All posts by Dr. Vrajlal Sapovadia

https://www.techandtrain.com/mentors/vraj.html
Education

21st Century Business Management Education: Neither Content nor Pedagogy, Essence is Integration with Triple Bottom Line

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Abstract:

In 21st century, many questions clutter our horizon. The way business is passing through sudden and continuous changes; the new business management norms are created every day. The business management education should follow the suit; rather provide lead to the business. Visibly the technological disruptions, social expectations and globalization demands better understanding of its impacts on economy, society and environment in terms of the costs, and benefits. This paper constructs argument emphasizing on core principle of business management education that any course delivery should integrate with triple bottom line. The opportunity costs are enormous if industry or institute fails to do so. Spender rightly poised the questions in his research. What are business schools, and what should they be? What are the social, business, or personal purposes of management education? And how might management education evolve next to meet society’s present needs (J.C. Spender, 2016)? The key questions business school should address revolves around subjects that should be included in syllabus, content of the courses, teaching pedagogy or learning mechanism and recruiting students with right aptitude, attitude and temperament. Business schools attempts to address one or all those ingredients. The more important missing element in business management education is integration of course delivery with the triple bottom line. The existence of business is economic or accounting profit. To be a sustainable business; social and environmental performance of business cannot be ignored. Each course included in the syllabus has a purpose; to help business to enhance triple bottom line. 


Each business problem is unique and one must be able to find solution optimally that fits to the unique situation. There is no one solution which is best to solve a problem. You must be persistent to solve problems on a continuous basis until desired result is obtained. Teacher can guide student, can ignite student’s mind to think beyond horizon. A teacher can expand thinking horizon of the student. In real life situation, a teacher will not accompany student. Student must equip himself to solve the business problem. Therefore, the business management pedagogy seeks involvement of the student while learning. Your solution must be feasible and acceptable to your economic and social surrounding. To solve a problem, you must have information. Scarcity of information is not a problem but abundance information is rather a big challenge. The current age is full of information accessible on public platforms, big challenge is to recognize and extract relevant and reliable information from the information ocean. The next level challenge in this endeavor is to identify real life problems, the application information, and to solve them efficiently and effectively. Learning is a lifelong process. You have to improve your skills on a continuous basis. Without mastering ability to learn new skills, one will become irrelevant. Technology guides you but it also misguides you. It is your ability to judge veracity, relevance, reliability and usefulness of information to churn out the right information.  


Business problems are now seen from prism of economic, social and environmental aspects. Twenty-first-century learning encompass mastery in content producing, synthesizing, and evaluating information from a wide variety of subjects and sources with an understanding of and respect for diverse cultures beside economic understanding. The pillar of success in the 21st century is about knowing how to learn independently. The learning now is eventually be “learner-driven.” The 21st century learning builds upon such past conceptions of learning as “core knowledge in subject areas” and recasts them for today’s world, where a global perspective and collaboration skills are dynamic, critical and focused. It’s no longer enough to “know things”, but to know things to find solution that is economically feasible, socially acceptable and environment friendly. It’s even more important to stay curious about finding out things. We have powerful learning tools at our disposal that allow us to locate, acquire, and even create knowledge much more quickly than our predecessors. Ability to recognize and acquire skills to fit in ever-changing environments is sine-qua-non. No one will tell you what skills are required and the way to acquire it. The self-management is the key to succeed in 21st century. We strongly argue that business management course content should be rich, contemporary, reflecting real life situation, relevant and providing lead to future industry. The course delivery should be learner centered and integrated with triple bottom line.

Vrajlal Sapovadia (Ph.D.)
United States


Mathematics

Mystery of ‘Perfect Numbers’ Resolved – Perfect Number is Always Even and Predictable

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Mystery of ‘Perfect Numbers’ Resolved – Perfect Number is Always Even and Predictable

Vrajlal Sapovadia (Ph.D.)
Sweta Patel (Ph.D.)

Introduction

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. In other words, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n. To explain in practical terms, we elaborate first few Perfect Numbers. It may be noted that ‘Perfect Numbers’ are sparse are thinly dispersed. Starting from 3rd Century BC, mathematicians are working on Perfect Numbers. Till April 2018, i.e. during last 2300 years active research, researchers could recognize only 50 perfect numbers. There are 2 perfect numbers in first 100 and 4 in first million. Absolute distance between two perfect numbers increase exponentially as you go higher to the next perfect number[1]. One can find at least one perfect number till 4 digit numbers, and then it becomes a real rarity. Subsequent perfect numbers appears at 8, 10, 12 and 19 digits. 15th perfect number has 770 digits while 16th have 1327 digits. 25th perfect number has 13066 digits. 50th perfect number has 46,498,850 digits.

The current literature is still debating on two issues:

  1. Can perfect number is predictable?
  2. Can perfect number be odd?

We argue that perfect number is predictable and we have developed a formula which answers both lead questions as follow:

  1. Perfect number is predictable
  2. Perfect number is always even

Predictability

Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Euclid, Prop. IX.36). the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number, as follows:

for p = 2:   21(22 − 1) = 6

for p = 3:   22(23 − 1) = 28

for p = 5:   24(25 − 1) = 496

for p = 7:   26(27 − 1) = 8128.

Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number.[11] In fact, Mersenne primes are very rare—of the 2,270,720 prime numbers p up to 37,156,667,[12] 2p − 1 is prime for only 45 of them.

Nicomachus (60–120 AD) conjectured that every perfect number is of the form 2p−1(2p − 1) where 2p − 1 is prime.[13] Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of that form.[14] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem. As of January 2018, 50 Mersenne primes are known,[15] and therefore 50 even perfect numbers (the largest of which is 277232916 × (277232917 − 1) with 46,498,850 digits).

Owing to their form, 2p−1(2p − 1), every even perfect number is represented in binary as p ones followed by p − 1  zeros. Interestingly, when a perfect number is converted into binary, it is not only a pernicious number, but binary sequence is spectacular having all 1 on the left side followed by all 0. Interestingly count of 1 is a prime number (p) and 0 is p-1.

610 = 1102

1 (p = 2) and 0 (1)

2810 = 111002

1 (p = 3) 0 (2)

49610 = 1111100002

1 (p = 5) 0 (4)

812810 = 11111110000002

1 (p = 7) 0 (6)

3355033610 = 11111111111110000000000002

1 (p = 13) 0 (12)

858986905610 = 1111111111111111100000000000000002

1 (p = 17) 0 (16)

13743869132810 = 11111111111111111110000000000000000002

1 (p = 19) 0 (18)

230584300813995212810 = 11111111111111111111111111111110000000000000000000000000000002

1 (p = 31) 0 (30)

Thus every even perfect number is a pernicious number. Note that every even perfect number is also a practical number. Therefore a formula to find a perfect number can be developed as 1….(p) 0…(p-1), where 1 (p) and 0 (p-1) are binary symbol. Thus, a binary number so written equal to a PRIME (p) ‘1’ followed by p-1 ‘0’ could be a perfect number. It may be noted that all prime count does not result into perfect number. Therefore, it is pertinent to test each prime number with formula[2] will establish whether resultant number is perfect number or not. But in any case, this will reduce substantially the experiment time to find next perfect number or this formula provides a lead to find perfect number with less experiment time.

Odd vs. Even

It is unknown whether there is any odd perfect number, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid’s rule gives all perfect numbers, thus implying that no odd perfect number exists. More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also Ore’s harmonic numbers, and it has been conjectured as well that there are no odd Ore’s harmonic numbers other than 1. An exhaustive search by the GIMPS[3] distributed computing project has shown that the first 46 are all even numbers represented by 2p−1(2p − 1).

First 50 perfect numbers listed[4] are all even and their last one or two digits are always 6 or 28. In support of arguments made by various researchers, we found that a perfect number can be presented as binary with formula 1….(p) 0…(p-1). Any binary pattern as 1 (n) 0 (n-1) will always result into even number. Therefore any perfect number is always a even number.

[1] 6 (1), 28 (2), 496 (3), 8128 (4), 33550336 (8), 8589869056 (10), 137438691328 (12), 2305843008139952128 (19)
[2] Binary numbers 1….(p) 0…(p-1)
[3] The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
[4] https://en.wikipedia.org/wiki/List_of_perfect_numbers

Education

Do educational institutes fulfill their obligation to supply proper workforce?

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This question is negatively answered for most of the institutes across the globe. India is not an exception.

There is wide gap in the skills required by the industry vis-à-vis supplied by educational institutes. Over 75 per cent graduates are not employable.

What are the reasons for this gap, quantum and the nature of gap? It is observed that there is no link between the institutes and industry. Institutes are lagging to industry. Industry is actively working to beat the competition and hence brings innovations in the products, processes, strategies, business models and techniques.

Industry creates knowledge through research; it uses research in real life to solve problems. Industry pays for solving problems, this
motivates employee to perform. It is a question of survival and growth for industry and employee.

Institutes are designing curricula based on what has happened in the industry, rather than what will happen in the industry or market. Institutes have very narrow focus while they teach students. They are governed by rules rather than reality. The curriculum focuses on prescribed and sink knowledge.

Industry requires dynamism. Institutes teach to solve existing problems. Industry requires people who can solve new problems. Teacher and students’ performance is measured on wrong calibration of examinations.

Examination systems are built to test memories rather than ability to solve real life problems. Subject knowledge is imparted in isolation.

A good student may be capable to solve a problem in isolation, but real life situation in industry is complex; and hardly in isolation.  Real life problems are complex, dynamic and resolved through team work. Students are prepared and tested in isolation.

Team work and communication in industry is typical and sine-qua-non for success in industry. Education institutes fail to realize and gauge this problem.