# Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

10,336
questions

**0**

votes

**2**answers

109 views

### Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?

This is a follow-up on the previous question.
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb ...

**1**

vote

**0**answers

37 views

### Reference for the following flow equation

I'm looking for reference on the following partial differential equation. $\partial_tF(t,x) = G(t)((\partial_xF(t,x))^2 + \partial_x^2F(t,x))$, where G(t) is a fixed Schwartz function. If possible, I ...

**4**

votes

**1**answer

180 views

### Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?

Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...

**2**

votes

**0**answers

38 views

### Vertex operator algebras and isomorphism of graded vector spaces

I have two vertex operator algebras and I would like to show that as graded vector spaces, they are isomorphic, rather than as algebras.
The issue is I have not found anything in the literature that ...

**3**

votes

**1**answer

93 views

### reference request: unbounded operators on normed spaces

I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...

**3**

votes

**2**answers

242 views

### Examples of plane algebraic curves

There are many interesting sequences of polynomials which contain
polynomials of arbitrarily high degree, for example classical
orthogonal polynomials. Most of them arise as characteristic polynomials
...

**4**

votes

**0**answers

117 views

### Eigenvalues of structured matrices

Let $A=(a_{i,j})$ be an $n\times n$ matrix with $a_{j,j+1}>0,\; 1\leq j\leq n-1,$ and $a_{j,j-2}>0,\; 3\leq j\leq n$, the rest of the entries are zeros.
Is the following fact known:
All ...

**2**

votes

**1**answer

83 views

### Flatness directions of the operator norm

It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...

**0**

votes

**0**answers

46 views

### Quasi-group classes

There are natural classes of finite groups like cyclic groups, abelian groups, nilpotent groups, solvable groups etc. I means to say that cyclic group class is inside of abelian group class and ...

**3**

votes

**0**answers

87 views

### Nascent formal group law

The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x + a_3 x^2 + ...$ and its formal compositional inverse, perhaps ...

**1**

vote

**2**answers

170 views

### Faithfully flat modules over a group algebra

Suppose we have the following data:
1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.
2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...

**2**

votes

**1**answer

93 views

### References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...

**2**

votes

**0**answers

140 views

+100

### What is rigidity of Hirzebruch, and Witten genera?

I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that ...

**2**

votes

**2**answers

135 views

### For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that
If an automorphic representation on $GL(2)$ is ramified at a ...

**4**

votes

**1**answer

101 views

### A conjecture (or theorem?) on unit vectors in a Euclidean space

I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?).
Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...

**0**

votes

**0**answers

19 views

### Reference request: numerical methods for HJB free boundary problems

Suppose $r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d$ and $ \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2$, and consider an optimal ...

**14**

votes

**2**answers

532 views

### Reference to a conjecture on unit vectors in Euclidean space

I have heard that there exists the following conjecture (if I am not mistaken).
Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...

**3**

votes

**0**answers

48 views

### Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation
$$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$
where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...

**33**

votes

**2**answers

922 views

### Fermat's Last Theorem for integer matrices

Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...

**-1**

votes

**0**answers

59 views

### Asking for reference request to study the proof of a result which is used in atleast 4 papers to prove existance of irrational odd zeta values

I am studying a research paper of T. Rivoal and Wadim Zudilin , "a note on odd zeta values " and I am unable to think how a result implies the theorem to be proved. So, I began to read other paper of ...

**5**

votes

**1**answer

85 views

### Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces.
Given two $\Cst$-...

**0**

votes

**0**answers

51 views

### “Chapman-Kolmogorov”-convolution vs. smoothness

Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...

**4**

votes

**0**answers

77 views

### Scalar curvature in terms of second fundamental form, reference request

I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...

**6**

votes

**1**answer

135 views

### “Sub-logarithmic” zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let:
$\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$),
$\beta_n$ the largest real zero of $L(s,\chi_n)$,
$\delta_n := (1-\beta_n)\...

**0**

votes

**1**answer

45 views

### Monotonicity of $\mathbf{P} ( \bar{X}_N > 0 )$ in $N$

Let $X$ be a real-valued random variable with positive expectation (wlog, $\mathbf{E}[X] = 1$, say).
For $N \in \mathbf{N}$, let $X_1, \cdots, X_N$ be independent, identically-distributed copies of $...

**4**

votes

**0**answers

156 views

### analytic approximations of the min and max operators

Question:
What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here.
For any $\...

**3**

votes

**1**answer

109 views

### Reference for Dedekind's problem

Dedekind's problem is about enumerating antichains in the Boolean lattice.
Is there an explicit reference where Dedekind stated this problem?
Is there a good motivation to study this problem except ...

**4**

votes

**1**answer

173 views

### Invariants of symmetric forms with respect to the symplectic group

Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...

**3**

votes

**1**answer

100 views

### References for systems of elliptic PDEs

I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...

**0**

votes

**0**answers

108 views

### Neukirch's theorem on absolute Galois groups in English [duplicate]

Is there a paper or book available in English that proves the result of Neukirch on absolute Galois groups of number fields? I'm having a hell of a time with the German originals.

**3**

votes

**1**answer

142 views

### Finite models for torsion-free lattices

Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...

**2**

votes

**1**answer

84 views

### Explicit construction of Kakeya sets using Perron tree

I have found many excellent notes online that illustrate how to construct a Kakeya needle set (with measure $<\varepsilon$.) Yet none of them gives full argument about the construction of a Kakeya ...

**7**

votes

**0**answers

208 views

### Category of metric spaces

Is there a standard/good reference text that does category of metric spaces?
Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- ...

**11**

votes

**4**answers

830 views

### Book on manifolds from a sheaf-theoretic/locally ringed space PoV

I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...

**0**

votes

**0**answers

26 views

### Relation between the covariance of a random variable taking values in an embedded submanifold and the covariance matrix in the ambient Euclidean space

Let $M^m \subset \mathbb{R}^d, m < d $ be an $m$-dimensional embedded submanifold. Let $X: \Omega \to M^m$ be a manifold valued random variable. Then we've apparently two different notions of ...

**1**

vote

**0**answers

106 views

### Where can I find a table of the exponents of the sporadic groups?

Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ...

**2**

votes

**0**answers

63 views

### Embedding random variables in infinite-dimensional spaces

Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...

**7**

votes

**2**answers

402 views

### Representation theorem for matrices (reference request)

Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...

**6**

votes

**0**answers

198 views

### What is known about “almost orthogonal vectors”?

Motivation:
Suppose we have a kernel $k(a,b)$ defined over the natural numbers.
Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...

**2**

votes

**1**answer

74 views

### Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the ...

**1**

vote

**0**answers

30 views

### Lower bound for the chromatic number in terms of minimum feedback vertex set

Let $MFVS(G)$ denote the size of minimum feedback vertex set of $G$.
We believe we proved $\chi(G) \ge (|G| - MFVS(\overline{G}))/2$
and this bound is sharp.
Is this known or trivial result?
This ...

**5**

votes

**1**answer

163 views

### Classification of $\operatorname{Rep}D(H)$

Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...

**1**

vote

**0**answers

42 views

### Yet another graph characteristic

I wonder if the following graph-theoretical concepts have been considered before, and if so, under which name.
Consider a directed graph $G$ with $n$ nodes.
Let the cycle number $\gamma(\nu)$ be ...

**4**

votes

**1**answer

118 views

### Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...

**3**

votes

**0**answers

119 views

### Accuracy of Richardson's error estimate in the presence of rounding errors

Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $h$. Common ...

**9**

votes

**1**answer

172 views

### Weyl map for $SU(n)$

Let $G= SU(n)$ and let $\mathbb{T}$ be the maximal torus in $G$ given by diagonal matrices. We have
$$
H^*(G,\mathbb{Q}) \cong \Lambda_{\mathbb{Q}}[x_3, x_5, \dots, x_{2n-1}] \ .
$$
Now consider the ...

**0**

votes

**0**answers

62 views

### Neat expresion for an anti-symmetric matrix

Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write
$$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$
for ...

**6**

votes

**2**answers

216 views

### Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...

**7**

votes

**0**answers

155 views

### Kan extensions inside a monoidal category

Every monoidal category $(\mathcal{C},\otimes)$ can be seen as a one-object bicategory: the morphisms are the objects of $\mathcal{C}$, and the $2$-morphisms are the morphisms of $\mathcal{C}$. In ...

**5**

votes

**0**answers

449 views

### Spectral norm bound on smooth primary matrix function perturbation

Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...