Mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Mathematical modelling in mathematics education and instruction

This paper aims at giving a concise survey of the present state-of-the-art of mathematical modelling in mathematics education and instruction. It will consist of four parts. In part 1, some basic concepts relevant to the topic will be clarified and, in particular, mathematical modelling will be defined in a broad, comprehensive sense. Part 2 will review arguments for the inclusion of modelling in mathematics teaching at schools and universities, and identify certain schools of thought within mathematics education. Part 3 will describe the role of modelling in present mathematics curricula and in everyday teaching practice. Some obstacles for mathematical modelling in the classroom will be analysed, as well as the opportunities and risks of computer usage. In part 4, selected materials and resources for teaching mathematical modelling, developed in the last few years in America, Australia and Europe, will be presented. The examples will demonstrate many promising directions of development.

Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Verbal proficiency as fitness indicator: Experimental and comparative research on the evolutionary psychology of language and verbal displays

Summary: Recent research on the evolution of language and verbal displays (e.g., Miller, 1999, 2000a, 2000b, 2002) indicated that language is not only the result of natural selection but serves as a sexually-selected fitness indicator that is an adaptation showing an individual’s suitability as a reproductive mate. Thus, language could be placed within the framework of concepts such as the handicap principle (Zahavi, 1975). There are several reasons for this position: Many linguistic traits are highly heritable (Stromswold, 2001, 2005), while naturally-selected traits are only marginally heritable (Miller, 2000a); men are more prone to verbal displays than women, who in turn judge the displays (Dunbar, 1996; Locke & Bogin, 2006; Lange, in press; Miller, 2000a; Rosenberg & Tunney, 2008); verbal proficiency universally raises especially male status (Brown, 1991); many linguistic features are handicaps (Miller, 2000a) in the Zahavian sense; most literature is produced by men at reproduction-relevant age (Miller, 1999). However, neither an experimental study investigating the causal relation between verbal proficiency and attractiveness, nor a study showing a correlation between markers of literary and mating success existed. In the current studies, it was aimed to fill these gaps. In the first one, I conducted a laboratory experiment. Videos in which an actor and an actress performed verbal self-presentations were the stimuli for counter-sex participants. Content was always alike, but the videos differed on three levels of verbal proficiency. Predictions were, among others, that (1) verbal proficiency increases mate value, but that (2) this applies more to male than to female mate value due to assumed past sex-different selection pressures causing women to be very demanding in mate choice (Trivers, 1972). After running a two-factorial analysis of variance with the variables sex and verbal proficiency as factors, the first hypothesis was supported with high effect size. For the second hypothesis, there was only a trend going in the predicted direction. Furthermore, it became evident that verbal proficiency affects long-term more than short-term mate value. In the second study, verbal proficiency as a menstrual cycle-dependent mate choice criterion was investigated. Basically the same materials as in the former study were used with only marginal changes in the used questionnaire. The hypothesis was that fertile women rate high verbal proficiency in men higher than non-fertile women because of verbal proficiency being a potential indicator of “good genes”. However, no significant result could be obtained in support of the hypothesis in the current study. In the third study, the hypotheses were: (1) most literature is produced by men at reproduction-relevant age. (2) The more works of high literary quality a male writer produces, the more mates and children he has. (3) Lyricists have higher mating success than non-lyric writers because of poetic language being a larger handicap than other forms of language. (4) Writing literature increases a man’s status insofar that his offspring shows a significantly higher male-to-female sex ratio than in the general population, as the Trivers-Willard hypothesis (Trivers & Willard, 1973) applied to literature predicts. In order to test these hypotheses, two famous literary canons were chosen. Extensive biographical research was conducted on the writers’ mating successes. The first hypothesis was confirmed; the second one, controlling for life age, only for number of mates but not entirely regarding number of children. The latter finding was discussed with respect to, among others, the availability of effective contraception especially in the 20th century. The third hypothesis was not satisfactorily supported. The fourth hypothesis was partially supported. For the 20th century part of the German list, the secondary sex ratio differed with high statistical significance from the ratio assumed to be valid for a general population.

Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications

Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.