Leinigen And Brain Essays - Leiningen Versus The Ants, Leiningen

Leinigen And Brain The human brain needs only to become fully aware of its powers to conquer even the elements. The human brain is powerful and controls all of a person's body. It easily compared to the central processing unit in a computer; all information is received, transferred, and sent back out. Without it nothing would work. Leiningen, a plantation farmer, was persistent and followed this motto to help him overcome many elements, or "acts of God." Leiningen had met and defeated these elements which had come against him unlike his fellow settlers who had little or no resistance. Something terrible was coming, about a hundred yards wide, a flood of ants. All people except Leiningen, who stood his ground as the ants quickly approached his plantation, feared them. He had built this plantation thinking that someday the ants may come. Leiningen thought to himself that he would be ready for them. He incorporated a horseshoe shaped ditch around his plantation. The forth side is a river that can be used quickly to fill the ditch. Toward the middle of the plantation lay another ditch that encircled the barn, house, stables, and other buildings. This ditch was made of concrete, and the inflow pipes of three great petrol tanks could easily be emptied here. If the ants had miraculously made it through the first ditch the second one could be filled with gas which was sure to stop them. This intricate defense system was thought of by Leiningen and built to stop one of the elements, ants. This firm man stayed calm as the ants advanced toward him and his land. Leiningen used his cool brain to calm his many workers. He reassured them that these ants could be easily defeated. The Indians trusted this plantation planter, who guided them through many other "acts of God," wholeheartedly. When one of there fellow workers had slacked off from his duties he was eaten alive by the ants. Leiningen realizing this casualty might plunge his men into confusion and destroy their confidence he quickly yelled loader than the screams of the dying man. An observer would have estimated Leiningen's odds of overcoming the ants a thousand to one, but still Leiningen stood his ground. As the ants started to enter the plantation the dam broke preventing the water to rush in and wipe out the ants. All Leiningen's men had fled to beyond the petrol ditch to seek shelter. That was not enough, the petrol ditch was filled but the ants still crossed. Leiningen scourged his brain until it rolled. Was there anything he could do? (Then out of the he got an idea.) Yes, one hope remained. He thought it might be possible to dam the great river completely, so that the water would not fill only the ditch but overflow into the circle of land which made the plantation. In little time the army of ants would be flooded and killed. It was possible, but he needed to get to the dam, which was two miles away. Leiningen knew none of his workers would make the trip so he would have to do it himself. Leiningen told his men he would return. "I called the tune, and now I'm going to pay the piper," he told them. He started the journey and was quickly covered with ants. Leiningen was so determined to reach the dam he hardly felt the venomous bites. As he reached his destination ants covered his face and were under his clothes. He successfully lowered the dam and the river immediately started to overflow. Leiningen could no longer see and knew if he stumbled he would be quickly eaten alive. This determined man, to weak to walk, tripped over a rock and fell to the ground. He began having flashbacks of the stag he saw the ants devour. He thought to himself he could not die like this and something outside him brought him to his feet and he began to stager forward again. Leiningen leaped through the fire the workers had set to the petrol. He suddenly became unconscious for the first time in his life. There were wounds on his body so deep the bone could be seen. When Leiningen regained consciousness he said to the men, "told you I would return." Everyone knew he would be alright. The human brain needs only to become fully aware of its powers to conquer even the elements. This motto has proved true for Leiningen for he has met and defeated drought, flood, plague,

amazon vs. Barnes essays

amazon vs. Barnes essays Amazon.com vs. Barnes The race to soak up as much of the market as possible has been highly intensified by the process of eCommerce. Some companies choose to base their entire operations on the Internet, for example, Amazon.com. On the other hand there are already brick and mortar storefronts that also wish to deal business on the Internet, for example, Barnes they sell books (and much more, but I will only concentrate on the book aspect of operations). Amazon.com had the head start in the eCommerce division, but Barnes In 1994 Jeff Bezos in his garage in Seattle, Washington founded Amazon.com. To this day, it continues to run as a virtual organization having not one physical storefront. This is how they keep their overhead and inventory costs down, that leads to savings for the consumer. With a total of 7600 employees, Amazon.com was first to successfully conduct operations on the Internet. Also, they were the first enterprise to sell traditional consumer goods online which opened the eyes to other industries wishing to conduct business through the World Wide Web. Amazon.com takes pride in providing the customer satisfaction through an easy and helpful website. They believe if one has a good experience buying a book on the Internet then they will return to that site again and again for future business. By having access to over 3 million titles, they have become very successful over the years. On the other hand there is Barnes & Noble, a century old storefront on Fifth Avenue that was bought out by Leonard Riggio in 1971. Riggio, being an entrepreneur looked to expand his business by merging with other bookstores to create a storefront, which can have virtually any book. B&N operates more than a 1000 stores ...

LA SUITE EAST PLC EXTRACTS FROM THE CHIEF EXECUTIVE'S CIRCULAR Assignment

LA SUITE EAST PLC EXTRACTS FROM THE CHIEF EXECUTIVE'S CIRCULAR - Assignment Example This means that the company uses more in production and other expenses to increase the sales, thus reducing the profit, which undermines the company’s performance. The project being started by the company has not been well planned for, which means that the money being spent on the project may lead to the downfall of the company. While planning for the building to be constructed, the company did not consider that there will be depreciation of the building with time. There is no improvement in the financial performance of the company because the increase in sales has increased the cost, so the project will not be the best idea. The difficulties of the company are tackled in various ways. First the company should consider ways of reducing the cost to raise the profits and minimize losses. The other solution to the company’s poor performance is to plan better for upcoming projects, considering all facts and risks, as well as future plans. The current project should not be carried out since it will lead to very serious

Citi bank Assignment Example | Topics and Well Written Essays - 4000 words

Citi bank - Assignment Example Citigroup operates in six major regions in the world which are North America, Latin America, Asia, Europe, Middle East and Africa. An institution connecting millions of people from more than 1000 cities covering 160 countries, they represent themselves as a global bank. After achieving tremendous success domestically since its inception in the year 1812, they expanded globally with a sole purpose of serving their clients and shareholders effectively (Citigroup, 2013b). Citibank’s main objective of internationalization is to enter new markets and structure a banking relationship with a dedicated team of country specific officials who have thorough understanding of foreign markets and have a wealth of experience working with Citibank divisions across the globe. One of the primary global solution services of Citibank includes quick and easy opening of account in any of the 39 currencies from a single Citibank branch. Thus it spares the need for opening accounts country by country ; currency by currency. They also offer foreign exchange services to international clients thereby providing personalized one-to-one guidance from a foreign exchange specialist. They also offer 24 hours foreign exchange trading in 135 different currencies. In addition to that, Citibank also provides World Link service by means of which their international clients can make payments such as wires, automatic clearing houses or checks in any of these 135 currencies. Their multicurrency payment system offers a one-stop solution for their international clients to make payments using their preferred currency without opening accounts in different currency which they want to trade in (Citigroup, 2013c). The Citibank global business solution division helps its clients who are involved in internal business (importing materials or expanding overseas) by providing paperwork-oriented services thereby mitigating the risk. CitiBusiness trade services and manage trade transactions on behalf of their international clients thereby helping them to minimize their risk when dealing with counterparties from different countries (Citigroup, 2013d). As far as the Citibank’s international business operations in the Asia pacific region is concerned, its history dates back to 1902. They provide more services in the market, with a large base of clients, compared to any other financial institutions in this region through its institutional clients group and global consumer banking business. They employ more than 60,000 officials across 18 countries in this region. Citibank has a rich history of innovation and customer service in this region and has been as the region’s leading retail bank (Citibank, 2013e). As far as Europe, the Middle East and Africa is concerned, Citibank operates in 116 countries. However, it maintains physical presence only in 55 of them. Operating in this region has proven to be of great benefits for the organization, primarily because this region includes a thick combination of both developed and emerging markets. The main rationale for preferring Citibank over any other financial institutions in this region, as explained by the clients, is because of the bank’s global footprint, market position, in-country relationships and availability of wide

Pugh v. Locke Case Research Paper Example | Topics and Well Written Essays - 1000 words

Pugh v. Locke Case - Research Paper Example On February 26, 1974, an inmate of G. K. Fountain Correctional Center filed a complaint concerning the state of inmates confined by the Alabama board of corrections or those who may be confined later (Robbins & Michael, 1977). The court found that those actions were maintained as class actions under Federal Rule 23(a) and (b) (2) (Gerald, 1978). Notably, the court investigated and found out that the defendants in both cases acted and refused to act on the ground set for the class. The defendants were sued in their official and individual capacities including the Governor of Alabama, the Commissioner of Alabama Board of Corrections, Deputy Commissioner of the Alabama Board of Corrections, the board members, the Warden of G. K. Fountain Correctional Center and the Warden of Kilby Corrections Facility. These people were retained as individual defendants (Robbins & Michael, 1977). On April 16, 1974, the court-appointed counsel filed the amendment to the case. This complaint was filed on behalf of all inmates of the state penal system confined to, G. K. Fountain Correctional Center and those who underwent such violence (Gerald, 1978). The Alabama Board of Corrections was charged with the responsibility for managing these state penal institutions. The board operated four large institutions for male inmates, which were, G. K. Fountain Correctional Center, Holman Unit Prison, Kilby Corrections Facility and Draper Correctional Center (Robbins & Michael, 1977). They also managed Julia Tutwiler Prison for women and the Frank Lee Youth Center for young men. The inmate populations of these institutions were in excess by 5000. The overcrowding of these institutions heightened, and inmates were crowded so much that they had to sleep on mattresses on hallways and even near urinals (Robbins & Michael, 1977). These made sanitation and security impossible to maintain. Alabama’s penal institutions were filthy. In a research carried out by a public health officer, he found r oaches, flies, mosquitoes and other vermin in all stages of development. The sanitary rooms were terrible and constituted to poor drainage systems. Strange odors emanated from these facilities due to the gross under maintenance of hygiene. Moreover, personal hygiene was not observed in these facilities. The parties predetermined that the state only provided for razor blades and soap to the inmates (Gerald, 1978). Items such as shampoo, toothbrushes, toothpaste, shaving cream and combs were unavailable to the inmates since the state did not provide. However, inmates who could afford the products were required buy them. On the other hand, catering services were of poor quality in these facilities. Food was stored in unsanitary conditions. The storage units were dirty and infested with insects. The food service personnel who mostly comprised of inmates were unskilled on how to handle and prepare food. In addition, the inmates were not supplied with eating utensils. This forced them to use, old, dirty, tin cans. The food was unappetizing and unwholesome while at the same time dangerous for human consumption. An expert witness once toured the facilities. Shockingly enough, he concluded that the conditions in the facilities were unfit for human habitation in every criterion (Robbins & Michael, 1977). The prison officials did not dispute evidence that most inmates are in a terrifyingly poor condition. Consequently, a

Theorems Related To Mersenne Primes Mathematics Essay

Theorems Related To Mersenne Primes Mathematics Essay Introduction: In the past many use to consider that the numbers of the type 2p-1 were prime for all primes numbers which is p, but when Hudalricus Regius (1536) clearly established that 211-1 = 2047 was not prime because it was divisible by 23 and 83 and later on Pietro Cataldi (1603) had properly confirmed about 217-1 and 219-1 as both give prime numbers but also inaccurately declared that 2p-1 for 23, 29, 31 and 37 gave prime numbers. Then Fermat (1640) proved Cataldi was wrong about 23 and 37 and Euler (1738) showed Cataldi was also incorrect regarding 29 but made an accurate conjecture about 31. Then after this extensive history of this dilemma with no accurate result we saw the entry of Martin Mersenne who declared in the introduction of his Cogitata Physica-Mathematica (1644) that the numbers 2p-1 were prime for:- p= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 and for  other positive integers where p So simply the definition is when 2p-1 forms a prime number it is recognized to be a Mersenne prime. Many years later with new numbers being discovered belonging to Mersenne Primes there are still many fundamental questions about Mersenne primes which remain unresolved. It is still not identified whether Mersenne primes is infinite or finite. There are still many aspects, functions it performs and applications of Mersenne primes that are still unfamiliar With this concept in mind the focus of my extended essay would be: What are Mersenne Primes and it related functions? The reason I choose this topic was because while researching on my extended essay topics and I came across this part which from the beginning intrigued me and it gave me the opportunity to fill this gap as very little was taught about these aspects in our school and at the same time my enthusiasm to learn something new through research on this topic. Through this paper I will explain what are Mersenne primes and certain theorems, related to other aspects and its application that are related with it. Theorems Related to Mersenne Primes: p is prime only if 2p  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is prime. Proof: If p is composite then it can be written as p=x*y with x, y > 1. 2xy-1= (2x-1)*(1+2x+22x+23x+à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦..+2(b-1)a) Thus we have got 2xy à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 as a product of integers > 1. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be 1 plus a multiple of 2n. This holds even when 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is prime. Examples: Example I: 25 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 31 is prime, and 31 is multiple of (2ÃÆ'-5) +1 Example II: 211 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 23ÃÆ'-89, where 23 = 1 + 2ÃÆ'-11, and 89 = 1 + 8ÃÆ'-11. Proof: If m divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 then 2n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). By Fermats Theorem we know that 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). Assume n and m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 are comparatively prime which is similar to Fermats Theorem that states that (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)(n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n). Hence there is a number x à ¢Ã¢â‚¬ °Ã‚ ¡ (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)(n à ¢Ã‹â€ Ã¢â‚¬â„¢ 2) for which (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) ·x à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n), and thus a number k for which (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) ·x à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = kn. Since 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m), raising both sides of the congruence to the power x gives 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã¢â‚¬ °Ã‚ ¡ 1, and since 2n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m), raising both sides of the congruence to the power k gives 2kn à ¢Ã¢â‚¬ °Ã‚ ¡ 1. Thus 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x/2kn = 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã‹â€ Ã¢â‚¬â„¢ kn à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). But by meaning, ( m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã‹â€ Ã¢â‚¬â„¢ kn = 1 which implies that 21 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m) which means that m divides 1. Thus the first conjecture that n and m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 are relatively prime is unsustainable. Since n is prime m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 have to be a multiple of n. Note: This information provides a confirmation of the infinitude of primes different from Euclids Theorem which states that if there were finitely many primes, with n being the largest, we have a contradiction because every prime dividing 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be larger than n. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be congruent to +/-1 (mod 8). Proof: 2n + 1 = 2(mod m), so 2(n + 1) / 2 is a square root of 2 modulo m. By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to +/-1 (mod 8). A Mersenne prime cannot be a Wieferich prime. Proof: We show if p = 2m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is a Mersenne prime, then the congruence does not satisfy. By Fermats Little theorem, m | p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1. Now write, p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = mÃŽÂ ». If the given congruence satisfies, then p2 | 2mÃŽÂ » à ¢Ã‹â€ Ã¢â‚¬â„¢ 1, therefore Hence 2m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 | ÃŽÂ », and therefore . This leads to , which is impossible since . The Lucas-Lehmer Test Mersenne prime are found using the following theorem: For n an odd prime, the Mersenne number 2n-1 is a prime if and only if 2n -1 divides S(p-1) where S(p+1) = S(p)2-2, and S(1) = 4. The assumption for this test was initiated by Lucas (1870) and then made into this straightforward experiment by Lehmer (1930). The progression S(n) is calculated modulo 2n-1 to conserve time.   This test is perfect for binary computers since the division by 2n-1 (in binary) can only be completed using rotation and addition. Lists of Known Mersenne Primes: After the discovery of the first few Mersenne Primes it took more than two centuries with rigorous verification to obtain 47 Mersenne primes. The following table below lists all recognized Mersenne primes:- It is not well-known whether any undiscovered Mersenne primes present between the 39th and the 47th from the above table; the position is consequently temporary as these numbers werent always discovered in their increasing order. The following graph shows the number of digits of the largest known Mersenne primes year wise. Note: The vertical scale is logarithmic. Factorization The factorization of a prime number is by meaning itself the prime number itself. Now if talk about composite numbers. Mersenne numbers are excellent investigation cases for the particular number field sieve algorithm, so frequently that the largest figure they have factorized with this has been a Mersenne number. 21039 1 (2007) is the record-holder after estimating took with the help of a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland and yet the time period for calculation was about a year. The special number field sieve can factorize figures with more than one large factor. If a number has one huge factor then other algorithms can factorize larger figures by initially finding the answer of small factors and after that making a primality test on the cofactor. In 2008 the largest Mersenne number with confirmed prime factors is 217029 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 418879343 ÃÆ'- p, where p was prime which was confirmed with ECPP. The largest with possible pr ime factors allowed is 2684127 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 23765203727 ÃÆ'- q, where q is a likely prime. Generalization: The binary depiction of 2p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is the digit 1 repeated p times. A Mersenne prime is the base 2 repunit primes. The base 2 depiction of a Mersenne number demonstrates the factorization example for composite exponent. Examples in binary notation of the Mersenne prime would be: 25à ¢Ã‹â€ Ã¢â‚¬â„¢1 = 111112 235à ¢Ã‹â€ Ã¢â‚¬â„¢1 = (111111111111111111111111111111)2 Mersenne Primes and Perfect Numbers Many were anxious with the relationship of a two sets of different numbers as two how they can be interconnected. One such connection that many people are concerned still today is Mersenne primes and Perfect Numbers. When a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself then is it said to be known as Perfect Numbers. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors. There are said to be two types of perfect numbers: 1) Even perfect numbers- Euclid revealed that the first four perfect numbers are generated by the formula 2nà ¢Ã‹â€ Ã¢â‚¬â„¢1(2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1): n = 2:    2(4 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 6 n = 3:    4(8 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 28 n = 5:    16(32 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 496 n = 7:    64(128 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 8128. Noticing that 2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is a prime number in each instance, Euclid proved that the formula 2nà ¢Ã‹â€ Ã¢â‚¬â„¢1(2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1) gives an even perfect number whenever 2p  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is prime 2) Odd perfect numbers- It is unidentified if there might be any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. An example would be the first perfect number that is 6. The reason for this is so since 1, 2, and 3 are its proper positive divisors, and 1  +  2  +  3  =  6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: (1  +  2  +  3  +  6)  /  2  =  6. Few Theorems related with Perfect numbers and Mersenne primes: Theorem One: z is an even perfect number if and only if it has the form 2n-1(2n-1) and 2n-1 is a prime. Suppose first that   p = 2n-1 is a prime number, and set l = 2n-1(2n-1).   To show l is perfect we need only show sigma(l) = 2l.   Since sigma is multiplicative and sigma(p) = p+1 = 2n, we know sigma(n) = sigma(2n-1).sigma(p) =  (2n-1)2n = 2l. This shows that l is a perfect number. On the other hand, suppose l is any even perfect number and write l as 2n-1m where m is an odd integer and n>2.   Again sigma is multiplicative so sigma(2n-1m) = sigma(2n-1).sigma(m) = (2n-1).sigma(m). Since l is perfect we also know that sigma(l) = 2l = 2nm. Together these two criteria give 2nm = (2n-1).sigma(m), so 2n-1 divides 2nm hence 2n-1 divides m, say m = (2n-1)M.   Now substitute this back into the equation above and divide by 2n-1 to get 2nM = sigma(m).   Since m and M are both divisors of m we know that 2nM = sigma(m) > m + M = 2nM, so sigma(m) = m + M.   This means that m is prime and its only two divisors are itself (m) and one (M).   Thus m = 2n-1 is a prime and we have prove that the number l has the prescribed form. Theorem Two: n will also be a prime if 2n-1 is a prime. Proof: Let r and s be positive integers, then the polynomial xrs-1 is xs-1 times xs(r-1) + xs(r-2) + + xs + 1.   So if n is composite (say r.s with 1 Theorem Three:   Let n and m be primes. If q divides Mn = 2n-1, then q = +/-1 (mod 8)  Ã‚  Ã‚  Ã‚   and  q = 2kn + 1 for some integer k. Proof: If p divides Mq, then 2q  =  1 (mod p) and the order of 2 (mod p) divides the prime q, so it must be q.   By Fermats Little Theorem the order of 2 also divides p-1, so p-1  =  2kq.   This gives 2(p-1)/2 = 2qk = 1 (mod p) so 2 is a quadratic residue mod p and it follows p = +/-1 (mod 8), which completes the proof. Theorem Four: If p = 3 (mod 4) be prime and then 2p+1 is also prime only if 2p+1 divides 2p-1. Proof: Suppose q = 2p+1 is prime. q  =  7 (mod  8) so 2 is a quadratic residue modulo q and it follows that there is an integer n such that n2  =  2 (mod  q). This shows 2p = 2(q-1)/2 = nq-1 = 1 (mod q), showing q divides Mp.       Conversely, let 2p+1 be a factor of Mp. Suppose, for proof by contradiction, that 2p+1 is composite and let q be its least prime factor. Then 2p  =  1 (mod  q) and the order of 2 modulo q divides both p and q-1, hence p divides q-1. This shows q  >  p and it follows (2p+1) + 1 > q2 > p2 which is a contradiction since p > 2. Theorem Five: When we add the digits of any even perfect number with the exception of 6 and then sum the digits of the resulting number and keep doing it again until we get a single digit which will be one. Examples. 28  ¬10  ¬ 1, 496  ¬ 19  ¬ 10  ¬ 1, and 8128  ¬ 19  ¬10  ¬ 1 Proof: Let s(n) be the sum of the digits of n. It is easy to see that s(n) = n (mod 9). So to prove the theorem, we need only show that perfect numbers are congruent to one modulo nine. If n is a perfect number, then n has the form 2p-1(2p-1) where p is prime which see in the above theorem one. So p is either 2, 3, or is congruent to 1 or 5 modulo 6. Note that we have excluded the case p=2 (n=6). Finally, modulo nine, the powers of 2 repeat with period 6 (that is, 26 = 1 (mod 9)), so modulo nine n is congruent to one of the three numbers 21-1(21-1), 23-1(23-1), or 25-1(25-1), which are all 1 (mod 9). Conjectures and Unsolved Problems: Does an odd perfect number exist?   We have so far known that even perfect numbers are 2n-1(2n-1)from the Theorem One above, but what about odd perfect numbers?   If there is an odd perfect number, then it has to follow certain conditions:- To be a perfect square times an odd power of a single prime; It is divisible by at least eight primes and has to have at least 75 prime factors with at least 9 distinct It has at least 300 decimal digits and it has a prime divisor greater that 1020. Are there infinite numbers of Mersenne primes?   The answer is probably yes because of the harmonic sequence deviation. The New Mersenne Conjecture: P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., have conjectured the following:- Let n be any odd natural number. If two of the following statements hold, subsequently so does the third: n = 2p+/-1  Ã‚   or  Ã‚   n = 4p+/-3 2n-1 is a prime (2n+1)/3 is a prime. Are all Mersenne number 2n-1 square free? This is kind of like an open question to which the answer is still not known and hence it cannot be called a conjecture. It is simple to illustrate that if the square of a prime n divides a Mersenne, then p is a Wieferich prime which are uncommon!   Only two are acknowledged lower than 4,000,000,000,000 and none of these squared divide a Mersenne.    If C0 = 2, then let C1 = 2C0-1, C2 = 2C1-1, C3 = 2C2-1à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦ then are all of these prime numbers?   Dickson Catalan (1876) responded to Lucas stating 2127-1 (which is C4) being a prime with this sequence: C0 = 2 (which is a prime) C1 = 3 (which is a prime) C2 = 7 (which is a prime) C3 = 127 (which is a prime) C4 = 170141183460469231731687303715884105727 (which is a prime) C5 > 1051217599719369681875006054625051616349 (is C5 a prime or not?) It looks as if it will not be very likely that C5 or further larger terms would be prime number.   If there is a single composite term in this series, then by theorem one each and every one of the following terms would be composite.   Are there more double-Mersenne primes? Another general misunderstanding was that if n=Mp is prime, then so is Mn; Lets assume this number Mn to be MMp which would be a double-Mersenne.  As we apply this to the first four such numbers we get prime numbers: MM2 = 2(4  -1) -1= 23-1  Ã‚   =  7 MM3 =  2(8-1)-1  Ã‚   =  127 MM5 =  2(32-1)-1  =  2147483647, MM7 =  2(128-1)-1 =  170141183460469231731687303715884105727. Application of Mersenne Prime: In computer science, unspecified p-bit integers can be utilized to express numbers up to Mp. In the mathematical problem Tower of Hanoi is where the Mersenne primes are used. It is a mathematical puzzle consisting of three rods, and a number of disks of different sizes, which can slide onto any rod. The puzzle begins with the disks in ascending order of size on the first rod, the largest at the bottom to the smallest at the top. A diagram given below illustrates the Tower of Hanoi. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod. No disk may be placed on top of a smaller disk. Now to solve this game with a p-disc tower needs the minimum of Mp no of steps, where p is the no of disc used in the Tower of Hanoi and if we use the formula of Mersenne then we get the required result. An example of this would be if there were 5 discs involved in this Tower of Hanoi then the least number of steps required to finish this game would be 31 steps minimum. Conclusion After investigating the entire aspects, functions, and few applications of Mersenne Primes I believe that there is still many unsolved theories when it comes to Mersenne primes. These primes are also useful to investigates much further and deeper into the number system and help us to understand more sets of numbers such as Fermat prime, Wieferich prime, Wagstaff prime, Solinas prime etc.

Adaptive Thresholding :: essays research papers

Summary We have to develop an adaptive thresholding system for greyscale image binarisation. The simplest way to use image binarisation is to choose a threshold value, and classify all pixels with values above this threshold value as white and all other pixels as black. Thresholding essentially involves turning a colour or greyscale image into a 1-bit binary image. If, say, the left half of an image had a lower brightness range than the right half, we make use of Adaptive Thresholding. Global thresholding uses a fixed threshold for all pixels in the image and therefore works only if the intensity histogram of the input image contains distinct peaks corresponding to the desired subject and background. Hence, it cannot deal with images containing, for example, a strong illumination gradient. Local adaptive thresholding, on the other hand, selects an individual threshold for each pixel based on the range of intensity values in its local neighbourhood. This allows for thresholding of an image whose global intensity histogram doesn't contain distinctive peaks. The assumption behind method is that smaller image regions are more likely to have approximately uniform illumination, thus being more suitable for thresholding. Firstly, we develop a method based on the local row average to binarise the current line using that threshold. We then extend this technique to a moving window of different sizes. Method For the first part of the assignment, we develop a method based on the local row average to binarise the current line using that threshold. We consider each individual row at a time; calculate the average brightness value for that row based on the brightness values of all the pixels in that row. We then use this average value to binarise that row. We then proceed to the next row and so on. In this way we binarise the whole image. For the second part of the assignment, we make a window of user defined size around the centre pixel under consideration, calculate the average value for all the pixels in this window and then binarise that centre pixel using this average value as the threshold value. We continue this procedure till we binarise the whole image. For the pixels towards the edges of the image, we check for the number of pixels preceding the centre pixel. If this number is less than half the window size, we modify our code accordingly to take care so that we calculate the average value for that centre pixel.

Gender Differences in Mathematics Essay

Throughout the first half of the 20th century and into the second, women studying or working in engineering were popularly perceived as oddities at best, outcasts at worst, defying traditional gender norms. Female engineers created systems of social, psychological, and financial mutual support, through such strategies, conditions for female engineers changed noticeably over just a few decades, although many challenges remain. Engineering education in the United States has had a gendered history, one that until relatively recently prevented women from finding a place in the predominantly male technical world. For decades, Americans treated the professional study of technology as men’s territory. At places where engineering’s macho culture had become most ingrained, talk of women engineers seemed ridiculous (Sax, 2005). For years it’s been assumed that young women avoid careers in mathematics-based fields, like engineering and physics, because they lack confidence in their math skills. But a new study finds that it’s not a lack of confidence in their math skills that drives girls from those fields; it’s a desire to work in people-oriented professions. It has been found that young women who are strong in math tend to seek careers in the biological sciences. They value working with and for people, they don’t perceive engineering as a profession that meets that need. The environment at many tech schools is hostile toward helping students achieve a degree and is more geared toward weeding out those who are struggling. It’s difficult to come up with alternative engineering solutions if everybody in the room looks alike. That’s the initial reason why automakers and suppliers are busy trying to identify and hire minority and women engineers. The business case is that if more than half of an automaker’s customers are either female and/or people of color, which they are, then those groups need to be represented in every sector of the company. One of the most important areas for automakers to get a range of views is in product development. With that diversity mission in mind, DaimlerChrysler Corp. , Ford Motor Co. and General Motors Corp. , all have mounted aggressive programs to identify and hire minority and women engineers. At GM the story is the same. To attract minority and women engineers, the automaker proclaims that innovation comes from the people who see the world in a different way than everyone else. One women and minorities enter into the automotive engineering ranks, they need to be challenged and encouraged to develop their careers or they’ll be gone (Sax, 2005). It’s not just the Big Three that are working to create a more diverse engineering workforce. Suppliers and engineering support organizations such as the Society of Automotive Engineers are trying to draw more women and minorities into the profession. Faced with chronically small percentages of minorities and women in virtually every segment of engineering, companies are going to great lengths to attract them to the world of automotive engineering. Harvard President Lawrence Summers ignited a firestorm recently when he suggested more men than women are scientists because of differences between males and females in â€Å"intrinsic aptitude. † Many scientists-both men and women-expressed outrage at Summer’s remarks and blamed any lag in math among girls mainly on discrimination and socialization (Dean, 2006). They point out that girls have closed the gap in average scores on most standardized math tests in elementary and high school. Today women constitute almost half of college math majors and more than half of biology majors. But Summer’s supporters say he courageously raised a legitimate question for scientific inquiry. Indeed, in recent years some researchers have been pursuing a scientific explanation for the discrepancies in math and science aptitude and achievement among boys and girls and have found differences, including biological ones. Summer’s suggestion that women are biologically inferior in math infuriated many female scientists. Some asserted that the other two factors he mentioned were far more important in keeping women out of science: sex discrimination and the way girls are taught to view math as male territory. Some differences are well established. Girls do better on tests of content learned in class and score much higher on reading and writing tests than boys. Boys score higher on standardized tests with math and science problems not directly tied to their school curriculum. On tests of spatial awareness, boys do better on tests that involve navigation through space. Girls are better at remembering objects and landmarks. Studies show differences in brain structure and hormonal levels that appear to influence spatial reasoning. But the implications of these differences for real world math and science achievement remain unclear. â€Å"There is evidence that male and female brains differ anatomically is subtle ways, but no one knows how these anatomical differences relate to cognitive performance,† (Dean, 2006). At the heart of the current controversy is a societal implication-that the failure of an institution like Harvard to tenure even one woman mathematician can be blamed on the lack of top-flight women mathematicians, which in turn can be blamed on too-few top female minds in math. As evidence of intrinsic aptitude differences, Summers pointed out that more boys than girls receive top scores on standardized math tests. Today girls receive better grades than boys in math and science through high school, have closed the gap on average scores on most standardized math tests and take more advantage high school classes than boys in almost every category except physics and high-level calculus. In college they constitute nearly half the math majors and more than half the biology majors. Indeed, today a growing number of researchers contend boys are the ones who are shortchanged-judging by the larger proportion of boys in special-education classes and the declining proportion attending college. Women now make up 56 percent of students enrolled in college; by 2012, the Department of Education projects they will account for about 60 percent of bachelor’s degrees (2002). The fact that more boys than girls make top scores on standardized math tests is often invoked as evidence that boys possess an innate superiority in high-level math. Experts on both sides of the divide agree gender differences are real, even if they disagree bout how much is socially learned and how much biologically based. Girls do better on writing and on algebra problems, probably because algebraic equations are similar to sentences, and girls excel in language processing. Boys are better at mathematical word problems; girls are better at mathematical calculation. Boys and girls also differ on spatial skills, and experts are divided over how innate or important these differences are. A recent study of the Graduate Record Exam, for instance, found men did better on math problems where a spatially based solution was an advantage (Gallagher, & Kaufman, 2005). Sex hormones have been shown in several studies to affect the ability to envision an object rotating in space. Females who take male hormones to prepare for a sex-change operation improve on tests of 3-D rotation and get worse on tests of verbal fluency, at which women typically excel. During their menstrual cycle, women do better on 3-D rotation when levels of the female hormone estrogen are low; they do better on verbal fluency when estrogen levels are high. If science be taught directly with a hands-on, inquiry-based approach, it sustains girl’s interest in science. Girls like to work in cooperative teams, a lot of science was taught in a competitive mode. Women scientists also earn less than men. But it’s only fair that women who work fewer hours face the economic consequences of lower salaries and less status. References: Dean, Cornelia. (2006). â€Å"Dismissing ‘Sexist Opinions’ About Women’s Place in Science†. A Conversation with Ben A. Barres. The New York Times. July 18, 2006, pp. 1-5. Gallagher, Ann M. , & Kaufman, James M. (2005). â€Å"Gender Differences in Mathematics: An Integrative Psychological Approach. Cambridge University Press. National Center for Education Statistics, â€Å"Projections of Education Statistics To 2012†. (2002). Available on-line: http://nces. ed. gov/pubs2002/proj. 2012/ch_2. asp.. Sax, Leonard. (2005). Too Few Women- â€Å"Figure It Out†. Los Angeles Times. Jan. 23, 2005.

building suburbia essays

building suburbia essays Building Suburbia: Green fields and Urban Growth, 1820-2000 Since World War Two, American cities have gone through enormous changes. Industrial decline, crumbling homes and schools, overcrowded neighborhoods, rigid segregation and racial trauma, rising crime and violence and an alarming revenues have all contributed to a troubled urban landscape. For a short period of time in American developments, large scale government interventions seemed to point the way to urban salvation. But in the wake of massive urban renewal, expressway construction, and public housing projects, cities seem worse off than ever. Meanwhile, many sought refuge in the supposed safety of the vast new suburbs that encircled the old cities. But many found that suburban life brought new kinds of problems, such as auto dependency, increased pollution, and a loss of public life. Today, scholars, journalists, and citizens increasingly realize that urban issues are not confined to inner cities, but are broadly metropolitan and national because they involve everyone in an u rban region and the nation. Growing Suburbia is an excellent historical analysis book about suburbs in America. Dolores Hayden, author and a professor at Yale, clearly demonstrates the origin of the suburb and emphasizes the role of the federal government played in building suburbia in America. Examples Hayden uses in the book include policies to subsidizing suburbia by massively funding highways or providing many generous tax benefits to homeowners. Upon reading various chapters, Hayden gives me the impression that she is not very happy about the overly developments and the increasing social crime rates in the urban region. However, Hayden still holds an optimist mind that many issues concerning various urban experiences will improve over time. It is not surprise to find plenty of women and ethnic issues in the book. She points out many controversial topics such as womens role, class...

Essay on Branding part 2Essay Writing Service

Essay on Branding part 2Essay Writing Service Essay on Branding part 2 Essay on Branding part 2Essay on Branding part  1Challenges and opportunitiesAt the moment, Natural Kitchen faces the problem of several design challenges. First of all, the company has to focus on the improvement of the consumer trust which is essential for the creation of a reliable and reputable brand. If customers do not trust with the brand, the company cannot gain any considerable success in the market and improve its competitive position (Brown, 2003). As the company can expand its business nationwide as well as internationally, it can use its brand to expand but still keep core identity. The core identity is essential for the maintenance of the brand that means that the company should not change its brand along with the expansion of its business. The creation of a stable, recognizable and popular brand should become priorities of the company. Finally, at the moment, the company faces the problem of the co-operation with other organisations (environmental, fair-trade, lifest yle, charities, etc.). Such cooperation can influence kids and shoppers to think about environmental and sustainable issues. The interaction with public organisations, wider involvement of the company into community activities and other forms of the cooperation of Natural Kitchen with other organisations contributes to the improvement of the public image of its brand and the overall formation of the positive attitude of the public to the company’s brand.The process for selecting a suitable branding agencyThe selection of a suitable branding agency is very responsible and important process. In this regard, Natural Kitchen should elaborate key criteria for the selection of the branding agency (Peters, 2007). At this point, the selection of the branding agency should match the marketing goals of the company and its marketing strategy. Taking into consideration the opportunity of the development of the company’s operations nationwide and internationally, Natural Kitchen sh ould select the agency that operates in the UK and takes one of the leading positions in the UK branding industry (Mohrman, 2009). At the same time, the company should not refer to the branding agency which has a large staff and multiple customers because, in such a case, the company may face the problem of the standard approach used by the branding company to the creation of Natural Kitchen brand. Instead, Natural Kitchen needs the branding agency that uses the personalized approach to each customer and is capable to meet specific needs of the company and understand its uniqueness. Therefore, a relatively small agency may be the right choice for Natural Kitchen. In this regard, Rareform is the right choice of the branding agency for Natural Kitchen because this is one of the most successful branding agencies in the UK, although its annual revenue does not exceed $1 million that means that the company does not involve a large number of serious projects that may distract the best hum an resources of the branding agency. Instead, Natural Kitchen may count on the involvement of the best professionals of the successful branding agency for the creation of the new brand of the company or the improvement of the existing one.Chosen branding agency and the approach to the briefThe chosen agency, Rareform, is the successful branding agency which main approach is based on the personalized, customer-centred approach to each customer (Gitlow, 2009). In such a way, Natural Kitchen can gain considerable benefits from the cooperation with Rareform because the agency will take into consideration specificities of the company and meet its branding needs and goals (Bamberg, 2000). Rarefrom can provide Natural Kitchen with an opportunity to create the unique brand that mirrors the vision and mission of the company and helps Natural Kitchen to implement successfully its marketing strategy oriented on the national and international market expansion.Plan  Ã‚  Ã‚  Ã‚   The rollout of the new brand, phases and plan and budgetThe new brand of Natural Kitchen is the brand oriented on the delivery of healthy food products to its customers. The primary concern of the brand is the customer health. This is why the company is supplying organic food products that are healthy and safe for human health as well as natural environment. The first stage of the development of the new brand is the identification of the brand philosophy, which is the philosophy based on the healthy food for mass customers (Viardot, 2001). The next stage is the identification of the target customer group, which is customers in the UK and this customer group may expand steadily internationally. The next step is the elaboration of the methods of the brand promotion. In this regard, the company should identify tools and methods that may be applied successfully. In case of Natural Kitchen, the company should focus on the development of close company-community ties that means that the company can prom ote its brand within local communities throughout the UK. For instance, the company can sponsor health programs informing the public of the importance of the healthy food for their health. The company can sponsor meetings of health care professionals with local community members. In such a way, the company will create the image of the socially responsible company that takes care of customers’ health and offers products that may be useful for their health. The budget of the project (See App.) will need to raise funds for the implementation of the plan of the new branding policy of the company but the project will start brining return on investment during first years of the implementation of the project.  Ã‚  Ã‚  Ã‚   ROI and testingThe company can start receive return on investments during the second and third years of the implementation of the new branding policy. The first year will be the most difficult one since the company will need to invest in the creation and promoti on of its brand. The overall success of the project and its effectiveness can be measured with the help of the assessment of the recognition of the brand by customers in the UK. The assessment of the recognition of the brand of the company can be conducted with the help of surveys conducted online or via phones.ConclusionsThus, Natural Kitchen has extensive opportunities to develop its business successfully. However, the company needs to enhance its brand image because the brand of the company is very important for its marketing success. The brand influences the perception of the company by customers and influences its competitive position. Natural Kitchen should use the branding agency’s services to reach the target customer group and create a strong and attractive brand that is recognised nationwide.

My Teaching Philosophy Essay Example | Topics and Well Written Essays - 250 words

My Teaching Philosophy - Essay Example I don’t want to produce a workforce that is well educated but not competent enough to handle the complexities of the real life problems. Using moving image is a very important part of my teaching style which makes it all the more convenient for the students to understand the concept. I believe that human tendency to learn is at its best when it requires the humans to use maximum senses. Thus, if I only deliver the lecture without showing the students videos, they would only engage their hearing sense to understand me in the class. On the other hand, when I supplement my theoretical demonstration with the moving image, the students not only engage their hearing sense in the lecture, but also make use of their viewing sense. Thus, learning occurs at two levels. I also need the multimedia demonstration in order to show the videos of concepts being applied on the real life cases. Therefore, multimedia is a very essential component of my

Leap motion Essay Example | Topics and Well Written Essays - 750 words

Leap motion - Essay Example Before developing Leap Motion, Holz and Buckwald first developed a small device that resembled an iPod with an end that can be plugged into the USB port of a computer and enhance motion detection so that â€Å"the exact movements of individual fingers and rotations of the wrist can be accurately detected and processed with no latency† (Spiegelmock 2013). Leap Motion itself is 80mm long and 12.7mm wide and connects to the computer or Mac allowing a person to interact with the computer via simple hand movements. In other words, Leap Motion is a sensor that identifies each of an individual’s movement and changes them into a specific action. â€Å"Leap Motion constitutes of two LEDs and three infrared cameras that analyze al the movements of the hand† (Design, User Experience, and Usability, User Experience Design for Diverse Interaction Platforms and Environments 2012). In analyzing the movement of the hand, the device covers a radius of 1 meter. Its accuracy is also very high; it is 1/100th. Leap Motion has the potential to detect fingers, hands as well as pencils by coming up with a 3D environment. Although it works like a mouse, Leap Motion does not in any way aim at replacing the keyboard and the mouse. It is an additional tool aimed to improve the user experience as well as interaction with the computer. The use of Leap Motion in any computer calls for some things. It detects on the type of the operator system that a computer should have. For instance, the device works best and only in the computers installed with Windows 7, 8 or Mac OS X 10.6 Snow Leopard. The hardware of the computer also plays a very significant role. â€Å"Leap Motion only works in computers with either AMD Phenom (tm) II or Intel  ®core (TM) i3, i5, i7 processor, and with a 2GB of RAM† (Spiegelmock 2013). Additionally, the computer needs to have an internet connection in addition to a