| viXra.org > Functions and Analysis > 2503 |
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| viXra.org > Functions and Analysis > 2503 |
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[4] viXra:2503.0136 [pdf] submitted on 2025-03-22 08:10:22
Authors: Harry Willow
Comments: 7 Pages.
This paper explores the applications of generating functions in solving various recurrence relations. We present the explicit formulas for recurrence relations of different forms, demonstrating step-by-step transformations and manipulations using generating functions. Several cases are examined, including linear and nonlinear recurrences, factorial-based sequences, and Fibonacci-related expressions. The derivations leverage algebraic techniques and characteristic equations to obtain closed-form solutions. The results highlight the power of generating functions in simplifying complex recurrence relations and deriving explicit formulas efficiently.
Category: Functions and Analysis
[3] viXra:2503.0111 [pdf] submitted on 2025-03-18 21:02:38
Authors: Loïc Renaut, Laurent Bruneau
Comments: 65 Pages. In French
The aim of this thesis is to study the basic concepts and the mathematical formalism of quantum mechanics. To do so, we will first examine Newtonian mechanics. This was the prevailing theory until the early 20th century for describing the motion of objects of all sizes. However, it was later discovered that this theory had its limitations. Consequently, at the beginning of the 20th century, two new theories emerged to address the shortcomings of Newtonian mechanics: quantum mechanics, which is the focus of this thesis and describes the physics of atomic and subatomic-scale objects, and special relativity, which deals with the physics of objects on a very large scale.In the second part, we will study quantum mechanics and attempt to identify the similarities and differences between the two theories. The foundational principle of quantum theory is wave-particle duality. This principle states that the objects that make up matter (such as electrons and photons) exhibit behavior characteristic of a particle in some situations and behavior characteristic of a wave in others. However, they are neither purely waves nor purely particles—they are something else entirely. This is one of the fundamental differences between the two theories. In Newtonian mechanics, objects are considered exclusively as particles. To illustrate the intellectual revolution brought about by this principle, we can recall the major debate that took place in the 17th century regarding the nature of light (at the time, photons were unknown). Two opposing camps emerged: one led by Isaac Newton, who believed that light was composed of particles, and the other led by Christiaan Huygens, who argued that light was a wave. Newton’s influence was so great that his view was widely accepted by most scientists of the time. No one had yet considered that light might be neither a particle nor a wave. Moreover, the necessary mathematical tools to explore this idea did not yet exist.
Category: Functions and Analysis
[2] viXra:2503.0100 [pdf] submitted on 2025-03-17 22:33:44
Authors: Edgar Valdebenito
Comments: 3 Pages.
We give two sequences for Pi.
Category: Functions and Analysis
[1] viXra:2503.0051 [pdf] submitted on 2025-03-09 17:59:27
Authors: Sigrid M.-L. Obenland
Comments: 2 Pages.
In 1991, Gilbert Strang published a proof of Euler's formula using polar coordinates in Calculus, Wellesley-Cambridge, p. 389. In the following we show that this alleged proof is not valid.
Category: Functions and Analysis
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