#Numerical Methods

[a-lesson-you-never-asked-for]

A class of curves

During my thesis, researching ways of flow around rocks and ships near the surface of water, I found a set of equations that can be used to model these weirder shapes. One example was the lemon curve, which you saw as the cricket ball simulation I showed earlier. The other is a generalised version of these that allows one to customise the location of bumps on the surface of the solid. The following plots were made in python.

Lemon curves (name courtesy of my fabulous sister @mrunmione) - the shape used to model a cricket ball

Crocodile curves (because they literally look like crocodiles)

Why are they important? ↓

It is important that when you model obstacles, you do so with at least second-differentiable functions, which means that they can be differentiated at least twice without behaving badly.

This is due to the nature of equations used to model fluids, called the Navier Stokes equations, which is based on Newton's 2nd law of motion, the force applied on an object is proportional to the acceleration (which is where the 2nd derivative comes into play).

[boredphysicsstudent]

Been working a lot of simulations!

Also posted the yt link

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 2

Let's continue the series of Early Projects. The next one was from the sphere of analytical mechanics. The task was to simulate the rotational motion of bodies using quaternions. Quaternions make it possible to describe such motions quite succinctly in formulas. Special attention should be paid to modeling the Dzhanibekov effect, aka Tennis racket theorem.

P.S.: Below is a test of the operation of this simulator with a spin of the simple screw around its axis.

P.S.2: Here is a simulation of the Dzhanibekov effect. As it can be seen when the body have three distinct principal moments of inertia, rotation around one of the principal axes (associated with the second-order inertia moment) unstable. That's why the wing screw turns over.

[dolcechild]

first year ever doing numerical methods and matlab so it's kicking my ass please tell me there's good resources out there for them before i start despairing

[aromanrom]

Percolation in fortran

Introduction I thought I should add another project1 to my GitHub repositories and since I was mainly extending the Quantum Computing Simulator project I decided to implement something very simple, but still very rich, a model that despite being simple is still useful for many physical phenomena and also exhibits a phase transition… that is, something on Percolation theory. Because it is so…

[bitstitchbitch]

first math homework is done, now I need to do the second one because yes, both are due today

I was mostly done with the first one. The second one? Not so much. I’m delusional though and feel like it won’t take me very long. I do have some of the code I need to write done, and the last question is just summarizing a video so, we shall see.

I think the hard proofs were all on hw 1 though

[mathmedicine]

ALEVEL PURE MATHEMATICS 3 (PAPER 3)

PURE MATHEMATICS 2/3 (2002-2010) C1 Algebra C2 Logarithmic and Exponential Functions C3 Trigonometry C4 Differentiation C5 Integration C6 Numerical Methods C7 Vectors C8 Differential Equations C9 Complex Numbers

[mushroom-winners-proof]

If only i could put in one more character then it’d be perfect- I finally got the bitch ass bug so yay :) when I finally got money to buy timer balls (cus poor and was trying to save for the adm) it wants to go into the dusk ball- this asshole just LIVES to spite me

Figured out where to put in mystery gifts so I got all the stuff I need for other legendaries :) except celbie(?) cus- where tf do I get spiky eared pichu and what lvl is he because I cannot be bringing a lvl 40-/+ to a battle with a lvl 70+ Pokémon- also here’s everyone so far :) I’m trying to sort it by region just to make it near but 1 where is fucking jirachi from 2 where would melmetal go and 3 alola needs its own damn box because of the tapus also being in the game apparently- other than that I have some spots open because of specific Pokemon I haven’t caught yet like the slot under mewtwo being for mew and under reshiram and zekrom for kyruem- it’s a lot-

omg bug got :) not in the ideal ball tho smh but we got there : ]]

omg box organization .... we love that .....

[taohun]

GOT A 100 ON MY FIRST QUIZ OF THE YEAR 🥳

[mortaltravailed]

stares wistfully at his baby brother from behind the bar counter. because if that isn't a classic diluc ragnvindr "i miss you but i don't know how to say it" moment, then what is ?

as ever, the cavalry captain is the life and soul of the party.

flitting from table to table as though the angel's share were a ballroom in place of a tavern, he moves with grace and confidence, charming and smiling his way through most of the bar's occupants with ease. with death after noon in low supply, he's paradoxically no less jolly and rosy-cheeked than usual.

but when he finds his mark, he sticks to them like glue for the rest of the night until they're stumbling out of the door together by the end, a mess of slurred speech and entangled limbs.

on their way out, his eye catches diluc's gaze, ever-watchful from his perch behind the bar. as he waves his goodbye, the smile on kaeya's lips is telling: he's got no intention of following through on the sweet nothings he'd been whispering into all-too eager ears just minutes prior. ignorance is bliss, it seems, for he appears too focused on achieving his own ends to notice his brother's unspoken distress.

kaeya's own recognition of the truth remains equally unspoken, in kind.

if he allowed it to show, people would see the both of them for what they were: two broken-hearted boys, no less caring, and hurting bitterly in one another's absence. yet it was safer to keep one another at arm's length, to maintain their frosty indifference and play the parts they've assigned to each other.

everything was a front: the alluring smiles, the cheerfully painted door to his townhouse -- even the very walls of mondstadt themselves hid his ugly, painful truths from the world, stuffed deep within the recesses of loose brickwork.

and kaeya preferred it remain that way.

come daybreak, he would attach the night's findings to the leg of the carrier pigeon at his window and let it loose. it was one of many letters that would find its way to the winery, and with it came an inability to distance diluc in absolutely everything he did, which seemed much too inhumane in sharp contrast to what had unfolded behind the locked doors of his home.

words that he had not the courage to speak out loud, but could just about manage under the safety of anonymity:

"be sure to take care of yourself."

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 1

While the work is underway, and there is not much to show, it was decided to present examples of past projects in the field of computational physics. They were manufactured in the same way as the planned project: an integration solver on C++ language, a visualizer on Python… It's not appropriate to leave this blog for a long time without any content at all.

The first simple project from the field of applied physics was devoted to modeling a double pendulum using Lagrangians. This is a good example from chaos theory, representing the effect when the slightest changes in the initial conditions lead to a completely different behavior of the system.

P.S.: Below is an animation of two double mathematical pendulums with a bit different initial conditions (the second pendulum has a more elevated second end).

P.S.2: Here is an animation of two double pendulums, but with a much heavier second end. Apparently, this causes a more predictable behavior of the double pendulum, allowing it to be interpreted as a pendulum inside a pendulum.

[dolcechild]

when ao3 goes down and you have two exams within the next week then yeah maybe it is a sign from the gods

[aromanrom]

Machine Learning from scratch

Introduction This is the second project I already had when I posted Updates to project. Here is its repository: Machine Learning project on GitHub1. I started it as the Artificial Intelligence hype was going stronger, just to have a project on a domain that’s of big interest nowadays. At that point I was thinking to continue it with convolutional networks and at least recurrent networks, not…

[nxghtshades]

now that I'm no longer drowning in coursework, I've been writing again!

[dnieperrobot69420]

What have I been up to? Well more or less more optics.

I started to learn about the Index Ellipsoid, and I am beginning to understand how we can quantify a materials allowed polarizations using the "indicatrix" (not going to lie this is a hard word to spell lol)

I now understand that there are two possible electric flux densities (basically the quadratic equations +/- gives two roots, so when you solve for two possible refractive indices, you also get two D solutions) and that these two fluxes are orthogonal.

The index Ellipsoid let's us characterize anisotropic crystals into biaxial (two defined optical axes) and uniaxial (two axes are equal)

But honestly, I'm hitting a bit of a wall here. I just want to be familiar with the material before grad school starts. Since I'm self studying now, I decided to venture off into a related topic: Computational electromagnetics!!

I took numerical methods this winter so when I heard about Finite Difference Method I didn't look like a 🦌 staring at 🚗 lol (honestly I find Runge-Kutta methods scarier haha)

So I learned a bit about collocated grids, recalled Dirichlet boundary conditions, periodic boundary conditions as well as Neuman boundary conditions.

My plan is to use COMSOL multi physics to do a little simulation for my masters thesis, so I'm betting that having a grasp on the numerical side of emag is something I need to accomplish to bear fruitful results.

And of course I get hit with the realization exactly *why* plane waves are used to teach us optics: because they are simply a stepping stone towards us learning about Gaussian Beams!

The 🐇 hole just keeps getting deeper and deeper... 🕳️

[reasonsforhope]

"A medical technology company in Australia is aiming for a world-first: it wants to launch a blood test for endometriosis (sometimes called 'endo' for short) within the first half of this year [2025].

In a recent peer-reviewed trial, its novel test proved 99.7 percent accurate at distinguishing severe cases of endometriosis from patients without the disease but with similar symptoms.

Even in the early stages of the disease, when blood markers may be harder to pick out, the test's accuracy remained over 85 percent.

The company behind the patent, Proteomics International, says it is currently adapting the method "for use in a clinical environment," with a target launch date in Australia for the second quarter of this year [2025].

The test is called PromarkerEndo.

"This advancement marks a significant step toward non-invasive, personalized care for a condition that has long been underserved by current medical approaches," managing director of Proteomics International Richard Lipscombe said in a press release from December 30.

Endometriosis is a common inflammatory disease that occurs when tissue similar to the lining of the uterus grows in other parts of the body, forming lesions. The disease can be very painful, and yet the average patient often suffers debilitating symptoms for up to seven years before they are properly diagnosed.

While there are numerous reasons for such a long delay, symptoms of endometriosis are often highly variable, unpredictable, difficult to measure or describe, and dismissed or overlooked by doctors.

Today, the only definitive way to diagnose endometriosis is via keyhole surgery called a laparoscopy, which is expensive, invasive, and carries risks.

Proteomics International is hoping to change that.

In collaboration with researchers at the University of Melbourne and the Royal Women's Hospital, the company compared the bloodwork data from 749 participants of mostly European descent.

Some had endometriosis and others had symptoms that were similar to endo but without the lesions. All participants had a laparoscopy to confirm the presence or absence of the disease.

Sifting through the bloodwork, researchers ran several different algorithms to figure out which proteins in the blood were best at predicting endometriosis of varying stages.

Building on previous research, a panel of 10 proteins showed a "clear association" with endometriosis.

For years now, scientists have investigated possible blood biomarkers of endometriosis to see if they could differentiate between those who have endo and those who do not. Similar to cancerous tumors, endo lesions can establish their own blood supply, and if cervical cancer can be diagnosed via a blood test, it seemed possible that endometriosis could be, too...

Proteomics International claims patents for PromarkerEndo are "pending in all major jurisdictions," starting first in Australia.

It remains to be seen if the company's blood test lives up to the hype and is approved by the Australian Therapeutic Goods Administration (TGA). But that's not outside the realm of possibility.

In November of 2023, some researchers predicted that a "reliable non-invasive biomarker for endometriosis is highly likely in the coming years."

Perhaps this is the year."

-via ScienceAlert, January 9, 2025

--

Note: As someone with endometriosis, let me say that this is a HUGE deal. The condition is incredibly common, incredibly understudied, and incredibly often dismissed. Massive sexism at work here.

I got very lucky and got diagnosed after about 6 months of chronic pain (and extra extra lucky, because my pain went away with medication). But as the article says, the average time to diagnosis is seven years.

Being able to confirm endometriosis diagnoses/rates without invasive surgery will also lead to huge progress in studying/creating treatments for endo.

And fyi: If you have a period that is so painful that you can't stand up, or have to go home from school/work, or vomit, or anything else debilitating (or if any of those things apply if you forget to take pain meds), that is NOT NORMAL, and you should talk to a competent gynecologist asap.