Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
Special functions play a significant role in mathematical physics, especially in boundary value problems. Usually a function is called “special” when, like the logarithm, the exponential and trigonometric functions, it belongs to the toolbox of the applied mathematician, the physicist or the engineer. The function usually has a particular notation and a number of properties of the function are known.
Certain special functions have interesting properties where they vanish or become zero. One well-known property of the zeros of orthogonal polynomials is that in between any two zeros of the one polynomial, there is exactly one zero of the next polynomial in the sequence. This is known as the interlacing of the zeros and is useful in many contexts, most notably numerical quadrature.
Collaborative research with Prof Kathy Driver of the University of Cape Town and Dr Norbert Mbuyi has examined conditions under which the zeros of combinations of the simplest and most useful orthogonal polynomials retain this interlacing property.
Contact person: Dr KH Jordaan.
This work provides valuation formulae of financial instruments called derivatives traded in the energy markets in large volumes. The class of spread contracts (spread between two refined energy products) is huge, including important examples like the spark and dark spread. The spark spread is the difference between spot electricity and gas, whereas “dark" refers to the difference between electricity and coal. This research analyses the pricing of options for a class of arithmetic spot price models.
The price dynamics in typical energy markets possesses several specific features that must be accounted for in modelling, and which complicate the valuation of financial derivatives. Prices may show a strong seasonal influence, say due to a high demand for electricity or gas for heating in the winter. The prices are mean-reverting, but with significant spikes occurring, possibly seasonally. These spikes may, in electricity markets say, occur due to a sudden drop in temperature not matched by sufficient production volume, or an unexpected failure of a power plant. To get reliable prices on derivatives, one needs to have accessible spot price models that incorporate these features of energy prices.
The researchers considered an arithmetic model for the spot price dynamics of an energy, which was proposed by Prof Fred Benth, Prof Jan Kallsen and Dr Thilo Meyer-Brandis in 2007, and which includes seasonality of prices, mean-reversion at different speeds and seasonal spikes. These researchers derived analytical prices for energy forwards, along with expressions for the price of call and put options on these forwards.
The analysis done by the researchers was generalised in this research project to cover average-type options and spread contracts of various kinds. It turns out that this spot price model is very flexible, and easily computable expressions come out for these derivatives. This is a direct consequence of the arithmetic nature of the underlying spot price process that was chosen.
Contact person: Dr R Kufakunesu.
A collaborative research programme between the Department of Physics and the Department of Mathematics and Applied Mathematics focuses on the mathematical structure of quantum dynamical systems. In particular, the long-term behaviour (so-called ergodic theory) of such systems is studied by means of mathematical objects known as “operator algebras”, and more generally by the techniques of functional analysis. The basic physical motivation is quantum statistical mechanics. The goal of the work is to identify general structures that are present in wide classes of quantum dynamical systems, and that can be used to analyse these systems mathematically. In the most recent and ongoing work, the focus is especially on how a given system’s properties can be classified by considering subsystems of it, as well as by investigating in which ways the system can be combined with other systems to give larger systems (roughly put, by viewing the given system as a subsystem of larger systems). Subsequently, this work has been used in a substantial paper of a recent Fields Medallist, Prof Terrance Tao and his co-workers at the University of California, Los Angeles.
The idea is to extend classical multiple ergodic theories to quantum dynamical systems with the aim of understanding the recurrence properties of such systems. Classical recurrence theorems were used by Tao and Green to prove the famous Green-Tao theorem in number theory, which states that the set of prime numbers contains arbitrarily long arithmetic progressions.
Contact person: Dr RdV Duvenhage.
A collaborative research programme between the Department of Physics and the Department of Mathematics and Applied Mathematics focuses on the mathematical structure of quantum dynamical systems. In particular, the long-term behaviour (so-called ergodic theory) of such systems is studied by means of mathematical objects known as “operator algebras”, and more generally by the techniques of functional analysis. The basic physical motivation is quantum statistical mechanics. The goal of the work is to identify general structures that are present in wide classes of quantum dynamical systems, and that can be used to analyse these systems mathematically. In the most recent and ongoing work, the focus is especially on how a given system’s properties can be classified by considering subsystems of it, as well as by investigating in which ways the system can be combined with other systems to give larger systems (roughly put, by viewing the given system as a subsystem of larger systems). Subsequently, this work has been used in a substantial paper of a recent Fields Medallist, Prof Terrance Tao and his co-workers at the University of California, Los Angeles.
The idea is to extend classical multiple ergodic theories to quantum dynamical systems with the aim of understanding the recurrence properties of such systems. Classical recurrence theorems were used by Tao and Green to prove the famous Green-Tao theorem in number theory, which states that the set of prime numbers contains arbitrarily long arithmetic progressions.
Contact person: Dr FJC Beyers.
A collaborative research programme between the Department of Physics and the Department of Mathematics and Applied Mathematics focuses on the mathematical structure of quantum dynamical systems. In particular, the long-term behaviour (so-called ergodic theory) of such systems is studied by means of mathematical objects known as “operator algebras”, and more generally by the techniques of functional analysis. The basic physical motivation is quantum statistical mechanics. The goal of the work is to identify general structures that are present in wide classes of quantum dynamical systems, and that can be used to analyse these systems mathematically. In the most recent and ongoing work, the focus is especially on how a given system’s properties can be classified by considering subsystems of it, as well as by investigating in which ways the system can be combined with other systems to give larger systems (roughly put, by viewing the given system as a subsystem of larger systems). Subsequently, this work has been used in a substantial paper of a recent Fields Medallist, Prof Terrance Tao and his co-workers at the University of California, Los Angeles.
The idea is to extend classical multiple ergodic theories to quantum dynamical systems with the aim of understanding the recurrence properties of such systems. Classical recurrence theorems were used by Tao and Green to prove the famous Green-Tao theorem in number theory, which states that the set of prime numbers contains arbitrarily long arithmetic progressions.
Contact person: Prof A Ströh.
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