Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho/Moment via Getty ImagesCourtney Gibbons, Hamilton College You scrambled up a Rubik’s cube, and now you want to put it back in order. What sequence of moves should you make? Surprise: You can answer this question with modern algebra. Most folks who have been through high school mathematics courses will have taken a class called algebra – maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x. The word “algebra” may evoke memories of complicated-looking polynomial equations like ax² + bx + c = 0 or plots of polynomial functions like y = ax² + bx + c. You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x-axis, too.Graph of a quadratic equation and its roots via the quadratic formula.Jacob Rus, CC BY-SA Equations and plots like these are part of algebra, but they’re not the whole story. What unifies algebra is the practice of studying things – like the moves you can make on a Rubik’s cube or the numbers on a clock face you use to tell time – and the way they behave when you put them together in different ways. What happens when you string together the Rubik’s cube moves or add up numbers on a clock? In my work as a mathematician, I’ve learned that many algebra questions come down to classifying objects by their similarities.
Sets and groups
How did equations like ax² + bx + c = 0 and their solutions lead to abstract algebra? The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x was three or four. But they couldn’t do it for five. It took mathematician Évariste Galois and techniques he developed – now called group theory – to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more. So what is a group, anyway? It starts with a set, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don’t have too many properties – that is, characteristics – but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.In clock addition, 3 + 12 = 3.OpenStax, CC BY-SA Let’s call this set of numbers 1 through 12 “clock numbers.” Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say “3 + 11 = 2” is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock. Clock addition has some nice properties. It satisfies:
closure, where adding things in the set gives you something else in the set,
identity, where there’s an element that doesn’t change the value of other elements in the set when added – adding 12 to any number will equal that same number,
associativity, where you can add wherever you want in the set,
inverses, where you can undo whatever an element does, and
commutativity, where you can change the order of which clock numbers you add up without changing the outcome: a + b = b + a.
By satisfying all these properties, mathematicians can consider clock numbers with clock addition a group. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can’t be made into a group easily – what’s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.
Rings and fields
Along with groups, the two other fundamental types of algebraic objects you would study in an introduction to modern algebra are rings and fields. We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call a ring. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where a(b + c) = ab + ac. Lastly, fields are rings that satisfy even more conditions. At the turn of the 20th century, mathematicians David Hilbert and Emmy Noether – who were interested in understanding how the principles in Einstein’s relativity worked mathematically – unified algebra and showed the utility of studying groups, rings and fields.
It’s all fun and games until you do the math
Groups, rings and fields are abstract, but they have many useful applications. For example, the symmetries of molecular structures are categorized by different point groups. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.The water molecule H₂O can be flipped horizontally and the end result is indistinguishable from the original position.Courtney Gibbons, CC BY-SA But let’s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to describe a Sudoku puzzle: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board. To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using x₃₅ to represent the cell in the third row and fifth column. The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x₁₁ – 1)(x₁₁ – 2)(x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that’s already 81 equations just to say, “Don’t plug in anything other than 1 through 9.” The rule “1 through 9 each appear exactly once in the top row” can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say x₁₁ + x₁₂ + ⋅⋅⋅ + x₁₉ – 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say x₁₁ x₁₂ ⋅⋅⋅ x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero. If you’re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you’re not wrong.Turning Sudoku into algebra takes a fair bit of work.Courtney Gibbons What do we get by doing this complicated translation into algebra? Well, we get to use late-20th century algorithms to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring – called an ideal – these game board clues make within the larger ring. The algorithms will tell you if there’s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all. This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle problems in artificial intelligence, robotics, cryptography, quantum computing and so much more – all with the same bag of tricks you’d use to solve the Sudoku puzzle or Rubik’s cube.Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
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The 8,000-Year History of Pecans: How America’s Only Native Nut Became a Holiday Staple
Discover how pecans went from ignored trees to holiday staples over 8,000 years. Learn about Native American pecan use, the enslaved man who revolutionized pecan grafting, George Washington’s pecan obsession, and why the US produces 80% of the world’s pecans.
How pecans went from ignored trees to a holiday staple – the 8,000-year history of America’s only native major nut crop
Shelley Mitchell, Oklahoma State University Pecans have a storied history in the United States. Today, American trees produce hundreds of million of pounds of pecans – 80% of the world’s pecan crop. Most of that crop stays here. Pecans are used to produce pecan milk, butter and oil, but many of the nuts end up in pecan pies. Throughout history, pecans have been overlooked, poached, cultivated and improved. As they have spread throughout the United States, they have been eaten raw and in recipes. Pecans have grown more popular over the decades, and you will probably encounter them in some form this holiday season. I’m an extension specialist in Oklahoma, a state consistently ranked fifth in pecan production, behind Georgia, New Mexico, Arizona and Texas. I’ll admit that I am not a fan of the taste of pecans, which leaves more for the squirrels, crows and enthusiastic pecan lovers.
The spread of pecans
The pecan is a nut related to the hickory. Actually, though we call them nuts, pecans are actually a type of fruit called a drupe. Drupes have pits, like the peach and cherry.Three pecan fruits, which ripen and split open to release pecan nuts, clustered on a pecan tree.IAISI/Moment via Getty Images The pecan nuts that look like little brown footballs are actually the seed that starts inside the pecan fruit – until the fruit ripens and splits open to release the pecan. They are usually the size of your thumb, and you may need a nutcracker to open them. You can eat them raw or as part of a cooked dish. The pecan derives its name from the Algonquin “pakani,” which means “a nut too hard to crack by hand.” Rich in fat and easy to transport, pecans traveled with Native Americans throughout what is now the southern United States. They were used for food, medicine and trade as early as 8,000 years ago.Pecans are native to the southern United States.Elbert L. Little Jr. of the U.S. Department of Agriculture, Forest Service Pecans are native to the southern United States, and while they had previously spread along travel and trade routes, the first documented purposeful planting of a pecan tree was in New York in 1722. Three years later, George Washington’s estate, Mount Vernon, had some planted pecans. Washington loved pecans, and Revolutionary War soldiers said he was constantly eating them. Meanwhile, no one needed to plant pecans in the South, since they naturally grew along riverbanks and in groves. Pecan trees are alternate bearing: They will have a very large crop one year, followed by one or two very small crops. But because they naturally produced a harvest with no input from farmers, people did not need to actively cultivate them. Locals would harvest nuts for themselves but otherwise ignored the self-sufficient trees. It wasn’t until the late 1800s that people in the pecan’s native range realized the pecan’s potential worth for income and trade. Harvesting pecans became competitive, and young boys would climb onto precarious tree branches. One girl was lifted by a hot air balloon so she could beat on the upper branches of trees and let them fall to collectors below. Pecan poaching was a problem in natural groves on private property.
Pecan cultivation begins
Even with so obvious a demand, cultivated orchards in the South were still rare into the 1900s. Pecan trees don’t produce nuts for several years after planting, so their future quality is unknown.An orchard of pecan trees.Jon Frederick/iStock via Getty Images To guarantee quality nuts, farmers began using a technique called grafting; they’d join branches from quality trees to another pecan tree’s trunk. The first attempt at grafting pecans was in 1822, but the attempts weren’t very successful. Grafting pecans became popular after an enslaved man named Antoine who lived on a Louisiana plantation successfully produced large pecans with tender shells by grafting, around 1846. His pecans became the first widely available improved pecan variety.Grafting is a technique that involves connecting the branch of one tree to the trunk of another.Orest Lyzhechka/iStock via Getty Images The variety was named Centennial because it was introduced to the public 30 years later at the Philadelphia Centennial Expedition in 1876, alongside the telephone, Heinz ketchup and the right arm of the Statue of Liberty. This technique also sped up the production process. To keep pecan quality up and produce consistent annual harvests, today’s pecan growers shake the trees while the nuts are still growing, until about half of the pecans fall off. This reduces the number of nuts so that the tree can put more energy into fewer pecans, which leads to better quality. Shaking also evens out the yield, so that the alternate-bearing characteristic doesn’t create a boom-bust cycle.
US pecan consumption
The French brought praline dessert with them when they immigrated to Louisiana in the early 1700s. A praline is a flat, creamy candy made with nuts, sugar, butter and cream. Their original recipe used almonds, but at the time, the only nut available in America was the pecan, so pecan pralines were born.Pralines were originally a French dessert, but Americans began making them with pecans.Jupiterimages/The Image Bank via Getty Images During the Civil War and world wars, Americans consumed pecans in large quantities because they were a protein-packed alternative when meat was expensive and scarce. One cup of pecan halves has about 9 grams of protein. After the wars, pecan demand declined, resulting in millions of excess pounds at harvest. One effort to increase demand was a national pecan recipe contest in 1924. Over 21,000 submissions came from over 5,000 cooks, with 800 of them published in a book. Pecan consumption went up with the inclusion of pecans in commercially prepared foods and the start of the mail-order industry in the 1870s, as pecans can be shipped and stored at room temperature. That characteristic also put them on some Apollo missions. Small amounts of pecans contain many vitamins and minerals. They became commonplace in cereals, which touted their health benefits. In 1938, the federal government published the pamphlet Nuts and How to Use Them, which touted pecans’ nutritional value and came with recipes. Food writers suggested using pecans as shortening because they are composed mostly of fat. The government even put a price ceiling on pecans to encourage consumption, but consumers weren’t buying them. The government ended up buying the surplus pecans and integrating them into the National School Lunch Program.Today, pecan producers use machines called tree shakers to shake pecans out of the trees.Christine_Kohler/iStock via Getty Images While you are sitting around the Thanksgiving table this year, you can discuss one of the biggest controversies in the pecan industry: Are they PEE-cans or puh-KAHNS? Editor’s note: This article was updated to include the amount of protein in a cup of pecans.Shelley Mitchell, Senior Extension Specialist in Horticulture and Landscape Architecture, Oklahoma State University This article is republished from The Conversation under a Creative Commons license. Read the original article.
Why the chemtrail conspiracy theory lingers and grows – and why Tucker Carlson is talking about it
The chemtrail conspiracy theory has surged despite being thoroughly debunked. Learn why people believe contrails are chemical weapons, how Tucker Carlson amplified the theory, and what psychology reveals about conspiracy thinking and our need for control.
Contrails have a simple explanation, but not everyone wants to believe it. AP Photo/Carolyn Kaster
Why the chemtrail conspiracy theory lingers and grows – and why Tucker Carlson is talking about it
Calum Lister Matheson, University of Pittsburgh Everyone has looked up at the clouds and seen faces, animals, objects. Human brains are hardwired for this kind of whimsy. But some people – perhaps a surprising number – look to the sky and see government plots and wicked deeds written there. Conspiracy theorists say that contrails – long streaks of condensation left by aircraft – are actually chemtrails, clouds of chemical or biological agents dumped on the unsuspecting public for nefarious purposes. Different motives are ascribed, from weather control to mass poisoning. The chemtrails theory has circulated since 1996, when conspiracy theorists misinterpreted a U.S. Air Force research paper about weather modification, a valid topic of research. Social media and conservative news outlets have since magnified the conspiracy theory. One recent study notes that X, formerly Twitter, is a particularly active node of this “broad online community of conspiracy.” I’m a communications researcher who studies conspiracy theories. The thoroughly debunked chemtrails theory provides a textbook example of how conspiracy theories work.
Boosted into the stratosphere
Conservative pundit Tucker Carlson, whose podcast averages over a million viewers per episode, recently interviewed Dane Wigington, a longtime opponent of what he calls “geoengineering.” While the interview has been extensivelydiscredited and mocked in other media coverage, it is only one example of the spike in chemtrail belief. Although chemtrail belief spans the political spectrum, it is particularly evident in Republican circles. U.S. Secretary of Health and Human Services Robert F. Kennedy Jr. has professed his support for the theory. U.S. Rep. Marjorie Taylor Greene of Georgia has written legislation to ban chemical weather control, and many state legislatures have done the same. Online influencers with millions of followers have promoted what was once a fringe theory to a large audience. It finds a ready audience among climate change deniers and anti-deep state agitators who fear government mind control.
Heads I win, tails you lose
Although research on weather modification is real, the overwhelming majority of qualifiedexperts deny that the chemtrail theory has any solid basis in fact. For example, geoengineering researcher David Keith’s lab posted a blunt statement on its website. A wealth of other resources exist online, and many of their conclusions are posted at contrailscience.com. But even without a deep dive into the science, the chemtrail theory has glaring logical problems. Two of them are falsifiability and parsimony.The philosopher Karl Popper explains that unless your conjecture can be proved false, it lies outside the realm of science. According to psychologist Rob Brotherton, conspiracy theories have a classic “heads I win, tails you lose” structure. Conspiracy theorists say that chemtrails are part of a nefarious government plot, but its existence has been covered up by the same villains. If there was any evidence that weather modification was actually happening, that would support the theory, but any evidence denying chemtrails also supports the theory – specifically, the part that alleges a cover-up. People who subscribe to the conspiracy theory consider anyone who confirms it to be a brave whistleblower and anyone who denies it to be foolish, evil or paid off. Therefore, no amount of information could even hypothetically disprove it for true believers. This denial makes the theory nonfalsifiable, meaning it’s impossible to disprove. By contrast, good theories are not false, but they must also be constructed in such a way that if they were false, evidence could show that. Nonfalsifiable theories are inherently suspect because they exist in a closed loop of self-confirmation. In practice, theories are not usually declared “false” based on a single test but are taken more or less seriously based on the preponderance of good evidence and scientific consensus. This approach is important because conspiracy theories and disinformation often claim to falsify mainstream theories, or at least exploit a poor understanding of what certainty means in scientific methods. Like most conspiracy theories, the chemtrail story tends not to meet the criteria of parsimony, also known as Occam’s razor, which suggests that the more suppositions a theory requires to be true, the less likely it actually is. While not perfect, this concept can be an important way to think about probability when it comes to conspiracy theories. Is it more likely that the government is covering up a massive weather program, mind-control program or both that involve thousands or millions of silent, complicit agents, from the local weather reporter to the Joint Chiefs of Staff, or that we’re seeing ice crystals from plane engines? Of course, calling something a “conspiracy theory” does not automatically invalidate it. After all, real conspiracies do exist. But it’s important to remember scientist and science communicator Carl Sagan’s adage that “extraordinary claims require extraordinary evidence.” In the case of chemtrails, the evidence just isn’t there.Scientists explain how humans are susceptible to believing conspiracy theories.
Psychology of conspiracy theory belief
If the evidence against it is so powerful and the logic is so weak, why do people believe the chemtrail conspiracy theory? As I have argued in my new book, “Post-Weird: Fragmentation, Community, and the Decline of the Mainstream,” conspiracy theorists create bonds with each other through shared practices of interpreting the world, seeing every detail and scrap of evidence as unshakable signs of a larger, hidden meaning. Uncertainty, ambiguity and chaos can be overwhelming. Conspiracy theories are symptoms, ad hoc attempts to deal with the anxiety caused by feelings of powerlessness in a chaotic and complicated world where awful things like tornadoes, hurricanes and wildfires can happen seemingly at random for reasons that even well-informed people struggle to understand. When people feel overwhelmed and helpless, they create fantasies that give an illusion of mastery and control. Although there are liberal chemtrail believers, aversion to uncertainty might explain why the theory has become so popular with Carlson’s audience: Researchers have longargued that authoritarian, right-wing beliefs have a similar underlying structure. On some level, chemtrail theorists would rather be targets of an evil conspiracy than face the limits of their knowledge and power, even though conspiracy beliefs are not completely satisfying. Sigmund Freud described a fort-da (“gone-here”) game played by his grandson where he threw away a toy and dragged it back on a string, something Freud interpreted as a simulation of control when the child had none. Conspiracy theories may serve a similar purpose, allowing their believers to feel that the world isn’t really random and that they, the ones who see through the charade, really have some control over it. The grander the conspiracy, the more brilliant and heroic the conspiracy theorists must be. Conspiracies are dramatic and exciting, with clear lines of good and evil, whereas real life is boring and sometimes scary. The chemtrail theory is ultimately prideful. It’s a way for theorists to feel powerful and smart when they face things beyond their comprehension and control. Conspiracy theories come and go, but responding to them in the long term means finding better ways to embrace uncertainty, ambiguity and our own limits alongside a new embrace of the tools we do have: logic, evidence and even humility. Calum Lister Matheson, Associate Professor of Communication, University of Pittsburgh This article is republished from The Conversation under a Creative Commons license. Read the original article.
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DoorDash Driver Arrested After Claiming Sexual Assault: What Really Happened?
A DoorDash driver who claimed she was sexually assaulted during a delivery is now facing felony charges after police say her viral video showed an unconscious, partially nude customer without consent. Here’s what investigators found and why the case is sparking national debate.
DoorDash delivery driver involved in a viral video controversy after claiming sexual assault; police say no assault occurred, and the driver now faces felony charges.
DoorDash Driver Arrested After Claiming Sexual Assault: What Really Happened?
A Viral Accusation Turns Into a Criminal Case
A routine food drop-off turned into a national controversy this month after a DoorDash delivery driver claimed she was sexually assaulted during a delivery — only to later be arrested herself following a police investigation. The incident, which quickly spread across TikTok and other platforms, has generated fierce debate over privacy, personal safety, and the power of viral video culture.
The driver, identified as Livie Rose Henderson, posted a video on social media in mid-October claiming that when she arrived at a customer’s home in Oswego, New York, she found the front door open and discovered a man “half-naked and unconscious” on his couch. She publicly described the moment as a sexual assault, saying she felt endangered and traumatized.
Her posts went viral almost immediately, drawing attention from millions of viewers and sparking outrage over the safety risks faced by gig workers — particularly women — who make deliveries to unfamiliar homes.
But the narrative took a dramatic turn.
Police: No Sexual Assault Occurred
According to the Oswego Police Department, an investigation found no evidence that Henderson was sexually assaulted. Instead, authorities say that she:
Entered the home without consent
Recorded the unconscious customer, who was partially nude
Posted the footage online, identifying him
Made claims police say were “false and misleading”
Investigators concluded the man was intoxicated and unconscious, not acting with intent or awareness. As a result, Henderson was arrested and charged with:
Second-degree unlawful surveillance (felony)
First-degree dissemination of unlawful surveillance images (felony)
Police emphasized that recording a person who is nude or partially nude inside their home — regardless of context — constitutes a violation of New York’s surveillance and privacy laws if done without permission.
DoorDash Drops the Driver
Henderson also claimed that DoorDash deactivated her account, something she described as retaliation for “exposing her assaulter.” But following her arrest, DoorDash stated that recording customers inside their homes violates company policy and local laws.
DoorDash said it cooperated with investigators but declined to comment further on personnel matters.
A Complicated Public Reaction
Social media reaction has been sharply divided:
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Sympathy for the driver
Many viewers initially supported Henderson, arguing that gig workers often deal with unsafe conditions and should not be forced to decide between finishing a delivery or backing away from a potentially threatening situation.
Backlash over privacy violations
Others argue that Henderson crossed legal and ethical boundaries by:
Entering a private residence
Recording a vulnerable, unconscious person
Posting it publicly
Accusing the individual of a crime without evidence
These actions, critics say, show the dangerous consequences of rushing to social media before police or professional investigators evaluate the facts.
The Larger Issue: Safety vs. Responsibility
This case highlights a broader tension in the era of app-based work and viral content:
Gig workers do indeed face unpredictable and sometimes unsafe situations.
Customers have a right to privacy in their homes.
Social media, meanwhile, rewards the fastest and most dramatic version of a story — even before the truth is known.
As the criminal process continues, Henderson’s case may set a new precedent for how privacy laws interact with the realities of delivery work and the instant visibility of online platforms.
Focuses on the legal perspective, privacy violations, and the potential consequences of posting sensitive content online.
STM Daily News is a vibrant news blog dedicated to sharing the brighter side of human experiences. Emphasizing positive, uplifting stories, the site focuses on delivering inspiring, informative, and well-researched content. With a commitment to accurate, fair, and responsible journalism, STM Daily News aims to foster a community of readers passionate about positive change and engaged in meaningful conversations. Join the movement and explore stories that celebrate the positive impacts shaping our world.
Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
