Newest Questions
1,698,193 questions
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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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missing use of pair-wise coprimality in proving $P(x)\equiv 0 \pmod n$ is also $0 \pmod {n_k}$ factors of $n$
Exercise (3.4).10 from Number Theory Step by Step (Singh) is
Show that if a polynomial $P (x)$ with integer coefficients satisfies $P (x) ≡ 0 \pmod n$ where $n = n_1 × n_2 × \ldots × n_r$ and $n_1, ...
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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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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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