Fractals: The Organization of Chaos
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Fractals are a relatively new concept in geometry. Most concepts for Euclidean geomtery, the division of geometry which deals with lines, circles, triangles, and other standard shapes, stem from the Late Greek and Early Rioman times. Considering the age of mathematics, the study of fractals is new becasue it dates to the beginning of this century. However, the age of computers brought about an explosion into this yet untamed universe of math. As Heinz-Otto Peitgen and Dietmar Saupe remark in the preface for The Science of Fractal Images, "Computer graphics has played an essential role both in its development and rapidly growing popularity" (V). Before this, mathematicians could only visualize what they were discussing (Mandelbrot, Fractals: Form, Chance, and Dimension 2). But now, fractals are the mathematician's answer to chaos and therefore can be used to help scientists better understand nuature and the universe. Scientists can define any structure from a snowflake to a mountain or even an entire planet with this new division in Mathematics. Thus, fractals define our universe.
Benoit B. Mandelbrot is a key figure behind the rise of this new science. A Professor of mathematical Sciences at Yale and an IBM Fellow, Mandelbrot is the man who coined the term "fractal" in 1975. Mathematicians, such as Gaston Julia, only defined them as sets before this and could only give properties of these sets. Also, there was no way for these early fractal researchers to see what they were hypothesizing about. As Mandelbrot states in The Fractal Geometry of Nature, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means 'to break...'" (4). Mandelbrot used this particular root because of how he defines fractals. Unlike Euclidean geometry, which has its figures in a particular dimension (e.g. a square is two-dimensional), fractals have fractional dimensions. They do not exist in just one dimension but can encompass part of another. For example, as Mort La Brecque states in his article on fractals in the Academic American Encyclopedia "a natural fractal of fractal dimension 2.8 ... would be a spongelike shape that is nearly three dimensional in its appearance. A natural fractal of fractal dimension 2.2 would be a much smoother object that just misses being flat" (105-106, Mandelbrot "Fractals").
Biodiesel is produced by subjecting the triglycerides found in vegetables oils and fats to transesterification, which is the exchanging of the alkoxy group of an ester compound by another alcohol. This creates an ox...
System [14] etc.display chaotic behavior. A hyper chaos system is considered as a chaotic attractor having more than one positive Lyapunov exponents which gives the randomness and higher unpredictability of the corresponding system so the hyper chaos may be more useful in some fields such as communication, encryption etc. On the other hand the area which attracted much attention is chaos synchronization since the seminal work of Pecora and Carroll [12] recently synchronization of fractional-order chaotic systems starts to attract increasing attention due to its potential applications in secure communication and control processing. There are many types of synchronization for the fractional-order chaotic systems which are investigated, such as Cs [17], Gs [15], PHs [16], As [18], Ps [19, 20, 22] etc. Amongst all projective synchronization, which was first reported by Mainieri and Rehacek [19], is one of the most noticeable one because it can obtain faster communication with its proportional feature [21, 23]. In PS, the responses of the two systems synchronized up to a constant scaling factor. Recently, on account of linear separation Wang and He [26] introduced projective synchronization of the fractional-order chaotic systems Then GPS of the fractional-order chaotic systems was studied in [20, 24]. However, in the above studies, all the states of the drive and response systems synchronize up to an identical constant scaling factor. In [25], Chen et al. proposed a new hyper chaotic system through adding a nonlinear controller of the three- dimensional autonomous chaotic system. More recently, by stability theory of fractional- order systems and tracking control technique the function projective synchronization between fractional-or...
The Fibonacci numbers are a sequence of numbers that begin with 0, 1 ... and then calculated each number from the sum of the previous two. The equation for this method is . Another theory he studied was a sequence that has a flower like pattern. Fibonacci's second work was the Practica geometriae and was composed in 1220-1221. The Practica geometriae draws heavily on the works of the ancient Greek masters i.e. Plato. Fibonacci made a dent in mathematics history.
Palumbo, Donald. "The monomyth as fractal pattern in Frank Herbert's Dune novels". Science Fiction Studies 25.3 (Nov. 1998): 433-58.
While the studies at Governor’s School are noticeably more advanced and require more effort than at regular public schools, I see this rigor as the key to my academic success. For me, the classes I take that constantly introduce new thoughts that test my capability to “think outside the box”, are the ones that capture all my attention and interest. For example, while working with the Sierpinski Triangle at the Johns Hopkins Center for Talented Youth geometry camp, I was struck with a strong determination to figure out the secret to the pattern. According to the Oxford Dictionary, the Sierpinski Triangle is “a fractal based on a triangle with four equal triangles inscribed in it. The central triangle is removed and each of the other three treated as the original was, and so on, creating an infinite regression in a finite space.” By constructing a table with the number black and white triangles in each figure, I realized that it was easier to see the relations between the numbers. At Governor’s School, I expect to be provided with stimulating concepts in order to challenge my exceptional thinking.
Clawson, Patrick. "Arab Human Development Report 2004: Towards Freedom in the Arab World." Middle East Quarterly 13.1 (2006): 85. Expanded Academic ASAP. Web. 2 Jan. 2010. .
Hansen, M., ABA Journal. Jul97, Vol. 83 Issue 7, p20. 2p. 1 Color Photograph, 1 Chart.
Attracting focus from firms in all industries, Big Data offers many benefits to those companies with the ability to harness its full potential. Firms using small data derive all of the data’s worth from its primary use, the purpose for which the data was initially collected. With Big Data, “data’s value shifts from its primary use towards its potential future uses” (Mayer-Schonberger & Cukier, 2013, p.99) thus leading to considerable increases in business efficiency. Employing Big Data analytics allows firms to increase their innovative capacity, and realize substantial cost reductions and time reductions. Moreover, Big Data techniques can be applied to support internal business decisions by identifying complex relationships within data. However, it is also important to recognize that much of Big Data’s value is “largely predicated on the public’s continued willingness to give data about themselves freely” (Brough, n.d., para. 11). As previously discussed, much of the content of Big Data is unstructured data from social media sites etc., and so if such data were to no longer be publically available due to regulation etc. the value of Big Data would be significantly diminished.
Big Data is a term used to describe the large volume of data whether structured or unstructured that inundates a given operation on a daily basis (http://www.SAS.com). Big Data consists of data sets that are so huge and complex that the customary data processing applications would not adequately handle them. Of late, the concept of Big Data has been used to describe the use of predictive analysis, user behaviour analytics and other complex data analytics techniques for the extraction value from data. The concept of Big Data can be understood through the description of the three V’s as advanced by Doug Laney, who is an industry analyst. First, Big Data can be understood in terms of Volume, whereby organizations collect large data from a variety
Emulsified Diesel fuels by 5, 10 and 15% water/Diesel ratios by volume, were utilized in a 6 cylinder, direct injection Diesel engine. Fuel usage of the emulsion systems was more than that of diesel, but the tiny droplets of water improved diesel combustion. Compared with diesel, a reduction in carbon monoxide (CO) and nitrogen oxide (NOx) emissions and a slight reduction in HC was observed for emolsified fuel. The existance of water particles and additives in the fuel decreased the heating value of the fuel, and despite the improvement of combustion as a result of microexplosion phenomena, was made a loss of engine power.
You may ask what big data analytics is. Well according to SAS, the leading company in business analytics software and services describes big data analytics as “the process of examining big data to uncover hidden patterns, unknown correlations and other useful information that can be used to make better decisions.” As the goal of many companies which is to seek insights into the massive amount of structured, unstructured, and binary data at their disposal to improve business decisions and outcomes, it is evident why big data analytics is a big deal. “Big data differs from traditional data gathering due to that it captures, manages, and processes the data with low-latency. It also one or more of the listed characteristics: high volume, high velocity, or high variety. Big data comes from sensors, devices, video/audio, networks, log files, web, and social media which much of it is generated in real time and in a very large scale.”(IBM) In other words, companies moving towards big data analytics are able to see faster results but it continues to reach exceptional levels moving faster than the average person can maintain.
This report draws from many publications written over the last twenty years exposing the unique situation in Saudi Arabia, while also utilizing recent headl...
Flynn et al. [4] studied a turbocharged, intercooled diesel engine using second law analysis to evaluate low-heat-rejection engine concepts. They were performed the simulation of engine cycle process. Primus and Flynn [5] reported a study demonstrating the benefits of using the second law in determining various energy losses in a diesel engine. Defining a thermodynamic system as outside the engine cylinder. Alkidas [6] applied energy and exergy balances to a diesel engine using experimental measurements. Shapiro and Van Gerpen [7] extended their earlier work to include a two-zone combustion model and applied this model to both a compression-ignition and a spark ignition engine. As before, their study included chemical exergy considerations. They presented the
To effectively sell a product or service, organizations have to really know how customers behave, regarding to what they buy. The study
Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.