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

A simple model of quantum trajectories

Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. In this paper I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different ``unravelings'' of a mixed state master equation. I also comment briefly on the relationship of the theory to the Consistent Histories formalism and to spontaneous collapse models.Comment: 42 pages RevTeX including four figures in encapsulated postscript. Submitted to special issue of American Journal of Physic

Differential analysis for the turbulent boundary layer on a compressor blade element (including boundary-layer separation)

A two-dimensional differential analysis is developed to approximate the turbulent boundary layer on a compressor blade element with strong adverse pressure gradients, including the separated region with reverse flow. The predicted turbulent boundary layer thicknesses and velocity profiles are in good agreement with experimental data for a cascade blade, even in the separated region

The high temperature creep deformation of Si3N4-6Y2O3-2Al2O3

The creep properties of silicon nitride containing 6 wt percent yttria and 2 wt percent alumina have been determined in the temperature range 1573 to 1673 K. The stress exponent, n, in the equation epsilon dot varies as sigma sup n, was determined to be 2.00 + or - 0.15 and the true activation energy was found to be 692 + or - 25 kJ/mol. Transmission electron microscopy studies showed that deformation occurred in the grain boundary glassy phase accompanied by microcrack formation and cavitation. The steady state creep results are consistent with a diffusion controlled creep mechanism involving nitrogen diffusion through the grain boundary glassy phase

An investigation of using an RQP based method to calculate parameter sensitivity derivatives

Estimation of the sensitivity of problem functions with respect to problem variables forms the basis for many of our modern day algorithms for engineering optimization. The most common application of problem sensitivities has been in the calculation of objective function and constraint partial derivatives for determining search directions and optimality conditions. A second form of sensitivity analysis, parameter sensitivity, has also become an important topic in recent years. By parameter sensitivity, researchers refer to the estimation of changes in the modeling functions and current design point due to small changes in the fixed parameters of the formulation. Methods for calculating these derivatives have been proposed by several authors (Armacost and Fiacco 1974, Sobieski et al 1981, Schmit and Chang 1984, and Vanderplaats and Yoshida 1985). Two drawbacks to estimating parameter sensitivities by current methods have been: (1) the need for second order information about the Lagrangian at the current point, and (2) the estimates assume no change in the active set of constraints. The first of these two problems is addressed here and a new algorithm is proposed that does not require explicit calculation of second order information

An investigation of new methods for estimating parameter sensitivities

The method proposed for estimating sensitivity derivatives is based on the Recursive Quadratic Programming (RQP) method and in conjunction a differencing formula to produce estimates of the sensitivities. This method is compared to existing methods and is shown to be very competitive in terms of the number of function evaluations required. In terms of accuracy, the method is shown to be equivalent to a modified version of the Kuhn-Tucker method, where the Hessian of the Lagrangian is estimated using the BFS method employed by the RQP algorithm. Initial testing on a test set with known sensitivities demonstrates that the method can accurately calculate the parameter sensitivity

Searches for Long-lived Particles at the Tevatron Collider

Several searches for long-lived particles have been performed using data from p-pbar collisions from Run II at the Tevatron. In most cases, new analysis techniques have been developed to carry out each search and/or estimate the backgrounds. These searches expand the discovery potential of the CDF and D0 experiments to new physics that may have been missed by traditional search techniques. This review discusses searches for (1) neutral, long-lived particles decaying to muons, (2) massive, neutral, long-lived particles decaying to a photon and missing energy, (3) stopped gluinos, and (4) charged massive stable particles. It summarizes some of the theoretical and experimental motivations for such searches.Comment: submitted to Mod. Phys. Lett.

Freire re-viewed

The work of Paulo Freire is associated with themes of oppression and liberation, and his critical pedagogy is visionary in its attempts to bring about social transformation. Freire has created a theory of education that embeds these issues within social relations that center around both ideological and material domination. In this review essay, Sue Jackson explores three books: Freire’s final work Pedagogy of Indignation; Cesar Augusto Rossatto’s Engaging Paulo Freire’s Pedagogy of Possibility, which attempts to engage Freire’s pedagogy of possibility; and C.A. Bowers and Frederique Apffel-Marglin’s edited collection Re-thinking Freire, which asks readers to reconsider Freire’s work in light of globalization and environmental crises. Jackson questions the extent to which Freire’s pedagogical approaches are useful to educators as well as to “the oppressed,” and whether challenges to re-think Freire can lead to new kinds of critical pedagogies

The Trouble with the MDGs: Confronting Expectations of Aid and Development Success

The Millennium Development Goals (MDGs) are unlikely to be met by 2015, even if huge increases in development assistance materialize. The MDGs are a set of quantitative, time-bound targets for indicators such as poverty, education and mortality in developing countries adopted unanimously by the UN in 2000. However, the rates of progress required by many of the goals are at the edges of or beyond historical precedent. At the same time, there appear to be limits to the degree to which aid can contribute to development outcomes. Estimates of the ‘cost’ of reaching the MDGs are nevertheless frequently misinterpreted to mean that a certain quantity of aid—such as the oft-cited $50 billion—could cause the Goals to be met. Despite many benefits of the MDGs, there has been little discussion so far of potential costs of the specific form taken by these goals, especially the creation of unreasonable expectations about what is achievable in a short time frame and about the role of aid in the development process. Many countries making extraordinarily rapid progress on MDG indicators, due in large part to aid, will nonetheless not reach the MDGs. Unrealistic targets thus may turn successes into perceptions of failure, serving to undermine future constituencies for aid (in donors) and reform (in recipients). This would be unfortunate given the vital role of aid and reform in the development process and the need for long-term, sustained aid commitments. Though goal-setting can be useful, these particular goals might be better viewed not as practical targets but instead as valuable reminders of the stark contrast between the world we have and the world we want, and as a call to redouble our search for interventions to close the gap more rapidly.Millennium Development Goals (MDGs), development assistance