Bernoulli’s theorem
i
Bernoulli’s theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar. First derived (1738) by the Swiss mathematicianDaniel Bernoulli, the theorem states, in effect, that the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure, the gravitational potential energy of elevation, and the kinetic energy of fluid motion, remains constant. Bernoulli’s theorem is the principle of energy conservation for ideal fluids in steady, or streamline, flow and is the basis for many engineering applications. Bernoulli’s theorem implies, therefore, that if the fluid flows horizontally so that no change in gravitational potential energy occurs, then a decrease in fluid pressure is associated with an increase in fluid velocity. If the fluid is flowing through a horizontal pipe of varying cross-sectional area, for example, the fluid speeds up in constricted areas so that the pressure the fluid exerts is least where the cross section is smallest. This phenomenon is sometimes called the Venturi effect, after the Italian scientist G.B. Venturi (1746–1822), who first noted the effects of constricted channels on fluid flow.

Application of Bernoulli's Theorem

When we blow air over a strip of paper as shown in the above figure, we find that the paper moves up. This is because, on blowing air, the velocity of air increases, creating low pressure above the paper and high pressure below the paper. This difference in pressure, lifts the paper

The working of spray-gun is based on Bernoulli's theorem

When the rubber bulb is squeezed, air is blown into the tube A, due to which, low pressure and high velocity is created. Since this pressure is less than the atmospheric pressure, the liquid is pushed up. This rising liquid is sprayed out of...

...Exposition of a New Theory on the Measurement of Risk
Daniel Bernoulli
Econometrica, Vol. 22, No. 1. (Jan., 1954), pp. 23-36.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28195401%2922%3A1%3C23%3AEOANTO%3E2.0.CO%3B2-X
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...Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to the Pythagorean Theorem, the sum of the areas of the two red...

...Historical Account:
Pythagoras, the namesake and supposed discoverer of the Pythagorean Theorem, was born on the Greek island of Samos in the early in the late 6th century. Not much is known about his early years of life, however, we do know that Pythagoras traveled through Egypt in the attempt to learn more about mathematics.
Besides his famous theorem, Pythagoras gained fame for founding a group, the Brotherhood of Pythagoreans, which was dedicated solely to study of mathematics and worship of numbers. Pythagoras passed on his belied that numbers are in fact the true "rulers of the universe".
While studying in Egypt, it is believed that Pythagoras studied with people known as the "rope-stretchers", the same people who engineered the pyramids. By using a special form of a rope tied in a circle with 12 evenly spaced knots, they discovered that if the rope was pegged to the ground in the dimensions of 3-4-5, the rope would create a right triangle. The rope stretchers used this principle to help accurately lay the foundations of for their buildings.
It was this fascination with the rope stretchers 3-4-5 triangle that ultimately led to the discovery of the Pythagorean theorem. While experimenting with this concept by drawing in the sand, Pythagoras found that if a square is drawn from each side of the 3-4-5 triangle, the area of the two smaller squares could be added together and equal the area of the large square....

...Binomial, Bernoulli and Poisson Distributions
The Binomial, Bernoulli and Poisson distributions are discrete probability distributions in which the values that might be observed are restricted to being within a pre-defined list of possible values. This list has either a finite number of members, or at most is countable.
* Binomial distribution
In many cases, it is appropriate to summarize a group of independent observations by the number of observations in the group that represent one of two outcomes. For example, the proportion of individuals in a random sample who support one of two political candidates fits this description. In this case, the statistic is the count X of voters who support the candidate divided by the total number of individuals in the group n. This provides an estimate of the parameter p, the proportion of individuals who support the candidate in the entire population.
The binomial distribution describes the behavior of a count variable X if the following conditions apply: the number of observations n is fixed, each observation is independent and represents one of two outcomes ("success" or "failure") and if the probability of "success" p is the same for each outcome.
If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p).
* Bernoulli distribution
In probability theory and statistics, the Bernoulli distribution, named after...

...-------------------------------------------------
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...

...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...

...The Sylow theoremsThe Sylow Theorems
Here is my version of the proof of the Sylow theorems. It is the result of
taking the proof in Gallian and trying to make it as digestible as possible. In
particular, I tried to break the long proof into bite-sized pieces. The main
goal here is to convey an overview of how the ingredients fit together, so I'll
skip lightly over some of the details.
The prerequisites are basically all of the group theory that came before the
Sylow theorems in this course, including: Lagrange's theorem, the first and
second isomorphism theorems, and the orbit-stabilizer theorem. I'll also use
Cauchy's theorem, even though the book lists it as a corollary to the Sylow
theorems (more on that later). I'll assume you know the definition of a
Sylow-subgroup and all the terms in the statements of the Sylow theorems.
From now on, G is a finite group and p is a prime number that divides the order
of G. Recall that a p-subgroup of G is a subgroup of G with order equal to a
power of p.
Definition. A maximal p-subgroup of G is a p-subgroup of G that is not contained
in any larger p-subgroup of G.
This is only a temporary definition, since it will turn out that a "maximal
p-subgroup" is just the same thing as a "Sylow p-subgroup". However it would be
circular logic to assume this. Hypothetically, you could...