#f:r

[scorpiolight-madd]

Ignore this post 💙

[nostalgebraist-autoresponder]

random matrices

The "eigenmode" picture of the statistical mechanics of disordered physical systems is familiar to us from a good number of examples in condensed matter theory. "Quenched disorder" is a phenomenon that has a well-known and very simple explanation in terms of the "eigenmodes" of a random Hamiltonian.

It is much harder to see why this is so from first principles. (Even one of the textbooks on modern statistical mechanics elides the issue).

The reason is this: the eigenvalues of a matrix don't determine the matrix. (Indeed, given a matrix you can add a number to all its eigenvalues without changing the matrix).

It's easy enough to construct matrices that are (1) of the same size as a square matrix but which have a nontrivial fraction of their eigenvalues (eigenvalues of unit modulus) of size 2, or of size 1 (so that the characteristic polynomial is of degree 2). It is easy to build these matrices in certain "natural" ways, for instance, by choosing to add random constants or random signs to elements of the matrix.

However, if you don't know the matrix in advance (e.g. you've never seen the matrix before), you don't have a "natural" way to construct it. But there's still a perfectly "natural" way, once you've defined what it means for a matrix to be "of the same size."

This is important: in a typical quenched disorder experiment, you've got a whole ensemble of matrices, each "quenched" in a very specific way. (This is true of many "real-world" quenched disorder systems -- see e.g. the quantum quench experiments used in cold atom physics).

So far, I have explained what this means, but not "how" this is a natural construction.

The natural way to understand the construction is as the analogue of "pushing your matrices together and multiplying them" (in the special case of two matrices of size 1).

This is the analogue of "getting a linear transformation of your matrices."

The point is that this can be done -- it is a natural operation, one that will "push your matrices together" -- but it's a very strange operation, and only makes intuitive sense if you start with the basic idea that a random matrix corresponds to a random linear transformation, even though "linear transformations" themselves are not random.

It's not just that "matrices of the same size" form a basis for a vector space, this is a basis for the operation of linear "push-together-and-multiply."

(Here are some relevant definitions)

Given a finite vector space E with a distinguished element 0, a random matrix P is jointly randomized to size n (with respect to the vector space) if there is a natural way to choose all the entries of the matrix, except the diagonal (0, 0) entry, such that the operation P(a1,...,an) for an element of E is a linear transformation in the vector space of E which takes 0 to 0, and each pair of elements (ai, bi) in the set (1,..., n) is mapped to a linear transformation in this way, and the matrix P is "jointly randomized to size n.""

From a related point of view, you can see how "linear operators on matrices" become "linear operators on linear operators." If Q(a) and R(b) are the operators that apply to a and b, and f:R(a)->Q(b) is the transformation, the definition of the matrix representation says that Q(a) = Q(a). So if you write Q(a) for a linear operator on matrices, Q(b) is really a linear transformation on linear operators.

But this can go one way and not the other. "Linear transformations" on linear operators are obviously not joint in the way described above.

(Actually, this is a rather technical point that the Wikipedia article might not be getting across, but this is because I'm trying to phrase the original point and the technical point in the same sentence.)

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[uponafieldofclover]

hate googling math things. yeah it's a fucking uhh plinko duplo over a chipdee if and only if f:R->R. what does that mean? go fuck yourself.

[nohicelageti]

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[s-f9]

[hungline]

round and round

pairings: jihope, side namgi and namgikook, minor taejin  genre: fluff, light angst, non au, friends to lovers au, rated pg15  warnings: implied sexual content   a/n: written for the @jihope-love exchange 2018 for my recipient flowerstems  word count: 1830 

summary: It takes a while for Hoseok to realize that he and Jimin are circling one another - have been circling one another - for a while now, but at least it all comes to an end one way or another. 

When Hoseok first saw Jimin, it felt like his entire life had come to a screeching halt.

He knew before then that he liked boys, that he liked boys and girls. His own family had lacked the tact to ask him about it discreetly when he began high school and he had known even then that he did not only like girls. It was no secret to the other trainees either, but Hoseok had not realized just how much he was able to like someone until he saw Jimin.

Admittedly, Jimin was not much. He was just a kid with chubby cheeks and the dream of becoming an idol, of becoming someone in this world, but he was hardworking. Jimin was a cute kid and he was always polite, always acting as perfect as others expected him to, but Hoseok did not know how that meant he had somehow already grown attached to him.

After that, it only continued to develop. Hoseok did his best to keep his hands to himself, to stop being so touchy when it came to Jimin, but that did nothing. It got even worse after they had debuted because Hoseok would tease him non-stop, would pinch his cheeks and call him cute, would fake trying to kiss him all in the hope that maybe Jimin liked him like that too.

Jimin did not like any of it though and he always slapped Hoseok's hands away, always leaned back and got the space he needed when it looked like Hoseok wouldn't stop anytime soon.

Despite that, they were the go-to duo for dancing. They spent a lot of time together and maybe it was thanks to the fact that they all slept in the same room, but even when they moved dorms and Hoseok found himself rooming with Taehyung and Jimin, things were still like that with Jimin. Hoseok pined and he did his best to not let on how much he pined when cameras were on them, but he did not believe that he was actually fooling anybody.

The company had already warned him, had told him that they do not care about who he dated, but he had to keep it under wraps. That if he liked someone in the group, then it could not affect the group dynamic in the slightest. They reminded him again in a group meeting that was meant to be directed at everyone, but Hoseok knew they meant it explicitly for him.

And he also knows that Namjoon and Yoongi took full advantage of this. They had been circling one another for as long as Hoseok could remember, but as soon as the company had given them the green light, there was really no stopping them. Hoseok could not even count the number of times he had walked into a seemingly empty recording studio only to find Yoongi on his knees between Namjoon's legs or Namjoon splayed out on the floor with Yoongi thrusting into him. The number of times they have laughed at him for it is plenty, but the number of times they have looked at him with hooded gazes and invited him to join is slight at least.

Hoseok almost panicked when they moved dorms once more and new roommates were picked out, but it was okay. Yoongi was still going to room with Seokjin and Namjoon was now rooming with Taehyung while Jeongguk got a room to himself. The youngest often snuck into Hoseok's bed most nights so it became normal for Hoseok and Jimin to share a bed. Not that Hoseok wouldn't have minded cuddling with Jeongguk, but the kid always punched whoever happened to be sleeping with him in the same bed and Hoseok did not want to wake up to another bleeding nose again.

Jeongguk being in their room at night did start to happen a lot less once Yoongi and Namjoon began circling around him and now Hoseok wondered just how much more time would Seokjin spend giving Jeongguk impromptu sex education lessons in the kitchen most mornings. Hoseok always swooped in to save him from Seokjin whenever Jeongguk gave him those sparkly puppy dog eyes that he had never been able to develop a resistance to. Jimin once laughed at them and made sly comments about Seokjin's lack of lessons sprung upon him and Taehyung, but shut up when Seokjin only raised his brows and said in a suggestive tone, "I'm already giving Taehyung private lessons and I know that someone is giving them to you too."

And while it had not been the first time that someone had insinuated something going on between Hoseok and Jimin that wasn't completely platonic, it had definitely made Hoseok take pause as he considered the fact that all of them were, more or less, involved with another member romantically or sexually at this point. Maybe Jimin and Hoseok were the strange ones for not having built something between them.

Except that, things between Hoseok and Jimin had been brewing for quite some time already.

When Hoseok pinched Jimin's cheeks, he was the first to notice that Jimin had stopped slapping his hands away. He was the first to notice when Jimin would bear through his teasing and prodding instead of running away or leaving like he used to. Hoseok was the first to notice when Jimin would seek out his touch and return the skimming fingertips across his skin. When Hoseok was stressed, Jimin would give him a massage without needing to be asked and Hoseok took his time to come to terms with what all of this could mean.

Mostly though, Hoseok took his time with accepting the fact that maybe this was just Jimin finally being comfortable with him. Maybe this was Jimin trying to be a good friend and be there for Hoseok as Hoseok had been there for him when they were all just some bumbling kids trying hard to impress the public that could make or break their careers. Maybe they were just growing closer as friends now that Jimin was older, now that Jimin finally had a grasp on who he was as a person.

But the precarious protection that Hoseok had constructed around his heart all came crashing down on one rare night that Jeongguk had not crept into their room and they were still sharing a bed anyway. Jimin and he had been lying under the covers, curled around the other like they usually do whenever they cuddle. Hoseok was toying with Jimin's fingers as he stared past his face doing his best to not focus on how close they were when Jimin took his hand and interlaced their fingers together. It shocked Hoseok enough to look back at the younger man and meet his gaze, a question on the tip of his tongue that he never got the chance to ask.

Jimin's other hand had come up to cup Hoseok's cheek as he leaned in slowly, their breaths intermingling in the space between them. Hoseok only dared to blink, not knowing at all what to do with his hands as Jimin watched him. He just kept coming closer and closer, going slow enough to drive Hoseok mad, but Hoseok knew that it was for him. Jimin was giving him the chance to pull away, to lean back and put a stop to whatever Jimin was planning to do next.

Hoseok did not want to pull away anymore. He was tired of ignoring the ball of feelings in his chest, the one that kept him up some nights when he really needed to sleep or the one that jumped into his throat whenever Jimin managed to catch him off guard by doing something that many others would consider as flirtatious. He was tired of running and now Jimin was a mere few inches away from him, waiting for Hoseok to come to a decision.

It was all too easy for Hoseok to close his eyes and lean in the rest of the way to kiss Jimin like he had wanted to for so many years already. He let his hands find their place on Jimin's waist and his face, moving his body closer when Jimin opened up for him like a flower in bloom.

Hoseok's hand gripped on hard to the dip of Jimin's waist as he pulled the younger closer. Everything around them dulled as Hoseok immersed himself in Jimin, the sound of their breaths and his raging heart the only things he could hear. Jimin tipped his head back and allowed for the angle of their kiss to become better, to become almost perfect in Hoseok's opinion.

Jimin's lips were soft, like clouds or maybe like flower petals instead. They were soft and insistent against Hoseok's own and when he felt Jimin's tongue prodding at the corner of his mouth, Hoseok allowed his lips to part. Jimin swept into his mouth and left Hoseok weak, left him clutching even harder to the smaller body against him as his hands moved up into Jimin's hair. He twisted his fingers into the soft locks and met Jimin's tongue with his own, giving him as much as he was taking.

A sound escaped past Hoseok's lips and Jimin swallowed it, savoring the quiet groan Hoseok had made. Hoseok could not muster up enough energy to be embarrassed about it though because Jimin only kissed him even harder, pulling himself even closer until Hoseok could feel every plane of Jimin's body pressed tightly against his own.

They were in their own little world then, kissing like they were men touch-starved for almost all their lives. Like they had been wandering out in a desert and were now greedily taking in all the water they could. Hoseok's chest felt tight and constricting until he remembered that he had not given himself the chance to take a breath yet.

A series of sudden thumps against their wall made them break apart, gasping for air as the thumps only increased in tempo. Hoseok knew that this wall, in particular, was the one they shared with Jeongguk and when he finally opened his eyes again and met Jimin's gaze, they both burst into a fit of giggles that stole their breath again.

Namjoon and Yoongi had finally stopped circling and maybe if Hoseok listened hard enough he would be able to hear Jeongguk's whines, but instead, he decided to pull Jimin close again and let their chests rumble with the force of their laughter.

Jimin snuggled into him and laid his head on Hoseok's chest, listening to the very erratic beat of Hoseok's heart. Their laughs died down soon enough and the thumping was still going on, but Hoseok had other things to focus intently on as Jimin looked up at him with a glint in his eyes that meant he was up to no good.

Suffice to say, Hoseok and Jimin had finally stopped circling one another that night too.

[fy-parksungjin]

R | do not edit (1/2)

[fromnine]

© Red Shoes | do not edit.

[winterfable]

Preguntas sobre matemática

¿Por qué el gradiente es la dirección de máximo crecimiento para una función f:R^n->R?https://math.stackexchange.com/questions/223252/why-is-gradient-the-direction-of-steepest-ascent

https://www.youtube.com/watch?v=TEB2z7ZlRAw&t=28s

[quizllc]

f(x)= 4x-1 where f:R→R then f is.

f(x)= 4x-1 where f:R→R then f is.

A. injective B. exponential C. composite D. none

[pius2017]

Let F= {f:R → R} - the set of all real-valued functions. Determine if the following statements are true or false. Explain why

Let F= {f:R → R} – the set of all real-valued functions. Determine if the following statements are true or false. Explain why

(5) 20 pts) Let F= {f:R → R} – the set of all real-valued functions. Determine if the following statements are true or false. Explain why. (a) (5pt) VEF 39 EF fog=go f = id. (b) (5pt) 3f EF Vg E F fog=go f = id. (C) (5pt) V EF Vg E F fog=go f = id. (d) (5pt) 3f EF 39 EF fog=go f = id. Explain you answers. (e) (+5pt) Negate and simplify VEF 39 EF fog=go f = id.

[postsofbabel]

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[blogfuncoes]

Função no dia-a-dia e sua

efinição por fórmula

Como podemos achar funções no dia-a-dia?

Funções podem ser encontradas facilmente em nosso dia-a-dia. Independentemente de hora,

lugar ou dia, elas se fazem presentes no cotidiano. Veja alguns exemplos:

Exemplos

Número de pães que vou comprar, com o preço a pagar.

· Número de questões que acertei num teste, com a nota que eu vou tirar.

· Valor do meu salário, com o valor do desconto do INSS.

· Medida de contorno do meu terreno, com a quantidade de metros de arame de que preciso

para cercá-lo.

Examinando o primeiro exemplo dado acima, a tabela a seguir mostra a relação entre o número de pães comprados e o correspondente preço a pagar. Número de pães Preço a pagar (R$) 1 0,12 2 0,24 3 0,36 4 0,48 5 0,60 6 0,72 Para fazer esta tabela, o dono da padaria fez o seguinte cálculo: Preço a pagar = 0,12 . número de pães Dizemos que o preço a pagar (y) é função do número de pães (x), pois cada quantidade x de pães existe um único preço y a pagar. Usando as letras x e y, podemos representar esse cálculo na expressão: y = 0,12 . x Se eu quiser saber, por exemplo, quantos pães posso comprar com R$ 6,00, basta fazer Y = 6 na expressão. Y = 0,12 . x 6 = 0,12 . x x = 6 / 0,12 = 50 R: Posso comprar 50 pães. Função definida por fórmula Toda função é definida por uma lei de formação, é dessa forma que relacionamos dois conjuntos A e B. As funções servem para expressar situações com base na álgebra, generalizando os problemas através de fórmulas. Por exemplo, a função y = 2x ou f(x) = 2x mostra que os valores de y dependem dos valores de x. Nesse caso temos que y corresponde ao dobro de x. Veja a relação entre alguns dos valores de x e y:

f:R→R tal que f(x) = 2x

Exemplos:

Bibliografia https://funcoesopcao1c.blogspot.com/p/funcoes-no-dia-dia.html https://brasilescola.uol.com.br/matematica/funcao-definida-por-formula.htm

[s-f9]

[locally-normal]

Messy Math Solutions

There’s a divide in math between what a problem’s solutions like when you’re working on it, and the nice polished proofs you’d read in a textbook. One is a beautiful elegant masterpiece, and the other is a jumbled mess, usually. Which can get disheartening and whatnot. And there’s a question I often have of, how messy is other people’s work?

So I figured I’d post some raw, unedited solutions to math problems. And whaddya know, the other day @the-real-numbers gave me some cool math problems! Awesome! 

1. Without using a calculator: if s(n) is the sum of the digits of n, what’s s(s(s(4444^4444)))? (hint: don’t you fucking dare try to actually find 4444^4444.)

2. Find the family of functions f:R->R such that f(x) takes on the value of the slope of a line perpendicular to the tangent line of f at x. (Easier than it sounds.) 3. Let f:R^2->R be a function such that f(x,y)+f(y,z)+f(z,x)=0. prove there exists a function g:R->R such that f(x,y)=g(x)-g(y). (A putnam problem, I think, but actually simple once you see what happens with the right inputs.) 4. Alice and Bob agree to meet at a place between 5:00 and 5:30. What is the probability they arrive within ten minutes of each other? (Assume time is continuous, not discrete.) 5. Find all non-trivial zeroes of the Riemann Zeta function that lie off the ½ li- oh wait no, that one’s supposed to be a secret.

I’ll put in my messy solutions to 3, and somewhat less messy ones for the others, only cause I have them and don’t see a reason not to (giving up on succinctness for this one).

I’ll start with 3. (It’s been a while since I’ve done sorta competition style problems)

So, simplest case, x=y=z. Then 3f(x,x) = 0, so f(x,x) = 0. Excellent!Now, z=y. Then, using the result above, f(x,y) = -f(y,x), also nice, some sorta symmetry going on here. This is starting to feel kinda distance like, we have some sorta reflexivity, symmetry-ish, and f(x,y) = f(z,y) + f(x,z), that's kinda like a triangle equality. Also notably f(x,y) = -f(y,x) is really a generalization of f(x,x) = 0, but, we'll write them separately anyway.

Rewriting, f(x,y) = f(z,y) - f(z,x). That is, f(x, y) = f( - , y) - f( - , x), and sadly these aren't the same function are they. It almost worked. Still seems important.

Negatives seem to be important. Let's try z = -y. Then f(x,y) + f(-y,y) + f(-y,x) = 0. Not very helpful. Let's have y=-x also, so more simply, z=x, y=-x. Then f(x,-x) + f(x,-x) + f(x,x) = 0, so f(x,-x) = 0 also. Cool. So going up a level, forgetting that y=-x constraint, we have f(x,y) = - f(-y,x) = f(x,-y). That's nifty too. So in summary we have f(x,y) = - f(y,x) = f(x,-y). Seems like this thing is symmetric (with a *-1) around every axis. Moreover, if you fix any two horizontal lines (w,z), and pick an x value on them, the difference between the function's value at these two points, f(x,z) - f(x,w), will be the function's value at f(w,z). Pretty bizarre. The same works for vertical lines. If the two lines are close together, then the result would be near 0, if f were continuous, but it doesn't have to be. At least I think that's true. I'm trying to transfer some algebraic intuition I have into geometric intuitions, which for me are naturally weaker.

Let h_y(x) = c | \forall z. c = f(z, y) - f(z,x), which is well defined. Just a restatement of currying f(x,y).

Right around here I stopped for about two days. Thought about it a bit in my head one of those days, unproductively. But this morning, I returned to it, prodded the equation f(x,y) = f(-, y) - f(-, x) a bit more. Doesn't that just look like a solution? What if we say h_x(z) = h_y(z) - h_y(x)? I prodded it a bit more. Still useless. But then my prodding got to something that felt useful. Even, instead of calling it h, I went ahead and used g: let g(y) = h_y(0) = f(0,y). We don't need that generality with the z. I basically needed g to return the function h_y... applying it to a dummy argument doesn't change anything. Now f(x,y) = g(y) - g(x), exactly as we wanted.

Now I agree, that was an easy Putnam problem. (If I got it wrong, consider my words retracted and replaced with ‘Oops’)

Posting work like this feels embarrassing. I spent two paragraphs in a blind alley after basically stating the answer. Aw well.

Anyways, moving on, the other problems.

1. Ehh. Ehhh. I’m pretty sure I recall that there’s a nice trick to these sums, but I forget, and fooling around with it didn’t get anywhere. Maybe I’ll get back to it.

2. looks like dy/dx = -1/y to me, not bad, gets a +- sqrt(c-2x). It’s been too long since I solved a differential equation.

4. I suck at probability, but it looks like if you arrive at t \in [0, 10) minutes, the other person has a window of x+10 minutes to arrive in, so integrating gets that there’s a 1/2 chance the other person arrives within 10 minutes of you. 

If you arrive in (20, 30], same thing, and otherwise there’s a 2/3 chance (10 minute window before, 10 minutes after), all in all, 5/9.

5. ;)