Mathematical & Real Analysis

Section- A

1-Discuss about the Riemann Stieltjes integrals.

The simplex method is a widely used algorithm for solving linear programming problems. Linear programming is a method used to optimize a linear objective function subject to constraints represented by linear equations or inequalities. The simplex method is based on the principle of iteratively improving the current solution by moving to an adjacent solution that has a better objective value.

The simplex method begins with an initial feasible solution and then repeatedly moves to an adjacent feasible solution until it reaches an optimal solution. The method uses a set of non-basic variables and a set of basic variables. Non-basic variables are the variables that are not currently part of the basis and are used to represent the objective function and the constraints.


 Basic variables are the variables that are currently part of the basis and are used to represent the current solution.

In each iteration of the simplex method, an entering non-basic variable is chosen and a leaving basic variable is chosen. The entering non-basic variable is the variable that will be added to the basis and the leaving basic variable is the variable that will be removed from the basis. This change in the basis results in an improvement in the objective value.

Artificial variables are a special type of non-basic variable that are used to represent constraints in the problem. They are introduced when there are constraints that are less than or equal to zero. They are used to represent the slack in the constraints, and are set to zero in the final solution. Artificial variables are used when the constraints are not in a standard form.

In summary, the simplex method is an algorithm used to solve linear programming problems by iteratively moving to an adjacent feasible solution that has a better objective value. It uses the principle of non-basic and basic variables. Non-basic variables are the variables that are not currently part of the basis and are used to represent the objective function and the constraints. Basic variables are the variables that are currently part of the basis and are used to represent the current solution. Artificial variables are a special type of non-basic variables that are used to represent constraints in the problem and are set to zero in the final solution.

 

2-Write a note on Convergence of the sequence.

Convergence of a sequence is a fundamental concept in mathematics that is used to study the behavior of sequences of numbers. A sequence is a set of numbers that are arranged in a specific order, and the convergence of a sequence refers to the way in which the sequence approaches a particular value or set of values. In other words, it is a measure of how closely the terms of a sequence approach a specific number or set of numbers.

There are several different types of convergence, each with its own characteristics and applications. For example, pointwise convergence refers to the convergence of a sequence of functions at a specific point, while uniform convergence refers to the convergence of a sequence of functions over an entire interval.

One of the most common types of convergence is called Cauchy Convergence. A sequence is said to be Cauchy if for any given positive real number "epsilon" there exists a natural number "N" such that for any two terms in the sequence which are greater than "N" the difference between them is less than "epsilon". This means that, as the terms of the sequence get larger, the difference between them gets smaller.

Another type of convergence is called almost everywhere convergence. A sequence of functions is said to be almost everywhere convergent if it converges at all points except for a set of measure zero. This means that the sequence of functions converges at almost every point.

Another important concept related to the convergence of a sequence is the limit. The limit of a sequence is the number or set of numbers that the sequence approaches as the terms of the sequence get larger. The limit is often represented by the symbol "L" and is written as lim n->infinity (a_n)=L.

In many cases, the limit of a sequence can be calculated using a formula or a set of rules. However, in some cases, it is not possible to calculate the limit exactly and it has to be approximated using numerical methods.

Convergence of a sequence is a powerful tool for understanding the behavior of sequences of numbers and can be used in a wide range of areas, including calculus, probability theory, statistics, and many other areas of mathematics. In actuarial science, sequences are used to model the behavior of insurance premiums, claims, and other financial data over time. Actuaries use the concepts of convergence and limits to better understand the patterns and trends in data and to make more accurate predictions about the future.

In addition, the concept of convergence is also used in optimization techniques. In optimization, the goal is to find the minimum or maximum of a function, which can often be represented by a sequence of values. The optimization algorithm aims to find the point where the sequence converges to the minimum or maximum of the function.

In conclusion, the concept of convergence of a sequence is a fundamental concept in mathematics that is used to study the behavior of sequences of numbers. It is used to understand the patterns and trends in data, make more accurate predictions about the future, and optimize the performance of different systems. Actuaries use the concepts of convergence and limits to better understand the patterns and trends in data and to make more accurate predictions about the future. The convergence of a sequence is a powerful tool that can be used in a wide range of areas, including calculus, probability theory, statistics, and many other areas of mathematics, and in actuarial science.

 

3-State and prove Baire’s theorem.

 

Baire's theorem, also known as Baire's category theorem, is a fundamental result in topology and functional analysis that provides conditions under which a complete metric space is non-empty. It was first proved by French mathematician René-Louis Baire in 1899.

The theorem states that if a complete metric space X is the countable union of nowhere dense sets, then X is not empty. A set is said to be nowhere dense if its closure has an empty interior. A set is said to be closed if it contains all its limit points.

The proof of Baire's theorem is based on the contrapositive statement: 

if X is empty, then X cannot be the countable union of nowhere dense sets. To prove this, let X be a complete metric space and suppose that X is the countable union of nowhere dense sets, denoted by {A_n}. To prove that X is not empty, we will show that the intersection of the sets A_n is not empty.

First, we will show that the intersection of the closures of the sets A_n is not empty. 

Let B_n be the closure of A_n. Since A_n is nowhere dense, B_n has an empty interior. Therefore, the intersection of the B_n is also nowhere dense. But since X is complete, the intersection of the B_n is closed. Therefore, the intersection of the B_n is not empty.

Now we will show that the intersection of the sets A_n is not empty. Since the intersection of the B_n is not empty, there exists a point x in the intersection of the B_n. Since x is in the closure of A_n, there exists a sequence of points in A_n that converges to x. Since X is complete, this sequence of points must also converge to x. Therefore, x is in the intersection of the A_n.

As a result, we have proved that if a complete metric space X is the countable union of nowhere dense sets, then X is not empty.

Baire's theorem has many important applications in mathematics and science, including in functional analysis, where it is used to prove that certain spaces are non-empty, and in topology, where it is used to prove that certain spaces are connected. It also has applications in computer science, where it is used to prove the existence of certain algorithms and in physics, where it is used to prove the existence of certain solutions to differential equations.

In actuarial science, Baire's theorem is used to prove the existence of certain types of insurance products and to estimate the risk of different types of claims. Actuaries use the theorem to better understand the patterns and trends in data and to make more accurate predictions about the future.

In conclusion, Baire's theorem is a fundamental result in topology and functional analysis that provides conditions under which a complete metric space is non-empty. It was first proved by French mathematician René-Louis Baire in 1899. The theorem has many important applications in mathematics and science, including in functional analysis, topology, computer science, physics, and actuarial science. Actuaries use Baire's theorem to better understand the patterns and trends in data and to make more accurate predictions about the future.

 

Section - B

 

1-Define about the Hahn & Jordan decomposition.

The Hahn-Jordan decomposition is a mathematical concept used in measure theory and probability theory. It is a method for decomposing a measure into two parts: a singular part and an absolutely continuous part. The decomposition is named after the mathematicians Hans Hahn and Camille Jordan, who independently developed the concept in the early 20th century.

The Hahn-Jordan decomposition can be applied to any measure, which is a function that assigns a non-negative value to subsets of a set. The decomposition is used to split the measure into two parts: one part that is singular with respect to another measure, and another part that is absolutely continuous with respect to that measure.

The singular part of the measure is the part that is not absolutely continuous with respect to the other measure. It is a measure that assigns a non-negative value to subsets of a set, but cannot be represented as the derivative of a non-negative function. In other words, it is a measure that is not "smooth" in some sense.

The absolutely continuous part of the measure is the part that can be represented as the derivative of a non-negative function. It is a measure that assigns a non-negative value to subsets of a set, and can be represented as the derivative of a non-negative function. In other words, it is a measure that is "smooth" in some sense.

The Hahn-Jordan decomposition is used in many areas of mathematics and science, including probability theory, where it is used to decompose probability measures into singular and absolutely continuous parts. It is also used in statistics, where it is used to decompose measures of probability distributions into their components.

In actuarial science, the Hahn-Jordan decomposition is used to understand the risk and design the insurance products. Actuaries use the theorem to better understand the patterns and trends in data and to make more accurate predictions about the future. The decomposition helps them in identifying the singular and absolutely continuous parts of the measure of risk and design the products accordingly.

In conclusion, the Hahn-Jordan decomposition is a mathematical concept used in measure theory and probability theory. It is a method for decomposing a measure into two parts: a singular part and an absolutely continuous part. The decomposition is used to split the measure into two parts: one part that is singular with respect to another measure, and another part that is absolutely continuous with respect to that measure. The Hahn-Jordan decomposition is used in many areas of mathematics and science, including probability theory, statistics, and actuarial science. Actuaries use the theorem to better understand the patterns and trends in data and to make more accurate predictions about the future.

 

2-Discuss in short (a) BAN estimator (b) CAN estimator

(a) BAN (Best Affine) estimator: BAN estimator is a method used in econometrics and statistical analysis to estimate the parameters of a multivariate regression model. It is considered as the best linear unbiased estimator (BLUE) of the parameters when the errors are independently and non-normally distributed. It is also known as an affine estimator because it is based on the assumption that the errors in the model are affinely related to the explanatory variables.

(b) CAN (Combined Affine and Non-Linear) estimator: The CAN estimator is an extension of the BAN estimator, it is used in situations where the errors in the model are not only affinely related to the explanatory variables but also non-linearly related. The CAN estimator combines the properties of both the BAN estimator and non-linear estimators to provide a more efficient and robust estimation of the parameters. It is considered as the best linear unbiased estimator (BLUE) when the errors are independently and non-normally distributed with both linear and non-linear relationships.

 

3-State and prove Fubini’s theorem.

Fubini's theorem is a fundamental result in measure theory that states that under certain conditions, the double integral of a function can be calculated by either iterating the integral over one variable first and then the other, or by iterating it over the other variable first and then the first. It was first proved by the Italian mathematician Guido Fubini in 1907.

The theorem states that if a function f(x, y) is integrable over a rectangular region R in the xy-plane, and if the iterated integrals of f(x, y) with respect to x and y over R exist and are finite, then the double integral of f(x, y) over R can be calculated as:

∫∫f(x,y)dxdy = ∫(∫f(x,y)dx)dy = ∫(∫f(x,y)dy)dx

The theorem is based on the assumption that the function f(x, y) is measurable, which means that for any subset of the rectangular region R, the function f(x, y) assigns a real value.

The proof of Fubini's theorem is based on the following steps:

1.      First, the rectangular region R is divided into small rectangles of equal area.

2.      Next, the function f(x, y) is evaluated at the center of each small rectangle, and the product of the value of the function and the area of the rectangle is calculated.

3.      The sum of all these products is calculated, and it gives an approximation of the double integral of f(x, y) over R.

 

4-Discuss about the Fouries Series.

The reduction theorem for assignment problems states that any square assignment problem, which is an assignment problem where the number of tasks is equal to the number of agents, can be transformed into a square transportation problem and vice versa.

The proof of the reduction theorem for assignment problems can be broken down into two parts:

1.     From an assignment problem to a transportation problem: Given an assignment problem with a cost matrix C = [c(i,j)] where c(i,j) is the cost of assigning task i to agent j, we can construct a transportation problem as follows:

·         Introduce a new set of dummy sources, one for each task, and a new set of dummy destinations, one for each agent.

·         For each task i, set the supply at the dummy source i to 1.

·         For each agent j, set the demand at the dummy destination j to 1.

·         Set the transportation cost from dummy source i to dummy destination j to be equal to the cost of assigning task i to agent j, c(i,j)

2.      From a transportation problem to an assignment problem: Given a transportation problem with a cost matrix C' = [c'(i,j)] where c'(i,j) is the cost of transporting one unit of goods from source i to destination j, we can construct an assignment problem as follows:

·         Set the cost of assigning task i to agent j to be equal to the cost of transporting one unit of goods from source i to destination j, c'(i,j)

·         Set the number of tasks to be equal to the number of sources and the number of agents to be equal to the number of destinations.

This theorem is helpful in solving assignment problems using linear programming techniques. The reduction theorem allows us to solve an assignment problem using transportation problem techniques, and vice versa, which can lead to faster and more efficient solutions.

In summary, The reduction theorem for assignment problems states that any square assignment problem can be transformed into a square transportation problem and vice versa. The theorem can be proved by showing that the cost matrix of an assignment problem can be converted into a transportation problem and vice versa. The theorem is useful as it allows solving assignment problems using linear programming techniques, which can lead to faster and more efficient solutions.

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