On the classification of integrable differential/difference equations in three dimensions

Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes them a very active research area. In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed over the last few years. This approach, is essentially a perturbation technique based on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations and scalar higher-order quasilinear PDEs. As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important, since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition of integrability, based on hydrodynamic reductions, is used throughout, which is a natural analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are given for all classification results obtained

Higher-Order Regularization in Computer Vision

At the core of many computer vision models lies the minimization of an objective function consisting of a sum of functions with few arguments. The order of the objective function is defined as the highest number of arguments of any summand. To reduce ambiguity and noise in the solution, regularization terms are included into the objective function, enforcing different properties of the solution. The most commonly used regularization is penalization of boundary length, which requires a second-order objective function. Most of this thesis is devoted to introducing higher-order regularization terms and presenting efficient minimization schemes. One of the topics of the thesis covers a reformulation of a large class of discrete functions into an equivalent form. The reformulation is shown, both in theory and practical experiments, to be advantageous for higher-order regularization models based on curvature and second-order derivatives. Another topic is the parametric max-flow problem. An analysis is given, showing its inherent limitations for large-scale problems which are common in computer vision. The thesis also introduces a segmentation approach for finding thin and elongated structures in 3D volumes. Using a line-graph formulation, it is shown how to efficiently regularize with respect to higher-order differential geometric properties such as curvature and torsion. Furthermore, an efficient optimization approach for a multi-region model is presented which, in addition to standard regularization, is able to enforce geometric constraints such as inclusion or exclusion of different regions. The final part of the thesis deals with dense stereo estimation. A new regularization model is introduced, penalizing the second-order derivatives of a depth or disparity map. Compared to previous second-order approaches to dense stereo estimation, the new regularization model is shown to be more easily optimized

Микола Федорович фон-Дітмар – редактор часопису "Записки Харьковского отделения Императорского Российского Технического Общества" (1907–1908 рр.) та голова Харківського відділення Російського технічного товариства (1909–1912 рр.)

Classification systems in general and the DSM-IV in particular have a important significance in psychiatric diagnostic.In my doctorial thesis, I was discussing questions of the structure of the DSM-IV.To discuss those questions, I ordered the diagnosis criteria of the diseases anxiety disorders and somatoform disorders of the DSM-IV after the principals of the Information-gain- theory. Afterwards I compared the new order with the order given by the DSM-IV. A patient-based evaluation was not performed.The new, information-gain-based, order was possible with an assistant system, written by Dominik Lübbers in 2001. Its primary goal is to help the physician with differential diagnosis.It was possible to establish a database with all diagnosis and to transform them into the assistant system. With the assistant system, it was possible to calculate the information-gain data.The result of the calculation was, that the information-gain-based order is only in few diagnosis more useful, most of the time, the order in the DSM-IV is more useful. In most cases, where the order in the DSM-IV was more useful, an criterion for exclusion had a higher level in the information-gain-calculation.But there were also situations, where the information-gain-based order was more useful . In those cases, the diagnosis criteria in the DSM-IV were only a listing without structur

Four Point High Order Compact Iterative Schemes For The Solution Of The Helmholtz Equation

Teknik-teknik yang lebih baik diperoleh daripada beza terhingga dalam grid piawai dan grid putaran telah dibangunkan sejak beberapa tahun kebelakangan ini dalam menyelesaikan sistem linear yang terhasil daripada pendiskretan persamaan pembezaan separa (PDEs). Selain itu, satu sistem dengan peringkat kejituan yang lebih tinggi boleh dihasilkan daripada pendiskretan skema beza terhingga dengan menggunakan satu skim padat dengan kejituan peringkat empat yang dihasilkan daripada beza memusat dengan kejituan peringkat kedua. Dengan menggunakan beza terhingga padat ini, satu skim titik putaran dengan kejituan peringkat empat bagi persamaan Helmholtz dua dimensi (2D) yang baru terbentuk. Skim peringkat empat dalam grid piawai dan grid putaran boleh dikembangkan menjadi skim kumpulan ataupun sistem yang berperingkat empat. Sehubungan itu, kaedah multigrid berskala-multi digabungkan dengan ekstrapolasi Richardson diperkenalkan oleh Zhang [18] untuk menyelesaikan persamaan Poisson 2D. Improved techniques derived from the standard and rotated finite difference operators have been developed over the last few years in solving linear systems that arise from the discretization of various partial differential equations (PDEs) [14]. Furthermore, a higher order system can be generated from discretization of the finite difference scheme by using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, new standard and rotated point schemes with fourth order accuracy for the two-dimensional (2D) Helmholtz equation are formulated in this thesis. The fourth order point schemes in both standard and rotated grids can be further applied to formulate a fourth order system to be used as group iterative method in their respective grid. On the other hand, the multiscale multigrid method combined with Richardson’s extrapolation is first introduced by Zhang [18] to solve the 2D Poisson equation

Gravitational instabilities and faster evolving density perturbations

The evolution of inhomogeneities in a spherical collapse model is studied by expanding the Einstein equation in powers of inverse radial parameter. In the linear regime, the density contrast is obtained for flat, closed and open universes. In addition to the usual modes, an infinite number of new growing modes are contained in the solutions for pressureless open and closed universes. In the nonlinear regime, we obtain the leading growing modes in closed forms for a flat universe and also, in the limits of small and large times, for an open universe.Comment: latex, 17 pages; electronic address for programs correcte