tools of magnetic seismology of solar atmospheric wave guides (i.e. flux tubes from photosphere to corona) in light of the present and near-future high spatial and temporal resolution space missions, e.g. Hinode, Solar Dynamics Observatory, or Solar Orbiter

Kink oscillations in magnetic tubes with twisted annulus

Aims.We study kink waves in a magnetic flux tube modelled as a straight core surrounded by a magnetically twisted annulus, both embedded in a straight ambient external field, and derive the dispersion relation for this configuration. Methods.The existence and behaviour of the kink modes are examined with specific attention to the effect that the addition of magnetic twist has on phase speeds and periods. Analytic expansions to the short and long wavelength approximations are also considered. Results.The magnetic twist is found to introduce of an infinite set of body modes into solutions of the dispersion relation not present in the untwisted case. Moreover, for the kink modes, the width of interval of this infinite set, generally found to occupy phase speeds around the annulus' longitudinal Alfvén speed, increases for longer wavelengths. Two surface modes are also present in the solution, one at each surface: the internal and the external edges of the annulus. The magnetic twist is found to increase or decrease the phase speeds of these surface modes that are depending on the ratio of internal and external Alfvén speeds in the flux tube. Conclusions.The magnetic twist of the annulus region of a flux tube is found to have a marked effect on the phase speeds of occurring modes. A straight annulus layer increased (or decreased) the periods of the surface modes for a tube modelled as a density (magnetic) enhancement. The addition of twist reduces the periods of the modes in both cases

The effect of elliptic shape on the period ratio P-1/P-2 of emerging coronal loops

Aims. We determine the effect of an elliptical shape on the period ratio for the standing transversal oscillations of a longitudinally stratified coronal loop throughout its emergence from the low solar atmosphere into the ubiquitously magnetised corona. Methods. Under the assumption that elliptical curvature has a negligible effect on eigenfrequencies, the equation that describes the projection of a density profile onto a magnetic flux tube with elliptical shape is obtained in a gravitationally stratified atmosphere. The effect of the elliptical shape on the period ratio of the fundamental mode to the first harmonic (P-1/P-2) at various stages of emergence is determined, assuming that the oscillation periods are much shorter than the characteristic time scale of loop emergence. Results. We find that there are two separate cases of elliptical shape that occur, the minor ellipse and the major ellipse. It is then shown how the period ratio P-1/P-2 is dependent upon the ellipticity (epsilon), the parameter characterising the stage of emergence (lambda) and the density scale height (H). Ellipticity is found to make an important contribution to P-1/P-2 for the minor ellipse when compared to its counterpart of standing oscillations of stratified loops with semi-circle or circle-arc shape. The major ellipse was found to have a lesser effect on the period ratio of standing oscillations. We also find the value of P-1/P-2 is dependent upon the stage of emergence of the loop, where the greatest contribution from emergence to the ratio of P-1/P-2 is when the loop is almost fully emerged. The important implication for magneto-seismological interpretations of the observations of oscillating coronal loops is that measurements of ellipticity and stage of emergence should supplement observations of oscillation periods and should be considered when applying observed frequencies of the fundamental mode and first harmonic to determine the diagnostic properties of these oscillating loops, e. g. the density scale height or strength of magnetic field. Neglecting the determination of ellipticity and stage of emergence may result in a 35% error in estimating density scale height

Magnetohydrodynamic waves in a compressible magnetic flux tube with elliptical cross-section

Aims. The propagation of magnetohydrodynamic (MHD) waves in a finite, compressible magnetic flux tube with an elliptical cross-section embedded in a magnetic environment is investigated. Methods. We present the derivation of the general dispersion relation of linear magneto-acoustic wave propagation for a compressible magnetic flux tube with elliptical cross-section in a plasma with finite beta. The wave modes of propagation for the n = 0 (symmetric) sausage and n = 1 (anti-symmetric) kink oscillations are then examined within the limit of the thin flux tube approximation. Results. It is shown that a compressible magnetic tube with elliptical cross-section supports slow and fast magneto-acoustic waves. In the thin tube approximation, the slow sausage mode and the slow and fast kink modes are found in analogue to a circular cross-section. However, the kink modes propagate with different phase speeds depending on whether the axial displacement takes place along the major or minor axis of the ellipse. This feature is present in both the slow and the fast bands, providing two infinite sets of slow kink modes and two infinite sets of fast kink modes, i.e. each corresponding cylindrical mode splits into two sets of modes due to the ellipticity. The difference between the phase speeds along the different axis is dependent on the ratio of the lengths of the two axes. Analytical expressions for the phase speeds are found. We show that the sausage modes do not split due to the introduced ellipticity and only the phase speed is modified when compared to the appropriate cylindrical counterpart. The percentage difference between the periods of the circular and elliptical cross-sections is also calculated, which reaches up to 21% for oscillations along the major axis. The level of difference in period could be very important in magneto-seismological applications, when observed periods are inverted into diagnostic properties (e. g. magnetic field strength, gravitational scale height, tube expansion parameter). Also shown is the perturbation of focal points of the elliptical cross-section for different modes. It is found that the focal points are unperturbed for the sausage mode, but are perturbed for all higher modes