New answers tagged

3 votes

Closed formula for number of ones in a proper factor tree

A simple generating function, though not a closed formula, for $\gamma(N)$ is given by $\sum_{N\geq 1}\frac{\gamma(N)}{N^s} = \frac{1}{2-\zeta(s)}$, where $\zeta(s)$ is the Riemann zeta function. See ...
Richard Stanley's user avatar
5 votes

Closed formula for number of ones in a proper factor tree

This seems to be Sloane's A002033, namely the number of perfect partitions of $n-1$. Since the OEIS doesn't give any closed formula, there probably isn't one, but it's probably worth checking the ...
Dave Benson's user avatar
  • 9,681
4 votes

Closed formula for number of ones in a proper factor tree

Take the formal product $g(x_1,x_2,\ldots)=\prod_{i\ge 1} (1-x_i)$ and define $$f(x_1,x_2,\ldots) = \frac{g(x_1,x_2,\ldots)}{2g(x_1,x_2,\ldots)-1}.$$ Then $\gamma(\prod_i p_i^{\alpha_i})$ is the ...
Brendan McKay's user avatar
2 votes

Variants of the Bonk-Schramm embedding

Your question (actually, questions) is a bit too vague for my taste. Here is an answer of sorts. If we replace $\mathbb{H}^k$ by a Hadamard manifold with variable (but bounded) curvature, can we ...
Moishe Kohan's user avatar
  • 9,128
0 votes

Countable chain condition in topology

A very nice example of a completely regular ccc non-separable space is the Pixley-Roy hyperspace of the reals. While it's not metrizable (and it can't be, as you already know) it is as close as it ...
Santi Spadaro's user avatar
0 votes

Survey of recent developments of the Gelfand-Kirillov dimension

Complementing Manuel Norman's excelent answer, recently I've found a very nice survey about the Gelfand-Kirillov dimension, from 2015, by Jason Bell, called Growth Functions. This survey discusses ...
jg1896's user avatar
  • 2,525
1 vote
Accepted

On faces of polytopes

Let $A_0\subset A$ be the set where $\ell$ attains a minimum on $A$. It is a face of some dimension $k<d$. If $k=d-1$, we are done. Assume that $k<d-1$. Without loss of generality, $0\in A_0$ ...
Fedor Petrov's user avatar
  • 98.6k
6 votes
Accepted

The origin of a planar graph theorem of Steinitz and Rademacher

According to Frank Lutz's article it's in paragraph 46 of the Steinitz-Rademacher book: "every triangulated 2-sphere can be reduced to the boundary of the tetrahedron by a sequence of edge ...
Carlo Beenakker's user avatar
2 votes

What does it mean that the Hessian is proportional to the metric?

Since the proof that such a function implies the metric is a warped product metric is fairly simple, I include a complete copy below. 1 Start with $\nabla^2 f = \lambda g$. For any vector field it ...
Willie Wong's user avatar
  • 35.4k
0 votes

Abel–Plana formula with fractional offset

I eventually did find a published derivation of the fractional-offset (2) of the Abel-Plana formula, in A Generalized Mode Summation Formula of the Zero-Point Energy in a Cavity by Norio Inui (2003): ...
Carlo Beenakker's user avatar
0 votes

Abel–Plana formula with fractional offset

https://www.semanticscholar.org/paper/The-Euler-Maclaurin-formula-revisited-Elliott/43a86582da7acc57944cee260d1654bfbc5f251c A while ago I was looking at using similar formulas for numerical ...
pdmclean's user avatar
  • 270
8 votes
Accepted

Are “most” bounded derivatives not Riemann integrable?

In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...
Dave L Renfro's user avatar
9 votes
Accepted

A result on symmetric closed monoidal categories

This is a special case of: Let $F\dashv G$ be an adjunction. If there exists an isomorphism $id \cong GF$, then the unit $id \to GF$ is an isomorphism. Indeed, $[-,A]$ is left adjoint to itself as ...
Maxime Ramzi's user avatar
  • 12.5k
2 votes

Injective hulls of metric spaces

Given a metric space $X$ and basepoint $x_0 \in X$, the Kuratowski embedding $$x \mapsto (w \mapsto d(x,w)-d(x_0,w))$$ provides an isometric embedding $\Theta$ of $X$ into the injective space $\ell^\...
AGenevois's user avatar
  • 7,093
4 votes
Accepted

Can we approximate a Hölder pdf by higher-order Hölder pdf's?

No, take any $p$ such that $p(x)=|x|^\alpha$ in some neighbourhood of the origin.
Martin Hairer's user avatar
8 votes

Countable chain condition in topology

Take a look at the table in the back of Steen and Seebach's book. You will find that Example 103 contains a completely regular space that is ccc but not separable: $\mathbb{N}^\lambda$, where $\lambda$...
KP Hart's user avatar
  • 9,530
4 votes
Accepted

Pontryagin product on the homology of cyclic groups

I recommend always looking at the canonical reference: Ken Brown's "Cohomology of Groups". Here Chapter V.5 is literally titled "The Pontraygin product" and then the very next ...
Chris Gerig's user avatar
  • 16.9k
4 votes

Types of generating functions (ordinary, exponential, ???) closed under substitution

Not a definitive answer, but the question is stale, and this could be at least somewhat useful. Note that we can rewrite the coefficient in the composition as $$ \left[\frac{z^n}{a_n}\right] F(G(z)) = ...
Oleksandr  Kulkov's user avatar
6 votes

Countable chain condition in topology

How about the book Chain Conditions in Topology by W. Comfort and S. Negrepontis, Cambridge University Press (1982)? There's a lot of stuff in here. For the third bullet point you can try Bell's A ...
Tyrone's user avatar
  • 4,494
1 vote

Ways to prove the fundamental theorem of algebra

In the recent paper Anton, R.; Mihalache, N.; Vigneron, F., A Short ODE Proof of the Fundamental Theorem of Algebra., Math. Intelligencer (2023). the authors give a proof based on the analysis of the ...
5 votes
Accepted

Perturbation of Wasserstein distance: looking for references

You can find this in Villani's "small book", theorem Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.] I can also recommend looking ...
leo monsaingeon's user avatar
8 votes

Wick rotation for Laplace and wave equations

The transformation to imaginary time is used to relate the Green's function of the Laplacian to the Green's function of the d'Alembertian (therefore relating Laplace equation and heat equation). See ...
Carlo Beenakker's user avatar
1 vote
Accepted

Maximizing a sum minus its maximal summand

It is true. The proof rests on several observations. The first one is that if you want to maximize $\sum_i i\pi_i$ under the condition $\pi_j\le B_i$ with any prescribed $B_i$, then your best bet is ...
fedja's user avatar
  • 58.2k
6 votes

Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces

There's a famous trick attributed to Minty (sometimes also to Browder and Lions) which allows to recover at least a.e. convergence when you're able to pass to the limit through an increasing non-...
Ayman Moussa's user avatar
  • 2,505
3 votes
Accepted

Isolated eigenvalues of a random matrix

I use results from Forrester's article (written for a rank-one perturbation of a matrix from the GOE, but more generally valid for an isotropic random-matrix ensemble). A. Largest eigenvalue Equation (...
Carlo Beenakker's user avatar
5 votes

Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)

A reference for the "pro-unipotent radical" of a parahoric subgroup is given in Moy, Allen; Prasad, Gopal; (1994). "Unrefined minimal K-types for p -adic groups." Inventiones ...
Paul Broussous's user avatar
8 votes
Accepted

Series with the smallest number whose square is divisible by $n$

I couldn't find a reference, but (as noted in the OEIS page) if we have $k = a b^2$ with squarefree $a$ then $a(k) = ab$, so $$\begin{align*} \sum_{k\leq x}\frac1{a(k)} &= \sum_{a b^2 \leq x} \...
Command Master's user avatar
2 votes

Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold

Yes, this is a special case of the first theorem of Section 3.6 of WHITNEY STRATIFIED CHAINS AND COCHAINS.
Connor Malin's user avatar
  • 4,819
2 votes

Finding lectures PDF "Four lectures on simple groups and singularities"

The notes were produced at the University Utrecht in 1980 as part of the series: Communications of the Mathematical Institute Rijksuniversiteit Utrecht Nr 11-(1980). I can ask my (=Utrecht) math ...
Dirk Siersma's user avatar
4 votes
Accepted

Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup

Check Chapter 4, more specifically Chapter 4.2 in Ouhabaz: Analysis of heat equations on domains (2005). ZBL1082.35003. The approach there goes by the form method which I personally perceive to be ...
Hannes's user avatar
  • 2,100
2 votes
Accepted

"On models of elementary elliptic geometry"

In the mentioned article, Schwabhäuser proves that all models of elementary elliptic geometry are isomorphic to elliptic Klein spaces over real closed fields. Actually, the paper only deals with the ...
F Zaldivar's user avatar
  • 1,364
4 votes

Two dimensional perfect sets

The answer to both questions is negative. A subset $S$ of a abelian group $G$ is called a Sidon set if $a,b,c,d\in S$ and $a-b=c-d$ implies $a=b$ or $a=c.$ You can find Sidon sets of any finite ...
Colin McQuillan's user avatar
0 votes

Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations

I only found the result for $p=2$ Herbert Amann, Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations, Theorem 3.1. Since the ...
Panrui Ni's user avatar
2 votes

Last term of repeating continued fraction expansion

This result is classical, but unfortunately many usual textbooks (e.g. Davenport's "The higher arithmetic") don't give a direct refference to the original paper. This theorem belongs to ...
Alexey Ustinov's user avatar
1 vote

When do faithfully semiinjective complexes exist?

Turns out it isn't that hard. We can do better and always find faithfully injective modules (I'm certain this is written down somewhere but I'm a "noob" and (i) didn't already know it and (...
FShrike's user avatar
  • 405
1 vote
Accepted

Reference to a variant of Abel's summation formula

In the book K. Chandrasekharan, Arithmetical functions, Grundlehren math. Wiss. 167, Springer, 1970. (page 22) Abel's summation formula is given in the following form: Let $0 \leqslant \lambda_1 \...
Alexey Ustinov's user avatar
1 vote
Accepted

Some folklore about crystaline rings of differential operators

Proposition 1* does follow from that $\mathcal D_c$ sheafifies. For if $k(X) \cong k(Y)$, then $X$ and $Y$ are birational, so there are affine open subsets $U \subseteq X$, $V \subseteq Y$ such that $...
Joshua Mundinger's user avatar
2 votes
Accepted

A question on biharmonic functions

No. Let $w = (1 - |x|^{2 - n})_+$, with $n > 2$. This is globally Lipschitz. It is harmonic when positive, so therefore also biharmonic. it is subharmonic everywhere (max of two harmonic functions ...
user378654's user avatar
2 votes

Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)

For an overview of the algebraic approach to local class field theory, I highly recommend Chapter 8 of Number Theory 2: Introduction to Class Field Theory by Kazuya Kato, Nobushige Kurokawa, and ...
Hagi's user avatar
  • 39
0 votes

Proof of the Dunford-Pettis theorem in the context of probability spaces

You might have a look at the first volume of Probabilités et Potentiel (translated as Probability and Potentials) by C. Dellacherie and P.-A. Meyer; Dunford-Pettis is discussed in II.25.
John Dawkins's user avatar
  • 1,714
2 votes

Does substitution on named terms correspond to substitution on de Bruijn terms?

See the 2021-present work of Joshua Grosso, who formalised the paper in Coq, correcting some errors in the process (last update 3 weeks ago). However, Grosso wrote: To our knowledge, all of the main ...
David Roberts's user avatar
  • 33.2k
2 votes
Accepted

Matrices and vectors of intervals

$\newcommand\R{\mathbb R}$Any operation you can define on intervals on the real line, you can define (entry-wise) on any arrays of such intervals. For any function $f\colon\R^n\to\R$, you can define ...
Iosif Pinelis's user avatar
4 votes
Accepted

Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I do not have a self-contained reference, but the key is Long, D. D.; Reid, A. W., Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8, No. 4, 443-455 (2001). ZBL0992.57023....
Moishe Kohan's user avatar
  • 9,128
3 votes

Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)

Maybe Section 7 of my History of class-field theory would be helpful. See references [3] and [18] there.
KConrad's user avatar
  • 48.8k
1 vote
Accepted

Reference for article that introduces and motivates different notions of subdifferentials

I found the article: J. Li, A. M. -C. So and W. -K. Ma, "Understanding Notions of Stationarity in Nonsmooth Optimization: A Guided Tour of Various Constructions of Subdifferential for Nonsmooth ...
AatG's user avatar
  • 912
5 votes
Accepted

Uniqueness of the $J$ invariant

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...
David Loeffler's user avatar
9 votes

When are two elliptic curves with zero j invariant isogenous?

The curves $E_B$ and $E_C$ are isogenous over $\mathbb{Q}$ if and only if $C=u^6B$ or $C=-27u^6B$ for some $u\in\mathbb{Q}^\times$. In other words, up to isomorphism there are exactly two curves ...
Jonathan Love's user avatar
2 votes

When are two elliptic curves with zero j invariant isogenous?

Two elliptic curves are isogoneous if and only if they generate isomorphic Galois representation. So after computing the conductor and then the Sturm bound you know how many $a_p$ to look at to ...
Watson Ladd's user avatar
  • 2,399
1 vote

Roadmap to Ergodic Theory

Concerning "landmark papers", Cosma Shalizi has documented their own pathway into ergodic theory, with an eye towards applications in statistical learning theory (but with many side branches)...
Carlo Beenakker's user avatar
1 vote
Accepted

"Potency set" for power set?

The following answer https://hsm.stackexchange.com/a/15912/8094 was offered by kimchi lover, and accepted, at History of Science and Mathematics: "The paper Rieger, L. "On the consistency of ...
Frode Alfson Bjørdal's user avatar

Top 50 recent answers are included