Newest Questions
1,698,182 questions
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Prove that $4\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}dx=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}$
I'd like to prove that
$$4\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}dx=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}$$
Ok, someone said that this holds, but I tried really hard to prove this, ...
- 79
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Term for bivariate homogeneous tensor
This is a homogeneous coordinate matrix:
$
\begin{bmatrix}
x_0 & y_0 & 1 \\
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
...
\end{bmatrix}
$
Is there standard nomenclature for a bivariate ...
- 216
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Is the topological type of the polygons with the following words orientable
For the word $a b c a^{-1}d e b d^{-1}c e^{-1}$, I think the surface of this word is not orientable because some edge appears normally and reversed like $a, ,d, e$, but some does not appear reversed ...
- 17.4k
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How to depict a positive normal vector at boundary point of an oriented manifold
Let $ X $ be an oriented manifold with boundary whose local model is the half space $ \mathbb H^n = \{x\in \mathbb R^n : x^n\geqq 0\} $.
Given a boundary point $ x\in \partial X $, let's say that a ...
- 1,015
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9
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Showing order $p^3$ groups are regular for odd $p$
As the title indicates, I'm trying to prove that, when $o(G) = p^3$ for $p$ an odd prime, $G$ must be regular, i.e. for any $a, b \in G$, $(ab)^p = a^pb^pc^p$ for some $c \in \langle a, b \rangle^1$. ...
- 41
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What is the probability that a bus will skip the first stop?
Suppose that there are $k$ stops and $n$ passengers on the bus, including yourself. You are going to get off at the second stop and want to know what the probability is that nobody on the bus needs to ...
- 3,403
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Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
- 1
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Conditions for a Fourier-windowed quadratic form to define a finite-range or exponentially decaying interaction?
Let $A: \mathbb T^4 \to \mathfrak{su}(2)$ be a smooth gauge potential on the 4-torus.We consider a quadratic functional $Q[A]$ defined via a decomposition using a wavelet-type frame $\{\psi_{j,k}\}$:$$...
- 3,228
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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
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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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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....
- 177
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a complete theory in a countable language with countably many types but uncountably many countable models [duplicate]
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 ...
- 22.7k
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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
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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 ...
