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:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
- 1
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0
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8
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How many vertices does $(\Bbb{T}^2)^{\#g}$ have?
Here is a description in a topology lecture note that I need help.
Since $(\Bbb{T}^2)^{\#g}$ can be represented by a $4g$-gon with edges identified according to the word $a_1b_1a_1^{-1}b_1^{-1}\cdots ...
- 17.4k
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0
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3
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Conditions for a Fourier-windowed quadratic form to define a finite-range or exponentially decaying interaction?
Let $A: \mathbb T^4 \to \mathfrak{su}(2)$ be a smooth gauge potential on the 4-torus.We consider a quadratic functional $Q[A]$ defined via a decomposition using a wavelet-type frame $\{\psi_{j,k}\}$:$$...
- 3,228
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9
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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. ...
- 1
0
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0
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10
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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 ...
- 27
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11
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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....
- 157
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18
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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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7
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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 ...
- 22.7k
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12
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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 ...
- 1
2
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answers
16
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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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0
answers
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 ...
- 349
-5
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36
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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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1
vote
1
answer
46
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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.
...
- 11
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35
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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.
...
- 21
1
vote
1
answer
81
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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 “...
- 11
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0
answers
15
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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 ...
- 11
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votes
2
answers
60
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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.
...
- 1
0
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0
answers
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)...
- 58
2
votes
1
answer
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
1
vote
0
answers
43
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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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0
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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
1
vote
0
answers
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 ...
- 1,716
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votes
1
answer
27
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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:...
- 795
1
vote
2
answers
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 ...
- 539
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30
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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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0
answers
22
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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
1
vote
0
answers
30
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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 ...
- 13
0
votes
1
answer
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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3
votes
1
answer
73
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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 $...
3
votes
0
answers
96
views
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:
$$ 2 \le ...
-6
votes
0
answers
39
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The Imitation Number Formula in the Collatz Conjecture is it novel [closed]
I have been analysing the Collatz Conjecture and have identified an infinite family of numbers, which I call 'Imitation Numbers' (N), that share an identical initial trajectory structure with a ...
2
votes
0
answers
38
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Area estimate of the intersection of ball and sphere
The following estimate arises in the proof of Tomas-Stein restriction theorem.
$$
\sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1}
$$
The estimate is very intuitive, and I have a ...
- 666
0
votes
1
answer
31
views
Base of the line integration [closed]
Does line integral integrate over the projection of a 3d curve onto the x-y plane or over the 3d curve itself as the base of the integration?
Thanks
0
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0
answers
33
views
Construction of Base Change/Comparison Map $g^*f_*F \to f'_* g'^*F$
Let us consider a cartesian diagram of schemes
$$
\require{AMScd}
\begin{CD}
X'=X \times_S S' @>{g'} >> X \\
@VVf'V @VVfV \\
Y' @>{g}>> Y
\end{CD}
$$
and let $F$ a sheaf on $X$.
...
- 10.1k
1
vote
0
answers
58
views
Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group
Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...
- 181
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0
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92
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Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
3
votes
3
answers
143
views
Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
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2
votes
0
answers
17
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Solution to a Constrained Optimal Control Problem: Is $v(t) = S/T$ Optimal
The problem is stated as:
$\min_{v}\int^T_0f(v(t),t)dt$
Subject to the following constraints:
$s'(t)=v(t)$,
$s(0)=0$,
$s(T)=S$,
$v_{\min}\le v(t),S/T\le v_{\max}$
where:
$T$ and $S$ are given ...
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1
vote
0
answers
106
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How to integrate $\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$ analytically?
I'm trying to solve the integral
$$\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$$
I do know that a similar integral
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
- 13
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votes
0
answers
33
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Find the locus that divides the line joining a point and any arbitrary point on the circumference of a circle in a fixed ratio
Given a point and a circle, find the locus of points that divide the line joining the given point and an arbitrary point on the circumference of the circle in a fixed ratio.
(If A is a point and C(O, ...
0
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0
answers
24
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A surjective homomorphism of $\mathbb R$-groups that is not surjective on $\mathbb R$-points.
$\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
$Consider a non-connected reductive group $G$ over the field $\R$ of real numbers.
Write $S=G^0$ for the identity component of $G$, and ...
- 1,408
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0
answers
22
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Lyapunov Approximation of Separation of Trajectories
I am self-studying dynamical systems, and I came across a property that I am unsure if I correctly identified how it is found. That is,
$$
\phi_t(x+\epsilon) - \phi_t(x) \approx \epsilon e^{t\lambda}
$...
- 1
0
votes
1
answer
50
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Fundamental group of planar topological graphs
I am trying to do the following exercise on homotopy theory:
“Prove that every finite, connected topological graph $\Gamma\subset \mathbb{R}^2$ is homotopically equivalent to the wedge sum (pointed ...
- 871
3
votes
1
answer
85
views
Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra
This problem appears in the book:
Linear Algebra and its applications - David C. Lay - Fourth Edition
It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18
$(4.7), \...
- 5,844
1
vote
0
answers
67
views
Solving second order differential equation graphically
I am stuck on the following problem, hoping that someone will be able to help me.
I have a following second order differential equation:
$$y''=-\sqrt{y}+0.5y'$$
with the following initial conditions: $...
- 71
0
votes
1
answer
81
views
Is there a canonical functor F : 𝒞ᴼᴬ ⟶ 𝒞ᴬᴼ from the usual objects-and-arrows definition of a category to its arrows-only formulation?
Is there a canonical functor
$F : \mathcal{C}^{\mathrm{O\!A}} \longrightarrow \mathcal{C}^{\mathrm{A\!O}}$
from the usual objects-and-arrows definition of a category to its
arrows-only formulation?
...
- 121
0
votes
0
answers
56
views
Example of the $\mathrm{dim}A > \mathrm{depth}A$
In commutative ring theory of Matsumura, Theorem 17.4
" If A is a Cohen-Macaulay ring with $\mathrm{dim}A=n$, the following is equivalent.
(1) A sequence $a_1,\cdots,a_n$ is a regular sequence.
(...
1
vote
0
answers
55
views
Difficulties in explicitly constructing the "cap" bordism
EDIT: I made a mistake and I cross-posted this question also on MathOverflow, where it already has an answer. Please refer to the MathOverflow post and ignore this one.
Freed's notes give the ...
- 1,015
0
votes
0
answers
105
views
Why do vectors behave as derivation on functions
I am learning about differentiation in the context of Manifolds and Tangent Space and am struggling with the idea that a vector operates as a differentiation operation on a function $f$. Here is a ...
- 21
-1
votes
3
answers
126
views
How does $ \frac{(x+1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}}$ become $\sqrt{x} + \frac{1}{\sqrt{x}} $?
I came across this problem on a JEE exam paper. Apparently, the first expression is equivalent to the second expression.
$$
\frac{(x+1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}} \quad\...
- 605