Newest Questions
1,698,195 questions
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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?
...
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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.
(...
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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 ...
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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 ...
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3
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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.
$$
\frac{(x+1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}} \quad\...
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True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]
I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges.
I guess it is false but I can't find a ...
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$\beta$ reduction rule for $\Pi$-types on free variables
The $\beta$-reduction rule for $\Pi$-types in dependent type thoery states that $(\lambda x:A.t)(u)=t[u/x]$ (provided $u:A$ in suitable contexts), which makes perfect sense to me.
However, due to some ...
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Discrete Fourier Transform Shift Theorem Help
I'm trying to prove that $\mathcal{\underline{F}}(\tau_p \underline{f}) = \omega^{-p} \mathcal{\underline{F}}(\underline{f})$ version for the discrete Fourier Transform, but I'm getting stuck at the ...
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Can I treat the limit as a constant? [closed]
Imagine that the limit as $h$ approaches infinity of $f(1 + hx)$ is $g(x)$.
$$\lim_{h\to \infty} f(1 + hx) = g(x)$$
Can I then say that $f(x)$ is equal to the limit as $h$ approaches infinity of $g\...
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2
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Is $\varphi = \frac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\frac{1}{13}$?
Is $\varphi = \dfrac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\dfrac{1}{13}$?
My calculator suggests that
$$\ln(\varphi) < -\...
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Basic Proportionality Theorem/ Thales Theorem [closed]
The Basic Proportionality Theorem seems so obvious but the construction to prove it (drooping perpendicular to equate areas) is not at all obvious to me. Can anyone tell how to prove this Theorem in a ...
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fisher LDA variants simple nuance (row vs columns) [closed]
please tell all variants of LDA
one i know is fisher's LDA
like J(w)
why we store data points as columns why not rows
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Order in an open set
I need help to prove a theorem.
In the following definition $\operatorname{fr}_X(A)$ means the boundary of $A$ in $X$
Definition. Let $(X, \tau_X)$ be a topological space, let $p \in X$, and let $\...
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Is my understanding correct about the inverse relationship between derivatives and integrals?
I've learned that derivatives and integrals are inverse operators, but am not completely sure why. I've looked at many resources to understand why, and here goes.
The integral gives a sum of the total ...
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Definition of hyperprojective sets
I've seen two definitions of hyperprojective sets:
sets that are both inductive and co-inductive (cf. p.315 of Moschovakis's book);
sets that belong to the smallest $\sigma$-algebra that contains ...
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