Graph Theory: The Four Coloring Theorem "Every planar map is four colorable," seems like a pretty basic and easily provable statement. However, this simple concept took over one hundred years and involved more than a dozen mathematicians to finally prove it. Throughout the century that many men pondered this idea, many other problems, solutions, and mathematical concepts were created. I find the Four Coloring Theorem to be very interesting because of it's apparent simplicity paired with it's
revolution in the solutions of mathematical conjectures. Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. Kempe's argument goes as follows. “First, if planar regions separated by the graph are not trian... ... middle of paper ... ..., numerous renditions of the proof were disproven and built upon. Additionally, the Four Color Theorem was the first proof to be solved with aid from a computational device, creating controversy
The Mathematics of Map Coloring The four-color conjecture has been one of several unsolved mathematical problems. From 1852 to this day, practically every mathematician has studied the problem long and hard, but to no avail. The conjecture looks as though it has been solved by Wolfgang Haken and Kenneth Appel, both of the University of Illinois. They have used computer technology to prove the conjecture. The calculation itself goes on for about 1200 hours. The staggering length of the computation
States and southern Canada to earn points as well as completing routes designated on the destination tickets. The game itself is not only a fun way to spend hours playing but it is also a good tool to showcase various concepts in graph theory and combinatorics. Graph theory may also be useful in creating or enhancing game play strategies. Set Up and Game Play The original version of the game has a map of the United States and southern Canada with 30 destination tickets. An expansion to the original
0.1 abstract In a graph theory the shortest path problem is nding a minimum path and distance between two vertices.The ap- plication in many areas of shortest path algorithms are such as geographical rout- ing, transportation, computer vision and VLSI design involve solving optimiza- tion problems on large planar graphs. To calculate the shortest path we need to know some algorithms like Kruskal's algorithm,Prim's algorithm,Dijkstra's algorithm,BellmanFord's algorithm. These algorithms have some
Direct marketing The definition of DM is very complicated but it is simply selling a product or service via direct advertising sent through the mail, and also via several Internet promotion methods. The direct selling method enables the consumers to bypass inefficient wholesale and retail distribution systems. Women who left business in order to have children is able to do part time business, and also a very attractive career for woman reentering the work force. According to the “Direct selling
different face in every work. There is never a clearly definitive picture that identifies Arthur's character. It is therefore necessary to look at a few different sources to get better insight into the character of Arthur, the once and future king. GRAPH Arthurian literature can be divided into two basic categories, pseudo-histories and romances. The main difference between the two is that pseudo-histories such as Wace and much of the Celtic work, for example, Geoffrey of Monmouth show Arthur as
society. Wireless Communications is in the grand scheme of technological development, a rather recent event. But the quickness to which the market of cellular phones has expanded shows that some people have definitely embraced it as a positive. The graph on the following page shows the rapidity of America's love affair with the cell phone. Radio Telephone technology started in 1977 when Motorola, American Radio Telephone, and AT&T were licensed by the FCC to develop a high capacity radio telephone
reaction was because the more particles there were to collide and break old bonds and make new bonds. Also if the temperature was higher the particles would move around faster because they have more energy and would also cause more effective collisions. GRAPH Fair Test: The experiment will ... ... middle of paper ... ... anomalies within the experiment and this may have been caused by the stirring or the timer being started and stopped wrongly. I think that our group had a good method because our
only 100 ml of water each time, then you must divide the atomic weight by 10 before multiplying it by one gram. You will be using a computer-interfaced Temperature Probe to monitor how much each salt decreases the freezing temperature of water, and a graph of your results will be plotted using the computer. *PURPOSE: To learn which type of salt lowers the Freezing Point of water the greatest amount. *PROBLEM: Which type of salt lowers the Freezing Point of water to the lowest point? Blank 2 *HYPOTHESIS:
The Hertzsprung-Russell Diagram or, the H-R Diagram for short, is a graph which plots stars according to their temperature and absolute magnitude. This graph reveals a pattern, which in fact is quite interesting. The H-R Diagram is named for the two astronomers, Ejnar Hertzsprung and Henry Russell, who discovered this pattern of stars. These two astronomers independently discovered that comparing magnitudes and spectral class (color) of stars yielded a lot of information about them. One key purpose
and other effects). These two predictions together suggest that the best fit for the two trends described will be a curve as shown.. Analysis (see graph overleaf) ============================= The graph shows the relationship between the distance of the fan from the plant, and the % change of mass in the plant. From the graph, I can see that:- 1. The plant is losing water/transpiring when placed near a fan, because all the % changes in mass are negative numbers, and this shows
of the wave is the number of wavelengths that pass each second. Frequency and period are reciprocals. T = 1/f. The speed of a wave is equal to its frequency times its wavelength. A displacement/position graph shows the displacement of the different sections of a medium. A displacement/time graph shows the displacement of one point of a medium as time elapses. The speed of the particles of a medium is a maximum when their displacement is zero. The speed of the particles of a medium is zero where the
provide information about the value of one product in relation to another. The shape of the Productions Possibility Frontier, (PPF), illustrates the principle of increasing opportunity costs (Graph 3). As more of one product is produced, increasingly larger amounts of the other product must be given up. In Graph 3, some factors of production are suited for producing both Product A and Product B, but as the production of one of the other brands increases, resources better suited to production of the
the boys within his English class to ripe out the introduction entitled “Understanding Poetry” by Dr. J. Evans Pritchard, Ph.D. Evans explains Poetry by being able to be graded on a graph to determan if a poem is good or not. Mr Keating gets the boys to understand that you can not tell if a poem is good or not by a graph but by how much it means to you and the way it makes you feel. This is becoming a free thinker not using a set structure but actually understanding and appreciating it. He also believes
treatment for my results. Introduction: The Refractive Index is how the much a material bends the light. In this experiment I will be looking at the how much the angle of incidence gets refracted and I will multiply my results by sine. I will plot a graph from my results and, using a line of best fit, I will calculate the size of the angle of incidence in order for the refracted angle to be equal to 900 (critical angle). I will then calculate the refractive index by using Sine I and Sine R. I will
smaller number. However, I cannot take this for granted and I think using one more shape will be useful in order to back up my theory. [IMAGE] Area = 52 500m This proves my theory regarding squares and I shall now put my results into a graph to show what I have found. Length (m) Width (m) Area (m) 400 100 40 000 300 200 60 000 250 250 62 500 150 350 52 500 I will now further my investigation by looking at shapes of a different nature: [IMAGE] Regular Pentagon
The ratio of the two is defined as the refractive index, symbol: n. Equations Refractive Index Sin I Speed of light in Perspex =a constant = Speed of light in light in air Sin R [IMAGE][IMAGE] I could also use my graph to calculate the refractive index Apparatus * Ray Box * Perspex D-Block * Protractor paper * Pen/ Pencil * Ruler [IMAGE] Diagram [IMAGE] Prediction My Prediction is that first of all the ray of light will travel in
of the clamp; and finally the time the ball traveled through the clamp. After we recorded these different figures we calculated the speed of the marble from the given distance traveled and the time. We repeated the step 14 times, then proceeded to graph the speed and the height. Next, we took the measurements of position of the clamp, height, and speed and calculated the potential energy, the kinetic energy, and the total energy. Total energy calculated as mentioned before. Potential energy is taking
• 100 cm3 de-chlorinated water for each beaker • 2 g sea salt for each beaker • Stirring rod • Access to refrigerator • Water baths or incubators (one for each temperature to be investigated) • Brine shrimp egg cysts • Sheet of graph paper 3 cm _ 4 cm • Magnifying glass • Pair of forceps • Bright light • Fine glass pipette • 40 cm3 beaker of salt water Procedure 1. Decide on a range of temperatures from 5 °C to 35 °C to be tested. 2. Place 2 g of sea salt