user43208
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1 answers
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78 views
Name for this map of power sets associated with a function?
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3 votes

(Adapted from an email I sent the OP, who was also asking about connections with "the six operations" of Grothendieck.) I guess it might depend which community you're talking to. I believe ...

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1 answers
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125 views
Continuous functions that are differentiable in only one point
3 votes

A much more general result was given by Zahorski: Zygmunt Zahorski, Sur l'ensemble des points de non-dérivabilité d'une fonction continue, Bulletin de la Société Mathématique de France, Volume 74 (...

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1 answers
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83 views
Coherence criteria for tricategories
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3 votes

While Nate and I have been corresponding about this and I was able to answer his questions, I'll post something here as well in case anyone else is interested. To answer question 1: the question is ...

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2 answers
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290 views
A monomorphism of groups which is not universal?
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Do I misunderstand? Take an embedding of a group $i: H \hookrightarrow G$ into a simple group $G$, and suppose we have a nontrivial quotient $q: H \to Q$. The pushout of a quotient $q$ along a map $i$ ...

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1 answers
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133 views
Left adjoint to forgetful functor between varieties of algebras
1 votes

Only just saw this question. I'll give two answers: the first is more abstract but more economizing on conceptual effort, the second is somewhat more concretely tied to Lawvere theories. The ...

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2 answers
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519 views
Are there any free or fascist boolean algebras?
1 votes

I only just saw this question, while hunting for solid evidence that Lawvere ever spoke of "fascist functors" (unable to find anything yet, although Mac Lane in a 1950 paper spoke of "fascist groups", ...

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75 views
Is the natural order relation on an idempotent semiring total/linear?
3 votes

A distributive lattice is an idempotent semiring (with addition $\vee$ and multiplication $\wedge$), but most lattices are not totally ordered.

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1 answers
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2k views
Proof of convergence of Dirichlet's Eta Function
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I think this succumbs to applications of Dirichlet's test and some estimates based on the mean value theorem. Dirichlet's test gives conditional convergence of a sum $\sum_{n \geq 1} a_n b_n$ provided ...

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233 views
Definition of enriched category as lax-monoidal functor
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5 votes

I only just got wind of this question. The map $$\comp \colon d(y,z) \otimes d(x,y) \to d(x,z)$$ is obtained from the family of maps $$\coprod_{y \in X}V(v,d(y,z)) \times V(v',d(x,y)) \to V(v \...

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132 views
A question on the Lawvere theory of vector spaces
1 votes

This answer is to supplement the answer already given by Hurkyl. It's true in this case that the Lawvere theory $L$ is equivalent to the category $\text{Vect}_{fd}$ of f.d. vector spaces (morally ...

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3 answers
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1k views
Noob question about $\int \frac{1}{\sin(x)}dx$
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1 votes

If you are considering antiderivatives as functions on connected domains $D$ that are open in $\mathbb{C}$ -- suitably chosen so that we don't have to worry about multivaluedness of the logarithm -- ...

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2 answers
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705 views
The plastic number, and Padovan and Perrin-like sequences
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5 votes

Just about every sequence $(a_n)_{n \geq 0}$ that satisfies a linear recurrence $a_{n+3} = a_{n+1} + a_n$ has the property that $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$, where $\rho$ is the ...

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2 answers
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635 views
Iterated duals of a vector space
6 votes

As a very partial answer, here are soft categorical arguments that the dimension is $1$ in case $i = 1$ or $j = 2$, or in case $i = 0$ or $j = 1$. The first result shows we can reduce to the cases $i =...

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87 views
Can somebody prove this infinite series?
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0 votes

The same type of method as used to prove that $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$ can be used here, as soon as you write $\frac1{(2n + 1)(2n + 5)} = \frac1{4}(\frac1{2n + 1} - \frac1{2n + 5})$. ...

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1 answers
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461 views
Modulo multiplication of cyclic groups that are divisors of 24
1 votes

If the bounty description means you'd like to know why the "self-inverse" condition (or what I will call the "involutory" condition) forces $m$ to be a divisor of $24$, then that question is easy to ...

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1 answers
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162 views
Moore closure and realizing the Kuratowski monoid
2 votes

Thanks to the link (provided by rschwieb in a comment) to "Kuratowski's Closure-Complement Cornucopia", I was able to locate what I consider a reasonably satisfactory example in Janusz Brzozowski, ...

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167 views
What is the minimum number of rotations about axes in a plane that can describe an arbitrary rotation in 3D?
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1 votes

My answer is that you can. If you know about quaternions, then you know that a rotation about a unit vector $v \in \mathbb{R}^3$ through an angle $\theta$ can be accomplished by regarding elements $...

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1 answers
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115 views
Does $d(A,B) := \sum\limits_{n=0}^{\infty} \frac{1}{2^n}\cdot \frac{d_n(A,B)}{1+d_n(A,B)}$ define a metric $d$?
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4 votes

Lemma: Suppose $d: X \times X \to [0, \infty)$ satisfies the triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in X$. Then so does the function $d': X \times X \to [0, \infty)$ ...

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2 answers
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339 views
What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?
4 votes

In answer to a question that came up in comments under Najib's answer, let me point out that the category of pseudotopological spaces is a locally small cartesian closed category that contains $\text{...

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1 answers
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303 views
core-compact but not locally compact
1 votes

I only saw this question today; it is indeed hard to find examples of this sort of thing. A somewhat elaborate example, essentially the spectrum of the distributive continuous lattice of lower ...

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1 answers
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236 views
$\displaystyle\frac{\langle A x_{\min}, x_{\min}\rangle }{\langle x_{\min}, x_{\min}\rangle }=\lambda_1 \not = \lambda_2$? - Explanations
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3 votes

You seem to be getting twisted in a knot. The actual problem is, as I'm sure you realize, pretty simple: you are trying to minimize $x^\top A x$ subject to the constraint $x^\top x = 1$. For that you ...

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2 answers
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238 views
A discontinuous function with smooth sections
2 votes

What about $f(x, y) = \frac{x y}{x^2 + y^2}$ if $(x, y) \neq (0, 0)$, and $f(0, 0) = 0$? It's clear that $x \mapsto f(x, y)$ is smooth if $y \neq 0$, and of course for $y = 0$ as well since $f(x, 0)$ ...

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189 views
Cantor's Theorem with Posets
1 votes

Brian Scott's proof is perfectly clear, but somewhat removed from the type of diagonalization argument one naturally associates with Cantor's theorem, Russell's paradox, and so on (see this paper by ...

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1 answers
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2k views
Is there a more rigorous way to show these two sums are exactly equal?
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3 votes

Even though one has cause to be a little bit wary around formal rearrangements of conditionally convergent sums (see the Riemann series theorem), it's not very difficult to validate the formal ...

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3 answers
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970 views
Help me prove equivalently of regular semigroup and group.
2 votes

Adding to what Jack Schmidt has written, let me prove that 2 implies 4. If for all $a$ there exists a unique $x$ such that $axa = a$, then the same $x$ satisfies $xax = x$ since $a(xax)a = (axa)xa = ...

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4 answers
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1k views
Patterns of the zeros of the Faulhaber polynomials (modified)
3 votes

As an addendum to Antonio Vargas's answer, let's prove that the roots of $S_p$ are indeed symmetrically distributed around $-1/2$, or in other words that if $r$ is a root, then so is $-1-r$. A ...

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2 answers
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705 views
group of units in a topological ring
9 votes

(I've never offered a bounty before; I hope it's not bad etiquette to post a solution now, even though technically we're in a "grace period". Since there has been no activity, I'm guessing it'll be ...

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4 answers
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184 views
Integral $\int\frac{dx}{x^5+1}$
4 votes

One might as well explain how to derive the factorization in Troy Woo's answer. (This is by hand, not software.) Yes, he's right that any polynomial with real coefficients can be factorized into ...

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3 answers
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6k views
Dimension of Hom(U,V)
1 votes

It's probably worth adding that in case $U$ has infinite dimension $m$, the space $\text{Hom}(U, V)$ has dimension given by $|V|^m$ where $|V|$ is the cardinality of $V$. For if $B$ is a basis of $U$, ...

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4 answers
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3k views
$A$ and $B$ are finite sets. How many Partial Functions exist between them?
3 votes

Another solution is to identify a partial function $f$ from $A$ to $B$ with a total function from $A$ to the disjoint union $B \sqcup \{\ast\}$ (send every element not in the domain of $f$ to $\ast$). ...

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