Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,545
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Matroids with no relaxations (~ weak maps)
There's an operation in matroid theory which is called "relaxation".
To keep things simple, let's consider a matroid $M$ with set of bases $\mathcal{B}$. If $M$ has a subset $H$ of $M$ that is both ...
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3
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636
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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 $\lambda_1,\dots,\lambda_n$...
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2
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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. ...
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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 ...
4
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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 ...
7
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1
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365
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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
...
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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 ...
3
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1
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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 ...
3
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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 ...
23
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Existence of subset with given Hausdorff dimension
Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
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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 ...
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2
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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 ...
2
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2
answers
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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 ...
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General Hoeffding inequality with two uncorrelated random vectors
Let $X_1,\ldots, X_n$ be independent sub-Gaussian variables with sub-Gaussian norm $K$. Let $a_1,\ldots,a_n$ be a random vector such that $\mathbb{E}[a_j X_i]=0$ for all $i,j\in \{1,\ldots ,n\}$ and ...
12
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Bi-orderability of Baumslag-Solitar group $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and of $\langle a,b \mid a^{-1} b a^m = b^n\rangle$
We say that a group $(A, \cdot)$ is bi-orderable if there exists a total order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$.
Let $m,n$ be non-zero ...
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2
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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 ...
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Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)
Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
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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 ...
2
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1
answer
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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 ...
4
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0
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Computing the volume of intersection between a ball and a box
$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
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Polylogarithm : reference request for proof of integral representation
On page 494 of the book Integrals and series, volume I : elementary functions, Gordon and Breach, 1986, by A. P. Prudnikov, Y. A. Brychkov, and O.I. Marichev, (perhaps translated from Russian), the ...
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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 ...
2
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0
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415
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An equivariant Hahn embedding theorem?
The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
5
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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 $\Cst$-...
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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 --- ...
4
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0
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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 $\...
33
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Hahn-Banach theorem with convex majorant
At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
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Is there a name for this kind of structure? (Not quite a lattice-ordered group)
I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties:
The partial order is invariant under ...
6
votes
1
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"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)\...
5
votes
2
answers
265
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Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
I am looking at a polynomial of the entries of a matrix, and this polynomial is invariant under permutation of the rows or columns of the matrix. Is there a similar characterization as in the case of ...
0
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0
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“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
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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 ...
2
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2
answers
366
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Why we use Caputo fractional derivative in application?
I'm working on some papers which use Caputo fractional evolution equation (see on Wikipedia) as application for thier main result:
For example:
$$\left\{\begin{matrix}
^CD^{\sigma}_tx(t)+Ax(t)=&...
0
votes
1
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Monotonicity of $\mathbf{P} ( \bar{X}_N > 0 )$ in $N$
Let $X$ be a real-valued random variable with positive expectation (wlog, $\mathbf{E}[X] = 1$, say).
For $N \in \mathbf{N}$, let $X_1, \cdots, X_N$ be independent, identically-distributed copies of $...
3
votes
1
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Reference for Dedekind's problem
Dedekind's problem is about enumerating antichains in the Boolean lattice.
Is there an explicit reference where Dedekind stated this problem?
Is there a good motivation to study this problem except ...
5
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1
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References for systems of elliptic PDEs
I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...
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4
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2k
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Book on manifolds from a sheaf-theoretic/locally ringed space PoV
I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
6
votes
1
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327
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Naive question on local cohomology
Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims:
$$...
2
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463
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Difference quotient for functions of bounded variation
Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
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1
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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-...
31
votes
5
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How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
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0
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81
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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 ...
6
votes
1
answer
284
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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
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0
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449
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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 ...
3
votes
1
answer
2k
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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 ...
7
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0
answers
343
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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 ...
4
votes
0
answers
81
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A Krull-Schmidt theorem for partially ordered groups
If $G$ is a po-group (ie. partially ordered group), we say that $G$ is po-indecomposable if it's not the direct product of two non trivial subgroups (such subgroups are necessary convex and normal).
...
2
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2
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Reference for weak*-semigroup
Let $X$ a dual Banach space (there exists a Banach space $Y$ such that $X=Y'$).
A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T_tx\to x$ in the ...
1
vote
1
answer
101
views
Name of the class of linearly ordered groups with no minimal positive element
Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?
4
votes
1
answer
480
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Strictly totally ordered semigroups - Looking for references
Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is linearly orderable if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < ...