Sharif Digital Repository

Invariant manifolds are important objects in the theory of dynamical systems. The stable manifold theorem is a very important theorem about this concept which proves the existence of stable and unstable manifolds in a wide range of dynamical systems. The importance of invariant manifolds encourages us to view their pictures. It helps us to understand their bahavior. For this purpose, at first we need to approximate the invariant manifold we want to visualize. There are several algorithms designed to approximate invariant manifolds. Those algorithms approximate a set of points on an invariant manifold and then provide an approximation of the manifold by the calculated points. Visualizing an... 

Brain-Computer Interface (BCI) presents a way for brain’s direct connection with external world. BCI system is composed of three parts: 1) Signal acquisition, 2) Signal processing and 3) External device control. The main part of this system is signal processing which includes three subparts: 1) Feature extraction, 2) Dimension reduction and 3) Signal separation and classification. In this thesis, we focus on the signal processing section in BCI systems. One of the most successful works done in signal processing is the use of covariance matrices in feature extraction from brain signals. Since covariance matrices are positive semi-definite and symmetric, they belong to certain manifolds called... 

Einstein's general theory of relativity is an admirable successful and unifier geometrical modeling among concepts of space, time and gravity. This theory along with special theory of relativity made a massive change in our physical view and this, especially in its historical context, has deep and interesting consequences in man's philosophical viewpoint which can be studied. Some parts of this thesis relates to these consequences. Some of these interesting consequences may be hidden by computional or mathematical viewpoint and often these equations do not contain enough intuition. in one part, we provide a formulizatin of Einstein's equaion that is intuitional which can be translated in... 

Morse 2-functions are higher-dimensional analogs to morse functions. A morse 2-function is a mapping to 2-dimensional disk with its critical points satisfying certain generality conditions. The goal of defining morse 2-functions and trisections is to generalize methods of 3-dimensional manifolds to dimension 4. A morse function on a 3-dimensional manifold leads to a decomposition of that manifold to two manifolds with boundary with common boundary. This decomposition is called Heegaard splitting. This decomposition is unique up to stabilization. Trisections are 4-dimensional analogs of Heegaard splittings and decompose manifold into three parts. The intersection of each pair is a solid genus... 

Traditional methods for data classification only make use of the labeled data. However, in most of the applications, labeling the unlabeled data is expensive, time consuming and requires expert knowledge. To overcome these problems, Semi-supervised Learning (SSL) methods have become an area of recent research that aim to effectively addressing the problem of limited labeled data.One of the recently introduced SSL methods is the classification based on geometric structure of the data, namely the data manifold. In this approach unlabeled data is utilized to recover the underlying structure of the data. The common assumption is that despite of being represented in a high dimensional space, data... 

The human heart is a complex system that contains many clues about its function in its electrocardiogram (ECG) signal. Due to the high mortality rate of heart diseases, detection and recognition of ECG arrhythmias is essential. The most difficult problem faced by ECG analysis is the vast variations among morphologies of ECG signals. In this study, we propose an approach for y detection of abnormal beats and data visualization with respect to ECG morphologies by using manifold learning. In order to do so, a nonlinear dimensionality reduction method based on the Laplacian Eigenmaps is used to reduce the high dimensions of the ECG signals, followed by the application of Bayesian and FLDA method... 

We consider a manifold M equipped with 1-forms eta(1)...eta(s) which satisfy certain contact like properties. We prove a generalization of the classical Darboux theorem for such manifolds  

We introduce a refinement of the Ozsváth-Szabó complex associated by Juhász [8] to a balanced sutured manifold (X; ). An algebra A is associated to the boundary of a sutured manifold. For a fixed class s of a Spinc structure over the manifold X, which is obtained from X by filling out the sutures, the Ozsváth-Szabó chain complex CF(X; ; s) is then defined as a chain complex with coefficients in A and filtered by the relative Spinc classes in Spinc(X; ). The filtered chain homotopy type of this chain complex is an invariant of (X; ) and the Spinc class s 2 Spinc(X). The construction generalizes the construction of Juhász. It plays the role of CF (X; s) when X is a closed three-manifold,...