Homogeneous function Puns

Journal of the American Chemical SocietyDOI: 10.1021/jacs.0c12448

Hongxing Li, Jun Zhang, Chuanjing An, and Suwei Dong

https://ift.tt/3pgiBYi

Hello everyone,

I am stuck on a very specific differential equation, and I think that the solution lies in Hypergeometric form, but I am new to this type of math and would appreciate the help. The differential equation that I am trying to solve is specifically:

(A^2*t^(3/2))T'' + (3A)T' + (-3A^2*m'^2*t^(3/2))T = 0, where T is a function of t and A and m' are constants.

Let me know if you need more information!

Greetings! First year graduate student, here. My question comes from my Thermodynamics class, regarding the derivation of the Euler Equation and the Gibbs-Duhem relation. My textbook (Greiner,Neise,Stoecker) assumes the internal energy to be homogenous of first order, i.e.

[;U(\alpha S,\alpha V,...)=\alpha U(S,V,...);]

for some constant [;\alpha;]. They then assume to [;\alpha;] to represent an infinitesimal increase of the system, [;\alpha\equiv 1+\epsilon;], with [;\epsilon<<1;]. Then they expand the left hand side of the above equation in a Taylor series:

[;U[(1+\epsilon)S,...)]=U+\frac{\partial U}{\partial S}\epsilon S+\frac{\partial U}{\partial V}\epsilon V+...;]

I'm having trouble understanding the above result. Obviously, I am familiar with Taylor Series, but I had to remind myself of the precise definition, for which I found

[;f(x)=\sum_{n=1}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n;]

i.e. a function [;f(x);] expanded about a point [;a;]. In the context of that definition, I can't seem to make sense of my textbook's result. About what point are we expanding? The higher order terms are dropped due to the smallness of [;\epsilon;], I imagine?

If anyone could offer a few clarifying statements, I would greatly appreciate it. Thanks in advance!

i tried looking around as i wanted to see his proof of the theorem... can anyone please give the link to the original paper? also... any tips on finding these papers would be apreciated.

What are the graphical implications of and/or intuitions behind a homogeneous function? They are functions with multiplicative scaling behavior but what does that look like graphically? What would the scaling behavior of a homogenous function look like compared to the that of a non-homogeneous function? I wish there were some pictures or videos online, but I can seem to find any that help me understand the bigger picture of homogeneous functions. Does any one know of visual resources for understanding the topic better?

Hi, I'm an economic student and I had a doubt while reading my textbook.

The textbook defines the homogeneous function:

f (tx1,...,txn) = t^kf(x1,...,xn), k being the order of the function, i-1,...,n

Then, to prove that the derivative of a homogeneous function has degree k-1, it derives both sides:

t * d (tx1,...,txn)/d xi = t^k * d f (x1,...,xn)/ d xi (read d / d as the partial derivative)

Then d (tx1,...,txn) / d xi = t^(k-1) * d f(x1,..,xn)/ d xi

My doubt is:

The definition of the chain rule is:

being f (y(xi),...,yn(xi) ), d f (y(xi),...,yn(xi) ) / d xi = d f (y(xi),...,yn(xi))/ d y * d y / d xi

So, if we take the homogeneous function f (tx1,...,txn), if we consider yi (xi)=txi, the derivative should be d f / dxi = d f/ dy * dy / dxi = d f/ d (txi) * d (txi)/ d xi = d f/ d (txi) * t, which isn't equal to d f/ d xi * t (which appears in the book)

My professor says is a notation mistake, but I saw this notation in more than one book.

I didn't have any problem using Greens function on initial value problems but then I hit a brick wall in using the function for boundary value problems.

Link to some of the problems: Imgur

Link to 27 and 28: Imgur

35 I guess I understand as I worked backword after looking up the answer but 39 just has me stuck in my tracks and not sure where to even start. I think my problem is with choosing y1 and y2 but once I solve for the homogeneous problem and get a y1 and y2 after that it still doesn't work out right.

Anyone know of any resources like Boundary Value Problems for Dummies?

Consider a function f:R*+n* -> R*+* that is separately convex, i.e. such that d^2 f/dx*i2* ≥ 0 for all direction i in {1, ..., n}. Furthermore, f is positively homogenous of degree one: f(tx)=tf(x) for all t ≥ 0.

Can we say anything about the quasi-convexity of f? From this article, it seems that if n=2 then f is convex but that this result does not generalize to n ≥ 3. Is there a result that shows that f is quasiconvex in these cases? Thanks a lot any help!

Let f: R^n → R be a differentiable function such that f(ax) = af(x), for every a in R. Prove that f is linear.

Sup savants. I must be stupid because I can't understand this. This is a direct quote from my calculus book, 'Calculus. A Complete Course' by Robert A. Adams, chapter 12.5. This is a paragraph before introducing Euler's Theorem.

A function f(x_1,......x_n) is said to be positively homogeneous of degree k if, for every point (x_1, x_2,...,x_n) in its domain and every real number t>0, we have f(tx_1, tx_2.... tx_n) = t^k f(x_1,...x_n)

For example:

1. f(x,y) = x^2 + xy - y^2 is positively homogeneous of degree 2.

2. f(x,y) = SQRT(x^2 + y^2) is positively homogeneous of degree 1.

3. f(x,y) = (2xy)/(x^2 + y^2) is positively homogeneous of degree 0.

4. f(x,y,z) = (x - y + 5z)/(yz-z^2) is positively homogeneous of degree -1.

5. f(x,y) = x^2 + y is not positively homogeneous.

It would be awesome if someone could shed a little light on this.

So this is a math problem embedded in a quantum mechanics problem. Basically I have the radial Schrödinger equation with a gaussian well, neverminding the initial expression i know i can reduce this equation to two forms, standard 2nd order ode and Sturm-Liouville problem:

u''(r)+ f(r) u(r) = 0

(p(r)u'(r))'+ (lambda r(r) - q(r)) u(r) = 0

Here you can find the definitions for the S-L problem, to write it in standard 2nd order ode just multiply by 2m/h^2 and it becomes clear.

Now, the idea is to find the energy spectrum (Lambda = E ), aka, the eigenvalues of this operator (seen in the first equation). I believe I can do this by solving the differential equation (if analytically solvable), I could do variation of parameters or I could see if S-L takes me somewhere.

My question is: How would you go about finding the different values of E (They should be discrete and finite, given the physics at least)? If this equation is not analytically solvable, what's the alternative approach?

Thanks in advance!

Can I recognize if a function is homogenous by looking at it? Or can I recognize if a function is not homogenous by looking at it? Seems a bit annoying but doable if need be to have to sub in f(tx,ty) to every function.

I’m looking for an R package or function that I can use to measure the relative homogeneity/heterogeneity of groups of character strings. There are about 50 groups, each with thousands of character strings, with each string consisting of one or two characters. There are about 100 possible character strings, and not all character strings appear in all groups. A much-abbreviated example:

group1 <- c("AA", "N", "W", "AJ", "KC", "KB", "KB") 
group2 <- c("N", "W", "W", "N", "N", "N", "W", "W", "N") 
group3 <- c("WA", "W", "NY", "AA", "KC", "AK")

Group 2 is obviously the most homogenous/least heterogenous. What's an R package or function I can use to quantify this, such that it would produce an index score for each group?

Can someone explain to me why the exponential function is not homogenous? I know that the natural log isnt homogenous, while it scales additively. For an example: f(x) = lnx, f(tx)=lntx=lnt+lnx, but how can you give the same argument with the exponential function? f(x)=exp(x), f(tx)=exp(tx), but how do you go from here?

I got the idea from this non-rigorously defined exercise (4b). It's in German, but I think anybody who knows the basics about homogenous functions will get the idea.

I'm pretty sure that the function is supposed to be differentiable and don't need help doing the actual exercise, but that got me wondering - is it possible to construct a homogenous function such that all partial derivatives exist in some point x, but Euler's theorem does not hold in x? I tried to come up with a counterexample, but I didn't even manage to construct any in x partially differentiable, non differentiable, homogenous function.

Say that a positive valued function f: R^n -> R+ is quasiconcave if for all x,y∈R^n and t∈[0,1] we have f(tx + (1-t)y) ≥ min{f(x),f(y)}. Say that it is homogenous of degree 1 if for all positive θ we have f(θx) = θf(x). Prove that any such quasiconcave and homogenous function is actually concave.

Through my following of IOTA since 2017, Serguei has really struck me as one of the greats. Serguei is one of the founders and on the board of directors, but recently stepped down from an executive role at the foundation so that he could just focus on research for IOTA.

This is a brilliant man who has been heavily involved in the crypto space since 2013, has a Ph. D in mathematics and has published dozens of articles (including the tangle whitepaper) and two books book2. On top of that, he continues to do academic research on Probability Theory and Stochastic Processes, both of which play a huge role when it comes to predicting possible attacks on a crypto network. If anyone, anywhere in the crypto world can be considered a “leading expert” on crypto currency theory it’s Serguei.

But the thing that really strikes me is that despite the drama, despite all the FUD and craziness that comes out of IOTA and the crypto world in general, he keeps his eyes on the thing that matters the most: making IOTA the best crypto protocol possible. Obviously, the whole development team has been fantastic throughout all of this, realizing fundamental flaws in the old protocol, and creating the chrysalis network from scratch, was certainly a difficult process to say the least. But I honestly appreciate that Serguei has kept his hands clean through it all and hasn’t fallen into the reddit or twitter traps that are so easy to fall into. It really shows me where his priorities are, and that’s the type of leadership I can appreciate.

Anyways, this is just my public thank you to Serguei for all the hard work he’s done.

link to his website

From the r/AskHistorians mod team:

On Tuesday, 16 March 2021, eight people were murdered in a series of attacks on massage parlors in and around Atlanta, Georgia (United States). Six of these victims were women of Asian descent. Their names are Daoyou Feng (冯道友), Hyun Jung Grant (김현정), Suncha Kim (김순자), Soon Chung Park (박순정), Xiaojie “Emily” Tan (谭小洁), and Yong Ae Yue (유용애). Two others, Delaina Ashley Yaun and Paul Andre Michels, were also murdered on Tuesday evening.

The brutality of these crimes has been met with expressions of shock and dismay across the globe; however, the Atlanta-area attacks are hardly unprecedented. Since the onset of the COVID-19 Pandemic, over four thousand incidents of anti-Asian violence have been reported across Canada and the United States.^([1]) While it is easy to ascribe the xenophobic hatred that fueled these attacks to the impact of Trumpian rhetoric, it is important to understand that the sentiments underpinning that rhetoric first originated in the white colonial empires of the nineteenth century. Anti-Asian racism is woven into the fabric of Canadian and American national history, and it is important to understand and acknowledge both the systematic othering of Asian Americans, Asian Canadians, and Asian immigrants to North America and the violence that such othering has historically inspired and, in many ways, excused.

The “Yellow Peril”

European states began colonizing parts of Asia in the sixteenth century in an attempt to control the production and movement of lucrative trade goods between Asia and Europe, Australia, New Zealand, Africa, and North and South America. In this early period of colonialism, European perceptions of Asia were generally positive, resulting in a characterization of the region as being at least as civilized as Europe. However, by the nineteenth century, European intentions in Asia had become transparently imperialistic. Trade-driven colonization in the region was dominated by the United Kingdom, but Germany, France, Russia, and the United States, among others also held imperial aspirations in Asia. These aspirations were built increasingly upon stereotypes that characterized Asian persons as physically, intellectually, culturally, and morally inferior to the white Europeans who sought to exploit and control Asian resources. Gone were the positive stereotypes about Asia and its people, which were replaced by the same kinds of stereotypes that Europeans had used to justify the c

Hello, This is the first post from a long time lurker.

Its my opinion that although atheism shall increase as a percentage of the population of the western societies, and the world population as a whole, it will begin to slow in its percentage increase after a relatively short period of time. Id postulate this will be due to the rapid increase in the number of atheist "coming out of the closest" as it were and being more vocal on the subject of religion within society thus encouraging more open dialogue (which, from what I've seen, appears to be the environment wherein skepticism flourishes and deities whither). However I unfortunately believe this gradient will begin to ebb.

The reasons for this is simple; inequality and governance.

In-homogeneous wealth distribution leaves many with inadequate standards of education and leads to not only ignorance but also anti-intellectualism in communities. The main reason for religion in the world is (in my opinion, and I'm sure many, especially the religious, will disagree) scientific ignorance on fundamental questions (e.g. the origin of stars, life, evolution, etc). Until this changes there can not be an "atheist peace".

However, strong regimes within developing countries often impose limitations on knowledge and perpetuate a cycle of ignorance to remain supported by the people of these countries. Thus inadequate education. This is obviously coupled with the inherent dangers of being a atheist where atheism is a fractional minority.

I also think that this will occur only in developing countries. People from less fortunate backgrounds are often more predisposed to religion not only due to education. People take comfort in the the idea of being overseen, having a plan and that when this life is over - they get a better one (side note; does anyone else find the idea of knowingly existing for eternity truly horrific?). People will fight tooth and nail with rationality before saying "i got dealt a shit hand, I'm having a shit life compared to other people, the world isn't fair and when i die its over". Understandably.

In short, religion and superstition will continue to the cling to humanity until humanity evens out. I think a good example of this may be the northern European countries (also making clear this doesn't necessarily have to lead to communism, but perhaps a more regulated capitalism).

What do you guys think? Is this something obvious that everyone agrees with already? What does this mean for i

Those past few days, I've been obsessed with this great game and more specifically its "Good Bad Ending" on the "World's Tallest Hospital".

The TLDR is at the end of this post.

Information Gathering

First, we may deduce in what year and place OMORI is taking place.

From what I can see, Faraway Town seems to be a typical suburban American town. The "All-American Guy" and Kel's Dad grilling burgers in the park, the Othermarket and the Christian symbols seems to support this idea.

As for the date, the use of CRT televisions, game catridges found in Hoobeez, the "brand new console system" which seems ROM cartridge-based instead of using an optical medium, the fact that Sunny suggests to play the CDs in a Jukebox and the fact that Omori's laptop seems to be using some sort of Windows NT 4.0 - Windows 2000 system and that Omori has a laptop at all supports the idea that the game takes place between the late 90s and the early 2000s.

Therefore, thanks to Wikipedia's tallest hospitals list, the hospital we may choose for the calculations is the Hospital for Special Surgery as it was USA's tallest hospital until 2009 and it was built in 1988. [1]

Then, we need Sunny's weight.

Fortunately, the Good Ending Scene has both a doorframe as well as Sunny. The doorframe is 132 px tall and Sunny is 107 px tall.

A typical doorframe is about 2m tall. Mine is exactly 202,6 cm tall and I've used this value in the calculations.

Therefore, Sunny is 1,64m tall.

The World Health Organization has a Global Database on Body Mass Index [2] which states that an underweight person has a BMI included between 16 (or less) and 18,5.

Knowing that Sunny is described as "thin" by Kel's Mom I believe, we could say that Sunny is in the Mild Thinness range.

Therefore, Sunny weighs between 46 and 50 kg.

I will assume Sunny's mass is 48 kg.

Finally, we will assume that the air is dry since there seems to be no sign of rain/thunder, fog or even clouds.

These calculations will be done under standard atmospheric conditions aka. a temperature of 25°C and a pressure of 1 bar (I know that the pressure decreases every 100m over sea level for small heights but I will assume the air density as constant).

Free fall with drag

• We will use the Earth Reference Frame assumed as a Galilean Reference Frame.

• We will use the Cartesian coordinate system and, more specifically, the Oz axis positioned head down following the direction of the gravitational field represented by the g vect

Hi there r/math. What are the graphical implications of and/or intuitions behind a homogeneous function? They are functions with multiplicative scaling behavior but what does that look like graphically? What would the scaling behavior of a homogenous function look like compared to the that of a non-homogeneous function? I wish there were some pictures or videos online, but I can seem to find any that help me understand the bigger picture of homogeneous functions. Sorry if this question is too vague.