Physics, Math and Science, Explained

Velocity Time Function

Revisiting the Velocity-Time Function

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Kinematics: The Velocity-Time Function Author’s Preface: A Collaborative Approach to Lesson Development The intention behind what I hope will be a series of articles on Kinematics (this one being the first) is to begin an open, collaborative process of lesson development on Physics Forums. In order to expedite this workflow, I am making deliberate use…

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ai problem solving

AI Enriched Problem Solving

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Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems…
gate system

Remote Operated Gate Control System

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Introduction This project began life in 2017 when the author had just begun experimentation with GPIO pins on a Raspberry Pi 3. At that point in time…

Inside the Scientific Box: History and Challenges Today

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In Memoriam In Memory of Dr. Thomas J. LeCompte (1964-2025), Detector Designer and Champion of Education and Science. Prologue Defining "the box" Someone…
Entangled Photon Polarization Qubits

Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality per Quantum Information Theory

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In her YouTube video Bell's Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell's inequality. In…
relativator

Relativator (Circular Slide-Rule) – Simulated with Desmos

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The Relativator (revisited) This is an update of my 2006 post (reconstructed in 2014) Relativator: The circular slide-rule for physicists. This is a…
dirac function

What Exactly is Dirac’s Delta Function?

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Introduction: "Convenient Notation"In Dirac's Principles of Quantum Mechanics published in 1930 he introduced a "convenient notation" he referred …
quantum entanglement

Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

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Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and…
complex numbers

Fixing Things Which Can Go Wrong With Complex Numbers

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Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2…
Fermat's Last Theorem

Fermat’s Last Theorem

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Abstract Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states…
vector spaces

Vector Spaces: Concepts, History, and Applications Guide

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The Concept A vector space is an additively written abelian group together with a field that operates on it.Intuitive picture vs abstract definition Vector…

Groups, The Path from a Simple Concept to Mysterious Results

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Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement…
digital audio guide

Hybrid AI Video Codecs and Modern Streaming Techniques

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Introduction AI and Hybrid Codecs Modern Video codecs are all based on similar principles. Recently, these have been complemented by AI techniques,…
no tidal bulge

Why the Tidal Bulge Doesn’t Exist – Tidal Theory Explained

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Introduction Overview That there is no tidal bulge is the key premise of this article. Upper-level oceanography undergraduates and above know this. Yet…
topology

The Many Faces of Topology

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Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see…
brownian motion

Brownian Motions and Quantifying Randomness in Physical Systems

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Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It's now a key player in data science,…
physics of reality

Quantum Reconstruction & Bell Correlations Explained

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Comment on PBS Space Time video PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their…
pvsnp

Aspects Behind the Concept of Dimension in Various Fields

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Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,…
complex numbers views

Views On Complex Numbers

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Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted…

Addition of Velocities (Velocity Composition) in Special Relativity

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The "Addition of Velocities" formula (more correctly, the "Composition of Velocities" formula) in Special Relativity[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c}…
Schrodinger's cat

Schrödinger’s Cat and the Qbit

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The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##\ddot{\text{o}}##dinger noted in 1935 writing [1],…
slinky drop experiment

The Slinky Drop Experiment Analysed

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The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye.…
infinite universe

Open, Flat & Closed Universes: Curvature Explained

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Standard cosmological models are classified by spatial curvature into three broad types: open, flat, and closed universes. These correspond to negative,…
einstein field equations

Einstein Field Equations – Definition and Fast Facts

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Definition/Summary The Einstein Field Equations (EFE) are a set of ten interrelated differential equations that form the core of Einstein's general theory…
Multi-Atwood Machine Assembly

How to Solve a Multi-Atwood Machine Assembly

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IntroductionThe figure on the right shows a "double-double" Atwood machine with three ideal pulleys and four masses.  All pulleys are released from…
Lambert W Function in Finance

The Lambert W Function in Finance

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Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…
infinity

Why Division by Zero Is Impossible: 10 Mathematical Reasons

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A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was…
digital audio guide

Sinc Function & Upsampling in Digital Audio Explained

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Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function Upsampling and Zero Insertion The first link…
digital audio intro

Modern DACs, Dither, DXD & High-Res Audio Explained

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Introduction First, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this article to be accessible…
geometric series

Series in Mathematics: From Zeno to Quantum Theory

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Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…
Epsilontic limits and continuity

Epsilontic – Limits and Continuity

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Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I…
Milli-Ohm

The Poor Man’s Milli-Ohm Meter

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Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article,…
Differential Equation Systems and Nature

Differential Equation Systems and Nature

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Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural…
calc precalc

Beginners Guide to Precalculus, Calculus and Infinitesimals

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Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this.In the US, they go through…
Variable Mass Systems

How to Apply Newton’s Second Law to Variable Mass Systems

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Introduction The applicability of Newton's second law in the oft-quoted "general form"  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$…
what are numbers

What Are Numbers?

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Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going…
world of algebras

Introduction to the World of Algebras

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Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…
chatgpt-reliable

Why ChatGPT Is Unreliable: Design Limits and Misconceptions

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ChatGPT's reliability I'll start with the simple fact: ChatGPT is not a reliable answerer to questions. Wolfram's explanation To try to explain why from…
Infinitesimals

What Are Infinitesimals – Simple Version

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Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
quantum information

How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics

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In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit…
Infinitesimals

What Are Infinitesimals – Advanced Version

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Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
Carl Sagan

Popular Science Communication: Bridging Science & Public

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Abstract Famous Science Communicators There is so much to say about the many endeavors by professional scientists to explain to us the world. The list…
art of integration

The Art of Integration

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Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…
teaching physics

A Lesson In Teaching Physics: You Can’t Give It Away

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A central principle of Physics Forums regarding homework help is not to provide solutions on demand but to guide students along a path to the answer. The…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

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Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments…
Measure Internal Resistance of Battery

How to Measure Internal Resistance of a Battery

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Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery - typically an AA or D…
lie group physics

When Lie Groups Became Physics

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Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
white dwarfs

Why There Are Maximum Mass Limits for Compact Objects

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In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Implications

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder…
tensors relativity

Tensors Explained: Beginner’s Essentials for Relativity

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If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational…
twin paradox

Twin Paradox Guide: Spacetime Geometry Explained Clearly

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Introduction This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads…
gravity collapse

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the…
object slide down ramp physics

Subtleties Overlooked in Friction Questions: Object Slides Down Ramp

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Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ' on the ramp,…
math classifications

Mathematics Fields & Applications: A Structured Guide

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Introduction I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what…
recursion

Reduction of Order For Recursions

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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…
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, he gave…
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

Program ATmega8A with Arduino: Step-by-Step Guide

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Introduction If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino…
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 Explained: Qubits, Algorithms & Applications

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Overview This article provides an accessible introduction to quantum computing, a cutting-edge technology that processes certain types of information…
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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Common mistaken argument It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
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

CUDA Regression Line: GPU Parallel Programming Guide

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

Raspberry Pi Cluster Guide: Parts, MPI & Slurm Tips

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INTRODUCTION As a long-time computer programmer and almost as long a High-Performance Computer (HPC) user, I didn't know anything about how these machines…
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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Double-slit Key PointsQuantum mechanics is known for its strangeness, including phenomena like wave–particle duality, which allows particles to…
safety

Safety Factors: Why Forums Avoid Design Advice Online

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

LCEVC and EVC: Efficient Video Encoding Guide for 8K

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Introduction Background: H.264 and H.265 First, for background, here is an overview of the most common video compression standard in use today, H264:…
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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Some of these could even lead to heavy debates within the scientific community, so maybe I should say: from my point of view. So before you get excited…
Fourier Series Riemann Zeta Function

Computing the Riemann Zeta Function Using Fourier Series

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5 Comments
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

Building a Numerical FEM Solver for Duality in EM Fields

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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

Ambiguous SSA Case in Triangles — Law of Sines Explained

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What is the ambiguous case? In U.S. high schools, students are often introduced to geometric proofs for the first time. A common task is to use a small…