Complex variable positive definite functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

Complex Variable and Integrals

Exercises and solutions for a third year engineering maths course. Diagrams for the question are all together in the support.zip file, as .eps files

Complex Variable Theory

Notes, exercises, exam questions and solutions for a second year complex variable course.

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

"First edition 1948. Second printing 1952."

Functions of real variables. Functions of a complex variable.

Mode of access: Internet.

The applications of the complex variable to kinematics /

Thesis (M.S.)--University of California, Berkeley, 1894.

Elements of the theory of functions of a complex variable with especial reference to the methods of Riemann,

Translation of Elemente der Theorie der Funktionen einer complexen veränderlichen Grösse.

Theory of functions of a complex variable,

Available on demand as hard copy or computer file from Cornell University Library.

Theory of functions of a complex variable /

Mode of access: Internet.

Dissolution of a spatially variable DNAPL (dense non aqueous phase liquid) source

Dissolution of non-aqueous phase liquids (NAPLs) or gases into groundwater is a key process, both for contamination problems originating from organic liquid sources, and for dissolution trapping in geological storage of CO2. Dissolution in natural systems typically will involve both high and low NAPL saturations and a wide range of pore water flow velocities within the same source zone for dissolution to groundwater. To correctly predict dissolution in such complex systems and as the NAPL saturations change over time, models must be capable of predicting dissolution under a range of saturations and flow conditions. To provide data to test and validate such models, an experiment was conducted in a two-dimensional sand tank, where the dissolution of a spatially variable, 5x5 cm**2 DNAPL tetrachloroethene source was carefully measured using x-ray attenuation techniques at a resolution of 0.2x0.2 cm**2. By continuously measuring the NAPL saturations, the temporal evolution of DNAPL mass loss by dissolution to groundwater could be measured at each pixel. Next, a general dissolution and solute transport code was written and several published rate-limited (RL) dissolution models and a local equilibrium (LE) approach were tested against the experimental data. It was found that none of the models could adequately predict the observed dissolution pattern, particularly in the zones of higher NAPL saturation. Combining these models with a model for NAPL pool dissolution produced qualitatively better agreement with experimental data, but the total matching error was not significantly improved. A sensitivity study of commonly used fitting parameters further showed that several combinations of these parameters could produce equally good fits to the experimental observations. The results indicate that common empirical model formulations for RL dissolution may be inadequate in complex, variable saturation NAPL source zones, and that further model developments and testing is desirable.

Using GeoGebra to study complex functions

The aim of this workshop to present some of the strategies studied to use GeoGebra in the analysis of complex functions. The proposed tasks focus on complex analysis topics target for students of the 1st year of higher education, which can be easily adapted to pre-university students. In the first part of this workshop we will illustrate how to use the two graphical windows of GeoGebra to represent complex functions of complex variable. The second part will present the use of the dynamic color Geogebra in order to obtain Coloring domains that correspond to the graphic representation of complex functions. Finally, we will use the threedimensional graphics window in GeoGebra to study the component functions of a complex function. During the workshop will be provided scripts orientation of the different tasks proposed to be held on computers with Geogebra version 5.0 or high.

On chaos, transient chaos and ghosts in single populations models with allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)(infinity) occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. (C) 2011 Elsevier Ltd. All rights reserved.

Empirical evidence on the difference between the effective and the statutory tax rates for listed groups in Germany

The difference between the statutory and effective tax rate for listed groups is a complex variable influenced by a variety of factors. This paper aims to analyze whether this difference exists for listed groups in the German market and tests which factors have an impact on it. Thus the sample consists of 130 corporations listed in the three major German stock indices. The findings suggest that the companies that pay less than the statutory rate clearly outweigh the ones that pay more, and that the income earned from associated companies has a significant impact on this difference.

Recurrence relations for Clifford algebra-valued orthogonal polynomials

The theory of orthogonal polynomials of one real or complex variable is well established as well as its generalization for the multidimensional case. Hypercomplex function theory (or Clifford analysis) provides an alternative approach to deal with higher dimensions. In this context, we study systems of orthogonal polynomials of a hypercomplex variable with values in a Clifford algebra and prove some of their properties.