De Morgan’s laws
In formal logic, De Morgan's laws are rules relating the logical operators "and" and "or" in terms of each other via negation, namely: (A U B)’=A’ ∩ B’
(A ∩ B)’=A’ U B’
The rules can be expressed in English as:
"The negation of a conjunction is the disjunction of the negations." and "The negation of a disjunction is the conjunction of the negations." The law is named after Augustus De Morgan (1806–1871) who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians (in the 14th century William of Ockham wrote down the words that would result by reading the laws out), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial. Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments. Logic gates invert all inputs to a gate reverses that gate's essential function from AND to OR, or vice versa, and also inverts the output. So, an OR gate with all inputs inverted (a Negative-OR gate) behaves the same as a NAND gate, and an AND gate with all inputs inverted (a Negative-AND gate) behaves the same as a NOR gate. DeMorgan's theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs

De Morgan’s theorem is used to simplify a lot expression of complicated logic gates. For example, (A + (BC)')'. The parentheses symbol is used in the example.

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The answer is A BC.

Let's apply the principles of DeMorgan's theorems to the simplification of a gate circuit:

...retrenchment in certain regions and products.
In our 2013 report, Survival of the Fittest
(https://www.bcgperspectives.com/content/articles/financial_institutions_corporate_strategy_portfolio_management_global_capital_markets_2013_survival_of_the_fittest/),we asked whether the CMIB industry could survive in the long term. Our answer then, as it is today, is yes, but tall challenges remain. In this, our third annual report on the global CMIB business, we take a deeper look at those challenges, particularly regarding the core dynamics—revenues, regulations, clients
—that influence the six business models that we feel are most viable. Our aim is to provide food for thought for senior management teams as they refine their strategies.
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Market Developments
We see two key challenges facing CMIB players. First, revenues have now become a scarce and volatile resource, pressuring short-term ROE. Second, regulation continues to take its toll and will remain a critical element as banks address cost structures, revenue models, and strategic positioning. Both revenue and regulatory dynamics will affect what we see as the six core business models in the CMIB industry.
The Revenue Challenge
ROE in the CMIB industry fell to 11 percent in 2013, a decline of 1 percentage point from the previous year. This level, which remains at or just below the typical cost of equity (10 to 12 percent), might seem relatively stable. But given the trend toward combined...

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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including...

...bernoulli's theorem
ABSTRACT / SUMMARY
The main purpose of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tape red duct and to measure the flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The apparatus used is Bernoulli’s Theorem Demonstration Apparatus, F1-15. In this experiment, the pressure difference taken is from h1- h5. The time to collect 3 L water in the tank was determined. Lastly the flow rate, velocity, dynamic head, and total head were calculated using the readings we got from the experiment and from the data given for both convergent and divergent flow. Based on the results taken, it has been analysed that the velocity of convergent flow is increasing, whereas the velocity of divergent flow is the opposite, whereby the velocity decreased, since the water flow from a narrow areato a wider area. Therefore, Bernoulli’s principle is valid for a steady flow in rigid convergent and divergent tube of known geometry for a range of steady flow rates, and the flow rates, static heads and total heads pressure are as well calculated. The experiment was completed and successfully conducted.
INTRODUCTION
In fluid dynamics, Bernoulli’s principle is best explained in the application that involves in viscid flow, whereby the speed of the moving fluid is increased...

...Negative Externalities and the Coase Theorem
As Adam Smith explained, selfishness leads markets to produce whatever people want. To get rich, you have to sell what the public wants to buy. Voluntary exchange will only take place if both parties perceive that they are better off. Positive externalities result in beneficial outcomes for others, whereas negative externalities impose costs on others. The Coase Theorem is most easily explained via an example
This paper addresses a classic example of a negative externality (pollution), and describes three possible solutions for the problem: government regulation, taxation and property right – a better solution to overcome the externality as described by economist Ronald Coase.
Imagine being a corn farmer and growing corn. What are the private costs that you face that help you determine production? Things like fuel, seed, fertilizer; these are your private costs. But it turns out that every spring and summer when you lay down the fertilizer some of this flows into the stream nearby and flows into a lake downstream, oftentimes resulting in large fish kills. All those downstream, the fisherman, the recreationist, and the landowners all incur a negative externality.
There are three ways in which we can address these externalities:
1- Government Regulation:
a) First, direct regulation is applied through technology-specific methods. This is where the government requires producers to use a...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that...

...BINOMIAL THEOREM :
AKSHAY MISHRA
XI A , K V 2 , GWALIOR
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set.
HISTORY :
HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century...

...The Coase Theorem
In “The Problem of Social Cost,” Ronald Coase introduced a different way of thinking about externalities, private property rights and government intervention. The student will briefly discuss how the Coase Theorem, as it would later become known, provides an alternative to government regulation and provision of services and the importance of private property in his theorem.
In his book The Economics of Welfare, Arthur C. Pigou, a British economist, asserted that the existence of externalities, which are benefits conferred or costs imposed on others that are not taken into account by the person taking the action (innocent bystander?), is sufficient justification for government intervention. He advocated subsidies for activities that created positive externalities and, when negative externalities existed, he advocated a tax on such activities to discourage them. (The Concise, n.d.). He asserted that when negative externalities are present, which indicated a divergence between private cost and social cost, the government had a role to tax and/or regulate activities that caused the externality to align the private cost with the social cost (Djerdingen, 2003, p. 2). He advocated that government regulation can enhance efficiency because it can correct imperfections, called “market failures” (McTeer, n.d.).
In contrast, Ronald Coase challenged the idea that the government had a role in taking action targeted...

...successfully employed in mechanism like the carburetor and the atomizer.
The study focuses on Bernoulli’s Theorem in Fluid Application. A fluid is any substance which when acted upon by a shear force, however small, cause a continuous or unlimited deformation, but at a rate proportional to the applied force. As a matter of fact, if a fluid is moving horizontally along a streamline, the increase in speed can be explained due the fluid that moves from a region of high pressure to a lower pressure region and so with the inverse condition with the decrease in speed.
Bernoulli’s Principle complies with the principle of conservation of energy. In a steady Flow, at all points of the streamline of a flowing fluid is the sum of all forms of mechanical energy along a streamline. It was first derived by the Swiss Mathematician Daniel Bernoulli; the theorem states that when a fluid flows from one place to another without friction, its total energy (kinetic+ potential+ pressure) remains constant.
Many of schools, academies or universities cannot provide their student an equipment which can help them in understanding fluid dynamics. They don’t have a “hands on” environment which can develop their knowledge and theoretical concepts.
Our Bernoulli’s Apparatus which is an instructional material purposes will provide for those interested viewer and learners a demonstration of related Bernoulli’s Theorem takes into effect.
Our research...