Questions tagged [reference-request]

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

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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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54 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 ...
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71 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 ...
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1answer
50 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 ...
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39 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 ...
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85 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 ...
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2answers
166 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 ...
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1answer
90 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 ...
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91 views

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. As far as I got, one of the consequences of this ...
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2answers
129 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 ...
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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 ...
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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 ...
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2answers
511 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 ...
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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 ...
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2answers
838 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 ...
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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 ...
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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 $\Cst$-...
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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 ...
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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 ...
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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)\...
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1answer
43 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 $...
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153 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 $\...
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1answer
108 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 ...
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1answer
172 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 ...
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1answer
99 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'...
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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.
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1answer
124 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. ...
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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 ...
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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 --- ...
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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 ...
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24 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 ...
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105 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 ...
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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 ...
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2answers
397 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 $\lambda_1,\dots,\lambda_n$...
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197 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 ...
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1answer
73 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 ...
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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 ...
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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-...
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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 ...
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1answer
116 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 ...
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118 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 ...
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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 ...
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61 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 ...
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2answers
215 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 ...
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0answers
154 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 ...
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447 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 $f$ ...
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116 views

Is this case of Barnette's Conjecture known?

Context: Barnette's Conjecture is that every bipartite cubic polyhedral graph is Hamiltonian. I have been interested by this problem for a long time, and I recently came up with a result. From my ...
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41 views

Exponential sums involving completely multiplicative functions

It is well-known that exponential sums is used as a tool from the analytic number theory to optimize or to compute asymptotic formulas. My question is the following: Given a completely multiplicative ...
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82 views

Median spaces as retracts of hypercubes

It is known (See e.g. here, Theorem 2.1) that median graphs are retracts of hypercubes. Question: Is it also known that median metric spaces are retract of some $l¹$ product of unit intervals? By ...
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1answer
103 views

Right approximation in certain subcategories

Let $A$ be an Artin algebra and $C$ a subcategory of mod-$A$ that contains all projective modules and is closed under finite direct sums (but not necessarily under direct summands). Let $T:=add(C)$. ...

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