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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
2answers
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
4answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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 $f$ ...
