Symmetry, Hamiltonian Problems and Wavelets in Accelerator Physics

In this paper we consider applications of methods from wavelet analysis to nonlinear dynamical problems related to accelerator physics. In our approach we take into account underlying algebraical, geometrical and topological structures of corresponding problems.Comment: LaTeX2e, aipproc.sty, 25 pages, typos correcte

Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis

The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.Comment: 252 pages, no figures. Originally published as a book by North-Holland, 1990. Reviewed by Robert Hermann in Bulletin of the AMS Vol. 28 #1, January 1993, pp. 130-132; see http://wavelets.co

Microlocal and Time-Frequency Analysis

The present volume collects the contributions selected for publication in the Special Issue entitled "Microlocal and Time-Frequency Analysis" of the journal Mathematics, edited by Elena Cordero and S. Ivan Trapasso over 2020 and 2021

Constant & time-varying hedge ratio for FBMKLCI stock index futures

This paper examines hedging strategy in stock index futures for Kuala Lumpur Composite Index futures of Malaysia. We employed two different econometric methods such as-vector error correction model (VECM) and bivariate generalized autoregressive conditional heteroskedasticity (BGARCH) models to estimate optimal hedge ratio by using daily data of KLCI index and KLCI futures for the period from January 2012 to June 2016 amounting to a total of 1107 observations. We found that VECM model provides better results with respect to estimating hedge ratio for spot month futures and one-month futures, while BGACH shows better for distance futures. While VECM estimates time invariant hedge ratio, the BGARCH shows that hedge ratio changes over time. As such, hedger should rebalance his/her position in futures contract time to time in order to reduce risk exposure

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Continuous Shearlet Frames and Resolution of the Wavefront Set

In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets - able to efficiently handle data with features along edges. The main result in [G. Kutyniok, D. Labate. Resolution of the Wavefront Set using continuous Shearlets, Trans. AMS 361 (2009), 2719-2754] confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions $\psi$ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution $f$ with respect to the shearlet $\psi$ can resolve the Wavefront Set of $f$. We demonstrate that the same result can be verified under much weaker assumptions on $\psi$, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for $L^2(\mathbb{R}^2)$ from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure

Schnelle Löser für partielle Differentialgleichungen

[no abstract available

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao