Trending questions
159,064 questions
14
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Who solved the Bring quintic using the Rogers-Ramanujan continued fraction $R(q)$ and how to find all five roots?
I. The octahedral group
Given the nome $q=e^{\pi i \tau}$, then the elliptic lambda function $\lambda(\tau)$ shown below with its Ramanujan–Selberg continued fraction,
\begin{align}
\big(\lambda(\tau)...
- 12.6k
3
votes
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answer
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"Essential values" of a function at a point?
Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, ...
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1
vote
1
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201
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Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
- 13
3
votes
2
answers
190
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Necessary condition for invertible knot concordance from both ends
It is clear that if $K_1$ and $K_2$ are two concordant knots by a concordance that only present ambient isotopic phenomena (no saddles, maxima, or minima) they are invertible concordant from both ends....
- 31
27
votes
3
answers
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The relevance of knowing "just facts"
When you are new to a research area, whether as a PhD student, a young postdoc or even a more experienced researcher, you have to absorb a lot of information. Of course you have to learn the math ...
- 3,535
4
votes
1
answer
320
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Functoriality of infinite suspension spectrum functor on infinity groupoids!
Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\...
- 167
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96
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Action of Adams operations on $K_*(K)$
Any k-theory cooperation $K_*(K)$ can be uniquely expressed as some finite Laurent series $f(u,v)$ with certain generators $u,v$ arising from the left and right action of $\pi_*(K)$. That I understand....
9
votes
1
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257
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Higher or lower? (#2)
$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
- 6,313
4
votes
1
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162
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Topology on $O_M$, the space of slowly increasing smooth functions?
A smooth function on $\mathbb{R}^n$ is called slowly increasing if each of its derivatives is polynomially bounded. It seems that the collection of such functions is denoted as $O_M$.
Obviously, $O_M$ ...
- 3,487
6
votes
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167
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Number of periodic points of subshift of finite type
Let $(X, \sigma)\subset (\{0, 1, 2, 3\}^\mathbb{N},\sigma)$ be a subshift of finite type. Let $P_n$ be the set of $n$-periodic points. If $|P_n|=2^n$ for all $n\ge 1$, then it is true that $(X, \sigma)...
- 675
1
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0
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140
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Estimation of the degree of a projective surface
Let $S \subset \mathbb{P}^{n}_{\mathbb{C}}$ a projective integral smooth surface lying in no hyperplane (I adopt the point of view of scheme). I will denote by $d$ its degree. The following questions ...
- 73
2
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1
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199
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Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
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3
votes
1
answer
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Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
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2
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Why does this theta function value yield such a good Riemann sum approximation?
Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e.,
$$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$
Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...
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2
votes
1
answer
124
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Choice of the eigenbasis for the Dirac operator on $S^d$
This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.
Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with ...
- 3,487
11
votes
3
answers
782
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Is every recursively axiomatizable and consistent theory interpretable in the true arithmetic (TA)?
I am looking for a scholarly text that discusses this issue in detail.
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2
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278
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Extending polynomial hierarchy above $\omega$
The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the ...
- 3,029
8
votes
1
answer
385
views
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalent to AC?
Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?
For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?
Note: ...
- 11.3k
4
votes
0
answers
64
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Why optional stopping theorems require continuity conditions of martingales?
If we want to prove some form of optional stopping theorem (with a stopping time $T$) for continuous time martingales $M_t$, a typical strategy is to assume that $\mathbb E[M_{T\wedge n}] = \mathbb E[...
- 1,755
7
votes
2
answers
399
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Numerical choice and reverse mathematics
Consider the following fragment of numerical choice in the language of second-order arithmetic:
for any arithmetical $\varphi$, we have:
$$
(\forall n\in \mathbb{N})(\exists m\in \mathbb{N})(\forall X\...
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0
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122
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Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,649
2
votes
1
answer
354
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Problem in Uchida's Theorem
I m studying relative class number of CM fields and i found a very interesting theorem from Kόji Uchida's article: class numbers of imaginary abelian fields (1970) but one argument of the ...
- 103
9
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0
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164
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
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13
votes
3
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672
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How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
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10
votes
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337
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Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
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3
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1
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130
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Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
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7
votes
2
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186
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Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 3,317
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
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0
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43
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A distribution defined via an ODE for its Laplace trnsform
Fix a parameter $0 < c < \infty$.
As the solution to a certain problem,
there is a probability density function $f_c(t)$ on $0 < t < \infty$ with mean $1$ and
whose Laplace transform $L(\...
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votes
0
answers
50
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Riemannian metrics realizable as hypersurfaces both in Euclidean and spherical spaces
I am interested in smooth Riemannian metrics on $n$-sphere, $n\geq 3$, which can be imbedded isometrically both to $n+1$-dimensional Euclidean space and $n+1$-dimensional standard sphere of radius $r$....
- 21.8k
1
vote
1
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101
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Is Nelson-Symanzik positivity compatible with fermionic statistics?
Let $\{ S_n \}_{n =0}^\infty$ be a sequence of tempered distributions where $S_n \in \mathcal{S}'(\mathbb{R}^{nd})$ where $d \in \{2,3,4\}$ is fixed. Moreover, we put three additional conditions:
$...
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votes
0
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59
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Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
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3
votes
1
answer
320
views
Is the Hilbert Mumford Criterion true over the reals?
The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says:
Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$.
For a vector $v \...
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5
votes
2
answers
199
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Handle decompositions subordinate to an open cover
Let $M$ be a compact smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover.
We say a handle decomposition of $M$ is subordinate to the open cover if each handle is contained in a $U_i$. Do such ...
- 2,797
2
votes
1
answer
240
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Cohomology of torsion points on elliptic curves
$\DeclareMathOperator\Gal{Gal}$Let $E$ be an elliptic curve defined over a number field $K$. Put $G:=\Gal(\bar{K}/K)$, and for each valuation $v$ of $K$, put $G_v:=\Gal(\bar{K_v}/K_v)$. Consider the ...
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In dimension $n=5$, does a subgroup of $O(n)$ satisfying these properties exist?
I asked a question where @YCor provided a construction that seems to enable a group construction satisfying some properties when $n\ne 5$. However, in the case $n=5$, I am starting to think no such ...
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506
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Group of diffeomorphisms and its tangent space i.e. its Lie algebra
So I feel like there are many questions and also many sources on what I am asking, but I still don't understand what I think is a very basic thing in my head:
It is known, that for a Lie group $G$ (...
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2
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1
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224
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Example of stable bundle whose pullback is polystable
Kempf (1992): "Pulling back bundles" has the following theorem:
Let $f: Y \rightarrow X$ be a finite morphism. If $\mathscr{W}$ is a bundle on $X$ that is stable with respect to an ample ...
- 577
3
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0
answers
101
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Tuple rearrangement: a combinatoric problem emerging from the Hurwitz action on Coxeter groups
I am working on Artin Groups, so called Dual Artin groups and the conjecture that they are isomorphic. Tuples of $n$ group elements can be acted on by the braid group $B_n$ in a particular way called ...
- 31
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51
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Linear decompositions using spanning Bessel sequences that are not frames
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
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101
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Generic objects characterized by objects in the ground model
Kibedi showed, in his doctoral thesis, Maximal saturated linear orders (Lemma 4.3.1), the equivalence of existence between a ccc poset that adds a new $\omega_1$-gap in a linear order in the ground ...
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116
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Universal picard variety of degree d
Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$...
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152
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Discriminants and lattices in Algebraic geometry vs Geometry of numbers
(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual ...
- 473
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84
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Relation between quot scheme of birational curve
I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...
- 613
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5
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Uniqueness results that follow from CH
Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\...
1
vote
1
answer
249
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Higher cohomology of line bundles and small modifications
I am a PhD student in algebraic geometry and I am blocked on a question that I asked to myself for my research. I am not yet very comfortable with the traditional tools. Also, I apologize in advance ...
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405
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Motives and ring stacks
In the lecture “Motives and ring stacks” Peter Scholze begins by saying that there are many ‘Weil-type’ cohomology theories for (separable and finite type) schemes over $\mathbb{Z}$ each of which can ...
- 217
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vote
0
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32
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Surjectivity of MLP functions
Suppose that $\mathcal{F}(H,L)$ is a function class of all functions implemented by a multilayer perceptron with ReLU activation, width $H$ and depth $L$. Also, there is a set of all possible value ...
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51
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Time periodic Euler flows
What are some examples of solutions to the incompressible Euler equation on the torus $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$ (with $d\in \{2,3\}$)
$$\partial_t u+u\cdot \nabla u +\...
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125
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Néron-Tate height on abelian varieties and PDEs
Let $A$ be an abelian variety over some number field $K$. We know the $\mathbb{C}$-points of $A$ form a complex analytic manifold, so $A(\mathbb{C})$ is a smooth manifold in fact. It then makes sense ...
