Newest Questions
1,698,195 questions
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Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
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8
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How many vertices does $(\Bbb{T}^2)^{\#g}$ have?
Here is a description in a topology lecture note that I need help.
Since $(\Bbb{T}^2)^{\#g}$ can be represented by a $4g$-gon with edges identified according to the word $a_1b_1a_1^{-1}b_1^{-1}\cdots ...
- 17.4k
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Conditions for a Fourier-windowed quadratic form to define a finite-range or exponentially decaying interaction?
Let $A: \mathbb T^4 \to \mathfrak{su}(2)$ be a smooth gauge potential on the 4-torus.We consider a quadratic functional $Q[A]$ defined via a decomposition using a wavelet-type frame $\{\psi_{j,k}\}$:$$...
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Restrictions of Knight's Tour on Circular Board
Does a closed knight’s tour exist on an n-vertex “circular” chessboard with wrap-around moves?
I’m interested in variants of the knight’s tour, but on a “circular board” rather than a rectangular one. ...
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Changing the order of integration in an iterated integral with a single varible function
I am trying to consider a double integral:
$$
\int_t^\infty \int_s^\infty f(r) dr ds <+\infty
$$
where $f:\mathbb{R} \to \mathbb{R}$ is a smooth function, but NOT a non-negaitive function. And the ...
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If $\varphi(f)$ is riemann integrable for each $\varphi \in E^*$, then is $f$ riemann integrable?
Let $I=[0,1]$, $E$ be a banach space and $f:I \rightarrow E$ be a map.
Suppose that for every continuous functional $\varphi\in E^*$, the map $\varphi(f):I\rightarrow \mathbb{R}$ is riemann integrable....
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a complete theory in a countable language with countably many types but uncountably many countable models
I am trying to find such a theory.
I have a nice example of a complete theory with an $\aleph_0$-saturated countable model. Namely, consider the theory in the graph language with the graph axioms that ...
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The set $\mathcal{S}(R)$ of all radical ideals of a ring $R$ might form a boolean ring with $\circ=$ ideal addition and $\oplus=$ a certain quotient.
Define $S \subset \Bbb{N}$ to be the square-free integers $s$ i.e. such that no $p^2 \mid s$ for any $p \in \Bbb{P}$ a prime number.
Then it is easy to see that $(S, \oplus)$ forms an boolean group ...
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Seeking resources about multiple directed acyclic graphs/topological orderings
I'm modeling knowledge as consisting of two pieces: the content itself, which we can call Nouns (N), and things you can do with that content, which we can call Verbs (V). For each of the collections ...
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Chern class of tangent bundle of hypersurface of CP^n viewed as complex of coherent sheaves
I would like to compute the total Chern class of the tangent bundle to a hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ by viewing the following short exact sequence as a complex of coherent sheaves ...
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37
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Tricks for Computing the Center of a Group
I was doing a homework question about computing the center of a group, and realized everytime I've ever computed the center, I am very explicitly writing down elements and finding restrictions.
I ...
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i need to understand the topic TRIGONOMETRIC RATIO [closed]
i do not understand trigonometry ratio.i tried solving some questions on it but it keeps giving me serious issues and i need some deep explanations on trigonometry ratio.
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Possible arrangements for any n number of distinct cubes
This problem has been bouncing around in my head for years, and I can't seem to make progress. I'll give the rules. Once I get a handle on
Cubes are all uniform in size with an edge length of 1 unit.
...
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Probabilistic behavior of Carmichael numbers
Based on shape of the graph from https://oeis.org/A002997/graph and the list of Carmichael numbers up to 10^16 (the first c. 250,000 Carmichael numbers), it looks like (very crudely) : C(n) ~ n^3.
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Why don’t textbooks mention the 2-dimensional metric completion model of $S^2$?
Most differential geometry and topology books introduce the 2-sphere as the surface
$$
S^2 = \{ x \in \mathbb{R}^3 : \|x\| = 1 \}.
$$
This is fine, but it often leaves the impression that the sphere “...
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isometry group of an orbifold is the quotient of the isometry group of its cover
I have a 2-dimensional orbifold that is the quotient of $\mathbb{R}^2$ under a group of isometries $\Gamma$ generated by $180^\circ$ rotations. I would like to say that $\text{Isom}(\Gamma \backslash ...
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Let $f$ be defined by $f(x) = 1$ for $x = 1/n, n \in \mathbb{N}^*$ and $f(x) = 0$ otherwise. Show that $f$ is Riemann integrable on $[0,1]$. [duplicate]
I am having trouble solving this problem. The first thing to note is that I would like to solve this without using Lebesgue's theorem, since on my exam I will not be able to use that theorem.
...
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71
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Proof of Euler identity using ODE
I am writing an article to prove Euler identity :$e^{i\pi}+1=0$
Here the main part:
Consider the function :$ \mathbb{R} \rightarrow \mathbb{C}, f(x)=e^{ix}$
Differentiating twice,we get : $f''(x)=-f(x)...
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80
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Book Recommendation for Vector Bundles
I am interested in learning more about general vector bundle theory. More specifically, vector bundles of class $C^k$ for $k\in\mathbb{N}$ or $C^\infty$ or real-analytic whose fibers can be given the ...
- 237
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The Intuition behind the nim game and the XOR?
The game of nim is played with two players againts each other ,by removing 1 or many stones from only one pile in each turn from n piles each pile with $n_1,...,n_k$ and a player cannot skip a turn.
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49
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
- 1,987
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28
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Any easier ways to prove an explicit form for a generated $\sigma$-algebra besides transfinite induction?
This question is a follow-up to an answer to a previous question, and motivated by my laziness in not wanting to learn about transfinite induction or how to write proofs using transfinite induction ...
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Decomposition of von Neumann algebra in semi-finite and purely infinite part
I am reading the proof of type decomposition of a von Neumann algebra $M$ in the book 'Lectures on von Neumann algebras' by Stratila and Zsido (Theorem 4.17).
The proof starts with the following claim:...
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149
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Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
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Showing that the trace of the difference of the square of the curvatures of two connections on a vector bundle is exact
Let $E$ be a rank 2 complex vector bundle over a 4-dimensional manifold $X$. (I believe that the argument below does not depend on the rank of $E$ and the dimension of $X$.)
I want to try to show, for ...
- 1,792
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What are applications of inprimitive Dirichlet characters and associated L-functions?
For simplicity, I'm gonna call L-functions associated with primitive Dirichlet characters 'primitive' and same with inprimitive.
GRH (Generalized Riemann Hypothesis) says that all non-trivial zeros of ...
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39
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Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
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Prove $\frac{x^{2^n}-1}{x^{2^m}-1}$ is not a perfect square when $n, m$ have different parity
I am trying to prove the following number theory problem:
Problem:
Let $n, m$ be positive integers with different parity (one is even, the other is odd) and $n > m$. Prove that there is no integer $...
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Number of integer solutions to $\sum_{i=1}^{10} x_i = 100$ with $2 \le x_i \le 21$
I am trying to solve a combinatorial problem involving finding the number of integer solutions to the following equation:
$$ x_1 + x_2 + \dots + x_{10} = 100 $$
Subject to the constraints:
$$ 2 \le ...