In the mode [pic]is such that the following two conditions on the random vector [pic]are met: 1. [pic]
2. [pic]
the best (minimum variance) linear (linear functions of the [pic]) unbiased estimator of [pic]is given by least squares estimator; that is, [pic]is the best linear unbiased estimator (BLUE) of [pic]. Proof:

Let [pic]be any [pic]constant matrix and let [pic]; [pic] is a general linear function of [pic], which we shall take as an estimator of [pic]. We must specify the elements of [pic]so that [pic]will be the best unbiased estimator of [pic]. Let [pic] Since [pic] is known, we must find [pic]in order to be able to specify [pic]. For unbiasedness, we have

[pic]

But, to be unbiased, [pic] must equal [pic], and this implies that [pic]for all [pic]. Thus, unbiasedness specifies that [pic]. For property of “best” we must find the matrix [pic]that minimized [pic], where [pic], subject to the restriction [pic]. To examine this, consider the covariance

[pic]

Let [pic]Then [pic]. The diagonal elements of [pic] are respective variances of the [pic]. To minimize each [pic], we must, therefore, minimize each diagonal elements of [pic]. Since [pic]and [pic]are constants, we must find a matrix [pic]such that each diagonal element of [pic]is a minimum. But [pic] is positive semidefinite; hence [pic] Thus the diagonal elements of [pic] will attain their minimum when [pic] for [pic]. But, if [pic]then [pic] Therefore, if [pic]is to equal 0 for all [pic], it must be true that [pic]for all [pic]and [pic]. This implies that [pic]. The condition[pic]is compatible with the condition of unbiasedness, [pic]. Therefore, [pic]and [pic]. This completes the proof.

...Universidad Autónoma de Querétaro
Facultad de Ingeniería
“Iterative Methods”
“Gauss and Gauss-Seidel”
Profesor | | Nieves Fonseca Ricardo |
Mentado Camacho Félix
Navarro Escamilla Erandy
Péloquin Blancas María José
Rubio Miranda Ana Luisa
Abstract
Many real life problems give us several simultaneous linear equations to solve. And we have to find a common solution for each of them. There are several techniques to use.
Instead of using methods that provide a solution to a set of linear equations after a finite number of steps, we can use a series of algorithms with fewer steps, but its accuracy depends on the number of times it is applied (also known as iterative methods). For large systems they may be a lot faster than direct methods.
We will expand on two important methods to find numerical solutions to linear systems of equations. There will be an introduction to each method, besides detailed explanations on each of them.
Normally each process is long, so they are ideal for programming.
Keywords
Iterative, algorithm, linear equation, convergence.
Objective
Understand the concepts of iterative methods, and convergence, besides the difference and usefulness between direct and iterative methods. To give a clear and understandable idea of Gauss
and Gauss-Seidel methods to solve systems of linear equations, and show how to apply them....

...Carl Gauss was a man who is known for making a great deal breakthroughs in the wide variety of his work in both mathematics and physics. He is responsible for immeasurable contributions to the fields of number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics, as well as many more. The concepts that he himself created have had an immense influence in many areas of the mathematic and scientific world.
Carl Gauss was born Johann Carl Friedrich Gauss, on the thirtieth of April, 1777, in Brunswick, Duchy of Brunswick (now Germany). Gauss was born into an impoverished family, raised as the only son of a bricklayer. Despite the hard living conditions, Gauss's brilliance shone through at a young age. At the age of only two years, the young Carl gradually learned from his parents how to pronounce the letters of the alphabet. Carl then set to teaching himself how to read by sounding out the combinations of the letters. Around the time that Carl was teaching himself to read aloud, he also taught himself the meanings of number symbols and learned to do arithmetical calculations.
When Carl Gauss reached the age of seven, he began elementary school. His potential for brilliance was recognized immediately. Gauss's teacher Herr Buttner, had assigned the class a difficult problem of addition in which the students were to find the sum of the integers from one to one hundred. While...

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

...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...

...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...

...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...

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