on the below diagram
#numerical-analysis
distant sky
would it be wrong to assume the velocity is nearly 0and the pressure is highest at the top of the curve, point 2
wide spear
You should ask this in the physics server linked in #old-network
obtuse furnace
I have the option to take a course on harmonic analysis or measure theory. Which would be a better choice if I am interested in studying numerical methods/computational maths?
wide spear
Both are very good
Do you know what area you want to do numerics/computational stuff in?
obtuse furnace
I don't really have a complete knowledge of the areas of numerics.
wide spear
If you are thinking about more engineering/hard sciences, I would suggest harmonic analysis
However, if you go into something that uses a lot of stats/probability, measure theory will be more useful
However, you won't wrong with either
obtuse furnace
sadly my university has a rule that applied math majors can't take more than 4 courses outside of applied math, such as FA, measure
obtuse furnace
However, if you go into something that uses a lot of stats/probability, measure ...
hmm, what kinds of numerics/comp math uses stats and probability? I know of monte carlo methods and stochastic DEs, what else is there?
fleet sail
how does error analysis work for system of ODE? using say runge kutta
is it same order? but just worse constant
also my second question is can a function f(u(t),t) be lipschitz w.r.t f(•,t) and not continuous on f(u(t),•)? for continuous u(t)
and if there can be such a function, would picard thm not work for it?
fleet sail
oh ur right
fleet sail
also my second question is can a function f(u(t),t) be lipschitz w.r.t f(•,t) an...
also do u know about this
wide spear
So you want it to be lipschitz in the first variable and not continuous in the second?
Let $u(t)=0$ and $g$ be a discontinuous function. Then let $f(u(t),t)=u(t)+g(t)$. This is Lipschitz in the $u(t)$ and discontinuous in $t$
pine jettyBOT
Paradise of Despair
fleet sail
Zero stability is confusing, what if I got $\frac{du}{dt} = 0$ and $u(0) = 1$ as the IVP
pine jettyBOT
Anticipation
fleet sail
then the general multistep scheme is $\sum_{i=0}^k a_i \xi^{n-i} = 0$
pine jettyBOT
Anticipation
fleet sail
but this just means the general solution that could appear from the scheme is $u_n = c_1\xi_1^n+\cdots+c_k\xi_k^n$ and I don't see any convergence to u(t) = 1 at all when $n \to \infty$
pine jettyBOT
Anticipation
fleet sail
does this just show that linear multistep method in general is bad
since for example euler method we got $u_n-u_{n-1} = 0$ as the lin difference equation, guessing solutions of form $u_n = \xi^n$ gives $\xi = 0,1$
pine jettyBOT
Anticipation
fleet sail
and so the general solution is $u_n = c 1^n$ which I guess luckily matches the ODE if we got a good constant $c_1 = u_0$
pine jettyBOT
Anticipation
fleet sail
so euler method just got lucky
wide spear
Lol yeah multistep methods are bad
fleet sail
ok i found that finite difference method must have coefficient sum up to 0 so there must be a root of 1
so things make sense
brave crypt
Ange, where do you get your epic names?
fleet sail
where does the first equation come from
like why is accumulated error roughly $\sum_{n=m}^N|u(t_n)-u_n|$
pine jettyBOT
Anticipation
fleet sail
So lets say the scheme is denote $H(u,\Delta t)$, then $u(t_1) \approx u(0)+H(u(0),\Delta t) = u_1$
pine jettyBOT
Anticipation
wide spear
I think the accumulated error is referring to the total error you pick up from steps m to N
fleet sail
and then $u(t_2) \approx u(t_1)+H(u(t_1),\Delta t) = \tilde{u_2}$ and $u_2 = u_1+H(u_1,\Delta t)$ so then theres some deviation between $\tilde{u_2}-u_2 = u(t_1)-u_1 + H(u(t_1),\Delta t) - H(u_1,\Delta t)$
yea ik that
pine jettyBOT
Anticipation
wide spear
So that's where the sum comes from
fleet sail
yea, but i just want to know why its approximately that sum equation there
wide spear
Oh
T/delta t is the number of steps from m to N I guess
And then you have the sum absolute value
fleet sail
i mean lets say we start at u(0) right
and the scheme is some H(u,delta t), idk could be anything
wide spear
Sure you assume that there is no error at u(0)
fleet sail
so ill say u_k as the approx as u(t_k) is exact
then u_1 = u(0)+H(u(0),delta t)
and U_2 = u(1)+H(u(1),delta t) but U_2 ≠ u_2
since u_2 = u_1+H(u_1,delta t)
fleet sail
then for error u(2) - u_2 = (u(2) - U_2) + U_2 - u_2 where u(2)-U_2 is part of the sum in the picture above
fleet sail
i think so? since thats the local truncation at u(2)
cuz U_2 is based on the exact u(1)
fleet sail
yea but my notation is cursed
wide spear
....
wide spear
What
fleet sail
ill call U_2 the accumulated error at 2nd point
wide spear
Is your notation consistent with the image
fleet sail
I think the notation in image is saying u_k is the approx using exact u_k-1
so ill go with that
wide spear
No I don't think so
Their u_n is the approximate solution using the approximate u_n-1
fleet sail
I mean like since u_n in the local truncation error for tau_n is local
and here u_n accumulates from all the past errors
wide spear
I see
fleet sail
so the way i interpreted equation is lets say U_n is the approx we got after n steps right, then u(t_n) - U_n is the accumulated error, and they r saying the accumulated error is that sum there
approximately
wide spear
Ok their proof is weird
You can't approximate global truncation error the way they have
Because global truncation error grows
fleet sail
i think my prof mentioned this is just outline of proof
maybe theres a way to show the global error is of order <that sum>
wide spear
You can't approximate global truncation error as T*abs(tau_n)
That's nonsense
Any bound is exponential in T
fleet sail
yea makes sense
time to die
tbh i might have figured something
global truncation error is O(that sum) i think
cuz any scheme that is stable should only involve evaluation at f
and f is lipschitz in u
but question is where does T gets its exponent
wide spear
No
Global truncation error is not asymptotically that sum
Global truncation error at $t_n$ is bounded $\abs{e_n}=\leq\frac{\max_j\tau_j}{hL}\qty(e^{L(t_n-t_0)}-1)$
pine jettyBOT
Apostle of Epilogue and Eternity
fleet sail
ah ok
fleet sail
oh i think i know why now thx
this eq make sense if u unroll something with factor e_n = something + (1+L*delta t)e_n-1
wide spear
Yes
hard escarp
Hi.
I'm sort of looking into getting into research in optimization methods. Any book I should check out? I know it's pretty vague (sorry). But maybe someone here has something to recommend...
I got Yuri Nesterov's work to look up.
wide spear
Tbh for the sort of applied math that optimization is, I'm not sure that books are the best source
I can give you more people to look at though
Fritz John, Dantzig, Kantorovich, Karush, Kuhn, Tucker, Pontryagin
hard escarp
Fritz John, the PDE book author person?
wide spear
Oh I guess he also did pdes
hard escarp
These names are mostly familiar (assuming I'm not confusing them hehehe with other people). I'll check out their stuff. Thank you.
wide spear
He also did some work on linear programming
The fritz john conditions are a necessary condition for solutions in optimization problems to be optimal
hard escarp
Ok...
I only knew him from the PDE book
Wikipedia has a list like that
Good think you told me to pay attention to that stuff.
I'll check it out, thanks.
wide spear
tall solar
I'm sort of looking into getting into research in optimization methods. Any book...
Check out "convex optimization theory" by Dimitri bertsekas. It's the most thorough and rigorous text I've seen on the subject
wide spear
Do note that there is a lot of non-convex optimization though
tall solar
Do note that there is a lot of non-convex optimization though
Very very true. I guess I recommend the text to rigorously learn convex optimization. With the caveat that convex opt theory has become the typical entry point into the theory the same way linearity is for pdes.
I think the text surveys duality well
wide spear
Yeah not saying the text is bad or anything, just encouraging phao to keep an open mind about the big wide world of optimization
hard escarp
Thank you.
tall solar
Has anyone ever heard of a star domain?
tall solar
Well I was curious about whether they are relevant to optimization. They look like a generalization of convex sets and so I was thinking there could be an optimization theory based upon them. I'm not sure what a "star domain function" would be like.
Given the interpretation of differentiable convex functions as always lying on one side (usually above) its tangents, I feel that a differentiable "star domain function" would be a general function with a unique optima, this seems very broad.
Anything going on here? Star domains seem like cool sets.
wide spear
The correspondence between convex functions and convex sets is not that strong
And I don't think there is a good notion of a star convex function
Also you can always just enclose a star convex set in an actual convex set
tall solar
Ah I see.
tall solar
Also you can always just enclose a star convex set in an actual convex set
General non convex sets can also be bounded inside a convex set. I suppose I was interested in a star domain as an interesting intermediate between convex sets and general non convex set. But yeah I don't know a def of a star convex function. Weh
wide spear
Convexity is good because of the Hahn-Banach hyperplane separation theorem
As well as Krein-Milman
wide spear
Does this have what you want: https://link.springer.com/article/10.1007/s40305-014-0039-x
If so I can get a pdf
wide spear
prime kraken
techniques with projections are used in other contexts, it's not super unexpected
brave crypt
Not too sure where to ask, but can anyone here explain the main difference between the Bellman-Ford algorithm and Dijkstra's for solving single-source shortest paths problems in optimisation? As far as I understand, Bellman-Ford determines the distance from a source node s to all other nodes in the network based on an iteration on the number of edges, until reaching the maximum number of edges allowed on a path (for an n vertex network, that would be n-1 edges). It then tests if we have a negative cycle by allowing n edges and check whether it gives a smaller weight. On the other hand, Dijkstra is more based on an exploration of the network ie. it first explores the vertices that are the closest to the source, then their neighbors etc. At each exploration, the optimal distance is calculated. How can we ensure that this exploration does take the shortest path ?
prime kraken
the original dijkstra alg is between two specified nodes, and as far as i recall, requires positive weights along edges
brave crypt
Yep it does, the fact that all weights are positive makes a shortest walk between s and v a shortest path
But my question is, how does the exploration yields an optimal path
prime kraken
there's a weird (to me) proof on wikipedia, but you can think of it as splitting the distance from source to destination into distance from source to intermediate point + distance from intermediate point to destination. the latter is a fixed quantity, but you can control (and minimize) the former. dijkstra's alg minimizes the former recursively by splitting it again, following the same procedure
brave crypt
Right
I think I did some research and that pretty much answered my question
At each iteration, the algorithm updates the array value d[v] for a node v based on the value of d[u] where u is an immediate neighbor of v which has been just explored and whose distance from s has been just calculated
hollow gate
https://sci-hub.do/10.1145/2157.2158
I'm trying to understand the following proof on wigderson's algorithm for k-coloring graphs.
it seems to jump through a lot of steps which is confusing
like how do we get at least log_k(n) colorings in step 3 of Algorithm D
and why are we splitting the proof into two phases for when |W| gets below |V|/log_k|V|?
The algorithm basically a Greedy Independent Set Algorithm where vertices with the minimum degree are colored first
fleet sail
curious, is this a hard problem, i have no idea what this is but just curious
like is this hard to implement or something
i dont think it should be hard since u can just follow psuedocode but this is just the entire hw for one of the hw seet
set*
prime kraken
wide spear
prime kraken
have you worked with resolvents and neumann series?
or the wave equation or helmholtz equation
wide spear
Yes I know what those are
prime kraken
aight
so i have a fredholm integral equation of the second kind that i'm currently solving with neumann series
wide spear
Oke
prime kraken
but this has very poor convergence properties
i've also taken a resummation of the neumann series to get some padé approximants which appear to have a larger radius of convergence, but not large enough
how else could i go about (approximately) solving the fredholm int. eq. of the second kind?
wide spear
prime kraken
the expression is something like
$f(r) = f_0(r) + \int_D G(r, r') k (r') \gamma (r') f(r') dr', ,, r',r \in D$
pine jettyBOT
Edd
prime kraken
for practical means and purposes, k, gamma, and f of r' in the integral can be treated as a single function, and G is a green's function
wide spear
What does a Fourier transform do
prime kraken
i expect G to be space variant, so it doesn't really turn into a pointwise product in (spatial)frequency domain
wide spear
What about a Fourier transform in r'
prime kraken
yeah that's what i meant
prime kraken
this is already in temporal freq domain, so the bins became orthogonal... gotta solve like 100 of these in parallel
after discretization in space, the current solution is kinda like
ayy lmao $f(r) \approx \sum_{n=0}^{+\infty} G(r,r')^n f_0(r') w(r')$
pine jettyBOT
Edd
prime kraken
(where everything is now matrices and vectors of appropriate sizes)
wide spear
How much do the terms vary
prime kraken
can truncate at about 7 usually... assuming it converges lol
idk if giving up on MoM and BEM is a better idea here and just doing finite differences or smth
ange
send help
wide spear
prime kraken
how do you mean?
which of all the newton methods 😛
it's ok, time to flex my b1 german
right, so, after discretizing and whatnot, my system is something of the form (I - B)f = f_0
and i wanna solve for f
i'm guessing you meant the newton method that takes the inverse of the jacobian
brave crypt
Say we want to store a (directed) graph (V,A) in a computer using an adjacency list. That is an array for vertices where each array entry consists of a list of neighboring vertices, eg. the array entry for node v consists of all nodes u such that (v,u) is an arc. What is the time complexity of an algorithm that looks up all the arcs in such an adjacency list? I first assume it's O(nxm) but after searching I found that it's O(n+m). Can someone explain why please?
indigo musk
an outer loop iterates over the vertices and an inner loop over the adjacency lists
the length of an adjacency list is the degree of the vertex so for each vertex v you do O(deg(v)) work, summing this up gives you O(|E|)
but you also have to consider what happens when deg(v) = 0, in this case you still have to do constant work to check that the adjacency list is empty which gives you another O(|V|)
@brave crypt
brave crypt
an outer loop iterates over the vertices and an inner loop over the adjacency li...
Right I thought when we had two "concatenated" loops then we multiplying their running times
echo iron
Are there any more efficient methods to approximate the square root of a prime number, other than Newton's method?
dawn viper
Are there any more efficient methods to approximate the square root of a prime n...
fast inverse square root + newton
prime kraken
if you're taking a ton of square roots at the same time, you could try to set up something like a bisection algorithm but taking the partitions randomly instead of in the middle of the intervals. doing this for several hundred square roots at the same time can be faster in some cases, in my experience
you need to start off with knowledge of an interval containing each of the roots though
ruby relic
what does 'efficient' mean in this context? if you want to optimize for wall-clock time on x86 its hard/impossible to beat sqrtsd afaik
otherwise generically i think you have to do newton (maybe with optimizations to reduce divisions if your computer is really that ancient)
echo iron
thx guys
brave crypt
Dijkstra's algorithm computes correct distances if the weights are all non-negative in the network. I'm curious, what if the distances instead are nonegative between any 2 nodes in the network?
wide spear
They're the same? Based on how you've written it
There can't be a negative weight because if there was, then the distance between the two nodes at the ends of the edge with negative weight would be negative
Unless I am misinterpreting
brave crypt
You're completely right
so there are no negative weight arcs, that means no negative cycle either
so dijsktra must in theory work no?
wide spear
For the problem as you've stated it, yes
brave crypt
How can I possibly modify the Bellman-Ford algorithm to compute a distance from a start node to a a finish node via some specific node of the network?
Without applying it twice
from the start to the intermediate node then from the intermediate to the end node
brave crypt
Yes
wooden ledge
How can I possibly modify the Bellman-Ford algorithm to compute a distance from ...
Apply Bellman-Ford algorithm starting at the intermediate vertex and add the distances to the start and finish node?
brave crypt
Yup thanks !
cosmic karma
Hi. I want to transform a complex valued function f(x) to f(e^i*theta). How do I do this in MATLAB? Thanks!
wide spear
cosmic karma
Should I ask my question there? But I'm not allowed to cross-post right?
cosmic karma
Okay thanks haha.
prime kraken
as an update to my problem, ange, i think i'll just stick to the differential equation instead of converting it to integral form 
might as well take the chance to learn pseudospectral methods
wide spear
Oh those are good
Why would you turn a differential equation into an integral equation
prime kraken
to do matrix shenanigans with MoM
using contrast terms w.r.t. the background medium properties, the inverse problem can be written as a compressed sensing problem, which my lab likes
wide spear
prime kraken
i wanted to do everything that way, but the integral equations that pop up in the middle are nasty, as you saw. you can mix and match the two things, tho. the final compressed sensing problem requires a matrix of green's functions for the background medium, a contrast term, and the total field in the medium at the previous estimate of the contrast. this total field is what i wanted to get earlier, but one can just use pseudospectral methods to solve the original diff eq with only partial knowledge of the medium
prime kraken
on a related note, does anyone have a good resource on perfectly matched layers/absorbing boundary conditions?
frosty abyss
When dealing with optimization algorithms that use random walks, does it matter that the parameters differ a lot in scale?
wide spear
By parameters are you referring to the components of the solution?
If so, you probably want the random walk to be at an appropriate scale for each component
Alternatively you can rescale everything
brave crypt
Hi I am trying to prove that the following result
$\int_{\mathbb{C}^n} \log|x_1x_3x_5 \ldots x_{2n - 1} - 1| \neq 0$ for $n \neq 1$
wide spear
You already asked about this in #advanced-analysis
copper abyssBOT
brave crypt
@wide spear I am not sure which section I should ask it in
wide spear
You already asked this in #advanced-analysis and it's fine there
wide spear
Yes
prime kraken
lame question (and mostly due to laziness), but what are the main differences between the gauss-seidel and jacobi methods? they seem to require the same conditions for convergence
slow niche
Dont know where to put this:
im looking for some closed form solution to calculate t for the constant jerk kinematic equation:
1/6jt^3 + 1/2at^2 + vt + x = 0
there is always one real root, but dont think there is a gurantee on +/-
or maybe something like newton's method would be faster?
fleet sail
lame question (and mostly due to laziness), but what are the main differences be...
i think gs converge for symmetric pos def too
and more powerful conv rate in general?
i wrote some code a while back and in general found spectral value is smaller for the iterative matrix in GS
spectral radius*
also in terms of heuristic its the same algorithm except GS is like "online" in the sense u use the updated values once they r available
fleet sail
Dont know where to put this:
im looking for some closed form solution to calcul...
i'd recommend newton for polynomial usually cuz u can use horners method to speed things, but for 3rd order cant u use a explicit formula to solve? unless u want speed and dont need exact solution
prime kraken
ah, thanks for the input
slow niche
i'd recommend newton for polynomial usually cuz u can use horners method to spee...
well i'm asking if there is an explicit formula for this specific case?
im trying to calculate t every frame for simulation so maybe newton's method is better.
fleet sail
yea, the formula is very complicated and is slow
u can just search cubic equation explicit solution and something like this should show https://math.vanderbilt.edu/schectex/courses/cubic/
judging from ur application newton's method is better
wide spear
Is there a corresponding Gram-Schmidt?
QR is closely related to Gram-Schmidt
The concern with doing Gram-Schmidt with the Minkowski form is that you can get <u,u>=0
In which case Gram-Schmidt clearly breaks down
Householder reflections also have <u,u> in the denominator
Maybe Givens rotations are the way to go
brave crypt
Hi
pine jettyBOT
Learner
wide spear
You already asked this in #advanced-analysis
copper abyssBOT
brave crypt
@wide spear sorry
digital scarab
not sure if this is the right place, but does anyone know an efficient way to compute all partial derivatives of a multivariate polynomial
dividing by coefficients when you take the derivatives is allowed,
for context, i am trying to write an algorithm that requires that computes any partial derivative of f (of any order) whose degree is 1.
wide spear
Are you evaluating them at a point
Or do you want all of them in whatever format
Like n-d array
Well
I guess it doesn't matter too much
What are you doing currently, and why isn't it sufficient?
digital scarab
i want to find one of the partial derivatives whose degree is 1. i edited my previous message to reflect this--originally i said something incorrect
so i dont need them in any format, i just need to find any such polynomial. what i have tried is a depth first search on f and its partial derivatives, and it can be quite slow. i am hoping for something bounded by the total degree of f.
wide spear
You only need a single partial derivative with smallest term order 1?
digital scarab
...sorry, i made another mixup and edited again. i am looking for a partial derivative with order 1.
wide spear
Ok so you want a partial derivative of order 1
There are n of these
Where n is the number of variables your polynomial is in
Will any of them do?
digital scarab
no, i would also further request that they are linear.
wide spear
Linear in each variable?
digital scarab
actually, im getting confused on this point, it might be better just to say any will do for now
wide spear
Well, you can always compute the n different first order partial derivatives
digital scarab
i feel like this is obvious, but how do we know there are exactly n
wide spear
You have n different variables right
So you can take a partial derivative in each one
digital scarab
oh, i left out an important detail. when i say degree 1, i mean it in the multivariate sense. maybe called total degree. so the monomial xyz would have degree 3.
wide spear
Yes this is how I was interpreting it
digital scarab
ah.. let me do some examples to sanity check some things
oh okay. .. i can look at these n first... im so dumb lol
wide spear
glossy yoke
These are three persistence diagrams produced by different datasets that represent the same underlying phenomenon. The points themselves are in a 512-dimensional space, but my computer doesn’t let me go that high. I know that these points lie roughly on a sphere in 512D, and the main question I’m interested in is how they are distributed on that sphere.
The 0D part of the diagram tells me that there are multiple connected components (usually 2, but potentially up to 4 in one of the plots) and no higher level structures (which is expected if they are distributed around a 512D sphere). I'm not sure what (if any) meaning the large gap in the 1D and 2D PH along the line x = y means.
Has anyone seen things like this before / have advice about interpreting these diagrams?
prime kraken
would be easier to understand with a little bit more input
what are h0, h1, and h2?
by 512D, it seems you mean a population of 512 samples, each one with an associated value for death and birth, and you have 3 such data sets, yeah?
wide spear
I think this is some sort of topological data analysis
H0 is the 0th homology group, H1 is the 1st homology group, H2 is the 2nd homology group
@timid kelp maybe you know something about visualizing TDA stuff?
prime kraken
i'll also need your help later, ange 
since i suck at diff eqns
wide spear
prime kraken
while max materializes, i'll just go ahead and ask :x
i have an inhomogeneous helmholtz equation i wanna solve numerically (in 2D)
so lagrangian of f + wavenumber squared times f = some excitation
wide spear
Ok
prime kraken
i've set up the lagrangian in 2D as some symmetric matrix with block structure. for the wavenumber k^2, since it may vary over space, i put its entries along a diagonal matrix, so that the differential operator is D = fancy block matrix + diagonal k^2 matrix
wide spear
Yes
prime kraken
then on the RHS goes the excitation. assuming i'm doing this in frequency domain and that i have a single pointlike source, the RHS should be a 1-sparse vector whose nonzero entry is the nth fourier coefficient of the excitation (and so the diagonal k^2 matrix has the wavenumbers for frequency bin n)
wide spear
Right
timid kelp
<@!158305429615673345> maybe you know something about visualizing TDA stuff?
never have i been pinged in this domain
prime kraken
my thought was that i could solve this system Df = s for each frequency bin, put all the f_n as rows in a matrix or smth, and ifft the columns
timid kelp
whats the Q
prime kraken
These are three persistence diagrams produced by different datasets that represe...
this one here
timid kelp
i see
uh
i in theory should be able to read this
but I have no idea how to
its very confusingly labeled
wide spear
It's persistent homology
wide spear
You know homology right
timid kelp
yes
wide spear
wide spear
@glossy yoke
timid kelp
im at the gym if you guys dont figure it out I will take a serious look when im home
glossy yoke
These are persistent homology diagrams. They're a topological data analysis tool for understanding the structure of high deimsnional data
glossy yoke
?
timid kelp
like these are all integers
this is what i would expect
I dont know how that graphic was generated and the axis labeling is practically useless :/
is there like
background on it i can read
when i get a chance?
glossy yoke
The x and y axes are radii
We are progressively growing spheres around the points
My numbers are less than 1 because these are embeddings produced by an ML algorithm that normalizes it
But the x and y axis are measured in "distance units"... you can multiply all the values by 100 and nothing changes
prime kraken
so ange, did my procedure seem to make sense for helmholtz eq? cuz my code isn't working lol
wide spear
Wait so you solve this for each frequency bin
How do you combine them
Also do you have boundary conditions
prime kraken
i treat each of the f_n vector entries as the field at some location x,y at frequency n. so i put the vectors as rows of a matrix and ifft the columns in hope of obtaining the behavior of the total field over time, for each location x,y
wide spear
Ok
prime kraken
i set the initial field and derivative to 0, with perfect reflections at the boundaries (i.e. everything is 0 outside of the region of interest)
prime kraken
yeah
wide spear
So shouldn't you add the vectors you get from each frequency bin?
And then ifft that?
prime kraken
hmm
if i add them up, don't i end up with a single vector that varies over space?
but no time dependence?
prime kraken
right
wide spear
So you shouldn't have time dependence in your solution either
prime kraken
sure, but these are each of the D_n f_n = s_n, solved for each of the wavenumbers i'm interested in
wide spear
Sure
prime kraken
each f_n is of length Nx * Ny
wide spear
Yes
prime kraken
and indeed, time independent
how could i get the time and space dependent field back from these?
wide spear
You want something that is time dependent?
But where does the time dependency come from
prime kraken
each wavenumber is related to a time domain frequency, right?
so i wanna use that, somehow
wide spear
Aren't you doing a FFT in the spatial variables
prime kraken
in my head, by taking the wave equation, doing separation of variables, and expressing the time dependent part as a superposition of complex exponentials, you end up with something like the helmholtz equation, but everything is multiplied by dirac deltas in frequency domain
not rn, no
that's what i'll try next
so rn i'm in space - temporal freq. domain
wide spear
Oh I see
prime kraken
but seeing as i apparently don't understand it well enough, the next one is wavenumber - time domain
wide spear
You start with the wave equation and get the Helmholtz equation + ODE out of it
prime kraken
yeah
doing everything with sums of complex exponentials and then transforming both sides, something like that
wide spear
Right ok
Don't you solve the ODE in time and then multiply the results to get your time-varying solution?
prime kraken
right, i could solve it in time domain the usual way, doing finite differences in space and time
but for whatever reason i thought i could do it in frequency domain instead, and invert the differential operator for each frequency i care about
which seemed reasonable since the D_n are symmetric block toeplitz and diagonally dominant
wide spear
Have you seen how FFT is used to solve the Laplacian?
prime kraken
i've seen how to do it for the opposite domain, wavenumber-time
that's the classic pseudospectral k-space method or some such
i will probably end up doing that anyway, i just had academic curiosity about what i tried, cuz i'm not sure where it's failing
wide spear
prime kraken
ah, part of the motivation was the eventual need for perfectly matched layers
wide spear
What are perfectly matched layers
prime kraken
i later wanna simulate an infinite medium
sommerfeld radiation conditions
so what some ppl do is add regions at the edge of the computational region that turn waves propagating outward into vanishing waves
and for people that are bad at math and differential equations like me, it tends to be easier to stay in the domain in which these PMLs are derived... except i can't get it to work even without them
wide spear
Have you tried randomly changing signs in your code
That's usually what I do when my code doesn't work
timid kelp
@glossy yoke the issue is that I really dont know how to read this because the data of persistent homology is like
like each homology group is represented by an integer
and scaling matters
like you cant scale it and maintain the information because its geometric
like I dont know, looking at this graph, what it means when theres a H^1 marker somewhere
what does that correspond to?
(also fwiw I understand persistent homology well and I have a lecture on it on youtube)
but even the concept of 2 radii doesn't really make sense
timid kelp
like the epsilon balls around the points should be growing in radius over time
but idk how you can make sense of 2 separate axes
I guess you can just adjust the metric
so that the balls are actually like ellipsoids or whatever
timid kelp
<:smug:784402518683156490>
not trying to be smug haha i just think there is miscommunication bc they keep responding w correct but basic explanations of what persistent homology is
so i figured id clarify
prime kraken
ignore the name of the emoji, i just meant the face the goat is making 😛
wide spear
fleet sail
can someone explain why minimizing
will separate the target and jamming component of signal
lol i think maybe i need to give more context, let me know if thats the case thanks
prime kraken
this is assuming A_T and A_J are distinct in some way
probably under some assumed degree of sparsity and some upper bound to the inner product between pairs of atoms taken from A_T and A_J
in finite dimensional stuff, this should be related to the "kruskal rank", "girth", or "spark" of the atomic sets (different names for the same thing). idk about the infinite dimensional case
fleet sail
ah thanks
prime kraken
they probably said something like "assume the jamming signal is independent and uncorrelated from the target signal", right?
so you can do stuff like easily find the expected value of the gramian or smth like that
fleet sail
hm hold on
so they have like N antenna and M timesteps, and basically a matrix to hold the data for each antenna at a given timestep. It says only that the jamming signal is spatially uncorrelated but the target signal is correlated in both space and time
prime kraken
that's ok too
but the antenna signals cannot be completely coherent or the problem does not have a unique solution
you need signal uncorr. to the jammer, and signals not perfectly coherent to eachother
if the signals are coherent, the atomic set of the signal only spans a line, and you cannot recover anything
it's ok if the signals become correlated in the medium, since you can then use the time steps, or snap shots, whatever you wanna call them, to do so-called "smoothing"
so we want something like the inner product between pairs of atoms in A_T to not be 1, the inner product between atoms in A_T and atoms in A_J to be zero if possible, since this makes stuff easier, and the jammer is usually unknown and assumed to be some nice white process with orthonormal atoms or something like that
it sounds like the conditions here require a ton of antennas and time steps so that you can "smooth" over space and time, most likely
fleet sail
thanks, ill have to think about it more and probably have more questions lol, this is new area im trying to understand
prime kraken
i've never done atomic norm minimization tbh, but it should be pretty closely related to the more conventional l-1 minimization stuff i work with, so i can help with some stuff, but not all
prime kraken
hmmm sorta
in some cases they can look the same
the steering vector is the response that a plane wave produces at an antenna array
if the antennas are omnidirectial in some 2D plane and regularly spaced, you can get a dft matrix, yeah
or at the very least, some vandermonde matrix of complex exponentials
if the antenna is not omnidirectional, this also goes into that matrix and it gets kinda nasty
i think in most papers about antenna stuff they call this the array manifold
some set of vectors of length M (number of antennas), whose value depends on the angle at which the waves arrive/depart
for omnidir antennas, the directional response is simply 1 in all directions, and a complex exp that gives the time shift w.r.t. some reference antenna
this is the classical shitty image you will see
in 3D they use spherical coords for it
the idea is that if the antenna is omnidir on this plane, the received signal from a distant source can be treated like an ideal plane wave, which yields just shifted copies of the same signal in all antennas, and the shift between antennas depends on their spacing and the angle of incidence
fleet sail
oh i see
that makes alot of sense, cuz the paper assumes omnidir antenna and stores the first value as 1 and the rest as e^i2pi*kf which r just phase diffs relative to first ig
prime kraken
yeah, they arbitrarily chose the first antenna as the "phase reference"
prime kraken
another dumb q from my side. what's a good way of solving a homogeneous system of linears eqs Ax=0 if i know one element from x and A is symmetric?
(A is also sparse and block toeplitz)
prime kraken
i don't know if it's pd tbh
that's what i would've thought, but since it's homogeneous, idk
in the meantime i thought i could just take the column from A for which i know the true value of x
then make something like By = -A_0 x_0, where B and y are A and x after removing the column A_0 and the entry x_0
and then just doing the usual stuff
wide spear
Yeah that sounds fine as well
But I’m not sure how much time savings that will provide
prime kraken
true enough, it's just a lame solution for now. can you drop me more hints on the FFT approach you had in mind? i know we can embed the original matrix into a larger circulant one so that we can turn the original Ax into an elementwise product of the fft of the circulant convolution kernel and x, yeah? so something like fft(big A) \odot fft(long x) = 0
wide spear
Yep
prime kraken
but then how could i enforce the knowledge of some x_n in the vector x?
wide spear
You don’t
I’m assuming your system is very big right
I don’t think that single entry will have that much of an impact
prime kraken
i wouldn't even know what to do then
cuz that just says the hadamard product of two vectors is 0
trivial solution moment
wide spear
Sparse LU and start the back substitution with the xn you know
prime kraken
yeah, that also came to mind, i can give that a shot later on
lofty tundra
in linear programs, the objective we want to maximize is the dot product c x, where x is unknown
is my interpretation correct that we basically want to maximize a continuous linear functional which is represented by vector c?
wide spear
Sure c defines a continuous linear functional
lofty tundra
Is there a theorem that implies that in LP checking feasibility is equivalent to solving it regarding time complexity?
wide spear
Feasibility is much easier than solving I should think
prime kraken
bruh ange
we finally going somewhere
the way i'm solving the helmholtz eq gives out only coefficients of the spatial waves
so to turn them into the actual stuff going on, one needs to return to space - time domain, which because of wave propagation stuff, happens to be sorta like an fft in space (the exponential is conjugated in wave prop) and an ifft in time
i'm still doing stuff wrong, cuz the boundaries and inhomogeneities aren't interacting with the field, but bruh
naive gradient descent worked surprisingly well with only a few iters. luckily some colleagues have set up a nice python library for dealing with huge structured matrices without making copies in memory and stuff
wide spear
Oh nice
prime kraken
ignore the temporal aliasing, i only used a handful of wavenumbers, so the pulse shape i chose became convolved with a sinc
wide spear
Exciting
fleet sail
edd i need help with indexing in dft 
fleet sail
ok so
fleet sail
like if u have a discrete fourier basis right, lets say up to $[cos(nx_1),...,cos(nx_{2n+1})]$ and $[sin(nx_1),...,sin(nx_{2n+1}})]$ where $x_1$ is left endpoint of $[-\pi,\pi]$ and $x_{2n+1}$ is like the point before the right endpoint then theres 2n+1 linear independent vectors right
pine jettyBOT
Anticipation
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
fleet sail
the question is how do you embed this into $C^n$ since its not even
pine jettyBOT
Anticipation
fleet sail
i guess its not that much of an issue
if I let the first vector just be vecto of 1/sqrt(2) [1,1,....,1]
and then the rest to be cos + i sin? so it lives in C^{n+1}
wide spear
You have two vectors in R^(2n+1) and you embed them into C^(2n+1) with x+iy
fleet sail
wait but i thought embedding works by having [cos(kx_j)] and [sin(kx_j)] 2 basis vectors embedded into cos(kx_j)+isin(x_j) which is one vector in C^{n+1}
wait nvm i am
i get it now
fleet sail
yea
fleet sail
Actually I'm still confused, so here is DFT
at least in R^2n form, where if I consider u^(0), ..., u^(2n-1) then its 2n-1 linear independent vectors
since u^(1) is actually 0 if using the definition here
so first my question is since the space is R^2n and theres only 2n-1 linear independent vectors then doesnt the {u} fail to be a basis, although they are orthogonal?
prime kraken
why is u1 0?
fleet sail
cuz u^(2*0+1) = sin(0x) = 0
fleet sail
lol yea i am so confused, it was my previous profs notes and he consistently makes typo
prime kraken
i don't even understand why they're using sines and cosines
lemme try to see if this can even be turned into the usual definition
no man, i give up
the notation is too dumb and i'm tired
i would just tell you to use the complex dft
like i don't know why some terms are cosines and others are sines
they should all be a sum of sine and cosine terms
what are you even doing with these u terms?
fleet sail
I think this is my understanding, u start with $f = \sum_{i=1}^n c_i \phi_i(x)$ where $\phi_1(x) = \frac{1}{\sqrt2}$, $\phi_2(x) = cos(x)$, $\phi_3(x) = sin(x)$, etc.
pine jettyBOT
Anticipation
fleet sail
like the fourier series
and then what they did was sample n points uniformly in the domain where f is periodic so that v \in \R^n stores the evaluation of f at these points
prime kraken
but in the fourier series also, each term has both a sine and a cosine
i have never seen it written this way
i guess they're putting it as a phase term, but what the crap
why not make it all cosines then?
fleet sail
ah ic, so you mean each term is acos(kx)+bsin(kx)?
prime kraken
yes
you CAN turn it all into sines, or all into cosines, or use complex exponentials
but why make some terms sines and others cosines
i have literally never seen it this way in my life
fleet sail
true combining them does make more sense lol
prime kraken
i don't understand what they mean with the u_j^(2k) either
all even terms? or when you have n even?
fleet sail
yea ill try to learn the standard way tbh
this is what he did to convert to complex, but i thought it didnt make alot of sense
since j ranges from 0 to 2n-1 so the v is in C^{2n} still...
prime kraken
i still don't get the 2k and 2k+1 thing
fleet sail
its just 2k is the cos(kx) terms and 2k+1 is the sin(kx) terms
prime kraken
yeah but if you put this in matrix form it still doesn't make sense to me
cuz you'd get a DFT matrix only for positive frequencies, for example
some matrix in C^{n by 2n}
which is not the standard DFT anyway
unless you have already defined what analytic signals are
they're randomly swapping out stuff in C^n and C^2n
fleet sail
yea
so im gonna try to start with acoskx + i*bsinkx instead i guess
scrap the sldies
slides
prime kraken
look at literally any other definition
fleet sail
Frick looks like i have to do a bit more analysis to understand fourier transform better, at least riemann integral
like i get how it works now through examples, etc. but would be nice to understand the theory
fleet sail
I don’t quite understand why it’s a linear relation between Doppler and spatial freq of clutter signal if the platform is movingly
Wait is it just Doppler effect
prime kraken
the doppler effect should be some shift in frequency domain, or complex exponential in time domain
i don't directly see how it's related to the spatial frequency tho
wikipedia gives these
in the expression below, you see the wavelength pop up
you can get the wavenumber from the wavelength, so i guess that's where you get the spatial freq from
fleet sail
Wait how do you get spatial frequency from wave number
And tbh I’m not very sure exactly how the spatial frequency is measured or like what it is
The antennae is emitting a sound wave I get that but what other wave is it emitting to get the spatial, I guess some form of light wave?
prime kraken
when you have a traveling wave of some frequency, it moves through space with some speed
so it has some number of oscillations per meter
that's the spatial frequency
each point in space oscillates with frequency fc if you plot it over time. if you take a chunk of space and use that as the axis instead, the frequency of the wave is different, and depends on the propagation speed
maybe it's easier to see if you look at the wave equation and do separation of variables
then you get that the solution is the product of spatial and temporal harmonics
fleet sail
Ah I see that explains it thanks
wide spear
Do you mind if I ask what your motivation for doing this is?
wide spear
I mean
I could have guessed that
Do Minkowski-orthogonal vectors have special meaning in Minkowski space?
And why are their eigenvalues important?
I see
Are you doing your phd right now
Ah ok
Your question inspired me and I might try to do a generalized QR
For arbitrary bilinear forms
I still don't think GS is the right way to do it
But yes I did investigate lorentz transforms
fleet sail
ok this is basics of kernel density estimation
my question is how does p_h(x) sum up to 1
like why do you need h^D and what are you summing/integrating over in the p_h(x)
frosty abyss
Is there a standard way to measure how well a search space has been explored?
prime kraken
any recommendations on how to solve dense systems of equations Ax = b with positive semidefinite A?
something less expensive than conjugate gradient
wide spear
Cholesky I guess
Do you know anything else about your problem
A isn't sparse so iterative is probably not the right solution
Are you solving this many different times with a fixed A?
Or does A change each time?
If A is fixed then you'll want to precompute the Cholesky factors and then store them so you just do back/forward substitution each time
prime kraken
there are several problems of this kind, say some 18 to 200 of them
wide spear
If A is changing then there are ways to compute rank-n updates to your Cholesky factors
prime kraken
in each one of the problems, one such A matrix is used to solve some ~20 systems
i know A is I - G diag{h}, where h is sparse but random
wide spear
Do you know anything about G
prime kraken
G is symmetric, but multiplying by h destroys the structure, since it basically sets several columns to 0
other than that, you'll recognize that if the problem is Ax = b, it's a discrete fredholm equation of the 2nd kind
but taking A^TA x = A^Tb makes it into your generic least squares problem with pos semi def A^TA
wide spear
Oh no this is the bad way to do least squares
prime kraken
that's the one i wrote above, letting A = A^TA and b = A^T b
wide spear
Computing A^TA is both inefficient and ruins the condition number
prime kraken
yep
i'm open to any suggestions 😛
the A^TA approach indeed takes a long time with CG
wide spear
If you want to do least squares, consider GMRES or QR
fleet sail
wait if ur solving Ax = b exactly with A PSD, then why do you need A^TA
prime kraken
it's only PSD after A^TA, i wrote a dumb substitution
fleet sail
ic
prime kraken
i'll try some QR then
limpid comet
How is a deterministic turing machine (DTM) supposed to simulate a non-deterministic turing machine (NTM) when the search space is infinite? For example, an NTM trying to find a string with a specific property by first generating a random string and by the virtue of getting lucky seeing if it is the correct string. When there is a solution the DTM eventually finds the string, but when there is no solution the DTM searches forever while the NTM just gets "unlucky" and correctly rejects the input.
wooden ledge
This isn't the right place but I'm not sure I understand your definition for NTM. DTMs are by definition NTMs with the transition function restricted to have only singletons as images. Also the usage of the word random is problematic.
limpid comet
well, it's an intuition. from wikipedia:
How does the NTM "know" which of these actions it should take? There are two ways of looking at it. One is to say that the machine is the "luckiest possible guesser"; it always picks a transition that eventually leads to an accepting state, if there is such a transition.
the intuition is probably the thing failing here and not the math, such a simple question shouldn't be breaking math 😄
wide spear
This is probably better asked in #foundations
Alternatively, the CS server linked in #old-network
limpid comet
okay, I'll repost in #foundations
fleet sail
does image compression SVD just involves doing PCA on the images?
or is there more things involved
wide spear
Image compression via SVD involves performing a partial SVD
So you find the bits that correspond to the largest singular values
And then ignore the rest
fleet sail
and you project to those subspaces right?
like the subspace of leading singular values
wide spear
Well
You don't need to do the projection separately
You have A=UDV^T and U and V^T have the projections
fleet sail
makes sense
thanks
ahh i was pretty naive in trying to extract eigenvalues then project, when probably SVD is much faster
wide spear
Of course, there is the choice of basis
jpeg uses a discrete cosine basis
jpeg2000 uses wavelets
fleet sail
so in those cases you still use SVD or like how does it matter
wide spear
You are performing a SVD type computation with a different choice of basis
fleet sail
so you mean depending on image format the SVD generated will be of different basis since the data itself is different
cuz i was thinking of fixed data then SVD is unique
prime kraken
these all fall under low rank approximation schemes, with different properties
the SVD one is the best one in the frobenius norm sense
the 2D DCT ones usually split images up into smaller chunks
scenic swift
Hey! I have a question in #help-7｜zen1thxyz that's all about inverse Fourier transforms
wide spear
Can you repost the question here
scenic swift
yeah sure
What do I need to do an inverse fourier transform?
I have spectrometry data with all the frequencies + the intensities
I also modelled all the data
wide spear
You have continuous time and continuous frequency data?
scenic swift
I dont have the time data
wide spear
Or like
scenic swift
I modelled it
wide spear
You want something continuous in time and not discrete
@prime kraken question for you
prime kraken
oi
wide spear
Computing IFT
prime kraken
what is this data from btw? because the inverse ft will look terribly alias3d in time domain
and do you NEED it to be continuous?
scenic swift
Spectrometry from burning strontium chloride
i can easily do the idft using python
i wanted to compare the result with my continuous graph
prime kraken
matlab, and so probably octave as well, have the ft and ift functions
in python, sympy probably has ft too
lemme go check
but be aware that this is all pretty arbitrary, and i don't think this type of data has a meaningful ift
scenic swift
i was reasaerching it and it should produce an interferogram?
scenic swift
can it plot the data too?
prime kraken
you could plot with matplotlib after evaluating the functions over some domain
scenic swift
Okay, I think i sorta get it
thank you so much
I'm also going to try compute at least 2 of the integrals by hand
bc i need to do like example calcs
scenic swift
Yeah I might just use functions from a transform table to medel it
and then it'll be easier to compute
thanks dude
prime kraken
aight
fleet sail
So I'm trying to apply linear PCA on mnist data for handwritten digits of 0, 1 as shown below, and wondering how to justify that the eigenfaces seems to not be picking up anything but the data is well split
so the first picture here is the mean on the top left, and the other pitch black are the leading eigenfaces
and here's the distribution of the coords of data on the 2 dimensional subspace, it looks well split
in fact none of the leading 100 components seem to be picking up anything so wondering how they do the separation
prime kraken
what's the size of the space, and how many eigenvectors did you take
subspace techniques only work if you know a good basis for your "signal"
i.e. the "correlation" or inner product between a small number of basis vectors and whatever you wanna represent is so large, that you can throw the rest away
you can run into two problems
you have few faces, meaning your null space is huge
and two: the faces(basis vectors) have a low inner product with what you wanna represent
then most of the info stays in the orthogonal complement and you lose it
it seems like the space where the numbers lie is close to orthogonal to the faces one
fleet sail
im taking 28*28 pixel images fyi aligned as columns, and took the 100 leading eigenfaces, basically examining all of them and seeing black void
prime kraken
,w 28*28
pine jettyBOT
prime kraken
how many faces do you have
ah, you said 100
yeah so
you wanna do something like
Ax = y
where y is your vectorized image
x is the sparse representation, and A is a matrix with atoms as columns (and they're orthonormal thanks to PCA)
your matrix A is size 784 x 100
your null space is huge
and it just so happens none of your leading faces can represent the numbers well
y is not in the image of A
fleet sail
intuitively what do the leading faces show, the location where the data varies most?/
prime kraken
they are the faces that are most different from each other
so the idea is that with a varied selection of faces, you can represent any other face modestly well
and well, the "faces" aren't really faces at all at this point, they're the largest eigen or singular vectors of the correlation matrix
i would suggest you forget about the faces and look at it as an eigendecomposition problem
but anyway yeah, you get some singular vectors that correspond to the largest singular values, so they represent the data set the best
these vectors are not any of the faces you originally had
fleet sail
ok so im still confused about this part, these only represent data set the best when you project on to them
but what do these vectors themself represent I'm still trying to understand
prime kraken
nothing physical in the real world
fleet sail
ic
prime kraken
have you tried plotting them?
the thing is faces are "highly structured"
so you would expect the basis vectors to more or less show this structure
but they aren't faces
they have stuff that represents faces well
which may or may not look like a face
fleet sail
so they are the "directions in which data get distributed the most widely" is all i can say
prime kraken
idk what you mean by widely
fleet sail
like with the most spread along the direction?
prime kraken
i wouldn't call that spread, but yeah
just like the eigenvector of the largest eigenvalue of any matrix
if you happen to be parallel to it, the matrix will scale you up the most
i really don't bother with assigning these things a meaning in the real world
unless you can tell me what it means for two faces to be orthogonal
cuz taking SVDs and EVDs of covariance matrices will inevitably yield orthonormal vectors
mathematically, this just means v^H w = 0
what does that mean for faces?
they are simply dissimilar
fleet sail
thanks, actually could you explain why for high dimensional data its harder to interpret those eigenvalues, I don't quite get it still
like why trying to interpret what part of the image they are picking up becomes more impossible as the dimension increase
prime kraken
the thing is that acquiring the coordinates in this basis is done via W^H v, yeah?
this'll give you the coordinates of v in the basis W
it's sort of like a change of basis
but W is usually rank deficient and does not span all of your desired R^n
so you project v onto each w_i
you get a large component if v and w_i are close to parallel
idk how you'd wanna visualize stuff being parallel in n dimensional space
maybe someone in the linalg channel can give you a better interpretation, but i doubt there is one
i just think of the inner product as a measure of similarity
and because the vectors given by the SVD or EVD of these symmetric matrices are orthonormal, you're guaranteed that if something is represented by one vector, no other vector has it
you could do something as simple as take a face and split it into 4 quadrants
those are 4 valid orthonormal vectors that will perfectly represent the face
fleet sail
thanks for explaining, ill take my time to digest this info cuz I was never good at linalg
prime kraken
unfortunately you're doing some sort of basic signal processing so...
it's all linalg and multivar calc
fleet sail
ye just need some time to familiarize and make it second nature
prime kraken
the stuff just happens to be dealing with pixelated faces, but you're doing some sorta Ax = b. i try to deal with stuff in terms of basic linalg constructions
and the context, i.e. face representation, only gives a structure to A
fleet sail
and because the vectors given by the SVD or EVD of these symmetric matrices are ...
ahh
starting to make sense thanks alot
fleet sail
the stuff just happens to be dealing with pixelated faces, but you're doing some...
i was misleading in my initial question 
I forgot that the imgrid function I used to plot the data normalized the parameter inputs wrt each other and so I didn't normalize the mean picture so it end up being ~200 times larger than W which is normalized, so W just dissapears
so actually what the eigenvectors displayed was more expected
prime kraken
ah, looks better
still, you'd expect some amount of info to be lost if the basis is not the right now
pine jettyBOT
Hawt
hidden minnow
for a Barycentric Interpolation definition is $p(x) = \frac{\sum^{n}{j=0}( \frac{\lambda{j}}{x - x_j} ) f_j }{ \sum^{n}{j=0}(\frac{\lambda{j}}{x-x_j})}$, and the equidistributed nodes to Barycentric Interpolation is $ x_j = -5 + j(10/n), j = 0, ..., n $ for $ n = 4, 8,$ and $12 \newline$.
So how come when I use $n = 4, j = 0,1,2,3,4$ and $x_j = -5, -2.5, 0, 2.5, 5$. Just seeing this I get 5 nodes instead of 4. Where $j =3, x_j = 0$ should be omitted for an equidistributed nodes? <@&681260373478735874>
pine jettyBOT
Hawt
gaunt crag
hidden minnow
what does pong mean?
clear gulch
Hi everyone, this is a question from my textbook (which also includes a solution). I've never before used phasors before, and after a couple web searches I kind of understand the concept behind them, but I still don't get how to apply them to this problem. Can anyone help?
So far, I understand about phasors:
- phasors are like tiny clocks
- the hands of the clock are the arrows representing amplitude (?)
- you can "add the arrows" or something to make linear combinations of sine/cosine functions easy (but the problem is I don't get how to do this part)
Also I'm sorry if this not the right channel; I don't actually know what category phasors are part of because I'm studying this for physics, not maths 👀
wide spear
Well if you don't get an answer here, you can always ask in the physics discord linked in #old-network
prime kraken
phasors are conplex numbers that represent a sinusoidal signal
the idea being that complex arithmetic can be simpler
for example, a simple differential eq involving sines and cosines has a solution that van be written as a sum of cosines or sines
if you pick cosines, you know that cos x is the rral part of exp(ix)
this cosine will carry over all the way, so you ignore it and focus on its complex amplitude
then integration and differentiation become multiplication by constants involving pi and i, and division and addition are straightfprward in polar coords
at the end, you take the result, multiply by exp(ix), and take the real part
you can probably find simple examples based on RLC circuits with alternating current
as for a category for the topic, either complex variables or ODEs
slow turtle
if you take a vector space of nice functions R -> C, say L^2 or whatever, a phasor is an element of this space of the form t \mapsto (Ae^{i\phi})e^{\omega t}. often \omega is just some fixed constant and the only variables are A and phi, and in this case the space of phasors can be identified with just C itself by taking the initial value Ae^{i\phi}
clear gulch
@prime kraken Thanks for the explanation! I see how to do it now
pliant laurel
I was told this question belongs here, so here goes. My friend is in the navy and thought itd be neat to give me a code, that turns into their return date, I've had no luck and was wondering if someone could help me. I have no other details aside from it being the return date and the code being this Vm10YVYxVXlSWGhTYms1WVlrWndZVlJVU2pSVU1WWnlWbTVPVDFKVk5YVlZSbEYzVTNkdlBRbz0K
So far I've tried basic rot and caeser shift. Any thoughts?
The furthest ive gotten is it has an absurd amount of v's to be random.
fleet sail
fleet sail
can someone explain what the kronecker product of temporal and spatial steering vector intuitively mean? so lets say I got [1, e^j2pi, ..., e^j(M-1)2pi] kronecker [1, e^j2pi, ..., e^j(N-1)2pi] where N is number of array elements (uniform) and M is the number of pulse intervals in the coherent processing interval
I know the kronecker product should be a M*Nx1 vector but apparantly it (to a constant factor) corresponds exactly with array response of a single target, if the target is both spatially and temporal correlated, (which I actually dont know what that means)
prime kraken
you sure one of those shouldn't be transposed?
fleet sail
they are both column vectors
so first i need to understand what exactly is an array response, is it just an MNx1 vector which each block of size M is the response for the ith time interval?
prime kraken
pretty much
array response is the same as what you asked last time
steering matrix
you get a response that depends on the relative position of the sensors, and the direction of arrival of a plane wave
for omnidir sensors in some plane, this is just complex exponentials w.r.t. some reference sensor
now imagine instead of 1 plane wave, you receive 2 identical plane waves
fleet sail
yea i get this bit, thought what does correlated mean? that the signal come uniformly in one angle (spatially)? but w/b temporal correlation, what does that mean
prime kraken
the second plane wave will yield an identical response to the first, but if you're still tracking the phase, everything is now multiplied by exp(-i w tau) for a time delay tau between the plane waves
temporally correlated would mean the same pulse shape
a lot of DOA stuff assumes spatial and temporal whiteness, or otherwise filters in such a way that induces this
temporally, this means one transmits "symbols" that modulate the plane waves, and these are chosen in such a wave that their correlation matrix is diagonal
i.e. the symbols transmitted at each time instant are set to be 0-mean and uncorrelated with any symbols before or after it
temporally correlated means you got the same (or very similar) symbols
fleet sail
ok so u mean they make the default correlation matrix in ambient setting diagonal?
prime kraken
yeah
fleet sail
covariance*
prime kraken
so the model is usual [steering mat size M x S] [excitation/symbols size S x 1], written as As or Ax
and the expectation of [A^Tx^T x A] is A^T Rx A, where Rx is the correlation matrix of the transmitted symbols, and it's diagonal
what i called S there is the "number of snapshots" or time samples you have
if 2 symbols in the vector x are correlated, Rx is not diagonal
fleet sail
ok maybe i just send u the paper since they didnt store the data in matrix form at first but a long ass vector, only made it a matrix for "SDP formulation" which im very lost in lol
prime kraken
hmmm wait i think i fudged up the symbols, shoulda been size S x N, with N as the number of snapshots
since you can have S simultaneous waves per snapshot
fleet sail
i remember once i asked u about the atomic norm minimization but i think get it now, i believe its somewhat heuristic since they r trying to promote sparsity using the 1-norm because they know theres not alot of target/jamming signal at each point in time
prime kraken
ah, i was wrong about the temporal component
it really is in time domain
it's a doppler shift
shift in frequency domain -> complex exp in time
fleet sail
shift in frequency domain -> complex exp in time
wait can you elaborate?
prime kraken
that's all, really
say you have a pulse P(f) in the frequency domain
and you shift it to P(f-f_d)
this is the same as a convolution P(f) * delta(f - f_d)
inverse transform into time domain
you get p(t) \cdot exp(i 2 pi f_d t)
if you know your pulse shape, you can work with the phasor alone
fleet sail
ohh
prime kraken
so over space you have a time delay due to sensor spacing and DOA/DOD
fleet sail
tbh i never formally learned changing time <-> frequency domains, but i think i got the idea
prime kraken
over time, you see these exponentials due to shift in frequency (i.e. due to motion and doppler)
fleet sail
or never learned basic signal processing in general
prime kraken
tbh i never formally learned changing time <-> frequency domains, but i think i ...
you're gonna have a bad time with this stuff if you don't brush that up
the whole point of these subspace techniques is that complex exponentials give you nice vandermonde decompositions
so you have to choose the right domain in which everything is complex exponentials
prime kraken
spectral analysis by randolph moses
fleet sail
thanks
prime kraken
the whole point is to turn this stuff into a matrix whose eigenvalues are complex exponentials whose phase gives you what you want
anyway
about the kronecker
say we have one doppler shifted pulse as above
p(t) exp( i 2 pi f_1 t)
this is the amplitude of a plane wave
so when the plane wave hits the array, you observe p(t) exp( i 2 pi f_1 t) exp( -i 2 pi f tau_m) at the mth sensor
for all sensors, this is a vector of length m
and then you consider all pulses, and that gives you the other dimension (i.e. we go over all the f_i, all the doppler shifts. assume each pulse has a different doppler shift)
this is a matrix of size M x N
then you vectorize it
this is the same as taking the kronecker product of two vectors with complex exponentials
might've gotten one or both of the signs wrong in what i wrote above, since i ignored that it's the exp of the sine of an angle
fleet sail
what is p(t)?
prime kraken
some generic pulse shape
in this type of scenario, because arrays are small and waves travel at the speed of light, once assumes the "narrow band assumption" holds
which means p(t) is basically a constant
you can just ignore it
fleet sail
ic
so the different doppler shifts would come from measurements in each antenna across the time intervals
and not in the different antenna given a fixed time interval?
prime kraken
that doesn't seem right
at a given time bin, you have some number of waves, each with their own doppler shift, arriving at the array
so at each time instant, you receive a matrix of data of size M x N, for N waves
and you do this for some number Nt of time bins
fleet sail
wait the paper says
"The space-time snapshot from a CPI datacube is a matrix whose (i,j)th entry corresponds to the data received by the ith antenna element in the jth PRI"
PRI they said is the pulse repetition interval and not the # waves
i think i see what u mean previously though, that if the doppler frequency throughout time is constant and just shifted, then u should be able to represent it using a steering vector?
just like how if the spatial signal arrive in a uniform direction then it can be represented in spatial steering vector
prime kraken
i wouldn't call it a steering vector in that case
looks the same, but it represents something else
not DOA
fleet sail
when can you use the steering vector then 
prime kraken
when it steers
for DOA and DOD
you have a kronecker product of a steering vector and the doppler response or whatever the radar people call it
and the overall thing is the array response
i think array response is generic and applies for everything. steering vectors are for DOA and DOD
it looks the same and does the same, just has a different name lol
fleet sail
thanks for the help
i might ask about SDP formulation later once i get this part.. have no clue its connection to the first minimization problem in II.2
i think thats cuz idk semidefinite programming lol
prime kraken
i'm bad at those too
iirc it means formulating the problem and constraints as a single chonky matrix-vector product
fleet sail
stupid question: why cant jamming signal/attack make its signal temporally correlated as well
so it blends in with the target signals
prime kraken
it can be if you want it to
that doesn't help at all
it would have to be temporally correlated with the signal you care about
fleet sail
ah right
prime kraken
also temporally uncorrelated is related to whiteness in spectral domain