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#### An introductory course on differentiable manifolds

, Book
Dover Publications, Inc
2016

Abstract

Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, elementary abstract algebra, and point set topology. More than 200 exercises offer students ample opportunity to gauge their skills and gain additional insights.

The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to...

The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to...

#### Einstein solvmanifolds and graphs

, Article Comptes Rendus Mathematique ; Volume 344, Issue 1 , 2007 , Pages 37-39 ; 1631073X (ISSN) ; Sharif University of Technology
2007

Abstract

In this Note, we obtain Einstein solvmanifolds using Abelian extension of two-step nilpotent Lie algebras associated with graphs. To cite this article: H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2006 Académie des sciences

#### The curious neglect of geometry in modern philosophies of mathematics

, Article Logic, Epistemology, and the Unity of Science ; Volume 49 , 2021 , Pages 379-389 ; 22149775 (ISSN) ; Sharif University of Technology
Springer Science and Business Media B.V
2021

Abstract

From ancient times to 19th century geometry symbolized the essence of mathematical thinking and method, but modern philosophy of mathematics seems to have marginalized the philosophical status of geometry. The roots of this transformation will be sought in the ascendance of logical foundations in place of intuitive primacy as the cornerstone of mathematical certainty in the late 19th century. Nevertheless, geometry and geometrical thinking, in multiple manifestations, have continued to occupy a central place in the practice of mathematics proper. We argue that this, together with advances in the neuroscience of mathematical processes, calls for an expansion of the present limited remit of...

#### Numerical Methods for Approximation and Visualization of Invariant Manifolds in Dynamical Systems

, M.Sc. Thesis Sharif University of Technology ; Razvan, Mohammad Reza (Supervisor)
Abstract

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...

####
Classification of Different Mental Activities Based on Riemannian Geometry

,
M.Sc. Thesis
Sharif University of Technology
;
Babaiezadeh, Massoud
(Supervisor)
Abstract

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...

#### Special Lagrangian sub-manifolds and super-symmetry

, Article International Journal of Geometric Methods in Modern Physics ; Volume 10, Issue 7 , 2013 ; 02198878 (ISSN) ; Sharif University of Technology
2013

Abstract

Special Lagrangian sub-manifolds of Calabi-Yau (CY) 3-folds are used to describe membrane instanton solutions of N = 1, 11-dimensional super-gravity theories. Super-symmetry is the essential ingredient that relates super-gravity branes to special Lagrangian sub-manifolds [K. Becker, M. Becker and A. Strominger, Fivebranes, membranes and non-perturbative string theory, Nucl. Phys. B 456(1-2) (1995) 135-152]. In this note we would like to explain this relation, more explicitly and with more details than in the current literature

#### On the deformation theory of Calabi-Yau structures in strongly pseudo-convex manifolds

, Article Bulletin of the Brazilian Mathematical Society ; Volume 41, Issue 3 , September , 2010 , Pages 409-420 ; 16787544 (ISSN) ; Sharif University of Technology
2010

Abstract

We study the deformation theory of Calabi-Yau structures in strongly pseudo-convex manifolds with trivial canonical bundles. Our approach could be considered as an alternative proof for a theorem of H. Laufer on the deformation of strongly pseudo-convex surfaces

#### Periodic solutions for a discrete time predator-prey system with monotone functional responses

, Article Comptes Rendus Mathematique ; Volume 345, Issue 4 , 2007 , Pages 199-202 ; 1631073X (ISSN) ; Hesaaraki, M ; Sharif University of Technology
2007

Abstract

In this Note, sharp sufficient conditions for the existence of periodic solutions of a nonautonomous discrete time semi-ratio-dependent predator-prey system with functional responses are derived. In our results this system with any monotone functional response bounded by polynomials in R+, always has at least one ω-periodic solution. In particular, this system with the most popular functional responses Michaelis-Menten, Holling type-II and III, sigmoidal, Ivlev and some other monotone response functions, always has at least one ω-periodic solution. To cite this article: M. Fazly, M. Hesaaraki, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences

#### Comparison of volumes of Riemannian manifolds

, Article Comptes Rendus Mathematique ; Volume 339, Issue 3 , 2004 , Pages 199-201 ; 1631073X (ISSN) ; Sharif University of Technology
2004

Abstract

Using the rigidity result of Besson, Courtois and Gallot, and also the notion of intersection of metrics, we compare volumes of Riemannian manifolds by means of lengths of their periodic geodesics. © 2004 Académie des sciences. Publié par Elsevier SAS. Tous droits réservés

#### About Space Time

, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract

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...

#### Cerf Theory and Trisections of Four Manifolds

, M.Sc. Thesis Sharif University of Technology ; Bahraini, Alireza (Supervisor)
Abstract

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...

#### From local similarity to global coding: An application to image classification

, Article Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Portland, OR ; 2013 , Pages 2794-2801 ; 10636919 (ISSN) ; Rabiee, H. R ; Farajtabar, M ; Ghazvininejad, M ; Sharif University of Technology
2013

Abstract

Bag of words models for feature extraction have demonstrated top-notch performance in image classification. These representations are usually accompanied by a coding method. Recently, methods that code a descriptor giving regard to its nearby bases have proved efficacious. These methods take into account the nonlinear structure of descriptors, since local similarities are a good approximation of global similarities. However, they confine their usage of the global similarities to nearby bases. In this paper, we propose a coding scheme that brings into focus the manifold structure of descriptors, and devise a method to compute the global similarities of descriptors to the bases. Given a local...

#### Semi-supervised Learning and its Application to Image Categorization

, M.Sc. Thesis Sharif University of Technology ; Rabiee, Hamid Reza (Supervisor)
Abstract

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...

#### Using Manifold Learning for ECG Processing

, M.Sc. Thesis Sharif University of Technology ; Jahed, Mehran (Supervisor) ; Hossein Khalaj, Babak (Supervisor)
Abstract

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...

#### A Darboux Theorem for Generalized Contact Manifolds

, M.Sc. Thesis Sharif University of Technology ; Fanaie, Hamid Reza (Supervisor)
Abstract

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

#### Heegaard Floer Homology and the Topology of Three Manifolds

, Ph.D. Dissertation Sharif University of Technology ; Bahraini, Alireza (Supervisor) ; Eftekhary, Eaman (Supervisor)
Abstract

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,...

#### Monocular 3D Human Pose Estimation with a Semi-supervised Graph-Based Method

, Article 2015 International Conference on 3D Vision, 3DV 2015, 19 October 2015 through 22 October 2015 ; October , 2015 , Pages 518-526 ; 9781467383325 (ISBN) ; Rabiee, H. R ; Gagne, C ; Brown M ; Kosecka J ; Theobalt C ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2015

Abstract

In this paper, a semi-supervised graph-based method for estimating 3D body pose from a sequence of silhouettes, is presented. The performance of graph-based methods is highly dependent on the quality of the constructed graph. In the case of the human pose estimation problem, the missing depth information from silhouettes intensifies the occurrence of shortcut edges within the graph. To identify and remove these shortcut edges, we measure the similarity of each pair of connected vertices through the use of sliding temporal windows. Furthermore, by exploiting the relationships between labeled and unlabeled data, the proposed method can estimate the 3D body poses, with a small set of labeled...

#### Efficient multi-modal fusion on supergraph for scalable image annotation

, Article Pattern Recognition ; Volume 48, Issue 7 , July , 2015 , Pages 2241-2253 ; 00313203 (ISSN) ; Jamzad, M ; Sharif University of Technology
Elsevier Ltd
2015

Abstract

Different types of visual features provide multi-modal representation for images in the annotation task. Conventional graph-based image annotation methods integrate various features into a single descriptor and consider one node for each descriptor on the learning graph. However, this graph does not capture the information of individual features, making it unsuitable for propagating the labels of annotated images. In this paper, we address this issue by proposing an approach for fusing the visual features such that a specific subgraph is constructed for each visual modality and then subgraphs are connected to form a supergraph. As the size of supergraph grows linearly with the number of...

#### Generalized rademacher-stepanov type theorem and applications

, Article Zeitschrift fur Analysis und ihre Anwendung ; Volume 28, Issue 3 , 2009 , Pages 249-275 ; 02322064 (ISSN) ; Sharif University of Technology
2009

Abstract

The main purpose of this article is to generalize a theorem of Stepanov which provides a necessary and sufficient condition for almost everywhere differentiability of functions over Euclidean spaces. We state and prove an Lp-type generalization of the Stepanov theorem and then we extend it to the context of Orlicz spaces. Then, this generalized Rademacher-Stepanov type theorem is applied to the Sobolev and bounded variation maps with values into a metric space. It is shown that several generalized differentiability type theorems are valid for the Sobolev maps from a Lipschitz manifold into a metric space. As a byproduct, it is shown that the Sobolev spaces of Korevaar-Schoen and Reshetnyak...

#### On harmonic maps from stochastically complete manifolds

, Article Archiv der Mathematik ; Volume 92, Issue 6 , 2009 , Pages 637-644 ; 0003889X (ISSN) ; Sharif University of Technology
2009

Abstract

The main purpose of this article is to generalize a theorem about the size of minimal submanifolds in Euclidean spaces. In fact, we state and prove a non-existence theorem about harmonic maps from a stochastically complete manifold into a cone type domain. The proof is based on a generalized version of the maximum principle applied to the Lapalace-Beltrami operator on Riemannian manifolds. © 2009 Birkhäuser Verlag Basel/Switzerland