Boost Your Spirits with Fun Math Strategies for Feeling Unpopular

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Your friend might have a larger circle of friends than you do, but don’t take it personally; it’s all about network structure.

Friendship dynamics can be illustrated through networks. By representing each person as a node and drawing lines between friends, you can visualize social connections effectively. This approach helps articulate the bonds of friendship in both offline and online environments.

Such networks enable examination of intriguing questions, like the concept of degrees of separation. For instance, if someone is a friend of your friend, they are two degrees separated from you, while their friends fall into the three-degree category.

How many connections must you traverse to link two individuals? Connections often form clusters. For example, consider a distinct group of friends, like those from your neighborhood, workplace, or a hobbyist club like astrophotography. It’s likely that many individuals within this group are interconnected as friends, thereby increasing your direct friendship circle.

However, long-distance connections exist as well. A long-lost friend living abroad may belong to a separate circle of friends, all passionate about soap carving. These individuals represent your second-degree connections, regardless of your unfamiliarity.

This concept relates to the well-known theory of six degrees of separation. By exploring connections beyond your immediate circle, you can swiftly extend the reach of your network. For instance, a former colleague in London could be engrossed in wargames with a barista located near the Houses of Parliament and may soon find themselves just a few degrees away from shaking hands with the prime minister.

What about influential individuals? Within friendship networks, some people naturally possess more connections. Consider a hypothetical group of 20 where 15 are friends with Sandy and only 5 with Charlie. If you randomly select someone, there’s a 75% chance they’re linked to Sandy but only 25% tied to Charlie. Therefore, your circle of friends is not randomly selected; you’re more likely to befriend popular individuals, creating an impression that your friends are more socially connected than you.

This scenario, referred to as the friendship paradox, serves a purpose in discovering prominent individuals. If you randomly survey a group, you’ll observe an average number of connections. However, if you ask them to name a friend, they will likely identify someone with a greater network. Thus, the connections in this newly identified group are typically above average.

So, when it seems your friends are attending more social events, have broader professional networks, or are engaged in more activities than you are, remember: it’s a matter of network dynamics, not personal inadequacy.

Peter Rowlett | Mathematics lecturer, podcaster, and author at Sheffield Hallam University, UK. Follow me on Twitter @peterrowlett

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Podcast: Paranormal Activity Celebrities Share Haunting Stories of Possessive Spirits

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Has the Olympics caused anyone else’s productivity to hit an all-time low? Like me, you’ve probably had that small but highly distracting iPlayer window open in the corner of your screen all day, gone straight home, watched athletics on TV, ignored the dirty dishes, and dozed off.

Surprisingly, what if you’re on the go? BBC Radio 5 Live and The official Olympic podcast. It’s great for catching up on the action away from the screen (you may have cried listening to the women’s triathlon on your commute to work), but these are just the tip of the iceberg when it comes to pods extending the Paris 2024 journey. As medals are awarded and sporting heroes make their appearance, there are shows that take you even deeper into the personalities themselves.

Simone Biles said she thought America hated her after the 2020 Tokyo Olympics. Call her daddy. Keeley Hodgkinson is confident of winning the gold medal. High Performance Podcast. Tom Daley’s Desert Island Discs Pleasant to listen to. Mondo Duplantis on being a pole vault champion Mindset Victory. Jess Thom, the lead psychologist for the British team, speaks candidly to The Guardian’s Science Weekly about how to prepare athletes for failure and success, and what happens when they have to return to normal life. Plus, Adam Peaty has a deep and meaningful conversation with his father-in-law, Gordon Ramsay. About the Olympics.

But if you’re completely bored of sports, don’t worry: this week’s best podcasts offer a lovely escape, with the paranormal, celebrity fantasies, and strolls through Borough Market. We’ve also rounded up the top five podcasts featuring A-list stars (which will likely include Olympic superstars and viral sensations at some point). Kim Ye-ji, South Korean shooting silver medalist.

Holly Richardson
TV assistant editor

This week’s picks




Dreamspace presenter Gemma Cairney. Photo: Katherine Ann Rose/Observer

Paranormal Activity: True Stories of Possession
All episodes available on Audible

Fifteen years after Katie Featherston and Micah Sloat terrified a generation with the film Paranormal Activity, they’re doing it again in podcast form. Sloat is a big fan of spiritual exploration, so it’s only fitting that he introduces the story of the Watseka Wonder, in which a 14-year-old girl claims to have been possessed by a dead woman for 16 weeks. “Dad of the Witch” Griffin Ceddo expands on the possession in a surprisingly moving account.
Hannah Verdier

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How certain songs can lift our spirits while others stir up excitement in our hearts

Music can stimulate emotions such as joy, sadness, and anger

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Scientists have discovered musical patterns that can make our hearts beat faster or make our stomachs feel like they're doing somersaults.

When a chord sequence of three or more notes played at the same time goes in a different direction than we expect, it seems to cause a strong sensation around the heart, but it follows an easily predictable pattern. The thing causes strong sensations around the heart. As if it hit us directly in the gut.

“Music has a unique power to stir emotions that cannot be expressed in words,” he says. Tatsuya Daikoku at the University of Tokyo in Japan. “It's not just an auditory experience, it's a physical experience. Sometimes when music comes on, my body shivers or I feel a warmth around my heart, and that's a feeling that's hard to describe in words.”

Researchers have already shown that music can evoke strong emotional responses, but pianist and composer Daikoku and his colleagues want to know where in the body people feel those emotions. I thought. To find out, they first used analysis and statistics software to analyze his 890 songs from the US Billboard Hot 100 chart.

The software determined that the song's chord-to-chord sequences were various variations with high or low levels of both surprise and uncertainty. For example, in some sequences, a low surprise, low uncertainty code may be followed by a low surprise, low uncertainty code, or a low surprise, low uncertainty code may be followed by a low surprise, low uncertainty code. , there are also sequences of codes with low uncertainty but high surprise.

From this, the researchers created 92 musical segments consisting of four chord sequences. Each segment represents one of eight different combinations of surprise and uncertainty. He then asked 527 volunteers to listen to various sets of all eight chord patterns while viewing a silhouette of a human body online.

Listeners were instructed to click where they felt a physical reaction within 10 seconds of listening to the music. They then completed an online survey about the emotions they felt when they heard the chords.

The researchers found that if the first three chords followed an easily predictable pattern, the main difference in physical sensation had a lot to do with what happened on the fourth chord. If that fourth chord follows the expected pattern, people will feel it in the abdomen, but if it deviates from the expected pattern, they will feel it around the heart.

Regarding emotions, participants reported greater feelings of calm, relief, contentment, nostalgia, and empathy when chord progressions followed a predictable pattern. If his first three chords were predictable and his fourth chord unsurprising, even if it was relatively difficult to predict, it was generally less awkward and less predictable compared to other chord arrangements. Feelings of anxiety were reduced.

The results of this study “revealed how music affects not only our ears, but also our bodies and minds,” Daikoku said. “Music has the power to elicit these strong, embodied emotions, leading us to understand our inner emotional landscape in a way that words cannot.” With this understanding, someday That could lead to better mental health interventions, he says.

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