## Probability Problems and Estimation Algorithms Associated with Symmetric Functions

In this chapter, we offer a simple yet powerful approach in which the independent variables 1,…, n in multiple symmetric functions and Vieta’s formulae are replaced by the indicator functions of the events Ai, I = 1,…, n, I = 1(Ai), I = 1,…, n. Both the random variable K, which counts the number of […]

## No Big Bang in the Non-Expanding Universe

Briefly stated and applied to cosmological models is a theory of gravity in flat space-time. These models begin with a gravitational field that is equally distributed yet devoid of substance. The whole energy of matter and gravity is preserved when gravitational energy is transformed to matter. There is no singularity (no great bang) in the […]

## Biquadratic Equation with Four Unknowns

Number theory is strongly reliant on diophantine equations, which can take many different forms. There are a variety of Diophantine equations that have no solution, trivial solutions, a finite number of solutions, and an infinite number of solutions. There are essentially two types of equations among higher degree Diophantine Equations. They are homogeneous and non-homogeneous […]

## Determination of Free Energy as Described by the New Axioms and Laws

Expanded Field Theory is used in this research. It transforms Classic Field Theory into a considerably broader theory with two additional axioms and eight rules. The researchers decided to take Einstein’s advise and try to change their way of thinking. Through new axioms and rules, the article describes a fresh new field type. It was […]

## Study about Transformations Formulae and Certain Transformations: A Mathematical Approach

In this part, we’ve tried to come up with several transformation equations involving various transforms. Using Bailey’s lemma to abstract many transforms such as Mellin, Fourier, and Laplace transforms to prove some new theorems that will be useful for future study. The theorems that have been derived are: Infinite series are used in the Mellin […]

## Improved Hoeffding’s Lemma and Hoeffding’s Tail Bounds: A Recent Study

This chapter aims to enhance Hoeffding’s lemma and, as a result, Hoeffding’s tail limits. To begin, we’ll offer Hoeffding’s lemma with a proof that differs from the original, and then show and prove the better Hoeffding’s lemma. The enhancement is for left skewed zero mean random variables X[a,b], with a0 and -a>b. The proof of […]

## Analysis of Methods for Calculating the Value of the Thermal Conductivity Coefficient in Numerical Modeling

Using the method of computational experiments on a computer, the effect of the method for calculating the values of the thermal conductivity coefficient at the nodes of the difference grid on the numerical solutions of a one-dimensional nonlinear problem of heat conduction according to explicit and implicit conservative difference schemes is investigated in this article. […]

## Applying the GSVD to the Analysis of the Augmented Lagrangian Method for Symmetric Saddle Point Problem

When using the iterative technique, the solution of the SPP needs appropriate computation in the case of the direct method and proper approximations of the inverse of the block and of the Schur’s complement in the case of the direct method. To enhance the numerical features of the (1,1) block, an Augmented Lagrangian approach was […]

## On Erdos – Lax and Turan Type Inequalities of a Polynomial

The above inequalities are respectively the well-known Erdӧs – Lax inequality and the Turẚn’s inequality. A natural question that follows is to investigate the extension of these inequalities for open or closed disk of radius In literature, we find extensions of Erdӧs – Lax inequality for a polynomial of degree having no zero in the […]

## An Efficient K-means Algorithm: Generating Clusters Dynamically in MapReduce Framework

Background: K-Means is a popular partition-based clustering technique that divides an input dataset into a collection of groups. K-Means is a popular technique because of its simplicity and quickness in grouping large amounts of data. Because of the massive quantity of electronic data generated, data clustering techniques have had to be modified in order to […]