Science and Math Articles

Here is the collection of expert STEM articles that span all the sciences and mathematics. Some examples include physics, mathematics, biology, chemistry, cosmology, and earth sciences. For math, we cover algebra to advanced calculus.  These articles are all peer-reviewed by the community.

SchwarzschildGeometry3

The Schwarzschild Geometry: Spacetime Diagrams

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12 Comments
 When we left off in part 2, we were looking at the metric for the Schwarzschild geometry in Kruskal-Szekeres coordinates:$$ ds^2 = \frac{32…
fakenews

Spotting Fake Science News: Wendelstein 7‑X & Media

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107 Comments
Fake News, Data, and the Need to Check Sources During and even after the recent US election campaign, numerous reports of fake or misleading news were…
quantumapplitudes

Learn About Quantum Amplitudes, Probabilities and EPR

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83 Comments
This is a little note about quantum amplitudes. Even though quantum probabilities seem very mysterious, with weird interference effects and seemingly nonlocal…
fishresearch

Real Research in Unexpected Places: Backyards, Barns, Bayous

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2 Comments
Challenging the Myth of Institutional Labs 21st-century science labors under the myth that good experimental science requires expensive lab facilities…
representations

Linear Representations and Why Precision is Important in Math

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2 Comments
First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space…
angledimensions

Can Angles be Assigned a Dimension?

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107 Comments
1. Some Background on Dimensional Analysis ... if you are not already familiar with it. 1.1 Dimensions Dimensional Analysis is a way of analyzing…
whystudyphysics

Why Study Physics? A Bit Goes a Long Way!

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6 Comments
Why Study Physics? If you don’t plan to become a scientist or an engineer, is there a good reason to study physics?  I say, “Yes,” because knowing…
vaccummyth

Explore The Vacuum Fluctuation Myth in Quantum Theory

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230 Comments
This Insight Article is a sequel of the Insight Articles ”The Physics of Virtual Particles” and “Misconceptions about Virtual Particles“ which…
anyon

Learn The Main Conceptual Ideas of Anyon Particles

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55 Comments
Every quantum physicist knows that all particles are either bosons or fermions. And the standard textbook argues that this so does not depend on the number…
frameofreference2

Frames of Reference: A Skateboarder’s View

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22 Comments
My essay Explaining Rolling Motion raised some commentary about frames of reference and their equivalence when solving physics problems. I wish to pursue…
relativitygauge

Exploring General Relativity as a Gauge Theory

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24 Comments
The fundamental interactions of the Standard Model are described by Yang-Mills theory. This is a gauge theory, which means the following: (1) Choose a…
govandteachers

Fixing Scientific Integrity: Teachers, Editors, and Data

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46 Comments
Introduction The article, "Governmental policy is wrecking science," makes some interesting points but is fundamentally in error, because government…
quantumparticle

Spectral Standard Model and String Compactifications

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29 Comments
 The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics -- as a classical action functional,…
cmbprimer4

A Poor Man’s CMB Primer: Cosmic Acoustics

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1 Comment
 Before decoupling, photons and charged particles were in good thermal contact: though the primordial plasma might have varied in temperature…
spacetimecon

Struggles with the Continuum: Spacetime Conclusion

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5 Comments
 We've been looking at how the continuum nature of spacetime poses problems for our favorite theories of physics --- problems with infinities.…
11dgravity

11d Gravity From Just the Torsion Constraint

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17 Comments
 It is familiar that Einstein gravity may be formulated in terms of a vielbein field together with a "spin connection", subject to the constraint…
spacetime7

Struggles with the Continuum: General Relativity

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2 Comments
 Combining electromagnetism with relativity and quantum mechanics led to QED. Last time we saw the immense struggles with the continuum this…
ohmslaw

Learn Why Ohm’s Law Is Not a Law

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41 Comments
At first, I wanted this title to say “Ohm’s law is not a Law.”  But someone else used that phrase in a recent PF thread, and a storm of protest…
carcapacity

Why Road Capacity Is Almost Independent of the Speed Limit

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30 Comments
Introduction Let us start with a familiar situation. Take a familiar part of a road, for example, the one from your home to work. How long will it take…
quantumhard

Why Quantum Mechanics Feels Difficult — Mathematical Bridge

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253 Comments
[CONTENT] QM's formalism Overview of the formalism Strangely enough, quantum mechanics (QM)'s formalism isn't any more difficult than other areas of…
Supersymmetry

Learn About Supersymmetry and Deligne’s Theorem

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55 Comments
 In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here…
LHC4

LHC Part 4: Searching for New Particles and Decays

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17 Comments
 The LHC experiments are in full swing collecting data this year (more information in the forum), and a while ago the collaboration of the…
acusticbeats

Mathematics of Acoustic Beats: Explanation & Examples

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11 Comments
Introduction Purpose and scope In late high school physics courses and first-year university courses, the phenomenon of acoustical 'beats' (digital audio)…
polymer

DNA Mapping & Nanopore Sequencing: Nanochannel Insights

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4 Comments
Introduction A single DNA molecule snaking its way through an array of cavities. This movie is from one of my later Ph.D. experiments. …
groupsandgeometry

Klein’s Erlangen Program: Groups Define Geometry

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7 Comments
Klein's Erlangen program: groups and geometry There is a very deep link between group theory and geometry. Sadly, this link is not emphasized in most…
computers

Ramsey Theory: Foundations, Generalizations, Key Results

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5 Comments
Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these six will either be meeting each other for the first…

Learn About Spacetime Diagrams of Light Clocks

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25 Comments
We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper that has been rotated by 45 degrees. Many quantitative…

A Black Hole Problem From Greg Egan’s “Incandescence”

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8 Comments
The following problem was inspired by reading Greg Egan's "Incadescence".  Consider a body, orbiting a Schwarzschild black hole and tide locked to it,…
softwarenotperfect

Why Your Software is Never Perfect

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52 Comments
Introduction We occasionally have students ask for help on software, "My software is perfect, but it doesn't work!" Your software is never perfect. My…
Photoacoustic

Light and Sound Interactions: Photoacoustic & Acousto-Optic

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6 Comments
Introduction I recently wrote a post on my blog about a fairly esoteric idea regarding sound propagating through light. This inspired me to write an…
logicp3

Scientific Inference: Balancing Predictive Success with Falsifiability

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9 Comments
  Bayes' Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning.…
logicp2

Scientific Inference: Do We Really Need Induction?

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14 Comments
  Part 2. "We don't need no stinkin' induction" -- Sir Karl Popper The traditional scientific method supposedly employs induction both in the…
antigravity

Roger Babson & the Gravity Research Foundation History

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13 Comments
Roger Babson and His War on Gravity In the early 20th century, an eccentric businessman named Roger Babson declared gravity to be mankind's greatest…
logic1

Hume and the Problem of Induction in Science Explained

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39 Comments
Click for Full SeriesPart 1: Logical induction ain’t logical Part 2: We don’t need no stinkin’ induction Part 3: Balancing Predictive Success…
virtualparticlesmyths

Top Misconceptions about Virtual Particles

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1 Comment
Introduction This Insight Article is a sequel to the Insight Article ''The Physics of Virtual Particles'', which contains an exposition of definitions…
acpoweranalysisp3

Power Grid Cyber Resilience: Risks and Defenses Explained

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10 Comments
Click for Series ListPart 1: The Basics Part 2: Network Analysis Part 3: Cyber ResilienceIntroduction Power system security is much too big…
nightsky

Why Night-Sky Stars Often Appear Brighter Than the Sun

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4 Comments
Are most visible stars brighter than the Sun? A couple of weeks back, a thread was started about whether or not most of the stars we see at night are…
encryption

Is It Possible to Design an Unbreakable Cipher?

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10 Comments
Is it possible to design an unbreakable cipher? Do methods of encryption exist that guarantee privacy from even the most capable and highly-resourced…
quantumentanglement

How the EPR Paradox Fits Between Entanglement and Nonlocality

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11 Comments
If we violate a Bell inequality, we know we have quantum entanglement, and it used to be thought that these two notions were one and the same.At least,…
acpoweranalysisp2

Intro to AC Power Analysis: Network Analysis

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26 Comments
 Let me use the terms “power grid” and “network” interchangeably. What is the Power Grid Required to Do?Deliver energy to customers,…
Newton-Cartan-Theory

Newton-Cartan Theory: Newtonian Gravity as Curved Spacetime

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12 Comments
Newtonian limit and spacetime curvature It's a calculation that you can find in any textbook on General Relativity (GR): when gravity is weak and does…
virtualparticles

Learn the Physics of Virtual Particles in Quantum Mechanics

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34 Comments
In discussions on the internet (including a number of Wikipedia pages) and in books and articles for non-experts in particle physics, there is considerable…
universeexpand

Big Bang Models: Expansion, Dark Energy, Inflation

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29 Comments
Big Bang and expansion (no inflation, no dark energy) Empty-universe solution (FLRW) Einstein's General Relativity allows a solution (the FLRW metric)…
acpoweranalysisp1

Intro to AC Power Analysis: Learn System Basics

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13 Comments
 Instantaneous PowerThe formula for power is: ##P=V\cdot I## , power= voltage*current.   We call that, instantaneous power.   Even…
cipher

The Monographic Substitution Cipher: From Julius Caesar to the KGB

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7 Comments
A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…
pet_scan

How PET Scans Work: Positrons in Medical Imaging Explained

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7 Comments
Practical applications of fundamental theoretical physics Positron Emission Tomography (PET) Less often, people ask about potential applications of anti-matter.…
tcp_ip

TCP/IP Protocols for Plug-and-Play Industrial Automation

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10 Comments
Background: Ethernet vs Fieldbus in the 1990s The 1990s saw the widespread adoption of network technology on two fronts: Office automation and factory…
Ethernettime

Network Time Synchronization Guide: NTP vs IEEE 1588

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18 Comments
Overview of Synchronization Needs Now and then you come across measurement problems that are tightly associated with the notion of synchronicity, meaning…
qminterpret

QM Interpretations: Semantics, Meaning & Probability

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12 Comments
Introduction Many times, we encounter ourselves tangled in some Quantum Mechanics (QM) interpretations debates. Unfortunately, the standard of discussion…
spacetime6

Struggles with the Continuum – Freeman Dyson and QED

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24 Comments
 Last time I sketched how physicists use quantum electrodynamics, or 'QED', to compute answers to physics problems as power series in the…
Mitochondria

When did Mitochondria Evolve?

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7 Comments
Eukaryote evolution and cellular complexity Historically, life is categorized into three broad domains: bacteria, archaea, and eukaryota.  Whereas bacteria…
islingmodel

High Temperature and Low Duality for the Ising Model on an Infinite Regular Tree

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1 Comment
Today I felt like sharing a computation that I thought was interesting. I'm sure I'm not the first to explore the Ising-Ising duality in this context,…
secondaryforces

Introduction to the Secondary Forces in Physics

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7 Comments
Many are familiar with the "fundamental forces" of nature: gravity, electromagnetism, and the strong and weak nuclear forces. Three of these tend to bind…
complexmath

Hear the Case for Learning Complex Math

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15 Comments
Resistance to complex math seems to never die out.  I see it frequently in PF posts.  Often it takes the form of challenges rather than questions. …
spacetime5

Struggles with the Continuum: Quantum Electrodynamics

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10 Comments
  Quantum field theory is the best method we have for describing particles and forces in a way that takes both quantum mechanics…
darkmatter

Knut Lundmark and the Forgotten Dark Matter Discovery

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16 Comments
Background: textbook accounts Look in your standard cosmology textbook (in my case, the closest one at hand is Barbara Ryden's Introduction to Cosmology)…
spacetime4

Struggles with the Continuum – Relativity and Quantum

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5 Comments
 In this series, we're looking at mathematical problems that arise in physics due to treating spacetime as a continuum---basically, problems…
complexnumbers

Things Which Can Go Wrong with Complex Numbers

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3 Comments
At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work:Example A [itex]-1=i^2=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1[/itex]Example…
theneedforinfinity

Why the Need of Infinity in Physics

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31 Comments
Infinity in Mathematics and Physics It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics, probably in a finite…
complexscience

Thoughts on the Complexity of Modern Science

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123 Comments
Science today is incredibly complex. Anyone who ever did scientific research knows that, but many in the general public seem to be completely unaware of…
MillenniumPrize

Millennium Problems: Poincaré, P vs NP, Riemann

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13 Comments
IntroductionWhat this Insight Covers In this Insight, I will go over the background information for the Millennium Prize problems and briefly describe…
rainbows

Exploration into the Physics of Rainbows

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88 Comments
For several years, I have been contemplating this beautiful picture by photographer Brian McPhee. I have a personal interest in the photograph because…
micro2

Peano Axioms: Construction of Natural Numbers and Properties

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0 Comments
Bloch Chapter 1.2The Peano system in Bloch has a special element ##1\in \mathbb{N}##. The intuitive idea here is that ##\mathbb{N} = \{1,2,3,...\}##.…
micro3

Set-Theoretic Foundations of Numbers and Functions

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1 Comment
Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets.…
earthgravity

Learn All About Earth’s Gravity

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18 Comments
Introduction Earth's gravitational field at the surface is approximately 9.8 Newtons/kilogram or 9.8 meters/second/second. But how does that change due…
greenhouse

Frequently Made Errors in Climate Science – The Greenhouse Effect

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87 Comments
1.What is meant by "The Greenhouse Effect"? Many gases, such as H2O, CO2, CH4, are transparent to visible light but absorb and emit parts of the infrared…
speedoflight1

Learn About the Speed of Light and Galilean Relativity

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39 Comments
Introduction In this article, I will discuss some experiments in the 19th and early 20th centuries that looked at how the velocity of a light source affects…
cancer

What Causes Cancer: Bad Luck or Bad Lifestyles?

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22 Comments
What causes cancer? For the most part, cancer is a disease that arises from mutations in the body that accumulate over time.  These mutations knock…
buoyancy

Understanding Buoyancy: The Limits of Archimedes

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18 Comments
It’s a principle as old as science itself, and remarkably, remains essentially unchanged in the light of revolution after revolution in our understanding…
physicsturnedmath

Trials and Tribulations of a Physicist who Became a Math Geek

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4 Comments
How did I go from the brink of changing my major from physics to ceramics (no more math) to the Math faculty of the Air Force Academy? How did I go from…
studymath

Math Self-Study Roadmap: Topics & Book Recommendations

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5 Comments
How to Study Mathematics How to Self-Study Basic High School Mathematics How to Self-Study CalculusIntroduction We often get questions…
studymath1

The Essential Guide to Self Study Mathematics

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12 Comments
How to Study Mathematics How to Self-Study Basic High School Mathematics How to Self-Study CalculusHow to self-study mathematics? People self-study…
seeelectron

Have Scientists Seen an Electron? How We Detect Them

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18 Comments
Have Scientists Seen an Electron? Why Seeing Isn't Enough This article is not about literally seeing a single electron with your eye; it addresses the…
peterblockuniverse

The Block Universe – Refuting a Common Argument

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154 Comments
The "block universe" interpretation of SR has come up repeatedly in threads here on PF. Rather than link to them, I want to summarize a common argument…
Retrocausality

Understanding Retrocausality and Blockworld

In an Insights series “Blockworld and its Foundational Implications,” I quoted Huw Price and Ken Wharton in several places advocating a blockworld…
blockuniverse5

Blockworld and its Foundational Implications: Delayed Choice and No Counterfactual Definiteness

 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity. In part 3, I introduced…
blockuniverse4

Blockworld and its Foundational Implications: General Relativity and Closed Timelike Curves

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2 Comments
 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). In part 3, I introduced…
blockuniverse3

Blockworld and its Foundational Implications: General Relativity and the Big Bang

 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). Geroch sums up the…
blockuniverse2

Blockworld and Its Foundational Implications: The Relativity of Simultaneity and Blockworld

 In part 1 of this 5-part Insights series, I introduced two consequences of the second postulate of special relativity (SR): time dilation…
blockuniverse1

Blockworld and Its Foundational Implications: Time Dilation and Length Contraction

 This is the first in a 5-part series of Insights that will introduce blockworld (aka "block universe”) and use it to address a puzzle…
DNA

Atomic Positioning with DNA Hinges

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13 Comments
I was perusing the Nature Nanotechnology website today when I came across an interesting article by Funke and Dietz, called "Placing molecules with Bohr…
prequantum4

Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories

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4 Comments
  At the end of this page, we will have come full circle back to the first article in the series 20 years ago -- The M-Theory Conjecture.We…
prequantum3

Examples of Prequantum Field Theories III: Chern-Simons-type Theories

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1 Comment
  After having constructed gauge fields and higher gauge fields in the previous article by systems of ##L_\infty##-algebroid-valued…
guagetheory

Examples of Prequantum Field Theories II: Higher Gauge Fields

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1 Comment
   After having recalled ordinary gauge fields from a dg-algebraic perspective in the previous article, here I discuss…
prequantumft

Examples of Prequantum Field Theories I: Gauge Fields

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3 Comments
  After having motivated the need for prequantum field theory and having laid out its principles (i. extremal action, ii. global…
animationwalk

Fun with Self-Avoiding Walks Simulations

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16 Comments
This post is about some simulations I did of self-avoiding random walks. These are what they sound like with each step, the position of the walk moves…
prequantumgeometry5

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

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1 Comment
 This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of…
prequantumgeometry4

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

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13 Comments
  The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement…
prequantumgeometry3

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

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9 Comments
  The previous article ended with the concept of classical locally variational field theories, of which a class of examples are…
blackbody

A Poor Man’s CMB Primer: Bumps on a Blackbody

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15 Comments
 Astronomers Arno Penzias and Robert Wilson discovered the cosmic microwave background in 1965. They were not looking for it.They were…
prequantumgeometry2

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

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1 Comment
  The previous article motivated the importance of considering "pre-quantum field theory" in-between classical and quantum field…
redshiftblackholes

Time Dilation & Redshift of Schwarzschild Black Holes

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76 Comments
Time dilation near a Schwarzschild black hole The following is an overview of the time-dilation and gravitational-redshift effects of a static (Schwarzschild)…
prequantumgeometry

Higher Prequantum Geometry I: The Need for Prequantum Geometry

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18 Comments
 Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article,…
atoms

Can We See an Atom?

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11 Comments
Introduction In this article I discuss how images of atoms are made, what exactly we are looking at, and what it means to "see" an atom. Over the decades…
gravity3

Does Gravity Gravitate: The Wave

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17 Comments
 In the first two posts in this series, we looked at different ways of interpreting the question "does gravity gravitate?" We left off at…
gravitate2

Does Gravity Gravitate? Part 2

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33 Comments
 In the first post of this series, I talked about two ways to answer the title question, one leading to the answer "no" and the other leading…
birthcmb

A Poor Man’s CMB Primer: The Birth of a Cosmic Background Radiation

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3 Comments
 The early universe was hot. So hot that nuclei boiled. The great thermal energy of the universe overwhelmed the confining efforts of the…
modulation

Mixing vs Beating: Superheterodyne & Beat Frequency

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36 Comments
Mixing vs. Beating: a brief history A long time ago I read a paper in the IEEE Proceedings recounting the history of the superheterodyne receiver. Overall…
balloonanalogy

Balloon Analogy Explained: Cosmology, Expansion & Limits

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63 Comments
The Balloon Analogy is a straightforward way to illustrate — without being complete — two important points of modern cosmology that many people find…
spacetime4

Struggles With the Continuum: Point Particles and the Electromagnetic Field

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20 Comments
 In these posts, we're seeing how our favorite theories of physics deal with the idea that space and time are a continuum, with points described…