Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,543
questions
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Buchi's conditional proof of the non-existence of finite algorithm to decide solubility of system of diagonal quadratic form equations in integers
I am doing some literature review regarding Buchi's problem. In particular, I am reading the relevant section in this survey paper by Mazur (Questions of Decidability and Undecidability in Number ...
0
votes
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36
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Generalized envelope theorems
I'm looking for references for two generalizations of Danskin/envelope-type theorems for convex optimization. The first is for when the parameters are functions on a space rather than numbers. A ...
5
votes
0
answers
126
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Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
9
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2
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632
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On martingale convergence
Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$.
Is it possible that there ...
4
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1
answer
261
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Examples of Borel probability measures on the Schwartz function space?
Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...
18
votes
1
answer
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Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
5
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answers
230
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Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
3
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1
answer
198
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Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
3
votes
1
answer
254
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Is the Fortissimo space on discrete $\omega_1$ radial?
Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.
A space is radial provided for every subset $A$ and ...
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160
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Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
8
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156
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An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
4
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0
answers
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Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
9
votes
1
answer
542
views
Irrationality of cubic threefold (before Clemens and Griffiths)
I came across this notice, which seems to say Fano proved that a general cubic threefold is irrational back in 1940s. I'm interested in seeing this work, especially a proof without intermediate ...
1
vote
1
answer
174
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The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$
In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
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answers
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Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
4
votes
1
answer
88
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Some questions about the definition of Chern classes in Cheeger--Simons differential characters
In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...
2
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answers
475
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Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
0
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answers
46
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Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...
3
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answers
172
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Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper
At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following:
"In some sense, the operator $\psi$ applied to a power series gives it "better
growth ...
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answers
89
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Weighted least squares regression: Iterative modeling of variance
In chemical analysis, the instrument's signal are plotted as a function of chemical concentration. In general, higher the concentration higher is the response and the relationship is linear. At ...
0
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201
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On characters of the symmetric group: Part 2
This question is related to my earlier MO quest.
For an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\...
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2
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634
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Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
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209
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Finding the DOI for a paper
How can we find the DOI of an old paper. For example, what are the DOI of the following papers?
Anderson, D.D. and Jayaram, C., 1995. Regular lattices. Studia Scientiarum Mathematicarum Hungarica, 30(...
4
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195
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A probabilistic proof of van der Waerden theorem
Is there an elementary proof of van der Waerden's theorem on arithmetic progressions using probabilistic methods?
9
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1
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499
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Reference request: number of antichains of a partially ordered set
Let $\mathbb{N}$ denote the set of all positive integers. For each $n \in \mathbb{N}$, define the set $$ P_n = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : 1 \leq a \leq b \leq n \} $$ and consider the ...
1
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0
answers
86
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Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology
The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...
5
votes
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106
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On Soergel's results concerning projectives modules in category $\mathcal{O}$
I am looking for a translated proof of two results of Soergel usually referred to as endomorphismensatz and struktursatz.
Both of those results were shown in the paper
Soergel, W. (1990). Kategorie 𝒪...
7
votes
2
answers
409
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On the existence of a real which is not set-generic over $L$
Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$.
I know that Jensen's ...
0
votes
1
answer
256
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Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
3
votes
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answers
147
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Defining ideal of a Schubert variety as a kernel
Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$...
11
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1
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686
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Smooth map between oriented manifolds
Let $f: M\rightarrow N$ be a smooth map between smooth closed oriented connected manifolds of same dimension.
Question: is it true that $f$ is smoothly homotopic to some smooth map $g: M\rightarrow N$...
2
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0
answers
325
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On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
2
votes
0
answers
91
views
Are covering families of localizations stable under pushouts?
For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...
5
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1
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515
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Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
17
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1
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1k
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Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
3
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3
answers
365
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Closed formula for number of ones in a proper factor tree
Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...
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0
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Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
5
votes
1
answer
123
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Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
2
votes
0
answers
82
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On dense subspaces of $L^p$-spaces of finitely additive measures
Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
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210
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Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
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131
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When does an iteration not add functions $\eta\to V$ at the final stage?
I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
2
votes
0
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89
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Self adjoint operators from energy functionals
It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
4
votes
0
answers
63
views
Pfaffian elements and anomalies
If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
4
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0
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244
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Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
4
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0
answers
87
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List of equivalent conditions for the invariant subalgebra to be polynomial
Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
9
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238
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Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
7
votes
1
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281
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The origin of a planar graph theorem of Steinitz and Rademacher
The subsequent statements are extracted from the article titled 'Generating r-regular graphs' (https://doi.org/10.1016/S0166-218X(02)00593-0).
A well-known classical theorem of Steinitz and ...
8
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0
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431
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Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
3
votes
2
answers
219
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
10
votes
1
answer
525
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...