Newest Questions
1,689,785 questions
0
votes
0
answers
8
views
Confusions about $\mathrm{Spec}(R)$ and algebraic groups
So I am studying about classical algebraic geometry (the one working with affine varieties instead of schemes) and I suppose I'm lost in a number of points. Note that I'm trying to study affine ...
- 1,622
0
votes
0
answers
7
views
Identity involving the polylogarithm and $\pi i/ 6$
Using the polylog multiplication formula, $\sum_{m=0}^{p-1} \operatorname{Li}_s(z e^{2\pi i m/p}) = p^{1-s} \operatorname{Li}_s(z^p)$
it's easy to see that the the real part of
$- \operatorname{Li}_2(...
- 390
2
votes
0
answers
14
views
What is the number of such circular arrangements?
I have been reading "Graph Theory with Applications to Engineering and Computer Science" by Narsingh Deo. And I found this interesting seating problem therein.
Nine members of a new club ...
- 2,311
0
votes
0
answers
13
views
Characterization of real numbers $ c $ for which $ \hat{\mu}(n) = e^{c i n^2}$ holds for a finite Borel measure $ \mu $ on $ \mathbb{T} $
Which real numbers $ c $ does the following hold for?
There exists a complex Borel measure $ \mu $ on the torus $ \mathbb{T} := \mathbb{R} / 2\pi \mathbb{Z} $ such that the Fourier coefficients of $ \...
- 1
0
votes
0
answers
7
views
Question about singular matrix in thin-plate spline interpolation
I'm trying to interpolate a function $f:\mathbb{R}^2\to\mathbb{R}$ given known values $y_i=f(\mathbf{x}_i)$, for $i=1,\dots,m$, to new points $\mathbf{x}$ using thin plate splines as described in this ...
- 960
0
votes
0
answers
18
views
Eigenvalues of $A+cd^T$ without determinants
Suppose that $\{\lambda_1, ..., \lambda_n\}$ are the eigenvalues for
$A_{n\times n}$ and let $(\lambda_k, c)$ be a particular eigenpair.
Let $d$ be a arbitrary vector, how to prove that the ...
- 460
0
votes
0
answers
7
views
If $f(x) = \sum_{b \in B(x)}f_b(x)$ and $B(k) \subset B(k+1)$ while $\sum_{b \in B(k)} f_b(x) \geq \sum_{b \in B(k+1)} f_b(x)$ what is $f^{-1}(A)$?
I'm having real trouble with finding a formula for the inverse image of a sum of functions whose summation bound depends upon the input of the function $x$.
Specifically, let $\{f_i\}$ be a family of ...
- 22.1k
3
votes
4
answers
56
views
Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$
Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$.
I tried substitution $t=e^{x}$, which led to the integral $$\int_{1}^{\infty}\frac{t^{n-1}}{\left(1+t^{...
- 314
0
votes
0
answers
7
views
Eigenvalues of $PA$ with $P$ a projection and $A$ having eigenvalues inside open unit disk
Given a projection matrix $P$ (not necessarily an orthogonal projection), and another matrix $A$ where $\mathrm{eig}(A)$ all have magnitude strictly less than $1$, and both $P,A$ are real-valued. Can ...
- 177
1
vote
2
answers
41
views
Find $O(b)$ given that $a^5=e$ and $ab^{-1}a=b^2$
In a mathematics group chat, somebody asked this question.
If in a Group G, $a^5=e$ and $ab^{-1}a=b^2$ then find O(b)
I believe he made an error, as this problem is similar to Problem 37 on p.49 in ...
- 45
0
votes
0
answers
8
views
does a 0 in the basic solution mean there's positive slack in the corresponding equation [closed]
enter image description here
This question has the solution to Y* as being (0,1), but from my understanding of the Complementary Slackness Theorem, there should be positive slack in the first primal ...
- 11
2
votes
0
answers
14
views
Zeros of $f(z) = z^5 - ae^z$ in $B(0;1)$ for $a<0$
In this question, they wanted to find the zeros of $f(z) = z^n - ae^z$ in $B(0;1)$ for $n\in\mathbb{N}$ and $a>0$. My question is, what about when $a<0$.
The case I am dealing with is when $n = ...
- 81
-3
votes
0
answers
14
views
Show that A is equicontinuos on [0,1]. [closed]
Let $\big(C([0,1]), ||\cdot||_{\infty}\big)$ the metric space of continuos functions on $[0,1]$ taking values in $\mathbb{R}$. Let $M>0$. How can i proof that $A\subset C([0,1])$ is equicontinuos ...
0
votes
1
answer
14
views
If ordinal $\alpha$ isomorphic to subset of ordinal $\beta$ then $\alpha \leq \beta$
If ordinal $\alpha$ isomorphic to subset of ordinal $\beta$ then
$\alpha \leq \beta$
My best attempt so far: Since any two well-ordered sets either two are isomorphic to each other or one is ...
- 578
0
votes
0
answers
17
views
Show that $∫_{X_1(t)}^{X_2(t)}u(t,x)dx=∫_{y_1}^{y_2}u(0,x)dx, (t\in \mathbb R)$.
Let $a\in C^1(\mathbb R)$ be a bounded function and suppose a and its derivative are both bounded. Let $u\in C^1(\mathbb R^2)$ be a solution to the following equation:
$$∂_tu(t,x)+∂_x(a(x)u(t,x))=0,((...
- 1
1
vote
0
answers
33
views
The smallest nonabelian finite simple group. [closed]
Here is the question I am thinking about:
What is the square of the size of the smallest nonabelian finite simple group which is not a subgroup of a permutation group on at most six letters.
Could ...
- 119
0
votes
2
answers
28
views
The order 4 isometry of the tetrahedron
Consider a tetrahedron centered at the origin. For example with the vertices $ (\pm1,0,-1/\sqrt{2}),(0,\pm1,1/\sqrt{2})$. The isometry group of the tetrahedron is $ S_4 $ and the subgroup of ...
- 9,878
1
vote
0
answers
17
views
Evans' PDE: proof theorem 2 section 5.9
The book proves the following $u(t)=u(s)+\int_{s}^{t}u'(\tau)d\tau$ and says this equality implies "easily" that $\max_{0\leq t\leq T}\lVert u(t)\rVert\leq C \lVert u\rVert_{W^{1,p}(0,T;X)}$ ...
- 11
0
votes
0
answers
9
views
Exact Triangle Sorting for Orthographic Rendering of a Triangulated Surface
I want a definitive front-to-back triangle drawing order under orthographic projection.
This is an X-post due to inactivity on the other: https://tex.stackexchange.com/q/735053/319072
I have a ...
- 239
3
votes
0
answers
17
views
What is the logic behind Householder decomposition? [closed]
I’ve been studying the Householder decomposition and have encountered a few conceptual questions I’m hoping to clarify:
Why are reflections used in this method instead of seemingly simpler operations ...
- 31
0
votes
0
answers
7
views
sensitivity of least squares solution subject to unit simplex constraint [closed]
The book by Golub/Van Loan covers the sensitivity of the least squares solution. However, I am interested in bounding $||\beta - \hat{\beta}||$ where both $\beta, \hat{
\beta}$ are nonnegative and sum ...
- 111
-1
votes
0
answers
33
views
How to visualize complex numbers without combining X and Y [closed]
I'm currently in high school, and I have been learning about the symmetrical distribution of roots of polynomials on the complex plane. What has been bothering me is that we were forced to combine $x$ ...
0
votes
0
answers
20
views
Null Space of Infinite Dimensional Matrix
Is there a general way to tell if the null space of an infinite dimensional matrix contains a vector which is not zero?
This question connects to the following problem:
If I know that
\begin{equation}
...
- 1
0
votes
0
answers
28
views
$1/|x|$ interpreted as a distribution and Dirac delta distribution
From Tao's blog:
Let ${d=1}$. For any ${r > 0}$, show that the functional ${\lambda_r}$ defined by the formula
$$\displaystyle \langle f, \lambda_r \rangle := \int_{|x| < r} \frac{f(x)-f(0)}{|x|}...
- 3,860
0
votes
0
answers
21
views
given the following limit equality, prove the following statement
given the equality
$ \displaystyle \lim_{ x\to 0}\left[ \frac{g(x)-g(0)}{x} \right] = \displaystyle \lim_{ x\to 0} \left( 1 - 2x \right)^{1/x} $
prove that
$ \displaystyle \lim_{ x\to 0} g(x) = g(0) $
...
- 241
0
votes
0
answers
16
views
Question on Einstein Notation for Distinct Expressions
For $n \in \mathbb{Z}_+$, $\alpha \in \{0, 1\}$, $\langle x_q \rangle_{q=1}^n \in \mathbb{R}^n$ and $f : \mathbb{R} \rightarrow \mathbb{R} $, let us consider the following possible expressions for a ...
0
votes
0
answers
34
views
Computing $\pi_{n+1}(S^n)$ with Pontryagin's construction, why can we restrict to positive matrices?
In Modern Geometry - Methods and Applications, by Dubrovin, Fomenko and Novikov, there is a calculation of $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$ using Pontryagin's construction. I use Pontryagin's ...
-2
votes
0
answers
30
views
Multivariable gauss integral [closed]
I have a question.
I don't understand how the result of the following equation is what's in the RHS:
$$\int_{-\infty}^{\infty}e^{-[\frac{1}{2}(\vec x,A\vec x)+ (\vec b,\vec x)+c]}\Pi_i\frac{dx_i}{\...
- 337
1
vote
0
answers
20
views
Existence of a harmonic function with prescribed partial derivative in $\mathbb{R}^3$
Question:
Let $\Omega \subseteq \mathbb{R}^3$ be a domain and $h: \Omega \to \mathbb{R}$ a harmonic function (i.e., $\Delta h = 0)$.
Under what conditions does there exist another harmonic function $H:...
- 11
2
votes
0
answers
47
views
An Inequality Conjecture in Harmonic Analysis
If $f(x)$ ($\Bbb{R} \to \Bbb{R}$) is an $L^1$ function and $\phi (t)$ is the Fourier transform, then it is well known that
$${\left\Vert {\phi}\right\Vert }_\infty \le {\left\Vert {f}\right\Vert }_1 $$...
- 21