I know we have to show that fg=0 implies f =0 or g=0. Let f be not zero . Then we have to show g=0.

f is not zero, so we can find a number a such that f(a) is not zero. Due to continuity we can find a neighborhood U of a where f is also not zero. So we have g_{|U} = 0.

Where I got some difficulties is to apply the identity theorem of holomorphic functions.

Let's define some set A:={x in C|g(x)=0} and let x in U. Since U is open we can assume that x+1/n is in U for all natural n . Let {x_n=x+1/n} be the sequence in U then x_n is in A for all n and x is accumulation point of A, so that g=0 on C.

I don’t need a proof, but I’m just a little confused about the statement. If a ring has unity 1, then x1=x. But if x is a zero divisor, then x1=0 for x=/=0, so there wouldn’t exist a unity. Am I thinking about this correctly ?

I'm tasked with showing that the ring P of polynomials with integer coefficients obeys the law of cancellation. I know in order to do this I need to show that P has no zero divisors, but I'm not sure how to approach this.

My initial reaction is to use the division algorithm somehow and maybe find a contradiction, but I'm not sure. Any hints or help would be appreciated.

In this pdf: here

On page three, the following can be found: https://imgur.com/a/PTGJTTJ

In the second step they reduce a sum to be 0. How is that the case? From my searching that is not a special property of the mobius function. Any insight is greatly appreciated.

In general, the set of zero-divisors is not closed under addition. For example (1,0) and (0,1) is a zero divisor but (1,1) isn't in the ring R^2 with coordinate-wise addition and multiiplication. When does the set of zero-divisors of a commutative ring form an ideal?(Here, 'zero-divisor' includes zero.)

Let f:R→S be a homomorphism of rings. If r in R is a zero divisor, is f(r) a zero divisor in S?

Here is an album of the text with pictures of the question and relevant definitions. https://imgur.com/a/M8wsf

My approach to the problem is to run the definition of a zero divisor through the function. That is to say, if r is a zero divisor, then there is an element, say s, such that r and s are both not zero and rs = 0 in R or sr = 0 in R.

It follows that f(rs) = f(r)f(s) . I need to be certain that f(r) and f(s) are both not the zero element in S. If I knew that this function was injective, I could be certain of that, because I know that the zero element of R would map to the zero element of S and nothing else would map to zero. However, I can't be sure that this function maps more than one thing to the zero element of S, can I?

So how can I prove that both f(r) and f(s) are not zero and their product is equal the zero element of S?

edit: clarification, wordy math is hard

Hello!

As the title says. I am having trouble with the following problem:

Prove or disporve the following statement: Let R be a commutative ring. Then the set of zero divisors in R is an ideal

Progress: I know what I have to prove that the set is non-empty, that if x,y belong in the set then so should x+y too, and if r belong in R then rx and xr belong in the set as well.

I know that, if x is a zero divisor then there is a nonzero element b in R such that xb=0.

That is pretty much how far I've gotten. Would appreciate some help!

I’ve always visualized hypercomplex numbers as representing geometrical rotation and scaling in higher dimensions. But two non-zero sedenions can multiply to zero; and interpreting that geometrically seems to imply that it’s possible to apply a series of rotations and non-zero scalings to an object in 16 dimensions in such a way that it disappears.

Is this really the way geometrical objects behave in 16 dimensions? Or is it inappropriate to visualize hypercomplex numbers this way?

Do any zero divisor exist in the ring of smooth functions over the whole real line? It's easy to see that they exist for n differentiable by piecewise functions. But for a function to be a zero divisor I'm pretty positive it needs uncountable many roots, which I'm not sure any smooth function has.

If there are, what about analytic? (My hunch is there aren't any in either)

Introduction

Hello again everyone! Welcome back.

If you have a good memory, you may remember my last post, about the testing I conducted on the performance effects of Skyrim's various graphical settings. The thread garnered far more attention than I was expecting, especially considering that the tests were only half-finished.

I spent the next week completing this testing, documenting the results, and creating images and GIFs for you to all enjoy.....

And then got completely burnt out by everything, and fell off the Skyrim bandwagon.

It's taken me a year to get through my work, my shitty mental state, and my procrastination, to finally post this conclusion to the testing, and for that, I apologize. I know that tech advances quickly, especially in the Gaming-PC space, and so waiting a year makes my data all the more obsolete.

For whatever it is still worth, I am happy to share my findings with you, as a way of giving back to the Skyrim Modding community for all the great mods I have enjoyed over the years. I am happy to report that thanks to all of your suggestions, I was able to gain NINTEEN FPS on my client, which brought me up from a sad 28 FPS to a playable 47 FPS with Rudy ENB, 4K landscape textures, obsidian weathers, etc., AT A 3440x1440p RESOLUTION!

I am now posting the results here to maybe help save other people's time, in case anyone is wondering how many frames they might stand to gain or lose by changing a game setting, or are wondering whether or not they'll really be able to see the difference with a setting on or off. I don't know if this data will ever make it into an official part of the subreddit's documentation, but I hope it remains easily found in the future.

For context, I'm playing on the following hardware, purchased in 2017:

• i7 7700k 4.2Ghz base clock Quad-core CPU
• GTX 1070 Graphics Card
• 16 GB DDR4 3400 MHz RAM
• With the entire game directory playing off a Samsung 970 Evo Plus M.2 nVME PCIe SSD. (The game directory is NOT installed under Program Files, as per the subreddit's guide.)

***Please note that I am displaying the game at a resolution of 3440x1440p. This is 2.5x the effective resolution of 1920x1080p. Yes, 2.5 times. If you're running your client on a 1080p screen, imagine having to also run another screen and a half. This is obviously a large part of why my starting FPS is so low, and if you are running 1080p, you will most likely have a much higher framerate than me. The r

Despite being somewhat of a renowned mathematician, I think I made a mistake.

Earlier today I stated that today is the 18th and I applied to 15 schools. 18+15=33. We just had a 3 day weekend, 33+3=36. I took the LSAT 3 times. 36 divided by 3 is 12. Essentially this led to something about dinosaurs and dinosaurs are extinct so I would be a part of zero waves.

HOWEVER. I went UR3 at UChicago today after these calculations were made. I did interview. Divide 12 by 3 (UR3) again, and we get 4.

4 is the smallest composite number, its proper divisors being 1 and 2. Does this mean I may actually be a part of 1-2 waves this week? I guess only time will tell. This is why you don’t rush your calculations folks.

It seems like many are still stuck on Day 24, so I thought I'd attempt to explain the decompilation process. You can follow along with your inputs and at the end, you will have some readable high-level code you can use to puzzle out the lowest and highest model numbers by hand.

In the prompt for Day 24, it is stated that there is no jump instruction. This means there is no looping and we should expect to see repeated logic that handles each digit of the 14-digit model number.

Indeed, each step is marked by inp w, which is used to read the next digit and begin processing it.

So we should treat the instructions from one inp to the next inp as a set of instructions to operate on an individual digit.

If we examine the instructions directly after inp w, we can notice some commonality between all 14 steps; namely, they all begin the same way:

inp w
mul x 0
add x z
mod x 26

Let's translate this to some pseudocode:

w = next_digit()
x *= 0
x += z
x %= 26

And then let's apply some simplification to make it easier to read.

Let's look at the first instruction, x *= 0. Anything multiplied by 0 is 0 (zero-product property). So we can simplify this to an assignment, i.e. x = 0.

The next instruction, x += z, adds the value stored in register z to the value in register x. We know this value to be 0, and 0 plus any other value is the other value (additive identity).

We can therefore combine these two instructions into one, x = z.

The last instruction, x %= 26, can also be combined with the previous instructions into one simple statement, x = z % 26.

After our work of decompiling the beginning instructions for every digit, we end up with:

w = next_digit()
x = z % 26

So far, we still can't figure much out, because the value of each register is 0. We can take note that in future steps, we should expect the value of z not to be 0. In fact, we should expect that the value of z, when moduloed by 26, yields some meaningful value.

If we examine the next instruction in each step, we will notice that each one performs a div on z. And there are only ever two possible divisors, 1 and 26 (seemingly a common number here). If we examine even closer, we will notice that 7 of the steps are div z 1 and the remaining 7 are div z 26.

This is the first instance of variability we have found. Let's add an explicit loop to the beginning of our decompiled code:

for step_number in 0..14:
    w = nex

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

This is the reference of both of them next to each other.

Black Hair and Red Hair (those are my terms, they'll be key to deciphering this madness)

General Observations

You can immediately tell that they're likely a part of the seven due to the placement of the leg holster/bag thing, and it's design. It's a leather bag with straps both down and up the thigh.

They also have the now iconic lined leather lower torso covering.

Their armor is more futuristic than Paradigm/Visitor, but it still matches with Scientist/Foundation/Origin

Many people, however, consider them too similar to be the last members of the seven, and if that were true, that'd be fair. Having two members of the most important group be the same would be terrible. I'm here to prove you (being the people who think this as true) wrong, and not observant. I love you, but it's true:

Black Hair (my favorite btw)

Her armor has the following unique characteristics:

• It is overall smoother in shape and design. Notably, look at the legs and shoulders.
• Her legs are fully armored. Obviously a big deal, but her leg armor wraps all the way around the thigh, and leaves minimal exposure to the knee joint.
• The pauldrons are shorter! For anyone uninitiated, pauldrons in this context means shoulder armor. Hers flow into the armor a bit more, and create a more angular shape (going back to the smooth design).
• Her lined leather torso is only on the front. The side (and likely back) of her torso is most likely supposed to be a carbon fiber mesh of some sort to provide more protection or stability. This will be more important later.
• Her chest plate is two-tone, most likely indicating multiple metals and parts! Not much relevance, just design.
• Her collarbone has a clear armored covering and wrap around to the back.
• Her belt has less items on it.
• The intricacy of her ankle armor flowing into the boot, is less present. It's a smoother and simpler transition, and is hidden by a guard on top of the boot.

Red Hair (damn she kinda looks like Penny Poledina)

Her armor has the following unique characteristics:

• It is overall sharper in shape and design. Notably, look at the lower legs and gloves.
• She has longer more defined pauldrons that stick out from the armor.
• Reminiscent of Visitor, she has a tube running from her upper chest to her ba

Hi! I(19M) have been doing some experimental studies with modelling quantum states through alternate mathematics using imaginary numbers without complex numbers(i.e. attempting to solve for sqrt(-x) rather than factor out sqrt(-1)). I think I've come up with a pretty interesting way to model relativity with pure mathematics and would love to engage with others so that I may further develop it. I know this is long, but I've lived my life asking these questions and this is what I've got so far!

From my understanding(and feel free to correct me if I'm wrong, I love to learn!), relativity suggests that light is released in quanta, which are little bits/packets of energy. These quanta for light are called photons, and they carry the electromagnetic force according to the standard model of particle physics. Similarly, I understand that light is expressed both as a line(constant vector quantity in linear algebra) and a wave and a photon would then be correlative of a point in Space with a constant instantaneous velocity. From my best attempt at conceptualizing the above, I am led to believe that we may then use imaginary numbers independently of complex compositions, but specifically for the pattern, they follow in iteration to model the movement of a wave function such as light/Energy.

The idea is this, according to Einstein's theory of General Relativity, we could detect whether or not a particle or force exists based on the presence of its Energy. In fact, the Standard Model of Particle physics defines the fundamental different Energies that makeup reality. Mass is gained from the Higgs field, the electromagnetic force from photons, W and Z bosons carry the Weak force, and gluons carry the strong force/gravity. Since those can all be defined in terms of Energy, perhaps our solution would come not from studying different forces and particles, but rather considering them all proofs of the existence of light! After all, E=mc^(2), and so rather than convert mass to energy, allow us to consider the existence of Energy proof of the existence of Light beside any given object or force in the Universe.

That may sound like a crazy idea, but crazy ideas tend to create discussions that lead to crazy knowledge. What if we considered Energy to be the rate at which Light is displaced? The viscosity of an object's Energy would be the very viscosity of its existence, contained inside its potential energy thanks to mass and momentum. At a certain point, specifical

É isso cara, não tem o que falar. O Bolsonaro é burro. As pessoas gostam de traçar paralelos com Hitler, Mussolini mas, por mais cruéis desprezíveis que eram estas pessoas, eram muito inteligentes e estrategistas. Qualquer pessoa sob uma ótica minimamente crítica e estratégica percebe que o Bolsonaro possui ZERO de pensamento estratégico. O Bolsonaro e seus asseclas. Tomar um golpe regido pelo Bolsonaro seria VERGONHOSO, porque ele não tem o mínimo de competência e inteligência pra tal. Nem ele, nem os militares aliados a ele.

A pandemia poderia ter sido o divisor de águas no mandato dele. Se ele fosse realmente inteligente e jogasse o tal xadrez 6391936d que os.bolsonaristas falam que ele.joga, ele teria feito TUDO diferente. No mínimo teria decretado estado de emergência e feito um comitê de enfrentamento de crise sanitária, porque já que ele tem o QI de uma porta ele teria deixado o trabalho de gerenciamento de crise para especialistas, e teria ENCHIDO O SACO com a vacina botando slogan em todo lugar com os dizeres "A VACINA DO PATRIOTA/BRASIL" e a cara escrota dele sorrindo segurando um frasquinho de vacina.

Qual o sentido de fazer o que ele fez na pandemia? Quando os governadores estavam decretando lockdown ele falou que estavam "puxando a corda". E o que ele ia fazer? Mandar os militares pra rua pra forçarem as pessoas a saírem de casa? Quer dizer que vai vir um fardado apontar um FAL pra minha cara e dizer "você vai sim pra balada seu comunista"? Porra, não faz sentido nenhum. A retomada da economia só seria possível com uma vacina, e se ele tivesse mais do que dois neurônios (ou escutasse as porras dos especialistas e ministros) ele teria feito uma campanha do caralho pra vacina.

E os ministros dele também. Aquele Guedes qualquer pessoa que acompanha liberais sérios e economistas sérios vão ouvir que o Guedes é incompetente, preso numa matriz ideológica ultrapassada, que não é levada a sério nem no exterior.

A única pessoa a ser culpada de derreter o apoio e a imagem pública dele, sendo que agora nem mesmo o mercado e o agronegócio querem ele, é o próprio Bolsonaro. Ele que se afundou em merda. E enquanto ele acha que tá arrasando lacrando mitando dando essas respostas tipo compra fuzil e desliga a luz, pessoas minimamente estrategistas estão morrendo de vergonha alheia de estarem associadas com esse BURRO.

Olha, eu nunca gostei do Lula, nunca gostei do PT, achei e acho uma puta sacanagem eles quererem sempre protagonismo dentro da esquerda e

Do your worst!

I'm surprised it hasn't decade.

Hey,

I've been trying to get League of Legends to run on Ubuntu 20.04.3 with a Radeon HD 7950, using the Radeon driver (xserver-xorg-video-radeon 1:19.1.0-1). However, every time I finish selecting my champion, I just get a pop-up saying "Error after champion select: Could not initialize graphics device".

This is the output I get from vulkaninfo:

ERROR: [Loader Message] Code 0 : /usr/lib/i386-linux-gnu/libvulkan_radeon.so: wrong ELF class: ELFCLASS32
ERROR: [Loader Message] Code 0 : /usr/lib/i386-linux-gnu/libvulkan_lvp.so: wrong ELF class: ELFCLASS32
ERROR: [Loader Message] Code 0 : /usr/lib/i386-linux-gnu/libvulkan_intel.so: wrong ELF class: ELFCLASS32
WARNING: lavapipe is not a conformant vulkan implementation, testing use only.
WARNING: lavapipe is not a conformant vulkan implementation, testing use only.
==========
VULKANINFO
==========

Vulkan Instance Version: 1.2.131


Instance Extensions: count = 18
====================
	VK_EXT_acquire_xlib_display            : extension revision 1
	VK_EXT_debug_report                    : extension revision 8
	VK_EXT_debug_utils                     : extension revision 1
	VK_EXT_direct_mode_display             : extension revision 1
	VK_EXT_display_surface_counter         : extension revision 1
	VK_KHR_device_group_creation           : extension revision 1
	VK_KHR_display                         : extension revision 23
	VK_KHR_external_fence_capabilities     : extension revision 1
	VK_KHR_external_memory_capabilities    : extension revision 1
	VK_KHR_external_semaphore_capabilities : extension revision 1
	VK_KHR_get_display_properties2         : extension revision 1
	VK_KHR_get_physical_device_properties2 : extension revision 1
	VK_KHR_get_surface_capabilities2       : extension revision 1
	VK_KHR_surface                         : extension revision 25
	VK_KHR_surface_protected_capabilities  : extension revision 1
	VK_KHR_wayland_surface                 : extension revision 6
	VK_KHR_xcb_surface                     : extension revision 6
	VK_KHR_xlib_surface                    : extension revision 6

Layers: count = 3
=======
VK_LAYER_LUNARG_standard_validation (LunarG Standard Validation Layer) Vulkan version 1.0.131, layer version 1:
	Layer Extensions: count = 0
	Devices: count = 1
		GPU id 	: 0 (llvmpipe (LLVM 12.0.0, 256 bits))
		Layer-Device Extens

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

Because she wanted to see the task manager.

Heard they've been doing some shady business.

BamBOO!

They’re on standbi

but then I remembered it was ground this morning.

Edit: Thank you guys for the awards, they're much nicer than the cardboard sleeve I've been using and reassures me that my jokes aren't stale

Edit 2: I have already been made aware that Men In Black 3 has told a version of this joke before. If the joke is not new to you, please enjoy any of the single origin puns in the comments

Pilot on me!!