All Questions
153,438
questions
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108
views
Constant in the estimate on the Green's function of the Laplacian
Given the Laplacian associated to a Riemannian manifold $(M^n, g)$, there is a Green's function $G(p,q): M \times M \to \mathbb{R}$ that satisfies an inequality of the form
$$|G(p,q)| \leq Ad(p,q)^{2-...
- 1,591
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votes
0
answers
237
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Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$
I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial.
The results was ...
- 671
5
votes
0
answers
201
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Quantum cup product and Dolbeault cohomology
Let $X$ be a smooth projective variety over $\mathbb{C}$. We consider the small quantum cup product $\star$ on the deRham cohomology ring $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...
- 461
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457
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Factorizations as a product of primes minus one
Let $x$ be a positive rational number. I am interested in factorizing $x$ as a product of primes minus one. In fact, I would also like make sure the primes in the decomposition are distinct, and I ...
- 591
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230
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Does Dijkgraaf-Witten theory have a time-reversal symmetry?
By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...
- 2,901
5
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0
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175
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
5
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0
answers
245
views
Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?
By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\...
user20948
5
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0
answers
90
views
Bound on the sum of projective and injective dimension
Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category.
In proposition 1.2. of https://link.springer.com/article/10....
- 26.2k
5
votes
0
answers
143
views
Are there torsion-free restricted simple Lie algebras?
It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the ...
- 137
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295
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How do these topological results imply the inverse function theorem?
In this MO question, Terrence Tao inquires about the everywhere differentiable inverse function theorem. This answer claims the theorem may be deduced from fairly intricate topological results of ...
- 10.3k
5
votes
0
answers
296
views
Polish groups with no small subgroups
Definitions.
A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space.
A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
- 933
5
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answers
140
views
Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?
Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
user130124
5
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0
answers
236
views
Polish transversals
A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$.
So a continuum has a composant transversal precisely when ...
- 3,055
5
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answers
223
views
The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula
I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...
5
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answers
89
views
Schur norm of weighted Cauchy matrix
The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value.
Let $a_1,\ldots, ...
- 81
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455
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Partitioning $\mathbb{R}^n$ into closed sets
Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected.
Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
- 45.5k
5
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440
views
Over what fields does the Mordell conjecture (Faltings's theorem) hold?
Inspired by this question, over what fields is the Mordel conjecture known to be true?
For instance, is it true over fields of finite type (that is, fields finitely generated over their prime ...
- 7,646
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327
views
Historically, how were Grothendieck topoi motivated?
The question is about how did the person who invented Grothendieck topoi (presumably Grothendieck) arrive at the necessity of a such a notion. I do not know much about the history of the subject. What ...
user141410
5
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0
answers
77
views
Polarization type of the complement abelian subvariety
Assume that $P$ is a Prym variety of a ramified double cover (hence not principally polarized). Let $A,B\subset P$ a complementary pair. Assume that the type of the polarization of $A$ is given by $\...
- 1,871
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answers
307
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Points of continuity of Kullback-Leibler divergence with respect to weak convergence
I know that the Kullback-Leibler
$D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$
over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
- 51
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162
views
Is there a representation theoretic way to define the pullback of densities and differential forms?
I find it convenient to define the bundle of densities of weight $\alpha$,say $\Omega_\alpha(M)$ over a smooth manifold $M$ as the associated vector bundle of the frame bundle $F(M)$ with the ...
- 151
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251
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On the definition of the Reeb foliation
To define the Reeb foliation on the sphere, one needs to fix two even functions of the from $f:(-1,1)\to\mathbb{R}$.
In the book I. Tamura, Topology of Foliations: An Introduction, the following is ...
- 51
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answers
281
views
Symmetry of the distribution of prime gaps
Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\...
- 6,932
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answers
122
views
Stable equivalence and stable Auslander algebras
Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
- 26.2k
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132
views
Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)
$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
- 513
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174
views
Homotopy of rigid analytic spaces
Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of ...
user140765
5
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0
answers
334
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,385
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answers
542
views
Concentration inequality for max component of a multivariate Gaussian in the general case
I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
- 271
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answers
155
views
Tiling rectangles using all squares of sides 1, 2, 3, ..., n
Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle?
- 3,677
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165
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Is there a prime $q$ smaller than a given prime $p>5$ such that the inverse of $q$ modulo $p$ is an integer square?
Let $p$ be a prime. For each $k=1,\ldots,p-1$ there is a unique $\bar k\in\{1,\ldots,p-1\}$ with $k\bar k\equiv1\pmod p$, and we call $\bar k$ the inverse of $k$ modulo $p$. In 2014 I investigated ...
- 14.5k
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365
views
Is there a 100...001 prime? [closed]
n=101 is prime, with only 1 zero among the two ones.
How many zeros do we need to add to n in order to become prime again?
1001 ...
- 187
5
votes
1
answer
239
views
Non-uniform property of sequences
Let us say a sequence $(x_n)_{n=1}^\infty$ in some Banach space $X$ has $S_C$ if there exist $k_1<k_2<\ldots$ such that for any $t\in \mathbb{N}$ and scalars $(a_n)_{n=1}^t$, $$\|\sum_{n=1}^t ...
user78375
5
votes
0
answers
51
views
Metric structures making the cohomology into a module over a Lie algebra
The cohomology of a closed Kaehler manifold is an $\mathfrak{sl}_2$-module. I think Verbitsky has shown that the cohomology of a closed hyperkaehler manifold is an $\mathfrak{so}_5$-module. For what ...
- 51
5
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131
views
Finite generation of the image of the induced homomorphism on homotopy groups of infinite loops spaces
Let $f:X\rightarrow Y$ be a map of infinite loop spaces such that image of homology groups $H_i(X,\mathbb{Z})$ for $i\geq 1$ under $f_*$ are finitely generated. Does this imply that the image of ...
- 5,851
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0
answers
156
views
Multivariable multilinear homogeneous polynomials with co-prime coefficients representing 1
Suppose $F(x_1,\dots,x_n)$ is a homogeneous polynomial in $n$ variables of degree $m$, which has degree $1$ in each of the variables. Suppose further that it has integer relatively prime coefficients. ...
- 255
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0
answers
210
views
Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
- 245
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272
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Is an algebraic number satisfying certain super-congruences a root of unity?
Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...
- 171
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202
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Is unitary group paracompact?
In this paper Martin Schottenloher notices that the unitary group $U(H)$ of a separable Hilbert space $H$ is metrizable in the strong operator topology. As a corollary (see R.Engelking, 5.1.3), it is ...
- 7,274
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answers
121
views
Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace
Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$.
Is there a characterization of the set of projections of $f$...
- 1,245
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0
answers
96
views
Kac-Moody groups for non-crystallographic root systems
Given a finite-dimensional crystallographic root system, we can construct an associated Kac-Moody group, with a corresponding flag variety and Littlewood-Richardson coefficients. Between a pair of ...
- 2,038
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0
answers
111
views
Is there a homogeneous compactum where non-empty $G_\delta$s have non-empty interior?
A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$.
Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$...
- 4,343
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answers
81
views
Minimize number of lattice paths below a given path
Every north-east lattice path (NE-path) $v$ from $(0,0)$ to $(k, a)$ can be identified with a sequence $0 \le \lambda_1 \le \lambda_2 \le . . . \le \lambda_k\le a$, that represent the hight of each ...
- 51
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329
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A compendium of weak factorization systems on $sSet$
A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying
Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...
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217
views
Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in the $n$-line of the $E_2$ of the ASS of the sphere?
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element $...
- 7,549
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votes
0
answers
75
views
Consequences of Ramsey-numbers of hypergraphs
We know that the (2-color) Ramsey-numbers for $3$-uniform hypergraphs are between roughly $2^{n^2}$ and $2^{2^n}$, and the situation is similar to $k$-uniform hypergraphs for every $k\ge 3$. (A recent ...
- 18.4k
5
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187
views
When is the graph of a morphism between smooth projective varieties nef?
Let $f: X \rightarrow Y$ be a morphism between smooth projective varieties over an algebraic closed field. The graph of $f$ namely $\Gamma_f$ is a cycle inside $X \times Y$. When is $\Gamma_f$ nef? ...
- 6,158
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245
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Legendrian surgery and invertible elements in zeroth degree symplectic cohomology
Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...
- 3,157
5
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229
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The bridge index and crookedness of a knot
I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much ...
- 492
5
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0
answers
479
views
Schemes admitting a cover by isomorphic affine opens
Let $X$ be a Noetherian integral scheme. When does $X$ have a cover by affine open subschemes that are isomorphic as abstract schemes?
Does there exist an example when we have a cover by $n$ ...
user140076
5
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0
answers
128
views
Algebraization of holomorphic functions of two variables
Let $f: \mathbb C ^2\to \mathbb C$ be (a germ at $0$ of) a holomorphic function. Does there exist a small neighborhood $U_0\in \mathbb{C}^2$ of $0$ and a holomorphic change of coordinates $g$ (i.e. ...
- 2,147