Selman Uğuz

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Papers by Selman Uğuz

A generalization of warped product manifolds with Spin (7) holonomy

by Selman Uğuz and Ayse Humeyra Bilge

A Generalization of Warped Product Manifolds with Spin(7) Holonomy. [AIP Conference Proceedings 1... more

Spin C Structures on 8-manifolds

Special Warped-Like Product Manifolds with (Weak) G 2 Holonomy

Ukrainian Mathematical Journal, 2014

ABSTRACT By using the fiber-base decompositions of manifolds, the definition of warped-like produ... more ABSTRACT By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form M = F × B; where the base B is a onedimensional Riemannian manifold and the fiber F has the form F = F 1 × F 2 where F i ; i = 1, 2, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then they are isometric to S 3 with constant curvature k &gt; 0 in the class of special warped-like product metrics admitting the (weak) G 2 holonomy determined by the fundamental 3-form.

Conformally parallel Spin(7) structures on solvmanifolds

TURKISH JOURNAL OF MATHEMATICS, 2014

ABSTRACT In this paper we review the Spin(7) geometry in relation to solvmanifolds. Starting from... more ABSTRACT In this paper we review the Spin(7) geometry in relation to solvmanifolds. Starting from a 7-dimensional nilpotent Lie group N endowed with an invariant G(2) structure, we present an example of a homogeneous conformally parallel Spin(7) metric on an associated solvmanifold. It is thought that this paper could lead to very interesting and exciting areas of research and new results in the direction of (locally conformally) parallel Spin(7) structures.

(3+3+2) warped-like product manifolds with holonomy

Journal of Geometry and Physics, 2011

3 + 3 + 2) warped-like product a b s t r a c t

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Lee form and special warped-like product manifolds with locally conformally parallel Spin(7) structure

Annals of Global Analysis and Geometry, 2013

ABSTRACT We study the cases of the Lee form on special warped-like product manifolds M with local... more ABSTRACT We study the cases of the Lee form on special warped-like product manifolds M with locally conformally parallel Spin(7) structure to determine the nature of the fibers. Using fiber-base decomposition, we prove that the connection on M is determined by the Bonan form and Lee one-form. Assuming that the fibers are complete, connected and simply connected, and choosing some classes of Lee form on M, we prove a main result that the fibers (or at least one of them) are isometric to S 3 with constant curvature k &gt; 0 in the class of (3 + 3 + 2) warped-like product metrics admitting a specific locally conformally parallel Spin(7) structure. We believe that the paper could help in producing new examples of (locally conformally) parallel Spin(7) structures.

Construction of Canonical Bases for Real Representations of Clifford Algebras

ABSTRACT

CONSTRCUTION OF WARPED-LIKE PRODUCT MANIFOLDS WITH SPIN (7) HOLONOMY

E. Cartan gave the holonomy group definition and the classification of irreducible, simply-connec... more E. Cartan gave the holonomy group definition and the classification of irreducible, simply-connected and symmetric manifolds in 1925. In 1955, Marcel Berger gave the holonomy group classification list of irreducible, simplyconnected and non-symmetric manifolds [6]. Berger's holonomy group list called the Berger list is short and includes two exceptional cases, G 2 holonomy on a 7-dimensional manifold and Spin(7) holonomy on an 8-manifold. These holonomy groups are called the exceptional holonomy groups and manifolds with these holonomy groups are called the exceptional holonomy manifolds. When M. Berger gave the possible holonomy groups list, he did not know whether G 2 and Spin(7) holonomy groups manifolds exist or not. In 1987, the existence question of exceptional holonomy manifolds was answered [1]. R. Bryant and S. Salomon gave the first complete example of Spin(7) holonomy manifold which was the total space of the spin bundle over the 4-sphere [2]. Dominic Joyce gave the first compact examples of exceptional holonomy manifolds in 1996 and wrote the methods of construction of manifolds as a book in [3]. He constructed manifold examples by resolving the singularities of the orbifold T 8 /Γ for some groups Γ. *

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A generalization of warped product manifolds with Spin (7) holonomy

A Generalization of Warped Product Manifolds with Spin(7) Holonomy. [AIP Conference Proceedings 1... more

Canonical bases for real representations of Clifford algebras

by Selman Uğuz and Ayse Humeyra Bilge

The well-known classification of the Clifford algebras Cl(r, s) leads to canonical forms of compl... more The well-known classification of the Clifford algebras Cl(r, s) leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For s 1 representations of Cl(r, s) on R 2N are obtained from representations on R N by adding two new generators while in passing from a representation of Cl(p, 0) on R N to a representation of Cl(r, 0) on R 2N the number of generators that can be added is either 1, 2 or 4, according as the Clifford algebra represented on R N is of real, complex or quaternionic type. We have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of Cl(r, s) to a canonical form. Our algorithm for the transformation of the representations of Cl(8d + c, 0), c 7 to canonical forms is based on finding an abelian subalgebra of Cl(8d + c, 0) and its invariant subspace. Computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions. The construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics.

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Edge detection with fuzzy cellular automata transition function optimized by PSO

Computers & Electrical Engineering

In this paper we discuss the application of two-dimensional linear cellular automata (CA) rules w... more In this paper we discuss the application of two-dimensional linear cellular automata (CA) rules with the help of fuzzy heuristic membership function to the problems of edge detection in image processing applications. We proposed an efficient and simple thresholding technique of edge detection based on fuzzy cellular automata transition rules optimized by Particle Swarm Optimization method (PSO). Finally, we present some results of the proposed linear rules for edge detection to the selected 22 images from the Berkeley Segmentation Dataset (BSDS) and compare with some classical Sobel and Canny results. Also, Baddeley Delta Metric (BDM) is used for the performance index to compare the results.

Fiber structures of special (4 + 3 + 1) warped-like manifolds with Spin(7) holonomy

by İbrahim Ünal and Selman Uğuz

International Journal of Geometric Methods in Modern Physics, 2014

ABSTRACT A generalization of 8-dimensional multiply-warped product manifolds is considered as a s... more ABSTRACT A generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uğuz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys.61 (2011) 1093-1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F × B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F1 × F2 where Fi&#39;s (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the Fi&#39;s are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S3 with constant curvature k &gt; 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied.

Multi Agent Biorobotic Laboratory Rochester Institute of Technology Rochester, NY, 14623 USA

The Transition Rules of 2D Linear Cellular Automata Over Ternary Field and Self-Replicating Patterns

International Journal of Bifurcation and Chaos, 2015

ABSTRACT In this paper we start with two-dimensional (2D) linear cellular automata (CA) in relati... more ABSTRACT In this paper we start with two-dimensional (2D) linear cellular automata (CA) in relation with basic mathematical structure. We investigate uniform linear 2D CA over ternary field, i.e. ℤ3. Present work is related to theoretical and imaginary investigations of 2D linear CA. Even though the basic construction of a CA is a discrete model, its macroscopic level behavior at large times and on large scales could be a close approximation to a continuous system. Considering some statistical properties, someone may also study geometrical aspects of patterns generated by cellular automaton evolution. After iteratively applying the linear rules, CA have been shown capable of producing interesting complex behaviors. Some examples of CA produce remarkably regular behavior on finite configurations. Using some simple initial configurations, the produced pattern can be self-replicating regarding some linear rules. Here we deal with the theory 2D uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) over the ternary field ℤ3 and the applications of image processing for patterns generation. From the visual appearance of the patterns, it is seen that some rules display sensitive dependence on boundary conditions and their rule numbers.

A Renormalization-Group Study of the Potts Model with Competing Ternary and Binary Interactions

LYAPUNOV EXPONENTS AND MODULATED PHASES OF AN ISING MODEL ON CAYLEY TREE OF ARBITRARY ORDER

International Journal of Modern Physics C, 2012

Behaviors of Phase Diagrams of an Ising Model on a Cayley Tree-Like Lattice: Rectangular Chandelier

In this work, we study an Ising model on a new lattice type which we called Rectangular Chandelie... more In this work, we study an Ising model on a new lattice type which we called Rectangular Chandelier, with competing nearest-neighbor interactions J1, prolonged ternary interactions Jt and one-level next-nearest-neighbor quinary interactions Jl1(5). We obtain the phase diagrams of the Ising model related to Hamiltonian system give above on a Rectangular Chandelier. The phase diagrams are presented in the Hamiltonian

Modulated Phase of an Ising System with Competing Interactions on a Cayley Tree of Arbitrary Order

We study the phase diagram for Ising model on a Cayley tree of arbitrary order k with competing n... more We study the phase diagram for Ising model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J1 and prolonged next-nearest-neighbor interactions Jp. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Vannimenus for k = 2.

Three-state von Neumann cellular automata and pattern generation

Applied Mathematical Modelling, 2014

ABSTRACT We study theoretical structure and classification of two-dimensional (2D) 3-states unifo... more ABSTRACT We study theoretical structure and classification of two-dimensional (2D) 3-states uniform cellular automata (CA) based on their visual behaviors. Although the basics of a CA is a discrete dynamic structure and modelled locally, the behavior at large times and spatial scales could be a close to a continuous system. Using some basic properties, it can be considered geometrical structures of patterns produced by cellular automaton iteration. After iteratively applying the rules, it is shown that CA are capable of producing complex behaviors. Some examples of CA show remarkably regular behavior on finite configurations. It is observed that with simple initial configurations, the generated pattern might be self replicating (SR), self similar (SS) or mixed type. Here we deal with the theory of 2D uniform, periodic boundary, adiabatic boundary and reflexive boundary CA (PB, AB and RB) of von Neumann neighborhood and applications of image analysis for patterns generation. We investigate a 2D CA under these boundary restrictions over 3-states cases, i.e. the ternary field image. We also study the applications of SR and SS patterns which correspond to the linear CA rules of 2D uniform CA with different boundary cases over image. The von Neumann neighborhood CA rule (i.e. rule 2460) is classified into SR and SS types depending upon the non-zero boundary values image of neighboring cells that influence the cells under consideration. It is also shown that, from the visual appearance of the patterns, sometimes the rule 2460 displays sensitive dependence on boundary conditions and chaotic behaviors. Finally we conclude the paper by analyzing some results about cellular automata defined by the rule number 2460NB, 2460PB, 2460AB and 2460RB for non-symmetric figures in detail.

Uniform Cellular Automata Linear Rules for Edge Detection

2013 IEEE International Conference on Systems, Man, and Cybernetics, 2013

ABSTRACT In this paper we discuss the application of two-dimensional linear cellular automata rul... more ABSTRACT In this paper we discuss the application of two-dimensional linear cellular automata rules to the problems of edge detection in monochromatic images. We proposed an efficient and simple method of edge detection based on uniform cellular automata transition matrix representation. We investigate of cellular automata linear rules for edge detection by using matrix representation, in some cases they are strong and some other rules are not in fact useful for practical edge detection. All rules give computationally effective result since the computation mechanism consist only matrix multiplication. Finally, we present some results of the proposed linear rules for edge detection and compare with some classical results.

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