I'd like to learn mathematics at a (semi-?)professional level.
I'm a studying Undergrad Medicine in South Asia. I'm looking to build up on my understanding of mathematics from scratch. This is to help me better comprehend the Physics and Chemistry that makes up much of Medicine (besides, I've t...
Lets say I have a group $G$ acting on $\mathbb{R}^2$ so that the orbit space $G \backslash \mathbb{R}^2$ is compact. Clearly, I can find an element $g \in G$ that acts nontrivially on $\mathbb{R}^2$. But can I also find an element that acts nontrivially in each component?
I have vector space of consisting of these "tentfunctions" $$ e_i(x) = \begin{cases} \frac{x-x_{i-1}}{h}, & x \in [x_{i-1},x_i] \\ \frac{x_{i+1}-x}{h}, & x \in [x_i,x_{i+1}] \\ 0 & \text{otherwise} \end{cases}. $$ where $V_d = span(e_1,\dots,e_n)$
Then I have function $f(x)=-x^2+1$, then I need to project this to subspace $V_d$
Using tent functions as basis vectors
Example using two tent functions as basis $$ \text{proj}_{V_d}=\frac{ \langle f , e_1 \rangle }{\langle e_1, e_1 \rangle} e_1 + \frac{ \langle f, e_2 \rangle }{\langle e_2 , e_2 \rangle}e_2 $$
I need to somehow compute these inner products but Im not sure how the inner product is defined in this case?
The excercise one before this was show that $\langle f , g \rangle := \int_a^b f(x)g(x) dx$ is inner product
yeah, I think the projection is basically $\sum_{i=1}^n\frac{\langle f, e_i\rangle}{\langle e_i, e_i\rangle}e_i$ because all tent functions are pairwise orthogonal in the above definition
Hello. I am pretty much confused with the concept of geometric interpretation of gradient, whether it points to normal of the surface and to the direction of steepest accent, throughout the internet I am getting both view?
Actually, In today class, my math prof told me that gradient points to the normal of the surface, but I had read earlier thing like gradient descent, which assumes that gradient points to the direction of steepest accent, how this descrepancy can be resolved. I told my prof that about this but he seems to not even heard of this idea of gradient pointing to the direction of steepest accent. What to do?
The level surfaces are for the function f(x,y,z), the gradients are always perpendicular to the level surfaces in the same manner as it is perpendicular to the level curves for a function f(x,y)
He is perfectly fine with the idea that $ \nabla f = f_x \hat i + f_y \hat j $ , but for this case If I say what is the interpretation, he replied that there are no interpretation and you cannot plot that. Now If I bring $z$ into the picture like $ z = f(x,y) $ and
say him to compute the gradient he do like that : $z - f(x,y) = g(x,y,z)$ and then compute the gradient, like $ f_x \hat i + f_y \hat k + f_z \hat k $ and if I ask him to prove this, he show me the equation of tangent plane, where indeed the normal vector is that gradient
@Tuki So Can you please give me the conclusion in one message?
Well, what your professor did there is expressing z=f(x,y) as a function of three variables, thus pulling the 2D function into 3D. The normal vectors thus show the gradient of that 3D function
in particular, you should find, other than the z component (which should become -1 or 1, I forgot which) of the normal vector for the surface 0=f(x,y) is identical to the gradient you calculate in the 2D case
and these normal vectors are indeed pointing to the direction of steepest descent of each of these functions, as one can check using the proof I linked earlier
To see this geometrically, find two level surfaces, say 0=g(x,y,z) and 1=g(x,y,z). The normal vector at 0=g(x,y,z) tells you which direction you should travel to 1=g(x,y,z) in order to maximise the change in the function value g(x,y,z)
The normal vector at 0=g(x,y,z) tells you which direction you should travel to 1=g(x,y,z) in order to maximise the change in the function value g(x,y,z)
2) What is difference between plot of 2D function and the plot of level surface like $f(x,y,z) = c$
But the basic idea is the same. You try to maximise a scalar function. A scalar function is maximum at the stationary points, thus its derivative there is zero. Compute and then expand the algebra, you recover the relation that the gradient is normal to the tangent vectors hence the level surfaces, thus proving what I said about the level surfaces
Hello, I'm sure at least one of you is familiar with the Vandermonde determinant/matrix
I'm looking at something that's almost the same and I'm trying to figure out what my options in terms of writing something as a determinant
What I have is the expression $\prod_1^N \prod_{j\neq i} (\lambda_i - \lambda_j+ic)$
The expansion for low $N$ appears to be related to the symmetric polynomials, but I have two indices instead of 1 and so far my research has not been conclusive as to whether this can be written as a determinant
The dream is that this can be written as a single $\operatorname{det}$, but I don't think it's possible :-(
The context allows me to go to $((\lambda_i - \lambda_j)^2+c^2)$, so if I could write this as a det instead that would also be nice
I've also skimmed through the Krattenthaler techniques to write it as a determinant but to no avail
Wait I'm stupid, 2.10 from Krattenthaler gives precisely the answer
I'm trying to find the possible transformations of an extended object in physics, so all transformations of a submanifold to a submanifold of the same topology
But I'm not 100% sure what the reasonable restrictions on such things should be
Can I transform the path of a point-particle into an extendible curve???
I mean in the case of point particles, I guess the expectation is that the transformations on a point particle as an embedding in spacetime should have the same set as the transformation as a point particle as an $\mathbb{R}^3$ bundle over $\mathbb{R}$
shouldnt hte sapce of embeddings be infinite dimensional? do you want "automorphisms of an embedding $X$" or "automorphisms of the space of embeddings"?
How could you possibly transform one into the other? One is a closed set and the other is not
The most natural geometric construction that produces automorphisms of the space of embeddings is pre- and post-composing with diffeomorphisms; that is, Diff(M) acts on Emb(M, N) on the right and Diff(N) acts on the left, $$(g \cdot i \cdot f)(m) = g(i(f(m))$$
@Slereah It's hard to say what I'd suggest, to a large degree you'd be better off trying to do the math here if you had some background in diffeomorphism groups --- your question is something I'd hope you could resolve by thinking about the circle or surfaces
GL_2(Z) acts on the torus by diffeomorphisms (act on R^2 and descend to R^2/Z^2), and the action of A in GL_2(Z) on pi_1 = Z^2 is what you think it is
In other words, there's a diffeomorphism taking some homotopy classes to others, but not all --- the null curve cannot be taken by a diffeomorphism to (1,0)
Nor can (1,0) be taken by a diffeomorphism to (2,0)
Hi. What does PSD mean in the context of a spectogram? I'm using a library that computes a spectogram for me, but it has several "modes", default one being PSD. From what I understand these modes refer to some sort of functions that modify the result from the short-time fourier transform before being returned.
Other modes are "complex", "magnitude", "angle", and "phase". Does anyone know what PSD means and could explain it in a simple manner? Ultimately, all I'd like to know is which mode I should use in order to extract note data from audio files.
@Tuki: A polynomial won't tell you anything. You need to know the kernel (nullspace) of the projection is orthogonal to the image (the subspace onto which you're projecting).
So we've talked about this before a few times. If you have an orthogonal basis for your subspace, you project onto the subspace by taking the sum of the individual projections onto those basis vectors.
You use the inner product to compute these projections, of course.
I have no knowledge. Is your adviser suggesting you try to give a 10- or 15-minute talk? You can write an abstract on work in progress without having a publication.
We have a interval $[a,b]$ which is divided into $h$-length segments, which we have total of $n-1$ and which "border points" form set $ \{x_1,\dots,x_n\}$
so $x_i$:s are the points which divide interval $[a,b]$ into these length-h segments
There are a lot of things which are defined which make this a big mess when I'm trying ask something
It would be great to just show the whole paper but unfortunately it's in different language
When these are plotted, it should look something like this
@user193319 the Lie algebra of $SO(n)$ is the skew-symmetric matrices ($A+A^T=0 \implies e^A (e^A)^T = I$) with trace zero ($\det(e^A) = e^{\operatorname{tr}(A)}$) (but trace zero is redundant), so the dimension is $\frac12n(n-1)$
yet the dimension of $\Bbb{RP}^n$ is, you've guessed it, $n$
@LeakyNun I'm not sure I follow. What is the relevance of looking at the Lie Algebras? $\Bbb{RP}^n$ isn't a Lie group, so it doesn't have an associated Lie algebra.
Is there any simple argument to show that $SO(n)$ is path-connected? I'm trying to prove this show that I can show that $\det^{-1}(0,\infty)$ is path-connected.
I'd like to learn mathematics at a (semi-?)professional level.
I'm a studying Undergrad Medicine in South Asia. I'm looking to build up on my understanding of mathematics from scratch. This is to help me better comprehend the Physics and Chemistry that makes up much of Medicine (besides, I've t...
