Integral operator Puns

When we have an integral operator we can have one that's like

I = int g(x,t) f(t) dt such that I[f(x)] = int g(x,t)f(t)dt

where g(x,t) is called a kernel. Now, there's a lot of kernels, there's kernel: the set of vectors mapped to the 0 element, kernel: what corn cobs are made of, kernel: lowest level of a layerd architecture of an OS, etc.... Now THIS kernel, why is it called kernel? does the word 'kernel' mean to communicate something about g(x,t) or is it just a name?

Hi r/learnmath

i m reading this article page 38 , an integral operator was defined like this where as:

• f dot : is a family of functions defined on the boundry of domain omega.

• k: is some sort of "kernel", defined like this with some nice properties

• P: a polynomial writen in termes of the functions f , on a side note similar to one veriable polynomials differentiating it acts like getting rid of the first term nd sliding the rest one degree back like this

basicly all it does it takes a set of functions defined on the boundry of omega , and gives u one function defined on the inside of omega.

now the heart of the matter is m trying to differentiate it m times , so my attempt was to introduce the differentiating sign into the integral and then apply leibniz formula like so

but on the other side his result was: this

what frustrates me is how he got that additional P(X,Z) term , and what does that Z even mean if it's an arbitrary point from the bondry , the author doesn't explain it or site any refrence about it nd i couldn't find anything similar to this.

any help would be appreciated, thanx. (also m sorry for my bad english)

Hello everyone, I suck at math altho I'm doing a Phys degree so kind of have to know my way around calculus!

Currently trying to teach myself calculus and I'm upto the part where integral theorems are introduced I've just learnt that if you consider a point **P**=(x,y,z) enclosed by volume V where the boundary of the surface is **S** (stuff in bold to show it's a vector) then there are some relations which hold. The first one being:

The gradient of a scalar field at point **P** is equal to 1/V multiplied by the closed loop integral of the scalar field multiplied by d**S**. The picture attached just shows what im on about.

Now my question is, and i really feel like this is definitely one of those dumb questions, but this formula thingy here is the integral definition of a gradient, can someone explain why you can have an integral definition of a gradient in the first place?? Basic calculus tells you differentiation is for gradients and integrals for areas/volumes, so what is going on here? Why are these integral definitions of gradients important too? Please assume I am idiot (which i am) and if possible use layman's terms as much as you can, also if you have any other nuggets of knowledge regarding integral theorems and stuff in general about divergence and curls and how to visualise them etc that would be much appreciated!! Ever since learning more advanced calculus I feel like my brain just can't deal or think about it clearly, as in actually knowing what's going on!

https://preview.redd.it/716zz38kalv61.png?width=734&format=png&auto=webp&s=17c5b148ecaf57f681e80ec7c32b8c3d85dcece7

I have this photon state:

https://preview.redd.it/apewoztjdpx51.png?width=304&format=png&auto=webp&s=6882fff5ea5d8dcbebe8a4d95ac8b383a1d04057

But I can't get how this creation operator works inside the integral. It depends on the frequency...

Like I know that a(dagger)|0>=|1>, but in this case it's found inside an integral? It's like creating states |w_i> and not |1>,|2>, etc?

In particular, if I want to normalize this state, what does this ket become as a bra? I've thought this:

<1|1>=integral(dw*conjugate(f(w))*e1*<0|a)*int(dw*f(w)*e1*a(dagger)|0> ??

Do both integrals combine somehow when multiplied? I can't really see it...

Thank you so much for the answers!!

Edit: New case

https://preview.redd.it/lbe51bcxysx51.png?width=410&format=png&auto=webp&s=c2dca08556e6814c5e67adbdf585d53403dfeee5

https://preview.redd.it/eyhx507yysx51.png?width=358&format=png&auto=webp&s=af27fe8e3963fa2c4cbf38222ed7988f79078d23

So i'm kind of stumped on some of these concepts, really the text i'm reading from is explaining things in a way i can't really conceptualize. Because of that, i cannot come up with a method of solving the following question:

Find the eigenvalues of the following nucleus of the Fredhom integral operator

K(s,t) = t*sqrt(s) - s*sqrt(t) (which is a function in L_2 ( [0,1] x [0,1] )

Are the systems of eigenvectors found a basis of the hilbert space? Could the inverse of a Sturm-Liouville operator be an [Fredholm] integral operator of degenerate nucleus? (Degenerate meaning it can be written as a sum of L.I. functions of t and s).

Attempt so far:

So i know that L_2 (square integrable functions in the Lebesgue sense))is complete and that this K is symmetrical as it IS hermitic (pretty arbitrary):

K(s,t) = K(t,s)*

It is also a degenerate nucleus since I can write

a_1(t) = t , a_2(t) = sqrt(t)

b_1(s) = sqrt(s) , b_2(s) = s

And thus K(s,t) = sum{i=1,2} a_i(t)*b_i(s)

All linearly independent accordingly. Until now this is just me identifying different qualities of the nucleus but i don't know how to proceed from here.

Thanks in advance for any help!

I am a graduate student in a Nuclear Engineering PhD program. Nuclear Engineers are interested in solving various forms of the transport equation. Typical approaches to solving these problems involve using Green’s functions and integral equations. Since the transport equation is integro-differential we also talk about linear operator theory quite a bit. Naturally adjoint operators are discussed when we want to compute sensitivity coefficients.

Unsurprisingly, the rigorous mathematical foundation for these subjects are glossed over in most Nuclear Engineering textbooks. So, I am looking for some general references to help me get a better understanding of these topics. Thanks in advance!

I have the integral operator on C[a,b] defined as

(Gf)(x) = integral from a to b of G(x,k)f(k)dk , where f in C[a,b] and G a cont fcn on [a,b] X [a,b].

Now, the problem I have gives the interval as [0,1] and G(x,k) = e^(x+k)

I have started by setting Gf = e^x integral a to b e^k f(k)dk = uf, u an eigenvalue of f.

How do I procceed from here?

Lets say I have a differential equation[; \frac{\partial^2 u}{\partial x^2} + sin(x) u;],

then we can define an operator [; L = \frac{\partial^2 }{\partial x^2} + sin(x) ;], and say that the differential equation is just [; Lu = 0 ;].

Now say I have an integral equation like [;1 = \int_0^1 k(x,y) u(y) dy ;], can I express this like function application in the same way? writing [; M = \int_0^1 k(x,y) dy, M u = 1;] feels wrong.

Sums and Integrals operate on f(x) and a set in f 's domain.

Is there an analogue to sums in which the set has cardinality |2^(R)| or greater?

I'm a bit skeptical since Integrals work fine in ℚ, which are the same size as ℕ.

Suppose [; U \subset \mathbb{R}^n ;] is bounded and open, with smooth boundary. Then for [; u \in C_c^\infty (U) ;] we have [; \int_U |\Delta u| = \sum_{i,j=1}^n \int_U |\partial_i \partial_j u| ;] .

I can see how [; \le ;] is true by the triangle inequality, but can't really see why it should be an equality. Any hints are appreciated!

Hello everyone!

I am currently working on an assignment in electromagnetism and I have some formula to prove. I managed to get to the desired equation, but I am not satisfied with the justification of the steps I've made (after all, I'm a physicist! :p ). So I want to know whether or not what I did is legit and why. Basically there are two steps I am worried about:

1. curl(dL)=0?: basically I used in my proof that curl(dL)=0. dL is an infinitesimal distance element (i.e. for cartesian coordinates it would be dx i+ dy j+ dz k). I am really not convinced with this, because taking the curl of an infinitesimal element doesn't make sense to me. I know that curl(r)=0 when r=x i+y j+ z k, and so by extension I applied the same to the infinitesimal distance element.

2. Suppose you have the closed integral of curl(f(u) du), is it legit to say that this equals the curl of the closed integral f(u) du ? I know about Leibniz integral rule for differentiating under the integral sign, and so I figured I could do the same for the curl operator (after all, the curl takes derivatives right?). So am I right in doing so ?

Thank you very much for clarifying !

I'm looking for help in how to find the eigenvalue(s) and eigenfunction(s) of the operator A defined by [; $(Af)(x) =\int (x+y)f(y)dy, \quad 0\leq y\leq 1.$ ;]

The operator [; $A$ ;] is a mapping from [; $L_2[0,1]\to L_2[0,1]$ ;] for reference.

(this is a just for fun thought exercise)

As in like a "title character" like Ceobe for Fungimist or Phantom for Crimson Solitaire.

I personally think Surtr would be an amazing choice, and I'm honestly neutral towards her. But I have noticed that some of her hate comes from a lack of lore and development; in which case, an IS is the perfect opportunity to explore her "having memories that are not her own" gimmick better than any profile or story could.

The endings could also play with this by actually giving her a detailed, credible-sounding backstory - but the catch is that they're wildly inconsistent from each other, so it's up to the reader's interpretation to decide which is real or not. Random events would be "flashbacks" that'd either confirm or deny, or even add a whole new thing to Surtr's lore entirely. If executed properly, it could be a hell of a mindfuck-y experience.

I feel like Surtr is also an obvious pick though. So, what are some other operators that'd fit a roguelike?

this seemed to me like the most important quote from the video. I wish he would've given a lot more detail into the inner workings of operation pheonix, i still don't really understand what operation pheonix is. excited for the future nonetheless.

Just think about it, it's the perfect setup. Anartatica dosent just have old lore, it has no lore. I doubt it's even mentioned in the game. Operation deep freeze also looks very high quality, on par with css. (Of course, will have to wait and see until it comes out) it's gotten a near universal positive reaction from the tno reddit community. With the announcement that german Antarctica will be getting released on steam (not sure if it's a demo or just first release) on their discord, it's the perfect opportunity for tno devs to check the mod out. If any devs are reading this, just think about it, the mod looks pretty good!

And no, I don't work for deep freeze lol