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:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
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How many vertices does $(\Bbb{T}^2)^{\#g}$ have?
Here is a description in a topology lecture note that I need help.
Since $(\Bbb{T}^2)^{\#g}$ can be represented by a $4g$-gon with edges identified according to the word $a_1b_1a_1^{-1}b_1^{-1}\cdots ...
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Conditions for a Fourier-windowed quadratic form to define a finite-range or exponentially decaying interaction?
Let $A: \mathbb T^4 \to \mathfrak{su}(2)$ be a smooth gauge potential on the 4-torus.We consider a quadratic functional $Q[A]$ defined via a decomposition using a wavelet-type frame $\{\psi_{j,k}\}$:$$...
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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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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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