Positive Definite Function Puns

Is this a fact ? If yes, can some link me to a paper where I can see it is true under what circumstances.

Not sure if this post belongs here.

I'm doing a least squares fit to data. I know that my model should be positive definite and smooth at a known scale but the functional form is arbitrary. I'm currently using an n-th order polynomial and an external positivity constraint.

Is there a basis set of polynomials that are constructed to be positive definite? It'll make the fitting much easier.

I thought I could use the square of a polynomial, but it's not clear to me that it covers the whole space.

In my machine learning class, my professor explained that a kernel function must be symmetric and psd. I understand that kernels represent the inner product of the feature vectors in some Hilbert space, so they need to be symmetric because inner product is symmetric, but I am having trouble understanding why do they need to be positive semi-definite.

I'm reading Sakurai, and I see that we writes the wave function for a particle with definite momentum as:

https://ibb.co/4FHyRNq

At the same time, when solving for the wave function of a free particle in 3D, he gets:

https://ibb.co/Vq8kcKp

Shouldn't these two be the same? If asked to write down the wave function for a state with definite momentum, wouldn't both be correct? Why is one of the waves not moving but the other one is?

Hello, I'm trying to run a simulation and have been receiving this error and I was wondering what I needed to do to correct this. The simulation itself is fine as I have ran it with other materials so I know it is something with the custom materials I'm adding, but I'm not sure how to correct this. Thank you.

I am struggling to set up a vector minimization equation and trying to figure out what my model, "A" is.

I know A must be a positive definite matrix which means x^(T)Ax > 0 for all x =/= 0.

What is x? What is the benefit of having/assuming a positive definite matrix? Where else are these useful? Wikipedia asserts that the matrix is also symmetric, why?

Hi, I hope this question fits here.

I have a question regarding the SVD and Eigen decomposition of a symmetric, positive semi-definite matrix. As far as I know, both decompositions should be the same, but I don't understand why. Can you guys help me understand this/find me a good explanation online?

Hey guys! I've been studying with the problems from my textbook and have been stuck for quite a while now. The problem asks to prove that if H is a real Hilbert space and T linear operator on H such that (Tx,x)>=0 for all x in H, where (-,-) denotes the inner product. I found the same question on Stack Exchange (https://math.stackexchange.com/questions/803293/show-that-t-is-continuous-with-langle-x-tx-rangle-geq-0?noredirect=1&lq=1), where somebody says that the problem is a special case of a more general theorem for Banach spaces (https://math.stackexchange.com/questions/216858/positive-operator-is-bounded?noredirect=1&lq=1) and mentions the Riesz Representation Theorem. I understand the proof for Banach spaces, what I don't get is how one would use the boundedness of A (using the notation of the second question) to conclude that T is bounded. Any hints are appreciated! Thanks!

If f(x) is a continuous even function, such that definite integral [0,2] f(x) dx = 8 and definite integral [0,6] f(x)dx =5, then what is definite integral [2,6] 1+|f(x)|^2?

I have attached the question and my solution in the link below. Any help is appreciated!

https://imgur.com/zqdjvNv

https://keenonmaths.org/a-powerful-symmetry-formula-for-definite-integrals-calculation/

I’m blessed with that Eastern European tendency to have a ton of body hair to go bald early. Ive been on fin for a year and a half maybe, and i just came across an old picture (2016) of me holding a praying mantis. I cannot believe how hairy my hands were back then... it’s a very clear shot, and my hand/knuckle hair has diminished by 50-75% I’d guess. I was always self conscious at how hairy my limbs were, and i didn’t even realize that i feel much better about them now since its been such a gradual change since i started.

Obviously this is personal preference/anecdotal, but if you have a lot of body hair it seems like fin definitely has the capacity to take that down a notch or 2 in some areas

I haven't played in like 5 years and was never a great player to begin with so it feels really intimidating trying to jump back in to everything.

I saw the tutorial about the Private Coaching function but do Coaches actually use it to help players?

Or should I really look into finding a coach to pay for lessons on some of the other sites like gamersensei?

This also goes for the little accidents that can happen to any dog in their adult life. A positive trainer told our class this years and years ago, and it really stuck with me and feels like a useful thing for people with new dogs ❤️

Let F be a primitive function of f, with the constant of integration being 0. Can this function always be defined as F(x)= ∫ₐˣf(t)dt?

This seems to make sense, since F(a)=0 (since it's ∫ₐᵃf(t)dt), so ∫ₐˣf(t)dt=F(x)-F(a)=F(x)-0=F(x).

But what about F(x)=eˣ. Going through all the steps above yealds F(a)=0, but no such real a exists. But, if eˣ= ∫ₐˣf(t)dt it's clear that eᵃ=0.

Is there some technical step I'm missing, or what's going on here?

If f(x) is an even function, does that mean: Integral(a to b) = Integral(-b to -a)