All Questions
23,894 questions
4
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1
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686
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Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 188k
7
votes
1
answer
237
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What is a Whitney Jet?
I'm currently reading Michor, Manifold of Mappings for Continuum Mechanics. In this paper he makes use of 'Whitney Jets' but takes it to be an already understood concept. I'm familiar with jets but ...
- 39.1k
16
votes
2
answers
737
views
Original reference for categories of presheaves as free cocompletions of small categories
It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A ...
- 10.7k
3
votes
1
answer
228
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Is compact-open topology stable with respect to injective limits?
Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
- 35.5k
2
votes
1
answer
98
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Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open
Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open?
I could not find any specific example for ...
10
votes
1
answer
521
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About Friedrichs historical contribution to QFT cited in Reed and Simon
In the Reed and Simon book, Appendix X.7, they mention that Friedrichs provided the first examples of inequivalent representations of the canonical commutation relations via the Weyl relations in the ...
- 188k
0
votes
1
answer
289
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Ellipsoid in $L^p([0,1],\lambda)$ spaces?
Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...
CommunityBot
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0
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116
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Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?
Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$.
According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
- 501
1
vote
0
answers
92
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The existence of such homomorphism [closed]
Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial ...
- 61
2
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0
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118
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What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
4
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1
answer
187
views
Topology on the space of compactly supported functions
Let $X$ be a locally compact Hausdorff topological space, and let $C_c(X)$ be the space of compactly supported $\mathbf{R}$-valued continuous functions. It is a well-known fact that this space is not ...
- 16.4k
7
votes
1
answer
432
views
Strict toposes as a finite limit theory
For some motivation I have been wondering about generalizing the topos of coalgebras theorem to relative monads in my previous question. This brought me to wonder about topos objects.
NLab on ...
- 10.7k
3
votes
1
answer
162
views
Counting equal covering sets
Definition. We call a set $C$ of sets to be an equal covering set of $S$ if the elements of $C$ are all the same size and each element of $S$ is contained an equal number of times throughout the sets ...
- 34.3k
4
votes
0
answers
177
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Recording of 2009 lecture on Harvey Friedman's work
On December 13--20 2009 at Bristol, there was a meeting devoted to thorough dissection of Harvey Friedman's work on the foundations of mathematics and his statements claimed to be equivalent to ...
- 869
3
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1
answer
182
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Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
- 2,385
2
votes
0
answers
177
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Eigenspaces of complex conjugation on étale cohomology of a smooth projective curve
Let $X$ denote a smooth projective curve defined over $\mathbb{Z}[1/N]$, and its base change $ \overline{X} $ to $ \overline{\mathbb{Q}} $. Let $ V $ be a $ p $-adic local system on $X$ ($p\mid N$), ...
5
votes
2
answers
266
views
Formal languages with non-unique interpretations of terms
In mathematical logic and model theory, one considers interpretations of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free ...
- 1,481
6
votes
1
answer
273
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Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
- 6,962
11
votes
3
answers
727
views
Application of Lie group analysis of PDE (beyond calculation of exact solutions)
I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide ...
- 113
2
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1
answer
182
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Reference request: uniformization theorem proof by Borel
This answer refers to a proof of the uniformization theorem via the PDE describing metrics of constant curvature (Liouville?) by Borel. I haven’t been able to find this reference, is anyone aware ...
- 869
3
votes
1
answer
140
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Generalizations of a theorem of Edmonds/Tutte on existence of a perfect matching in a graphs
It is well known that for a bipartite graph $G$ with bi-adjacency matrix $A$, then $\det A \neq 0$ (as a polynomial) iff $G$ has a perfect matching (there is a similar result for general graphs with ...
- 19k
34
votes
3
answers
5k
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A trigonometric equation: how hard could it be?
The following problem started out with a formulation in terms of complex numbers: let $\epsilon=e^{\frac{\pi i}3}$ and $z=e^{\frac{2\pi i}{3(2n-1)}}$. It's rather amusing that the following appears to ...
- 114k
4
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0
answers
123
views
proper : (proper + $\omega^\omega$-bounding) = generic : x
If $P$ is a forcing notion, $A \subseteq P$ (usually an antichain), $q\in P$, then I write $A\cap q$ for the set of all conditions in $A$ which are compatible with $q$.
For a proper forcing notion $P$,...
CommunityBot
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9
votes
1
answer
246
views
Is the standard model structure on reduced simplicial sets cofibrantly generated?
Let $\mathrm{sSet}_0$ be the category of simplicial sets with a single zero cell, also known as reduced simplicial sets. It is a well known fact (due to Quillen) that $\mathrm{sSet}_0$ supports a ...
- 37.8k
2
votes
0
answers
49
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Deformed preprojective algebras of generalized Dynkin type
Question 1:Is it true that the selfinjective (finite dimensional over an algebraically closed field K) algebras $A$ such that the stable module category of $A$ is 2-Calabi-Yau are exactly the deformed ...
- 26.5k
10
votes
1
answer
516
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Earliest proof of Solovay's theorem for successor cardinals
Solovay's partition theorem states that a stationary set over a regular cardinal $\kappa$ can be partitioned into $\kappa$-many disjoint stationary sets. The full theorem was proven by Solovay in 1971 ...
- 17.7k
0
votes
0
answers
73
views
Operator globally hypoelliptic
An operateor $T$ is globally hypoelliptic if :
$u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$.
My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$.
where $\...
- 531
13
votes
0
answers
174
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Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
2
votes
2
answers
2k
views
Composition of circle inversions
I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$
that results by iterating inversion in a unit circle.
Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle
centered on $q_1$,...
- 4,713
7
votes
2
answers
178
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Separating domains in $\mathbb{R}^{2n}$ by a real algebraic variety
Suppose $\Omega_1$ and $\Omega_2$ are two disjoint unbounded domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be ...
- 171
1
vote
0
answers
176
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Unitary operators in $\mathcal{U}(L^2(X,\mu))$ not coming from $\operatorname{Aut}(X,\mu)$
$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see ...
- 12.9k
6
votes
0
answers
111
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Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
6
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1
answer
274
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Subrepresentations of the $\text{SL}_n(k)$-representation $\mathfrak{gl}_n(k)$
$\DeclareMathOperator\SL{SL} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\sl}{\mathfrak{sl}}$One of my graduate students asked me for a reference for the following fact. Let $k$ be a general field (...
CommunityBot
- 1
2
votes
1
answer
111
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Structure of elements of a finite group not contained in any conjugate of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...
- 37.4k
6
votes
0
answers
145
views
Alternative proofs of the countable chain condition in forcing
Advance warning: This question is more about history and pedagogy than "hard" mathematics.
I am studying Cohen forcing with the forcing poset $(\operatorname{Fin}(E,2),\supseteq,0)$, and I ...
- 35.5k
1
vote
1
answer
269
views
Existence of a global solution to a differential inclusion that does not blow up
Let $\dot{x}(t) \in F(x(t))$ be a differential inclusion, with $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ an uppersemicontinuous, convex and compact valued set-valued map.
On Wikipedia it is ...
CommunityBot
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4
votes
0
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183
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Étale- or fppf-crystalline sites
I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
- 371
5
votes
2
answers
375
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Non-orientable real algebraic three-dimensional manifolds
Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-...
- 28.2k
5
votes
0
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180
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A variant of Schwarz's theorem on generators of smooth $G$-invariant functions
Let $V$ be a finite dimensional euclidean space and let $G\subset O(V)$ be a finite (or compact) group, let $\mathbb{R}[V]$ be the algebra of polynomial functions on $V$. If $E\subset \mathcal{C}^\...
- 7,817
4
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0
answers
820
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Calderón's complex interpolation: what is the corresponding classical theorem?
This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...
- 63.9k
3
votes
0
answers
161
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Lebesgue measure of the boundary of the positivity set of a function is zero?
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $C=\{w>0\}$;
$w$ is subharmonic ...
-2
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1
answer
241
views
Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
- 5,654
1
vote
0
answers
54
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Estimating the growth rate around singular points of the analytic continuation of functions of Nilsson class defined by an integral
In Lemma 8 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) the exponent $\alpha$ in the asymptotic expansion of a function of ...
- 360
2
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1
answer
215
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Forming real positive semidefinite matrices from complex matrices
I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices.
Let $Q \in \...
1
vote
4
answers
423
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Natural examples of functions $f$ in $L^1([0,1])$ such that any function $g$ in the class $[f]$ is discontinuous everywhere
I need many motivated examples of functions $f$ in $L^1([0,1])$
such that any function $g$ in the equivalence class $[f]$ is discontinuous everywhere.
Thank you in advance!
- 29.8k
5
votes
0
answers
174
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Effective Hecke Equidistribution
In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...
- 105k
2
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0
answers
71
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Complemented subspaces of $\mathcal s$
Crossposted from Math Stack Exchange
It is well known that a nuclear Fréchet space $X$ is isomorphic to a complemented subspace of $\mathcal s$ (the space of rapidly decreasing sequences) if and only ...
- 655
2
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0
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137
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Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
- 6,367
33
votes
24
answers
7k
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Early two-author math papers
The middle of the twentieth-century featured several famous papers with two authors. For example, Eilenberg and Mac Lane's papers introducing categories and Eilenberg-MacLane spaces appeared in 1945. ...
- 2,185
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0
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50
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On an Atiyah-Singer-Patodi like construction of the spectral flow
Let $\hat{\mathfrak{F}}$ be the space of selfadjoint Fredholm operators on a separable infinite-dimensional complex Hilbert space $H$, and let $\hat{\mathfrak{F}}_0\subset\hat{\mathfrak{F}}$ consist ...
