Newest Questions
1,689,785 questions
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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
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4
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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
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votes
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answers
14
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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
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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} $ with finite total variation (i.e., $ |\mu|...
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7
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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
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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
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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 ...
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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
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7
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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 ...
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2
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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 ...
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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 ...
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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 = ...
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answers
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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 ...
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answer
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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 ...
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16
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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,((...
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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
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2
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28
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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 ...
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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 $\max_{0\leq t\leq T}\lVert u(t)\rVert\leq C \lVert u\rVert_{W^{1,p}(0,T;X)}$ ...
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9
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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 ...
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answers
17
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What is the logic behind Householder decomposition?
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
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answers
7
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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 ...
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answers
33
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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 $x$ ...
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20
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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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28
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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
$$\displaystyle \langle f, \lambda_r \rangle := \int_{|x| < r} \frac{f(x)-f(0)}{|x|}...
- 3,860
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given the following limit equality, prove the following statement
given the equality
$ \displaystyle \lim_{ x\to 0}\left[ \frac{g(x)-g(0)}{x} \right] = \displaystyle \lim_{ x\to 0} \left( 1 - 2x \right)^{1/x} $
prove that
$ \displaystyle \lim_{ x\to 0} g(x) = g(0) $
...
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16
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Question on Einstein Notation for Distinct Expressions
For $n \in \mathbb{Z}_+$, $\alpha \in \{0, 1\}$, $\langle x_q \rangle_{q=1}^n \in \mathbb{R}^n$ and $f : \mathbb{R} \rightarrow \mathbb{R} $, let us consider the following possible expressions for a ...
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Computing $\pi_{n+1}(S^n)$ with Pontryagin's construction, why can we restrict to positive matrices?
In Modern Geometry - Methods and Applications, by Dubrovin, Fomenko and Novikov, there is a calculation of $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$ using Pontryagin's construction. I use Pontryagin's ...
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Multivariable gauss integral [closed]
I have a question.
I don't understand how the result of the following equation is what's in the RHS:
$$\int_{-\infty}^{\infty}e^{-[\frac{1}{2}(\vec x,A\vec x)+ (\vec b,\vec x)+c]}\Pi_i\frac{dx_i}{\...
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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
answer
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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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1
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85
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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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1
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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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What [is the minimal] differential equation is satisfied by $y = \sum_{n=0}^\infty {x^n}(n!)^{-k}$?
Specific Question
What [is the minimal] differential equation is satisfied by $y = \sum_{n=0}^\infty {x^n}(n!)^{-k}$?
That is, we are hunting for a monic differential operator of minimal order which ...
- 4,220
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Minimal polynomial of an algebraic integer [closed]
The definition I learned for the minimal polynomial is: the minimal polynomial of an algebraic integer $a$ is the monic integer polynomial $f$ with minimal degree such that $f(a)$ = 0.
Could there be ...
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Recurrence relation between $n^{-1}$ and $(n+1)^{-1}$ in GF(p)
In a prime field $GF(p)$ is there a recurrence relation between $n^{-1}$ and $(n+1)^{-1}$?
Both answers to this post explain how to obtain $(n+1)^{-1} \bmod n^2$, which made me wonder whether a ...
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1
answer
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Conditional expectation of a mixture distribution with measure zero event
Suppose $\theta \sim U[0,1]$. If $\theta = x$, you observe a signal $m = x$. If $\theta < t < x$, you also observe the signal $m = x$. How do you compute the conditional expectation of $m$? I ...
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37
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Question about an application of Hopf-Rinow Theorem
Let $\varphi: \Bbb{R}^2 \rightarrow \Bbb{R}^3, (x,y) \mapsto (x,2x,x^2+y)$ and $S= \varphi(\Bbb{R}^2)$. If we consider the Riemannian metric for $S$ induced by the usual product of $\Bbb{R}^3$, prove ...
- 585
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16
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Trees isomorphic [closed]
Let $T_1$ and $T_2$ be trees of order $n$ with degree sequences $D = (d_1, d_2, \ldots, d_n)$ and $X = (x_1, x_2, \ldots, x_n)$, where $d_n \geq \dots \geq d_1$ and $x_n \geq \dots \geq x_1$. We say ...
2
votes
0
answers
20
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Constructing the Hecke Theta Function for a Number Field
I'm working through writing down a proof for the analytic continuation of the Dedekind zeta function of a number field $K$, of degree $d$, signature $(r_{1},r_{2})$ and rank $r_{K} = r_{1}+r_{2}-1$, ...
- 305
1
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0
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22
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Justifying the interchange of limit and integral in a sequence involving a diffeomorphism with non-uniform determinant bounds
Let $f: \mathbb{R}^m \to \mathbb{R}^m$ be a diffeomorphism such that $f(B) \subseteq B$, where $B$ is the closed unit ball in $\mathbb{R}^m$. Suppose that for all $x \in B$, we have $|\det Df(x)| < ...
- 21
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0
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85
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Show that $\sqrt{a}+\sqrt{2\sqrt{a}}+\ldots \leq \frac{3}{2}\sqrt{a}$?
Let $(a_k)_k$ be the real sequence defined recursively by $a_0 = \sqrt{a}$ and $a_{k+1} = \sqrt{2a_k}$ for $k>1$. Show that $$\sum_{k=0}^\infty a_k \leq \frac{3}{2}\sqrt{a}$$
I tried to manipulate ...
- 1,119
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0
answers
33
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Intuition on when and why algebraic manipulations work with operator equations
I have written the following proof for the Euler-Maclaurin formula, taking references from many sources.
Let $f$ be an analytic function which has a Taylor series $f(y)=\sum_{n=0}^\infty \frac{f^{(n)}(...
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-3
votes
1
answer
36
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compute Cantor normal form of $(\omega k )^n$ [closed]
Recall the,Cantor normal form(CNF) in base $\omega$: Every ordinal $\alpha$ can be uniquely written in “base $\omega$” as
$$\alpha=\omega^{\alpha_1}n_1+\omega^{\alpha_2}n_2\cdots\omega^{\alpha_k}n_k,$$...
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1
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0
answers
28
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Existence of a function in L1(G) whose Fourier transform never vanish
I am reading Wiener Tauberian theorem on locally compact abelian group setting and there I have read that:
The closed linear span of the translates of $ f $ is $ L^1(G) $ if and only if $ \hat{f}$ ...
- 11
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0
answers
48
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Uni student needing help, where do I go from here? [closed]
As someone who has a decent grasp on college level algebra already from taking highschool courses, im struggling to see the importance in taking college algebra. How important is it? Does taking it ...
1
vote
1
answer
77
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Choiceless cardinal exponentiation
This question came up during one of the discussions among my friends and I, and we agree that intuitively this looks possible to construct, but we can't come up with a natural poset to force over: one ...
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