Newest Questions
1,689,788 questions
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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}
...
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$1/|x|$ 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
$$\displaystyle \langle f, \lambda_r \rangle := \int_{|x| < r} \frac{f(x)-f(0)}{|x|}...
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given the following limit equality, prove the following statement
given the equality
$ \displaystyle \lim_{ x\to 0}\left[ \frac{g(x)-g(0)}{x} \right] = \displaystyle \lim_{ x\to 0} \left( 1 - 2x \right)^{1/x} $
prove that
$ \displaystyle \lim_{ x\to 0} g(x) = g(0) $
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Question on Einstein Notation for Distinct Expressions
For $n \in \mathbb{Z}_+$, $\alpha \in \{0, 1\}$, $\langle x_q \rangle_{q=1}^n \in \mathbb{R}^n$ and $f : \mathbb{R} \rightarrow \mathbb{R} $, let us consider the following possible expressions for a ...
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Computing $\pi_{n+1}(S^n)$ with Pontryagin's construction, why can we restrict to positive matrices?
In Modern Geometry - Methods and Applications, by Dubrovin, Fomenko and Novikov, there is a calculation of $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$ using Pontryagin's construction. I use Pontryagin's ...
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Multivariable gauss integral
I have a question.
I don't understand how the result of the following equation is what's in the RHS:
$$\int_{-\infty}^{\infty}e^{-[\frac{1}{2}(\vec x,A\vec x)+ (\vec b,\vec x)+c]}\Pi_i\frac{dx_i}{\...
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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:...
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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
$${\left\Vert {\phi}\right\Vert }_\infty \le {\left\Vert {f}\right\Vert }_1 $$...
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What is (-1)^(³⁄₂)? (Fractional powers question) [closed]
In school, I was taught that exponents are commutative in regards to each other (EX: $(4^3)^7=(4^7)^3$ or $(\sqrt{2})^3=\sqrt{(2^3)}$), but $$\sqrt{(-1)^3}=\sqrt{-1}=i\neq\sqrt{-1}^3=i^3=-i$$ There ...
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Deforming Contour in Method Of Steepest Descent
I am trying to apply the method of steepest descent to approximate the integral
$${I = \int_{\mathcal{C}} \frac{z^{\alpha +n}}{(z-x)^{n+1}}e^{-nz}dz}$$
for large $n$ where $x>0$ and $\alpha > -1$...
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1
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Given 2 disjoint triangles in $\mathbb{R}^2$, there does not exist a matching that produces 3 non crossing lines
I will preface the problem by saying this is an intermediate step in a proof I have that is incredibly straightforward. This claim intuitively holds quite trivially and it could be proven ...
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Trying to understand $C([0,1])$ is separable. [closed]
I'm trying to understand this proof A proof that $C[0,1]$ is separable but I can't understand why $||f-g||_{\infty}\leq \epsilon$. I know that in the comments they were discussing about this, but I ...
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Optimization problem invloving inner and outer argmin [closed]
Can we use \arg\min_{y} { \arg\min_{x} F(x) } ? is this a valid optimization objective function?
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Examples of connected graphs that are hard to break (at least 3-connected) and have the following property
So I was trying to find examples of connected, simple graphs $G$ that have the following two properties:
$G$ is at least 3-connected (you need to remove at least $3$ vertices to disconnect it)
There ...
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Try to prove $e\pi$ is irrational
(My motivation for this try is from the idea in Irrationality of $\pi\ln(2)$)
Try to prove $e\pi$ is irrational
Let $e\pi=\frac{a}{b}$ where $a,b \in \mathbb{N}=\{1,2,3,...\}$ and $gcd(a,b)=1$.
Then $...
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