1. INTRODUTION. In advance analysis, the notion of 'Compact set' is of paramount importance. In $ \mathbb{R} $, Heine-Borel theorem provides a very simple characterization of compact sets. The definition and techniques used in connection with compactness of sets in $ \mathbb{R} $ are extremely...

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## Fourier Series

1. INTRODUCTION The study of a special type of series whose terms are trigonometric functions of a variable was started in the 18th century when J.B.J. Fourier (1768-1830) was successful in his attempt to prove that an arbitrary function given in the interval $[-\pi,\pi]$ can be expressed under...

## Permutations & Combinations Made Easy!

Combinatorics This lesson covers the topics of combinatorics (combinations, permutations with and without repetitions, circular and constrained permutations). It will equip you with all the necessary theoretical knowledge on the topic, and show how it is applied in standardized test problems....

## A Tutorial in Data Science: Lecture 1 – The Foundations of Data Science

Data Science is the study of the datum, which is the Being of a being in its there-being. The datum of a being is its having been measured, and thus observed in one of its states. The science of data is thus how to interrogate data such as to reveal the being that has been measured, and thus the...

## A Tutorial in Data Science: Lecture 2 – The Hermeneutic Nature of Scientific Data

The question of science itself has never been its particular object of inquiry but the existential nature, in its possibility and thereby the nature of its actuality. Science is power, and thus abstracts itself as the desired meta-good, although it is always itself about particularities as...

## A Tutorial in Data Science: Lecture 3 – Laplace’s Analytical Theory of Probability

This lecture serves as a philosophically informed mathematical introduction to the ideas and notation of probability theory from its most important historical theorist. It is part of an ongoing contemporary formal reconstruction of Laplace's Calculus of Probability from his english-translated...

## A Tutorial in Data Science: Lecture 4 – Statistical Inference by Systems of Conditions

We consider here how to estimate a parameter to a distribution given a sample space of observations. It is important to recognize that the testing of any hypothesis relies upon assumptions, which may be further explicated as obtaining validity based upon the further assumptions. The system of...

## A Tutorial in Data Science: Lecture 5 – Generating Distributions by Spectral Analysis

When two states, i.e. possibly measured outcomes, of the stochastic sampling process of the underlying statistical object communicate, there is a probability of one occurring after the other, perhaps within the internal time (i.e. indexical ordering) of the measurement process, $t \in \pi=(1,...