Konigsberg Bridge Problem

Konigsberg Bridge Problem The earliest paper on graph theory seems to be by Leonard Euler,Solutio problematic ad situs pertinentis,Commentarii Academetarii Scientiarum Imperialist Petropolitanae 8 (1736),128-140.Euler discusses whether or not it is possible to scroll around Konigsberg(later called the Pregolya exactly once. Euler gave the conditions which are necessary to permit such a stroll. Thomas Pennyngton Kirkman (1856) and Wiliam Roman Hamilton (1856) studied trips which certain sites exactly once. History of Euler paths and cycles An Eulerian path is a path in a graph which visits each edge exactly once in the theory graph .so, in the same way, an Eulerian circuit is an Eulerian path which starts and ends on the same vertex. They were first discussing by Leohard Eular while solving the famous Seven Bridges of Konigsberg problem in 1736. Mathematically the problem can be stated like this: Given the graph on the right, is it possible to construct a path (or a cycle for example, a path starting and ending on the same vertex) which visits each edge exactly once Graphs which allow the manufacture of so called Eulerian cycles are called Eulerian graphs. Euler observed that a necessary condition for the existence of Eulerian cycles is that all vertices in the graph have an even degree, and that for an Eulerian path either all, or all but two (i.e., the two endpoint) vertices have an even degree; this means the Konigsberg graph is not Eulerian. Carl Heierholzer published the first complete characterization of Eulerian graphs in 1873, by proving that in fact the Eulerian graphs are exactly the graphs which are connected and where every vertex has an even degree. Example using euler in our daily life is using in the teaching for set theory that widely use in the schools. Another example is to visualizing file system organization.it will allows files to appear in more than one directory in a computers file system. The history of the Konigsberg Leonhard Euler (1707-1783) is considered to have been the father of graph theory. His paper in 1736 on the seven bridges of Konigsberg is considered to have been the foundational paper in the subject. Konigsberg is a town, founded in 1256, that was originally in Prussia. After a stormy history, the town became part of Soviet Union and was renames Kaliningrad in 1946. In any event, during Eulers time the town had seven bridges (named Kramer, Schmiede, Holz, Hohe, Honig, Kottel, and Grunespanning) spanning the Pregel River. Figure 8.1 gives a simplified picture of how the bridges were originally configured (two of the bridges were later destroyed during World War II, and two other demolished by the Russians. History of Hamiltonian: Hamiltonian is introduced by Sir William Rowan Hamilton at 1857. He made a game called around the world and the originally in form of solid called dodecahedron. It has 20corners/for each corner, it called as town. The problem started when the travel started from one city to another city along the edge to arrive at city by only once arrived at one city. This is how the Hamiltonian is appearing. There is example of using Hamiltonian in life such as no-complete, n-cube and traveling salesman problem. Two types of Hamiltonian are Hamiltonian path and Hamiltonian cycle Introduction Path Path is the sequences of alternating vertices and edges. Which begin from a vertex and ended with a vertex. Each vertex is preceded and followed by its endpoints. Simple path Simple path is a path such that all its vertices and edges are distinct. Below is a graph that gives differences between path and simple path. Path 1: v,b,x,h,z(simple path) Path 2 : u,c, w,f,y,g,x,e,d,v(path) Example of path is the way of bus direction from one destination to another destination. In other hand, simple path is a path that no complicated for example the direction from faculty of FTSM to faculty of FUU. Cycle Circle is a circular sequence of alternating vertices and edges. Each edges is preceded and followed by its endpoints. Simple cycle Simple cycle is a cycle such that all its vertices and edges are distinct. Cycle: u,c,w,e,x,g,f,w,d,v,a,u Simple cycle:v,b,x,g,y,f,w,c,u,a,v, Example of using cycle in life is when we travel to another place then come back to our home with using the different ways. Another example in ukm is the bus ways for example bus zone 2 will make a circle to take student and will come back to the initial location where the busy will take a rest. For simple cycle, we always see in sport, such as the court for athletes running especially in event of 400 * 100 meters. Then, we also can see in power plant program that is simple cycle power plant (pp) program. It gives much benefit such as optimized design, reduced engineering costs, short lead times, increase availability and fast startups also high operational flexibility. Connected Graph Connected graph is a graph that there exists a path between all pairs of vertices. If a graph is a directed graph, there exist a path between vertexes to each other that in the graph, is called as strongly connected graph. The examples of disconnected graphs: Example of using connected graph is use in building. For example Menara Berkembar Petronas, there is a bridge to connect the two buildings. Another example is the bridge of Pulau Pinang. First use to connect the island and peninsular Malaysia. Example of disconnected graph is other hand than connected graph. For example the building of one employee is not connected by bridge with another employee. Next, the Island of Sipadan is not connected by a bridge with Borneo land. An Euler path in a graph is a path which traverses each edge of the graph exactly once. An Euler cycle is an Euler path which is contains cycle. If there are no loop graphs, without isolated vertices, the continuation of an Euler path implies the connected of the graph, since traversing every edge of such a graph requires visiting each vertex at least once. But, when the connected graph has an Euler path, one can be constructed by applying Fleurys algorithm. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle. The definition and properties of Euler paths, cycles and graphs are valid for multigraph as well. The seven bridge of Konigsberg In Konigsberg, Germany, a river ran through the city such that in its centre was an island, and after passing the island, the river broke into two parts. Seven bridges were built, so that the people of the city could get from one part to another part. A crude map of the centre of Konigsberg might look like this: The people wondered whether or not, one could walk around the city in a way that would involve crossing each bridge exactly once. Degree of vertex Term of degree of vertex in graph theory is the number of edges which connected to a vertex. Degree of vertex also known as local degree. The list of all degree of vertex is called as degree sequences. One way to find the number of vertex is count the number of degree for each vertex that endpoint. An easy way is draw a circle around the vertex and count the number of edges that cross the circle. The degree of vertex can be add or even. if the degree of vertex is even, it is known as degree vertex and the other hand, if the degree of vertex is odd, the vertex is called an odd vertex. To find out the degree of graph is by choose the largest degree of vertex. Example graph with have odd and even vertex: Example degree of vertex is application of roundabout because there are many roads that connected. Either the value is odd or even. The road can be representing as edges and the roundabout as the vertex. Another example is the number of use degree of vertex in electrical pole such as the number of wire connected to the one pole. Hamiltonian path: Hamiltonian path is also called as traceable path. Hamiltonian path is a path that visits each vertex exactly one and not repeated for each vertex in a graph. Hamiltonian graph us use to solve a problem when find a path that only visited each vertex only one in a graph. Hamiltonian cycle: Hamiltonian cycle is a cycle that goes through the entire city (vertex) only once for a graph. It cannot be repeated to reach a city for a one cycle except the starting and the ended city. Results of research and real world examples Graphs can be used to represent oil flow in pipes, traffic flow on motorways, transport of pollution by rivers, groundwater movement of contamination, biochemical pathways, and the underground network. The example of Euler path: There are many useful applications to Euler circuits and paths. Networks can be used to solve many difficult problems, like the Konigsberg Bridge problem. The can also used by mail carrier who wants to have a route where they do not retrace any of their previous steps. Other than that, Euler circuits and paths are also useful to painters, garbage collector, airplanes pilots and world navigators. Below are the examples of how Euler circuit and paths are useful in the real world. The maps that pilots use are called route maps. The route maps show the paths of the airplanes from one destination to another. Here is an example of actual route map. The centre for all travel with this airline is in Denver, Colorado. From there, we can travel to some of the major cities in the surrounding states. The Navigation below is a trip to see all different regions of the world. The above regions of the world have all been given different colors. Each region also has been given marked with a node or vertex and some (but not all) of the regions are connected with arcs. Conclusions and recommendations As the conclusion towards this particular project, the study of graphs and their properties is a classical subject in most computer science department around the world. Graph Theory can be further exploited by object-oriented software engineering, taking advantage of recent research in various fields. Other than that, Graph theory is one of the top reasons to learn linear Algebra. So, all graphs (included directed, weighted, and multi-graphs) can be represented intuitive by adjency matrices, and matrix operations often end up being meaningful in terms of graph they represent. Seeing the connection between a graph and its matrix helps to understand both of them, and being able to switch back and forth between mental models is often useful. For example, a person in many fields of modeling, are mostly easily thought of the weight graphs, and are most easily manipulated as matrices. By learning the entire graph, the student can get many benefits by it especially the computer science student. So, our recommendation towards this topic in order to make the student easy to learn and improve themselves are for example, ask the student to make a lot of exercise. Other than that, ask them to make an assignment about this topic. So that the student can search many information based on this topic and become more familiar and understand about graph theory

The Moon :: essays research papers

The moon   Ã‚  Ã‚  Ã‚  Ã‚  The moon is the only natural satellite of Earth. The moon orbits the Earth from 384,400 km and has an average speed of 3700 km per hour. It has a diameter of 3476 km, which is about  ¼ that of the Earth and has a mass of 7.35e22 kg. The moon is the second brightest object in the sky after the sun. The gravitational forces between the Earth and the moon cause some interesting effects; tides are the most obvious. The moon has no atmosphere, but there is evidence by the United States Department of Defense Clementine spacecraft shows that there maybe water ice in some deep craters near the moon's North and South Pole that are permanently shaded. Most of the moon's surface is covered with regolith, which is a mixture of fine dust and rocky debris produced by meteor impact. There are two types of terrain on the moon. One is the heavily cratered and very old highlands. The other is the relatively smooth and younger craters that were flooded with molten lava. Throughout the 19th and 20th centuries, visual exploration through powerful telescopes has yielded a fairly comprehensive picture of the visible side of the moon. The hitherto unseen far side of the moon was first revealed to the world in October 1959 through photographs made by the Soviet Lunik III spacecraft. These photographs showed that the far side of the moon is similar to the near side except that large lunar maria are absent. Craters are now known to cover the entire moon, ranging in size from huge, ringed maria to those of microscopic size. The entire moon has about 3 trillion craters larger than about 1 m in diameter. The moon shows different phases as it moves along its orbit around the earth. Half the moon is always in sunlight, just as half the earth has day while the other half has night. The phases of the moon depend on how much of the sunlit half can be seen at any one time. In the new moon, the face is completely in shadow. About a week later, the moon is in first quarter, resembling a half-circle; another week later, the full moon shows its fully lighted surface; a week afterward, in its last quarter, the moon appears as a half-circle again. The entire cycle is repeated each lunar month, which is approximately 29.

Hubspot Case Essay

The problem in this case is that Hubspot needed to make a transition from its initial start-up structure (organizational structure, target customers and pricing strategy) in order grow, and the dilemma was how to best approach this change. Hubspot faced three main issues for this: a) identify target customers, b) modify their pricing model and c) how to develop the growth strategy. Hubspot was good at building a community, e.g. over 300000 unique visitor in 2008, and thousands of freeware subscriptions in 2009. Nonetheless they had a diverse universe of customers, from small business owners (Ollies) to marketing professionals (Marys), different type of business ranging B2B or B2C, and size (over or under 25 employees). Table C shows there was a potential market evenly distributed among B2B and B2C. For Hubsport, the decision to identify a target customer was difficult. This is seen when contrasting exhibits 6 where 73% of customers were Ollies and exhibit 5 which indicated that Marys accounted for 68% of new customers from Sep-Dec 2008. Although the B2B customers were important for Ollies and Marys, there was an interesting growth of Marys in B2C. Thus a segmentation of customer was required to better assess their different needs. At the end of 2008, Hubspots’s products responded to the main two customers (Ollies and Marys), still its pricing model was similar for both, where Marys paid a slightly higher monthly amount as its software package included more features (exhibit 7). This was something Hubspot needed to analyze as Ollie and Marys had various pros and cons as customers. Ollies represented a lower cost to acquire ($1000) and where quick to sign in, but cancel subscription early, while Marys cost more to acquire ($5000) and took longer to sign in, by stayed for longer using the product. Assuming no churn rate an Ollie had to maintain subscription for 2 months and Marys had to maintain subscription for 9 months, to pay off their acquiring cost. The previous  scenario meant that HubSpot’s 2008 projections including the 100 paying customers from 2007 made the current pricing model not viable to support the high cost of Marys (see appendix 1). Another issued faced was the Hubsport was still a small company, seen in that it only had few engineers to build the software therefore it was hard to catch up with the sales team. Thus the product vs customer vs pricing situation presented an optimization and planning issue to keep the company growing. The previous two points require a growth strategy. At the same time it made the owners question their vision, i.e. to inbound or outbound. The strategy for growth had to clarify which customer to target, how to roll-out the respective products, whether to keep it a SaaS, and the transition into a new pricing structure to maintain current customer and capture more value from new ones. The objective of our proposed solutions is to keep Hubspot as the software-to-have for inbound marketing and grow financially from a start-up to an established business. For this we set out the following actions: Hubspot’s culture and vision should be maintained. Web 2.0 is continuing evolving as more businesses are using the various channels and HubSpot can differentiate itself as the inbound marketing which weighs more than outbound marketing (inbound represents 37% marketing budget while outbound 30%). HubSpot has the expertise to create traffic and analyze and qualify leads filling the respective demand of Ollies and Marys. At the same time we differentiate from our two main competitors by proving a lower price (Eloqua is more expensive) and focusing on inbound marketing (Marketo is a mix of inbound and outbound). Our conclusions are founded by overlaying HubSpot’s competitive field (exhibit 3) with customers’ needs a) traffic creation and b) leads analysis and qualifications, in line with HubSpot’s main strengths, as seen in appendix 2. Thus the company should not consider outbound as an alternative. As showed in appendix 4, our two segmented customers have showed different needs in terms of product features and consumption behavior. Based on the current churn out rate, we can estimate consumer lifetime value of Ollies is $4,750 and Marys $10,500 (see calculation in appendix 3). Therefore, according to our segmentation strategy, we propose following product bundles by differentiating product price and product features: 1) Product pricing: As Ollies have a shorter customer life and less marketing budget, we suggest keeping current up-front fee and a lower monthly fee. As suggestion, up-front $500 and monthly fee in the range of $150 to $250. As Marys have a longer customer life and lower price sensitivity we suggest increasing both up-front and monthly fee. As suggestion up-front $600 and monthly fee in the range of $600 to $750. Meanwhile, Marys are interested in deeper analytics, we suggest additional fee for each service of deeper analytics. As CMS system helps lower churn rate, we suggest initial fee of $300 covering 6 hours consulting to encourage both of them to use such service. 2) Product features: As Ollies prefer quick and simple solutions, we suggest tailor-made product focusing on generating leads. As Marys have a high demand of analytics, we suggest tailor-made product with more sophisticated tools to meet the needs of deeper analytics. As frequent log-in helps lower churn rate, we suggest to provide service update on a regular basis to encourage a continuous use of our service. After clearly identifying the segmentation of consumer and differentiation of products, we need ensure market-centered organizations that are capable of translating strategy into actions: 1) Engineering: To invest on product development and innovation to continuously provide with relevant service to enhance our competitive advantage of generating leads as well as analytics . 2) Sales force: To divide sales force to separately serve Maryer & Ollies by providing Maryer with long-term, more sophisticated support, providing Ollies with quick & simple service. 3) Marketing: To continue make a buzz for inbound marketing to create inbound marketing community rather than a simple business Finally the strategy has to be sensitive to our current customers, Appendix 5 indicates a tentative layout of the plan. Starting with the internal reorganization, then gradually change the product offering for consumers.

The Underlying Issues Of Cannabis Abuse - 3504 Words

The Underlying Issues of Cannabis Abuse Jeoffery Walton English 102 The International College of the Cayman Islands The topic outline 1) The cannabis plant and its ingredients 2) Abuse of cannabis a) Smoking Cannabis b) Addiction to cannabis c) Withdrawal symptoms d) Effects on the mental health cognitive development in adolescence e) Cancer and other illnesses 3) Cannabis for medicinal purposes 4) Law enforcement 5) Conclusion Abstract This paper examines the effects that cannabis abuse has on human health over an extended period. Over the years, some researchers have concluded that continuous use of the drug during adolescence, but not adulthood, could permanently impair cognition and increase the risk for psychiatric diseases. Other studies suggest that heavy use amongst users can cause changes in subjective mental status, brain functioning, and neuropsychological performance. Approximately 9% of those who experiment with cannabis will become addicted, and the treatment for addiction can be challenging, as it often leads to withdrawal symptoms. Consequently, most governments around the globe decided to enact legislation in the early to mid 20th century, to help control its abuse; however, recently, some countries in the Western World have begun to allow its use for medicinal and recreational purposes. All information for the writing of this paper has beenShow MoreRelatedShould Cannabis Be Legalised?1595 Word s   |  7 PagesShould cannabis be legalised in Australia? Your response should provide a considered discussion of the advantages and disadvantages of both legalisation and criminalisation which shows evidence of wide and varied reading. Cannabis originated from Central Asia generations ago, with references to Cannabis dating back to medical and religious Chinese and Indian texts. 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