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Riemann Zeta Function

Imagine a sum that keeps getting bigger: 1 + 2 + 3 + 4 + ... but somehow, its "value" is said to be  \[1+2+3+4+5… = -\frac{1}{12}\] This equation isn’t a joke. It isn’t a typo. And it definitely isn’t like one of those internet tricks that try to prove that 2 = 3. But how? Doesn’t calculus dictate that the sum should diverge? How can an infinite sum of positive integers result in a negative fraction? This equation is one of the paradoxical results of the famed Riemann Zeta function: \[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\] First Encounters My first encounter of the Riemann Zeta function was in a problem introduced to me in Math circle India:\(\newline\)               What is the probability that two random numbers x & y are coprime i.e. gcd(x,y) = 1 for \(  x,y \in \mathbb N\) At first glance, it seems like an impossible question to compute. We’re picking 2 numbers at random — how do you even define a probability over an...

How to Find and Understand Dinosaur Fossils

Paleontology doesn’t automatically mean dinosaurs but today, I am going to talk about dinosaurs. Paleontology is the study of any ancient animal through its fossils. Fossils can be any trace left behind by the extinct species that give us information on them. The word fossil means “dug up”. Bones? They are fossils. Imprints? They are fossils. Your cycle in a hundred years? Not exactly. You see, a fossil is a trace of a once-living organism and even though a cycle shows the evidence of a rider, it is not directly a remain of a living being. Now that what paleontology is has been clarified let's move into it. Today, most of the references of paleontology will go to a subsection that studies the dinosaur era namely, the Mesozoic or middle era.


How do paleontologists find and understand dinosaur fossils? Let’s start with finding the dinosaur fossil. The answer is, nobody knows. That’s disappointing, right? What I mean to say is that paleontologists cannot be sure about where to find fossils indefinitely but can make educated guesses. If we had a giant dinosaur scanning radar, we would have all the fossils there are in the world.


 

How to make an educated guess on where to start digging up for dinosaurs:

Don’t go digging up your backyard, you’ll make a mess


Find the geological area of the right age: 

You can get this information on a geological map. You can look for anything as old as 245 million years and also look for relatively new rocks of 66 million years


Find a desert: 

You need to find a desert because vegetation may cover up the dig and also destroy the fossil. An arid region also allows weathering to expose the fossil on the surface otherwise it would not be found


The rock should be sedimentary: 

This is because the other rock formation processes destroy fossils


Get as far away from other paleontologists as possible: 

Once you have followed the earlier rules, there is a high chance of finding dinosaur bones. Sure, you may find a T-Rex skull but that is only satisfaction for yourself. If you go to uncharted territories (paleontology wise) you have a higher chance of discovering a new species.


Give it a good name:

Unlike past names like the IRRITATOR or the DRINKER


There you go, a step to step guide to being an amateur paleontologist.


Now that you have a fossil, what do you do with it aside from keeping it as a showpiece?… You learn more about it.


First of all, you can find out about their habitat. You found the bones in the desert so they must be from there right? You’ve all seen Jurassic park so you know that dinosaurs live in a wide variety of habitats. Looking just at the fossils, you can’t figure out whether they lived in a desert, forest, swamp, etc. Here, you use a science called sedimentology. Sedimentology is the study of sed


By looking at the sediment rock that the fossil is found in, we can conclude what they used to be. See, sedimentary rocks are made when sediments or particles are squeezed together with immense force to form a rock. This can occur when sediments such as sand or gravel form a layer of say, a beach but over time they compress due to the weight of the new layers to form a sedimentary rock. Using the knowledge of sediments we can figure out what an area used to be like. For example, if a dinosaur fossil is found in sandstone, the sediment is sand and that indicates that the habitat was a beach, an ocean bed or a river channel. Shale or mudstone indicates mud and silt as sediments indicating a lake bottom environment. Coal is formed by the remains of dead animals and plants so a coal deposit indicates that it used to be a swamp or forest. Finding fossils in forests is very unlikely but we’ll see to that later. Finally, Limestone is made up of sediments of Calcium Carbonate which is found in the shells of crustaceans so it indicates a shallow marine environment. 


Now getting back to the topic of why the fossils of dinosaurs who lived in forests cannot be found. For a dead animal to fossilise, a lot of things need to go just right: the right environment, timing, and conditions and the fossils also need to survive millions of years and then resurface before natural processes dissolve the fossil. For example, lush jungles prevent fossilization as the chances that an animal will be buried here are minimal. Countless scavengers of all sizes break down freshly deceased animals extremely quickly and the soil is often so acidic that bones are dissolved. And so fossils of dinosaurs from jungles are practically nonexistent. This is sad as even today, HALF of all animals live in lush jungles that cover only 2% of the Earth’s surface. Back then, jungles covered a huge present of landmasses. This means that most of the dinosaurs that lived are lost to us forever.


Thinking of dinosaurs, we think big but what about the little ones. Dinosaurs had babies of course… they laid eggs. Now earlier, in the 1900s, all the museums wanted bigger and better dinosaurs. It took paleontologists a while- actually decades to ask the question- Where are the little ones? Throughout the 20th century, the common belief was that there were big dinosaurs and smaller versions of the dinosaur. But they could not find any little Dracorex Hogwartsia, or any little Torosauruses, they didn’t find many juveniles at all! Well, the problem had something to do with a scientist wanting to name something. You see, a scientist has a deep-seated want to name something so even when they saw a slightly different dinosaur, they named it differently. This caused a lot many species. The problem was that many dinosaurs had something called allometric cranial ontogeny. This means that different parts of their head grow at different speeds. So a juvenile triceratops will not have the same cranial proportions as a fully grown one. This is a problem because scientists that wanted to name stuff classified the same species as different because they ignored allometric cranial ontogeny. The only way to figure out if the dinosaur is juvenile or fully grown is to cut its bone. If you cut into a juvenile dinosaur’s bone, it is spongey but an adult's bone is compact and solid. But you can’t go to a museum and ask them if you can cut their dinosaur’s skull. It turns out that many dinosaurs that were classified as cousin species are the same animal. By looking at similar “cousins” and cutting their skulls we found out that Dracorex, Stygimoloch, and Pachesephalosaurus, despite having different features are the same animal. A palaeontologist named Jack Horner cut into a now classified as juvenile triceratops and found it to be spongey and then he cut into a sub-adult Triceratops and it was spongey but when he cut into an adult triceratops’s skull, it was still spongey! So he took a closely related dinosaur called Torosaurus and found it to be compact and adult. Now, many would say that Torosaurus has holes in its frill so it can’t be the same creature but when they saw the skulls, they saw micro holes in the triceratops’s skull and those were seeming to have to grow larger as evident from its cell pattern. So, even though we seem to focus on bigger dinosaurs, growth is something that we have to keep in mind.




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