paper presentation on topology

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Introduction to topology, course description.

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Mathematics > Dynamical Systems

Title: dynamics, cohomology and topology.

Abstract: For a smooth Morse-Smale vector field with Lyapunov constraints (Lyapunov function) one shows how and why the non-triviality of the additive structure of the cohomology detects rest points and the multiplicative structure of the cohomology detects instantons. The same remains true for Lyapunov closed one form but in this presentation this fact is discussed only informally. These observations are based on the smooth " manifold with corner structures" of the stable/unstable sets and of the set of trajectories of such vector fields. (This paper is a written version of two talks with the same title given at IMAR Bucharest in November 2023)

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What is Topology?

Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from a figure 8.

Here are some examples of typical questions in topology: How many holes are there in an object? How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point? Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900. The following are some of the subfields of topology.

• General Topology or Point Set Topology.  General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.
• Combinatorial Topology.  Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have an Euler characteristic of 2.
• Algebraic Topology.  Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.

Algebraic topology sometimes uses the combinatorial structure of a space to calculate the various groups associated to that space.

• Differential Topology.  Differential topology considers spaces with some kind of smoothness associated to each point. In this case, the square and the circle would not be smoothly (or differentiably) equivalent to each other. Differential topology is useful for studying properties of vector fields, such as a magnetic or electric fields.

Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics, and for describing the space-time structure of universe.

Computer Network Topologies

Jul 13, 2014

850 likes | 2.69k Views

Computer Network Topologies. Maninder Kaur [email protected]. What is a Topology?. Network topologies describe the ways in which the elements of a network are mapped. They describe the physical and logical arrangement of the network nodes.

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Computer Network Topologies Maninder Kaur [email protected]

What is a Topology? • Network topologies describe the ways in which the elements of a network are mapped. They describe the physical and logical arrangement of the network nodes. • The physical topology of a network refers to the configuration of cables, computers, and other peripherals

Different Types of Topologies • Bus Topology • Star Topology • Ring Topology • Mesh Topology • Tree Topology • Hybrid Topology

Bus Topology • All the nodes (file server, workstations, and peripherals) on a bus topology are connected by one single cable. • A bus topology consists of a main run of cable with a terminator at each end. All nodes (file server, workstations, and peripherals) are connected to the linear cable. • Popular on LANs because they are inexpensive and easy to install.

Bus Topology

Bus Topology Advantages of Bus Topology • It is Cheap, easy to handle and implement. • Require less cable • It is best suited for small networks. Disadvantages of Bus Topology • The cable length is limited. This limits the number of stations that can be connected. • This network topology can perform well only for a limited number of nodes.

Ring Topology • In a ring network, every device has exactly two neighbours for communication purposes. • All messages travel through a ring in the same direction. • A failure in any cable or device breaks the loop and can take down the entire network. • To implement a ring network we use the Token Ring technology • A token, or small data packet, is continuously passed around the network. When a device needs to transmit, it reserves the token for the next trip around, then attaches its data packet to it.

Ring Topology

Ring Topology Advantage of Ring Topology • Very orderly network where every device has access to the token and the opportunity to transmit. • Easier to Mange than a Bus Network • Good Communication over long distances • Handles high volume of traffic Disadvantages of Ring Topology • The failure of a single node of the network can cause the entire network to fail. • The movement or changes made to network nodes affects the performance of the entire network.

Star Topology • In a star network, each node (file server, workstations, and peripherals) is connected to a central device called a hub. • The hub takes a signal that comes from any node and passes it along to all the other nodes in the network. • Data on a star network passes through the hub, switch, or concentrator before continuing to its destination. • The hub, switch, or concentrator manages and controls all functions of the network. • The star topology reduces the chance of network failure by connecting all of the systems to a central node.

Star Topology

Star Topology Advantages of Star Topology • Easy to manage • Easy to locate problems (cable/workstations) • Easier to expand than a bus or ring topology. • Easy to install and wire. • Easy to detect faults and to remove parts. Disadvantages of Star Topology • Requires more cable length than a linear topology. • If the hub or concentrator fails, nodes attached are disabled. • More expensive because of the cost of the concentrators.

Tree Topology • A tree topology (hierarchical topology) can be viewed as a collection of star networks arranged in a hierarchy. • This tree has individual peripheral nodes which are required to transmit to and receive from one other only and are not required to act as repeaters or regenerators. • The tree topology arranges links and nodes into distinct hierarchies in order to allow greater control and easier troubleshooting. • This is particularly helpful for colleges, universities and schools so that each of the connect to the big network in some way.

Tree Topology

Tree Topology Advantages of a Tree Topology • Point-to-point wiring for individual segments. • Supported by several hardware and software vendors. • All the computers have access to the larger and their immediate networks. Disadvantages of a Tree Topology • Overall length of each segment is limited by the type of cabling used. • If the backbone line breaks, the entire segment goes down. • More difficult to configure and wire than other topologies.

Mesh Topology • In this topology, each node is connected to every other node in the network. • Implementing the mesh topology is expensive and difficult. • In this type of network, each node may send message to destination through multiple paths. • While the data is travelling on the Mesh Network it is automatically configured to reach the destination by taking the shortest route which means the least number of hops.

Mesh Topology

Mesh Topology Advantage of Mesh Topology • No traffic problem as there are dedicated links. • It has multiple links, so if one route is blocked then other routes can be used for data communication. • Points to point links make fault identification easy. Disadvantage of Mesh Topology • There is mesh of wiring which can be difficult to manage. • Installation is complex as each node is connected to every node. • Cabling cost is high.

Hybrid Topology • A combination of any two or more network topologies. • A hybrid topology always accrues when two different basic network topologies are connected. • It is a mixture of above mentioned topologies. Usually, a central computer is attached with sub-controllers which in turn participate in a variety of topologies

Hybrid Topology

Hybrid Topology Advantages of a Hybrid Topology • It is extremely flexible. • It is very reliable. Disadvantages of a Hybrid Topology • Expensive

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Network Topologies

Network Topologies. Objectives. Describe the basic and hybrid LAN physical topologies, and their uses, advantages and disadvantages Describe the backbone structures that form the foundation for most LANs. Simple Physical Topologies. Physical topology: physical layout of nodes on a network

877 views • 27 slides

Network Topologies. Objectives. Describe the basic and hybrid LAN physical topologies, their uses, advantages, and disadvantages Describe a variety of enterprise-wide and WAN physical topologies, their uses, advantages, and disadvantages

1.17k views • 44 slides

Fundamentals. A LAN (Local Area Network) is a

724 views • 41 slides

Network Security Topologies

Network Security Topologies. Chapter 11. Learning Objectives. Identify place and role of the demilitarized zone NAT and PAT Tunneling in network security Describe security features of VLANS Network perimeter’s importance to an organization’s security policies. Perimeter Security Topologies.

748 views • 35 slides

Network topologies

Network topologies. What is a network topology?. Physical arrangement of the devices in a communications network. Most commonly used are bus and star. Usually a combination of these two, referred to as hybrid or tree networks. Bus topology.

702 views • 11 slides

NETWORK TOPOLOGIES

NETWORK TOPOLOGIES. WHAT IS A NETWORK TOPOLOGY?. The number in which nodes are linked in a computer network A pure form of these basic topologies is seldom found in practice. TYPES OF CLIENT. - Dumb terminals - Thin client - Fat client. BUS TOPOLOGY.

275 views • 6 slides

Network Topologies. Topology – how nodes are connected – where there is a wire between 2 nodes. Routing – the path a message takes to get from one node to another. Network connections can be static or dynamic (ex. Bus). Network Properties. Diameter – Maximum distance between nodes

396 views • 23 slides

Network Topologies. Objectives. Describe the basic and hybrid LAN physical topologies, and their uses, advantages and disadvantages. Describe the backbone structures that form the foundation for most LANs. Simple Physical Topologies. Physical topology: physical layout of nodes on a network

444 views • 20 slides

Network Topologies. A network topology describes ways of configuring or laying out a network. The main topologies: Star Bus Hybrid or tree network. Bus Topology.

628 views • 7 slides

Network Topologies. Network Topology. The topology of a network specifies the geometric arrangement of the network. The complete physical structure of the cable (or data-transmission media) , workstations, nodes, routers and gateways are called the physical topology.

884 views • 13 slides

NETWORK TOPOLOGIES. There are three basic configurations used to connect computers they are the Bus Ring Star. Bus topology. This type of network was widely used in the 1980’s In this configuration every computer (node) shares the networks total bus capacities.

541 views • 27 slides

3-19. Network Topologies. T. T. Free Topology . 3-20. Ring, star or combination layout of network media All except ring are polarity insensitive One terminator installed anywhere on the segment Typical for device channels May be difficult to troubleshoot bad devices

810 views • 44 slides

Network Topologies. William M. Jones Assistant Professor Computer Science Department Coastal Carolina University. Network Classifications. Networks have two broad classifications Static Dynamic Static Networks

805 views • 41 slides

Network Security Topologies. Chapter 11. Learning Objectives. Explain network perimeter’s importance to an organization’s security policies Identify place and role of the demilitarized zone in the network Explain how network address translation is used to help secure networks

496 views • 35 slides

Network Security Topologies. Jason Kennedy March 23, 2004. Network Security Topics . Perimeter Security Topologies Demilitarized zone (DMZ) Network Address Translation Tunneling Virtual Local Area Networks. Perimeter Security Topologies.

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Network Topologies. LAN topologies WAN topologies. LAN Topologies(Physical). Bus Star Ring. Bus topology. All networked nodes are interconnected, peer to peer, using a single, open-ended cable Both ends of the bus must be terminated with a terminating resistor to prevent signal bounce.

184 views • 12 slides

343 views • 27 slides

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Network Topology

—This document talks in brief about Network Topolo-gies, types of Network Topologies and their advantages and disadvantages over each other. This document also covers about factors which should be kept in mind while choosing the topology for your network and the impacts due to those factors.

Related Papers

Muhammad Attique Anwar

What are the different network of topologies and why we should use them?

International Journal for Research in Applied Science & Engineering Technology (IJRASET)

IJRASET Publication

Network topologies is various components of network link nodes, link, peripherals are arranged. The way of connecting the computers is called as the topology.so depending on the manner of connecting the computers we can have different network topologies. Network topology is links and nodes of a network are arranged to related are arranged to each other. They describe the physical and logical arrangement of network nodes. The way in which different system and node are connect and communicate with each other is determined by topology of the network.

Amro Moussa

International Journal of Research

Rishabh Sharma

When two or more devices are connected to each other through connecting links, then it is known as a network.

Zeynel Misirli

IRJET Journal

A network is the interconnection of two or more devices. The study of arrangement or mapping of elements (links, nodes) of a network is known as network topology. For communication distribution of computers has become very important issue which deliver end to end performance at a low cost, hence distribution system performance is influenced by the technology adopted by network interconnection so distribution of computers is done according to communication network arranged in a geometrical manner known as network topology. This paper provides an analytical study of different types of basic network topologies on the basis of their advantages, disadvantages and different factors which differentiate them.

The classification of network topologies according to how they are implemented is examined in depth from a variety of perspectives. Each type of topology is used to accomplish a certain purpose and capture the system's structure at varying levels of detail. Distributed computer systems have been an important and popular topic in computing in recent years. It provides highend performance at an affordable price .In a distributed computing environment, autonomous computers are connected via a communication network that is arranged in a geometrical shape known as network topology.

Santanu Santra

In recent days for computing, distributed computer systems have become very important and popular issue. It delivers high end performance at a low cost. Autonomous computers are connected by means of a communication network in a distributed computing environment which is arranged in a geometrical shape called network topology. In the present paper a detailed study and analysis on network topologies is presented. Definitions of Physical and Logical Topologies are also provided.

International Journal of Advance Research in Computer Science and Management Studies [IJARCSMS] ijarcsms.com

Jules Degila

— This paper surveys important parameters for the design of large scale networks topology such as the YottaWeb topology [1][2][3]. First, a wide range of performance measures to evaluate the behavior of envisaged topologies are presented, discussed and classified according to their meaning and their effectiveness on large scale networks. Secondly, different types of topologies, from simple to more complex, are identified and the features of the called k−ary n−cube topologies [4] such as ring, torus and hy-percube topologies are surveyed and discussed. Full details and advantages of the recently introduced YottaWeb topology are pointed out, in the light of the predefined concepts. Finally, the application of the performance measure to the design of the topologies is surveyed.

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Highly tensile and sensitive strain sensors with micro-nano topology optimization

(note: the full text of this document is currently only available in the pdf version ).

Weixia Lan , Qiqi Ding , Tao Zhou , Zilong Guo , Wenbin Sun , Zhenghui Wu , Yingjie Liao , Bin Wei and Yuanyuan Liu

First published on 23rd August 2024

With the extensive application of flexible sensors in various wearable electronics being continuously explored, researchers are paying more and more attention to improving their sensitivity while ensuring high stretchability. In this study, a novel fiber strain sensor was proposed with micro-nano topology optimization, which was achieved through a simple, cost-effective and scalable method. The TPU/PEO substrate was prepared by electrospinning technology, then the fibrous membrane was subjected to deionized water to wash away the PEO and obtain the micro-nano topological structure. Carbon nanotubes (CNTs) and graphene were further adsorbed on the etched TPU fibrous membrane through ultrasonic treatment to get the TPU:PEO/CNTs and TPU:PEO/Graphene sensors, respectively. Both experimental and simulation results show that the optimization of the PEO ratio is crucial for the balance between wide deformation and high sensitivity. A wide detection range (0-650%) and high sensitivity (Max GF = 976.89) were obtained for the TPU:PEO/Graphene sensors, demonstrating its suitability for high-performance strain sensors. It can not only capture minor human movements, but also effectively be applied in fitness scenarios, which may contribute to personalized scientific training and reduce sports injuries.

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