Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,580
questions
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Is the following weak version of second Hardy-Littlewood conjecture already known?
Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...
5
votes
1
answer
424
views
consecutive prime gaps and explicit bound
I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
5
votes
1
answer
456
views
higher order analogues of sylvester's law of inertia?
Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
5
votes
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answer
374
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Variants of reflection principle
This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.
Let us state the first ...
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2
answers
774
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Was there ever proposed a theory where the value of Dirac Delta at zero had meaning on itself?
Was there ever proposed a theory where $\delta(0)$ has a meaningful value or used in a formal way outside integrals?
Particularly, following Fourier transforms, we can formally obtain
$$\pi\delta(0)=...
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433
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Nijmegen 1978 $p$-adic analysis proceedings
Anyone knows if there is a chance of getting a copy of the following:
Proceedings of the Conference on p-adic Analysis.
Held in Nijmegen, January 16–20, 1978. Report, 7806. Katholieke Universiteit, ...
5
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2
answers
474
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Wielandt automorphism tower theorem
I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939).
The theorem states the following:
Given a ...
5
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answer
412
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Primality test for $2p+1$
In 1750 Euler stated following theorem :
Let $p \equiv 3 \pmod 4$ be prime then $2p+1$ is prime iff $2p+1 \mid 2^p-1$ .
In 1775 Lagrange gave a proof of the theorem .
Recently I have formulated ...
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3
answers
413
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Exact bin packing the harmonic series: references?
Given $n$ and $B_n= \lceil H_n \rceil$, where the latter is the $n$th harmonic number $\sum^n_{i=1} 1/i$, for most $n$ it is easy to pack the first $n$ terms of the harmonic series into $B_n$ many ...
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A question on Cheeger-Gromov compactness theorem
The Cheeger-Gromov compactness theorem says the following. Let us fix $n\in \mathbb{N}$ and positive constants $K,D,v$. Let $\{(M_i^n,g_i)\}$ be a sequence of closed infinitely smooth $n$-dimensional ...
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Attaching an ideal whose square is zero: does this operation have a name and a notation?
I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define ...
5
votes
2
answers
535
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3-sphere bundles over 4-sphere bound smooth disc bundles
I saw in the answer of this post:
Is it true that all sphere bundles are boundaries of disk bundles?
that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is ...
5
votes
1
answer
304
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Coefficients of Ehrhart polynomials, in the binomial-coefficient basis
Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding ...
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Dynamics of the distribution of prime factorization types in increasing intervals
I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...
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703
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$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$
Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
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The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle
What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?
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Plucker embedding and tautological/universal quotient bundle
Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
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874
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solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
5
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2
answers
674
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Ricci curvature under rough convergence
From the work of Lott--Villani and Sturm, I know that the following fact holds:
(*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curvature ...
5
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2
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356
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Genus of Tutte-Coxeter Graph
What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...
5
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1
answer
409
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Perfect set property implies $\omega_1$ is a limit cardinal in $L$
Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...
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Line bundle ample iff induced morphism finite, looking for reference
Let $X$ be a proper scheme over a field $\Bbbk$. Let $\mathscr L\in\mathrm{Pic}(X)$ be a globally generated line bundle. If for some choice of global sections $V\subseteq\mathscr L(X)$, the induced ...
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912
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Methods to approximate the betweenness centrality on large networks
To calculate the between centrality wiki def:
$g(v) = \sum_{s\neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$
of a node in a graph/network;$\sigma_{st}$ is the ...
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434
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Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces
Is there any good reference for the Pontrjagin ring structure on
$$
H_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)?
$$
I am familiar with Serre's theorem ...
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455
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Heat Equation on $[0,T] \times \mathbb{R}^n$
I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.
Do you ...
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2
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782
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On Weil's characters of type (A)
In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite ...
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2
answers
692
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Examples of tilting objects that don't come from exceptional sequences
This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...
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658
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comparing Hodge structures on cohomology of conjugate varieties
What can one say about the relation between the Hodge decompositions
of $H^\*(X,C)$ and $H^{*}(X_\sigma,C)$
for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism
of the ...
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Why is this theorem attributed to J.-P. Serre?
Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.
$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}...
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Turing machines and Ising model
I have currently started a new research line aiming to prove a mapping between a 2-symbol Turing machine and the one dimensional Ising model. The connection is seen by recognizing that a set of ...
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Algebraic De Rham cup product versus Betti cup product
Let $X$ be a smooth projective variety over $\mathbf{C}$ of complex dimension $2n$. Let
$C_1,C_2\subseteq X$ be two closed subvarieties of complex dimension $n$.
Then we get two Betti homology ...
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434
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Pushforwards of stacks of algebras?
This is a refined/sheafified version of this previos question of mine.
Let $(X,\mathcal{O}_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}_X$ is a ...
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Borel–Weil theorem - reference request
I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
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Reference request: an algebraic stack whose closed points have no automorphisms is an algebraic space
The question is stated in the title. I think BCnrd states in a comment here
Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
that while the answer is not found in Laumon & ...
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reference request: John Baez on (-1)- and (-2)-categories and properties+structure+stuff
I vaguely recall reading a long time ago a 50-or-so page paper, either by John Baez or linked from his page (I think the former), which among other things gave a justification for his table of n-...
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Relationship between the focal locus and the cut locus
I am seeking
clarification of
the relationship between the
focal locus
and the
cut locus
of a curve $C$ in $\mathbb{R}^2$, and
of a surface $S$ in $\mathbb{R}^3$.
Essentially my question is,
Under ...
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Every real-valued continuous function on a closed set of compact Hausdorff space has an extension.
I've noted, that the following fact can be proven in a few lines using $C^*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
Suppose $X$ ...
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3
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Poincaré duality for smooth projective varieties over finite fields
What is exacly the statement of Poincaré duality for smooth projective varieties over finite fields and twisted constant $\mathbf{Z}_\ell$ sheaves? Where can I find a proof?
By twisted constant $\...
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1
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435
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Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
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336
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Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
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2
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450
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Burnside ring and zeroth G-equivariant stem for finite G
Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
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336
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Number of $k$-tuples of elements generating a cyclic group
Let $k$, $m$ be natural numbers, and $C_m:=\mathbb{Z}/ m \mathbb{Z}$ be the cyclic group of order $m$.
Let $N_{k, \, m}$ be the cardinality of the following set: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
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1
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What is the state of progress on this problem about continuous functions from spheres to Euclidean space?
In the 1954 paper Continuous Functions From Spheres to Euclidean Spaces, author Chung-Tao Yang cites the following problem:
Problem 1: Given a (continuous) map $f$ of an $(m+n-2)$-sphere $S^{m+n-2}$ ...
5
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1
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389
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Lower tail of random rank one sums?
Let $\{x_i\}_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$;
assume that $\|x_i\|_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x_1 \otimes x_1]$.
Define
$$
\Sigma_n =...
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205
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Terminology question: "core limit"
I am looking for the correct terminology and a reference to the following construction.
Let $F:\mathcal C\to\mathcal D$ be a functor. Consider "cones" $(G,\alpha)$ where $G:\mathcal C\to\...
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Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian ...
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463
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Reference for Calderon-Zygmund $L^p$ inequalities on the sphere
The following question is motivated from Chapter 2 (Generalized Hodge Systems in 2D), particularly Section 2.3 ($L^p$ theory for Hodge systems in 2D) of Christodoulou and Klainerman's book, The global ...
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387
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Making a submanifold transverse to a vector field by an isotopy
Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
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1
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166
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How strong is separation + reflection of unbounded quantifiers?
Consider a set theory with the following axioms:
separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
reflection: $\phi \to \exists u \phi^u$
...
5
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1
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432
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Looking for proof of Serre's mass formula
Let $F$ be a finite field extension of the $p$-adic numbers $\mathbb{Q}_p$, whose residue field has $q$ elements. Let $\mathfrak{p}$ be the prime ideal of $F$. Given a finite field extension $K/F$, ...