Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula $e^{j\theta}=\cos\theta+j\sin\theta$, and a matrix root of -1 isomorphic to the imaginary root $j$. The transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of -1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cl1,1 and Cl2,0. The significance of this result is both in the theoretical unification, and also in the scope it affords for insight into the structure of the various transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier transforms may be computed using standard matrix arithmetic packages without the need for a hypercomplex library, which is of importance in providing a reference implementation for verifying implementations based on hypercomplex code.Comment: The paper has been revised since the second version to make some of the reasons for the paper clearer, to include reviews of prior hypercomplex transforms, and to clarify some points in the conclusion

Põllumajandus rahvatarkuse ja teaduse vahel: folkloristi vaade

On Agricultural Sciences through the Eyes of aFolkloristAgriculture is based on understanding and respecting the patterns of nature, and the direct utilisation of nature. The ecological behavioural culture of a dweller of the land, which involves both communion with nature and taking responsibility for one’s part in the relationship, is rooted deep in our subconscious. Both a farmer and cattle breeder have harnessed the fruitfulness of the soil or the reproductionof cattle for their purposes. The scientific and technological development of agriculture is supported by basic and applied sciences. The sciences that served as the basis of agricultural technology began to evolve in the 18th and 19th century. Practical agricultural guidebooks in the first half of the 18th century still imparted old experiences. The guidelines provided often originated from Ancient Roman authorsand the Bible. As traditional and science-based agriculture began to develop separately, the 19th century brought along a breakthrough. The development of technology changes society as well as nature. Technology adapts to the needs and expectations of people and society, while it becomes easier to attain and operate. Education, upbringing,and culture as a whole will change in addition to the alteration of technology pursuant to the demands of people and culture.Science is becoming more important in order to provide enough food for all the inhabitants of the Earth in modernity as the population of the Earth is consistently growing, there is only a limited amount of agricultural land and climate change is causing regional production problems as well as uncertainty. Synthetic analogues are increasingly used in addition to the raw produce received from agriculture. As canbe said on the basis of analysing Internet forums—it is only habit that is stopping us from switching to artificial food, technologically it is already possible

Self-Interaction Correction to Black Hole Radiance

We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation. This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system. In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle. The radiation is no longer thermal, but is corrected in a definite way that we calculate. Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation.Comment: 20 pages, no figures, uses Phyzzx, IASSNS-HEP 94/6

Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts

A Metacognitive Instructional Approach and Self-Reflection: Reflective Practice From a Computer Science Perspective

Teaching is increasingly complex work which takes time to plan and continuous effort to ensure the highest standards of professional practice. There is a paradox in our rapidly changing society that educators are not very open to change in their practice (Hoban, 2002). The process for educators to learn and acquire effective teaching skills is a labour which has to be mastered over time. They would gain pedagogical knowledge and skills based on accumulation of new acquired knowledge and teaching methods and strategies to be added to their repertoire of existing knowledge and skills. In addition, it is vital for educators to utilize two aspects of reflective practice as suggested by Schon, which are ‘reflection in action’ and ‘reflection on action’ (Schon, 1983, 1987). Reflection in action refers to quick thinking action which takes places when one is teaching in the classroom. On the other hand, reflection on action usually takes place after the lesson out of the classroom when the educator reflects on his or her previous teaching and considers certain situations from the lesson again. Educational system in the 21st century encompasses reflection from both the teachers and students’ perspectives. In general, reflective practice in teaching and learning in undergraduate education focuses on the professional development of students and academics in an interdisciplinary education. Reflective practice has been in existence in most professional educational practices for several centuries. The use of reflective practice in computing education courses has significant benefits to enhance the knowledge of the students. However, there is some controversies on how this reflection was done and the manner in which this was done based on individual teaching practice. One of the main objectives of this study is to describe the various approaches used in teaching undergraduate students in a computing course. The study illustrates several modern approaches used during this classes. A qualitative research method was applied in gathering the feedback from the students using a general survey questions based on the course delivery. The study used statistical packages for the social sciences (SPSS) to analyze the data gathered. The results revealed the various level of acceptance of the teaching methods applied in the course. These results also demonstrate significant findings on the students' opinions and criticism which could help in future improvement of undergraduate computing curriculum. The study, firstly review literature on reflective practice. Secondly, discussed some of the good teaching practices and methods used in delivery the classes. Thirdly, the analysis and results obtained from the instrument questionnaires used for this study and finally, summary of the findings and further research directions

A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.Comment: First presented in Proc. of The 9th Int. Conf. on Clifford Algebras and their Applications, (2011

Evanescent quadrupole polariton

In the work we demonstrate the formation of new type of polariton on the interface between a cuprous oxide slab and a polystyrene micro-sphere placed on the slab. The evanescent field of the resonant whispering gallery mode (WGM) has a substantial gradient, and therefore effectively couples with the quadrupole $1S$ excitons in cuprous oxide. This evanescent polariton has a long life-time ($1.7 ns$), which is determined only by its excitonic component. The polariton lower branch has a well pronounced minimum. This suggests that this excitation can be utilized for BEC. The spatial coherence of the polariton can be improved by assembling the micro-spheres in a linear chain.Comment: 4 pages, 3 figures; APS/PREPRIN

Peter Josef Ell : discurs llegit a la cerimònia d'investidura celebrada a l'aula magna de Casa Convalescència el dia 5 d'abril de l'any 2005

Peter Josef Ell , nascut el 7 de maig de 1944, és professor de Medicina Nuclear a la Universitat de Londres des de 1987, director del Institute of Nuclear Medicine, UCL, des de 1986, Honorary Consultant Physician del Middlesex Hospital des de 1976, i Clinical Director del UCLH NHS Trust des de 2001. El professor Ell és una de les figures més rellevants del món en l'àrea de la medicina nuclear i del diagnòstic per la imatge. La seva formació, variada i europeista, inclou una llicenciatura en Medicina per la Universitat de Lisboa, un màster en Ciències per la Universitat de Londres i un doctorat per la Universitat de Berna. No solament parla diverses llengües, sinó que ha integrat les cultures portuguesa, alemanya i anglesa. És director del Institute of Nuclear Medicine del UCL, un dels serveis més grans d'Anglaterra en l'especialitat, i catedràtic de Medicina Nuclear en aquest mateix centre. És autor de més de cinc-centes publicacions científiques, moltes d'elles en les revistes de més factor d'impacte (Lancet, JCO, JACC, etc.), i és editor/autor de dotze llibres de text, i de múltiples comunicacions i presentacions a congressos. Ha estat convidat a pronunciar nombroses conferències arreu del món. La seva multiculturalitat li ha permès de tenir un paper central en la creació i formació de la Societat Europea de Medicina Nuclear (EANM), de la qual va ser autor de molts dels documents fundacionals, i de la qual ha estat president entre 1994 i 1996. Ha estat quinze anys redactor en cap del European Journal of Nuclear Medicine and Molecular Imaging, revista que, sota la seva direcció, s'ha consolidat com una de les més importants del món en aquesta àrea científica.Nomenament 10/11/2004. A proposta de Facultat de Medicina. Investidura 05/05/2005. Padrí: Ignasi Carri

Intensity invariant complex encoded colour correlation

Optical correlation has traditionally processed monochromatic grey scale images. This paper develops a new encoding mechanism that uses the chromaticy of the input signal. It is then not only possible to detect different coloured objects but the system is invariant to changes in the brightness of the lighting, including variations across the object