Sunday, January 26, 2020
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 à ¢Ã‹â€ ’ 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€ ’ 1 as a product of integers > 1. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹â€ ’ 1 must be 1 plus a multiple of 2n. This holds even when 2n à ¢Ã‹â€ ’ 1 is prime. Examples: Example I: 25 à ¢Ã‹â€ ’ 1 = 31 is prime, and 31 is multiple of (2ÃÆ'-5) +1 Example II: 211 à ¢Ã‹â€ ’ 1 = 23ÃÆ'-89, where 23 = 1 + 2ÃÆ'-11, and 89 = 1 + 8ÃÆ'-11. Proof: If m divides 2n à ¢Ã‹â€ ’ 1 then 2n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). By Fermats Theorem we know that 2(m à ¢Ã‹â€ ’ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). Assume n and m à ¢Ã‹â€ ’ 1 are comparatively prime which is similar to Fermats Theorem that states that (m à ¢Ã‹â€ ’ 1)(n à ¢Ã‹â€ ’ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n). Hence there is a number x à ¢Ã¢â‚¬ °Ã‚ ¡ (m à ¢Ã‹â€ ’ 1)(n à ¢Ã‹â€ ’ 2) for which (m à ¢Ã‹â€ ’ 1) ·x à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n), and thus a number k for which (m à ¢Ã‹â€ ’ 1) ·x à ¢Ã‹â€ ’ 1 = kn. Since 2(m à ¢Ã‹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€ ’ 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€ ’ 1)x/2kn = 2(m à ¢Ã‹â€ ’ 1)x à ¢Ã‹â€ ’ kn à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). But by meaning, ( m à ¢Ã‹â€ ’ 1)x à ¢Ã‹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€ ’ 1 are relatively prime is unsustainable. Since n is prime m à ¢Ã‹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€ ’ 1 must be larger than n. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹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€ ’ 1 is a Mersenne prime, then the congruence does not satisfy. By Fermats Little theorem, m | p à ¢Ã‹â€ ’ 1. Now write, p à ¢Ã‹â€ ’ 1 = mÃŽÂ ». If the given congruence satisfies, then p2 | 2mÃŽÂ » à ¢Ã‹â€ ’ 1, therefore Hence 2m à ¢Ã‹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€ ’ 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€ ’ 1 = 23765203727 ÃÆ'- q, where q is a likely prime. Generalization: The binary depiction of 2p à ¢Ã‹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€ ’1 = 111112 235à ¢Ã‹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€ ’1(2n à ¢Ã‹â€ ’ 1): n = 2:  2(4 à ¢Ã‹â€ ’ 1) = 6 n = 3:  4(8 à ¢Ã‹â€ ’ 1) = 28 n = 5:  16(32 à ¢Ã‹â€ ’ 1) = 496 n = 7:  64(128 à ¢Ã‹â€ ’ 1) = 8128. Noticing that 2n à ¢Ã‹â€ ’ 1 is a prime number in each instance, Euclid proved that the formula 2nà ¢Ã‹â€ ’1(2n à ¢Ã‹â€ ’ 1) gives an even perfect number whenever 2p à ¢Ã‹â€ ’ 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)    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  or  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  = 7 MM3 = 2(8-1)-1  = 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.
Saturday, January 18, 2020
General Motors Corporation General Motors Corporation
â€Å"General Motors is one of three leading automotive manufacturing companies in the United States. â€Å"General Motors merupakan salah satu dari tiga perusahaan manufaktur otomotif terkemuka di Amerika Serikat. Based in Michigan in 1903 by Henry ford and grew to reach revenue of $150 billion and more than 370,000 employees by 1996. Berbasis di Michigan pada tahun 1903 oleh Henry ford dan tumbuh untuk mencapai pendapatan sebesar $ 150 miliar dan lebih dari 370. 00 karyawan pada tahun 1996. In the 1970's, the automobile market for the major auto makers – General Motors (GM), Ford, and Chrysler- was crunched by competition from foreign manufactures such as Toyota and Honda. Pada 1970-an, pasar mobil untuk para pembuat mobil utama – General Motors (GM), Ford, dan Chrysler-adalah berderak oleh persaingan dari luar negeri manufaktur seperti Toyota dan Honda.In 1999, Ford acquired the Swedish Volvo model in an attempt to compete in the foreign market and expand to other regions. †Pada tahun 1999, model Ford mengakuisisi Volvo Swedia dalam upaya untuk bersaing di pasar asing dan memperluas ke daerah lain. †General Motors needs to use the business process reengineering for the information systems infrastructure to cut redundancies and requiring information process and the link among Ford centre in world wide.General Motors perlu menggunakan rekayasa ulang proses bisnis untuk infrastruktur sistem informasi untuk memotong redundancies dan memproses informasi membutuhkan dan link di antara pusat Ford di seluruh dunia. â€Å"General Motors implemented a 3-year plan to consolidate their multiple desktop systems into one. â€Å"General Motors mengimplementasikan rencana 3 tahun untuk mengkonsolidasikan beberapa sistem desktop mereka menjadi satu. This new process involved replacing the numerous brands of desktop systems, network Proses baru ini melibatkan berbagai merek menggantikan sistem
Friday, January 10, 2020
My Family History Essay
â€Å"In all of us there is a hunger, marrow deep, to know our heritage – to know who we are and where we came from. Without this enriching knowledge, there is a hollow yearning. No matter what our attainments in life, there is still a vacuum, emptiness, and the most disquieting loneliness. †–Alex Haley This quote explained to me the importance of my grandparent’s legacy and their history. A long twisting family tree inspires one who does not know where their roots originated. My grandfather Frank Douglas and my grandmother Delores Jones gave me a reason to find out where our legacy started. My grandfather Frank Kelow was adopted into a four person white family, which gave him the last name of Douglas. My grandfather was born on February 12, 1902. Frank was raised in Greenville, Mississippi with dozens of cousins, which gave him comfort. Frank’s biological parents did not attend college; in fact, they didn’t even graduate from high school. In Mississippi, â€Å"I was surrounded by racism, slavery, and poverty, which gave me the inspiration to give my father a better life†(Douglas). As a young kid Frank often hung out in the streets with his friends and partied a lot. He was a heavy smoker with a tiny taste for alcohol. â€Å"Growing up in a poor neighborhood I was introduced to a lot of bad things such as drugs, gambling, and fighting†(Douglas). Around the house, Frank was responsible for mowing the lawn, taking out the trash, and cleaning the pool. At the age of 21 my grandfather entered the army and decided to fight in World War II. After the war concluded, my grandfather married and moved to Queens, New York. Frank and his wife made history that day because they were the first black couple to move into the neighborhood, which they lived. This was the birthplace of my father Lance Douglas Sr. My grandmother Delores Jones was born on December 14, 1906, into a family of four. She was also raised through poverty, but with the help of her brother and cousins she found a way to stick it out. She was raised in New Orleans, Louisiana where her parents worked several jobs to maintain the tiny shack she was raised in. â€Å"Back in my day society consisted of smoking cigarettes, drinking beer, and partying heavily†(Jones). At the age of 13, she was required to work to earn extra money around the house. Some chores my grandmother had around the house was to clean the house, wash the dishes, wash clothes, and pull weeds from the lawn. The relationship between my grandmother and her parents was quite the opposite of mine with my parents. â€Å"After completing my chores, I was allowed to do basically whatever, as long as I was in the house at a reasonable hour†(Jones). Delores was a very social person. â€Å"I rarely spent time with my grandparents†(Jones). During her high school years she was often looked at as beautiful, ambitious, and persistent. At the age of 18 she was elected as prom queen for her senior dance. Although she was often free to do what she wanted, she was also held responsible. Delores was sometimes whooped and grounded for disobeying curfew rules and not completing her chores. This gave her everlasting the mentality of you must work for everything you want in life. I was born in Mississauga, Canada on the date of February 23, 1993. The name Kobie was given to me by my mother, it means warrior. Raised in a family with both parents, one-brother, and one sister, I was surrounded by people who loved me. My brother, Lance Douglas, was born four years earlier than me. Likewise, my sister was born two years prior of my birth. At the age of two, my parents decided to move to Plano, Texas, a beautiful city with the population of about 700,0000 people. As usual, around five I attended Kindergarten at the local school where my brother and sister attended elementary school. Being the youngest in the family provided both advantages and disadvantages. My brother and sister inherited my father’s gene of aggravating me to the point of physical confrontation, which later led to me getting beat up. Although women are usually on the feminine side, my sister was completely different. When my teenage years came around that’s when my siblings began to lighten up on the bullying. During my high school years I was considered the man on campus. I was an all-star at basketball, football, and baseball. During my senior year I only participated in football and was offered a scholarship to play for the Louisiana Lafayette, Ragin Cajuns. Now, as a freshman at the University of Louisiana at Lafayette I am living the life I once dreamed about; experiencing things I never thought I would. For example, going to the club on Mondays, Wednesdays, Fridays, and Saturdays. A student athlete who is enrolled in 17 hours and is also committed to football. Waking up at five a. m to workout on Mondays, Wednesdays, and Thursdays. Most of all, living the dream people told me wasn’t meant. All across the world there are families who have their own original legacies. In all of us there is a hunger, marrow deep, to know our heritage – to know who we are and where we came from. Without this enriching knowledge, there is a hollow yearning. No matter what our attainments in life, there is still a vacuum, emptiness, and the most disquieting loneliness (Haley page 1). Its up to one to figure out how and where their family started. My grandfather Frank Douglas and my grandmother Delores Jones gave me a reason to find out where our legacy started. My grandparents have told me many things I never thought I would know about which has expanded my knowledge for the better.
Thursday, January 2, 2020
Meaning and Scope of Managerial Economics - 2130 Words
MEANING SCOPE AND METHODS OF MANAGERIAL ECONOMICS INTRODUCTION Emergence of managerial economics as a separate course of management studies can be attributed to at least three factors.: (a) growing complexity of business decision making process due to changing market conditions and business environment (b) consequent upon, the increasing use of economic logic , concepts theories and tools o economic analysis in the process of business decision making (c) Rapid increase in demand for professionally trained managerial manpower The growing complexity of business decision- masking has inevitably increased the application of economic concepts, theories and tools of economic analysis in this area. The reason is that making an†¦show more content†¦Example Footwear Industry: Shoes may became outdated because fashions change very rapidly Dressmaking Industry : As fashion changes very often, the manager of a dress making industry will have to keep abreast of changing fashions. The Problem of forward planning When a manager has to plan for the future he has to make an accurate forecast of the position of raw materials, the prices of factors of production, the market prices of his products , the demand for then ,etc. SCOPE OF MANAGERIAL ECONOMICS In general, the scope of managerial economics comprises all those economic concepts, theories and tools of analysis related to demand prospects, production and cost, market structure, level of competition and general business environment to find solutions to practical business problems. In other words, managerial economics is economics applied to analysis of business problems and decision making. The areas of business issues to which economic theories can be directly applied can be broadly divided into two categories : 1. Microeconomics app The scope of managerial economics emanates from the solutions it offers to the manager who is confronted with problems such as : CHOICE OF COMMODITY what to produce CHOICE OF SIZE OF THE FIRM how much to produce CHOICE OF TECHNOLOGY choosing the factor combination CHOICE OFShow MoreRelatedWeek 4 Best Buy Case Study Homework Assignment904 Words  | 4 Pages09/03/2016. Through the course of the history of Best Buy as a firm, during and over the years, managerial changes have occurred, having being infrequent (meanings, rare, as uncommon, not occurring often, being exceptional) through internal leadership, had some-to-little impact about Best Buy’s overall strategy, because of promotions and managerial succession or managerial failure(s). Why did managerial have any succession or failure(s) have little impact? 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(2012) ‘Factors favoring managerial effectiveness: a study of select public and private sector organizations’, International Journal of Business Economics and Management Research, 2(1), Pp. 1-27. Adewuyi, D. (2002) ‘Comparison between school effectiveness characteristics and classroom instruction strategies in the United States and Nigeria’, Council for the Development of Social Science Research in Africa, XXVI, Pp. 263-287. Agarwal, V. (1983) ‘A study of stress pronenessRead MoreWeek 4 Best Buy Case Study Homework Assignment1164 Words  | 5 PagesJackson Sr. 09/03/2016. Through the course of the history of Best Buy as a firm, during and over the years, managerial changes have occurred, having being infrequent, like rare, uncommon, not occurring often, nor being exceptional, through internal leadership. For this purpose, had little to some impact about Best Buy’s overall strategy, because of promotions and managerial succession or managerial failure(s). In contrast, management does have either succession, or failure(s). Failures are consideredRead MoreA Comparative Analysis Between Aboriginal and Chinese Cultural Differences1504 Words  | 7 Pagesglobalisation and internationalisation of trade in resources, especially human capital, has made managerial of culture differences ever so important as to understanding the different variables (Hofstede G, 2010). In this paper, the Power Distance Index (PDI) of both Chinese and aboriginal traditional cultural value and belief in family, social life and workplace is going to be examined under the scope of Hofstede’s dimensions. To a truthful stereotype, both culture heavily emphasis upon power distanceRead MoreManagerial Questions On Managerial Economics1736 Words  | 7 Pages MANAGERIAL ECONOMICS MANDIP SINGH SETHI K1300050 TABLE OF CONTENTS INTRODUCTION†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦..3 TYPES OF DISECONOMIES†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.5 DECENTRALIZATION†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦6 CONCLUSION†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦..7 REFERENCE†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.8 What are managerial diseconomies of scale and what, if anything, can be done about this phenomenon? DecreasingRead MoreStrategic Fit1574 Words  | 7 Pagesits resources and capabilities with the opportunities in the external environment. The matching takes place through strategy and it is therefore vital that the company have the actual resources and capabilities to execute and support the strategy. Meaning of Strategic Fit The contingency theorist’s argument that performance outcomes are maximized when a firm achieves an alignment or â€Å"fit†between a firm’s external environment, its internal factors and its strategy has been well established in the literature
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