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recursion

Reduction of Order For Recursions

This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student. Most of us remember recursion relations from secondary school. We start with a number, say, 1. Then we add 3. That gives us 4. Now we that number and add 3 again…

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history of numbers

Counting to p-adic Calculus: All Number Systems That We Have

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An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim…
evariste galois

Évariste Galois and His Theory

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 * Oct. 25th, 1811  † May 31st, 1832   ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet,…
definition differences

Yardsticks to Metric Tensor Fields

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I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
ATmega8A

Programming an ATmega8A using Arduino

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If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino projects permanent,…
pvsnp

P vs. NP and what is a Turing Machine (TM)?

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P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and…
quantum computers 101

Quantum Computing for Beginners

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Introduction to Quantum Computing This introduction to quantum computing is intended for everyone and especially those who have no knowledge of this relatively…
gauss law misconceptions

A Physics Misconception with Gauss’ Law

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Introduction It is relatively common to see the following type of argument: The surface area is ##A## and the enclosed charge is ##Q##. The electric…
model magnet

How to Model a Magnet Falling Through a Conducting Pipe

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Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features…
model magnet

How to Model a Magnet Falling Through a Solenoid

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Introduction Modeling a magnet realistically is a task best done numerically.  Even the simplified model of two separated disks with uniform surface…
why can we jump

How Can We Jump When the Ground Does No Work?

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It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the ground, the ground does…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis HistoryRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to…
magnetism current

Symmetry Arguments and the Infinite Wire with a Current

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Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric…
cpu programming

Parallel Programming on a CPU with AVX-512

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This article is the second of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
programming gpu

Parallel Programming on an NVIDIA GPU

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This article is the first of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
hypercube integral

A Novel Technique of Calculating Unit Hypercube Integrals

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Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first…
Riemann Hypothesis

The Extended Riemann Hypothesis and Ramanujan’s Sum

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Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All…
Hyperbola

The Amazing Relationship Between Integration And Euler’s Number

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We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…
mermin device quantum entanglement

Superdeterminism and the Mermin Device

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Superdeterminism as a way to resolve the mystery of quantum entanglement is generally not taken seriously in the foundations community, as explained in…
relativity rotated graph

Relativity on Rotated Graph Paper (a graphical motivation)

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(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184…
probabilities virus testing

Probabilistic Factors Involved in Disease and Virus Testing

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Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,…
quantum mechanics new approach

Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

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This Insight article presents the main features of a conceptual foundation of quantum physics with the same characteristic features as classical physics…
trig special functions

A Trick to Memorizing Trig Special Angle Values Table

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In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we…
setup raspberry pi cluster

How to Setup a Raspberry Pi Cluster

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INTRODUCTION As a long-time computer programmer and almost as long a High-Performance Computer (HPC) user, I really didn't know anything about how these…
Theorema Primum tutorial

An Introduction to Theorema Primum

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Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…
python sympy module

Python’s Sympy Module and the Cayley-Hamilton Theorem

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Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a…
Kerr Spacetime

Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

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Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief…
tan rule

Physical Applications of the “Tan Rule”

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Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,…
MIT course corrections

Corrections to MIT Open Courseware: Systems of Varying Mass

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Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton's Laws to systems of Varying Mass PDF Course Link My concerns…
quantum information

How Quantum Information Theorists Revealed the Relativity Principle at the Foundation of Quantum Mechanics

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This Insight is a condensed version of No Preferred Reference Frame at the Foundation of Quantum Mechanics. Reference numbers here correspond to that paper.…
euler sums

Investigating Some Euler Sums

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So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that…
Quaternions in Projectile Motion

Quaternions in Projectile Motion

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Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our…
quantum double slit experiment

Quantum Mechanics and the Famous Double-slit Experiment

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Introduction Quantum mechanics is famously strange. But Schroedinger's dead and alive cat, Einstein's spooky action at a distance, and de Broglie's matter…
safety

When Science is Not Enough: Legal and Health Implications of Questions

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There are often questions on Physics Forums related to building machines or structures that are left unanswered because of this guideline (found in the…
LCEVC 8k tv

The LCEVC Coding Standard and 8K Television

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Introduction First, for background, here is an overview of the most common video compression standard in use today, H264: https://www.maketecheasier.com/how-video-compression-works/H265…
solving projectile motion

How to Solve Projectile Motion Problems in One or Two Lines

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Introduction We show how one can solve most, if not all, introductory-level projectile motion problems in one or maybe two lines. To this end, we forgo…
blackboard

10 Math Things We All Learnt Wrong At School

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The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot…
Fourier Series Riemann Zeta Function

Computing the Riemann Zeta Function Using Fourier Series

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Euler's amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous…
tolman law beginner

Tolman Law in a Nutshell

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The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic…
physics cannonball projectile

Maximizing Horizontal Range of a Projectile

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Introduction A recent homework problem that appeared in the forums was concerned with maximizing the horizontal range of a projectile subject to the launch…
electromagnetic computations duality

A Numerical Electromagnetic Solver Using Duality

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In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein…
Electric Field Seen by an Observer

The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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This Insight was inspired by the discussion in "electric field seen by an observer in motion", which tries to understand the relation between two expressions:…
valentines reflections graphs

Valentine’s Reflections: Mathematical Matters of the Heart

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Introduction Being a somewhat geeky Maths 'nerd', I spent days leading up to Valentine's day trying to find a Maths function appropriate to the day. In…
Bayesian statistics part 4

Posterior Predictive Distributions in Bayesian Statistics

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Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticiansRead part 1: How to Get Started with Bayesian StatisticsRead…
electromagnetic computations

How to Use Duality in Computational Electromagnetic Problems

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Some weeks ago I happened across a post that caught my eye. Dale asked a question about the number of photons in an electromagnetic field. His question…
impossible triangles

Geometry of Side-side-angle (SSA) and Impossible Triangles

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What is the ambiguous case? In high school geometry, the idea of proofs is often first introduced to American students. A common task is to use a basic…
equations of motion

Equations of Motion Revisited

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Introduction In any school Physics course, the Newtonian equations of motion are very much a 'stock' item. Students learn the equations and are given…
bayesian inference

How Bayesian Inference Works in the Context of Science

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Confessions of a moderate Bayesian part 3 Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian ProbabilityBayesian…
bayesian statistics part 2

Exploring Frequentist Probability vs Bayesian Probability

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Confessions of a moderate Bayesian, part 2 Read Part 1: Confessions of a moderate Bayesian, part 1Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13280237 Background One…
Split Electric Fields

Split Electric Fields in Electrodynamics: Capacitor and Antenna

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Abstract: The analysis of the two kinds of electric fields, namely the irrotational and non-conservative, is extended to electrodynamics, as exemplified…
bayesian statistics

How to Get Started with Bayesian Statistics

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Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13265286 Background I…
projectile motion

How to Master Projectile Motion Without Quadratics

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Introduction In a homework thread a while back a PF member expressed dismay along the lines of "oh no, not another boring projectile motion problem."…