# Questions tagged [reference-request]

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

10,929
questions

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### Rowmotion for general lattices

Let $L$ be a finite lattice and $x \in L$ with covers $r_1,...,r_l$ in $L$.
One can define $row(x):= \min \{ y | y \leq r_1 \lor \cdots \lor r_l $ and $ y \nleq r_1 \lor \cdots \lor \overline{r_t} \...

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51 views

### Convergence properties of related series

Let $u_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g_n\}_{n \in \mathbb{N}}$, $g_n \to 0$, such that
$$\sum\limits_{n \in \mathbb{N} } g_n = \infty, \ \ \ \ \...

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86 views

### Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?

Suppose that $\mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $\mathbb Z$-linear map $p:M\to M^*$ by
$$p(x)=\sum_{u\in R^*}(u,x)u$$
and proves many ...

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214 views

### Reference request on Gentzen's proof of the consistency of PA

I've always been interested in having a good understanding of Gentzen's proof of the consistency of arithmetic.
Being more precise, he showed that $PRA + WF(\epsilon_0) \vdash Con(PA)$.
I am ...

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49 views

### References on strongly orthogonal martingales and their Walsh and/or Haar expansions. Updated

I am looking for references on strongly orthogonal martingales and their Walsh and/or Haar expansions. In particular I am interested in products of such expansions in general. Since there are ...

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175 views

### When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?

Let $A$ be an $n \times n$ real symmetric matrix.
Let
$$
M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix}
$$
where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...

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2k views

### Why did Robertson and Seymour call their breakthrough result a “red herring”?

One of the major results in graph theory is the graph structure theorem from Robertson and Seymour
https://en.wikipedia.org/wiki/Graph_structure_theorem. It gives a deep and fundamental connection ...

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**1**answer

143 views

### Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.
I want to construct an example of an intersection ...

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47 views

### Pairing up vertices in a graph

Given a connected (undirected) graph with an even number of vertices, consider how many ways are there to pair up vertices so that each pair is connected by an edge. Is there a known classification of ...

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66 views

### Error rate implying regularity

My question is a bit general/vague.
It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...

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332 views

### Reference for graduate-level text or monograph with focus on “the continuum”

I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the ...

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52 views

### Slodowy slice intersecting a given orbit “minimally”?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $X\in\mathfrak{g}$, there exists an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak{g}$ such that
We have $X\in e+Z_{\...

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60 views

### Maximal order of $x^n-d$ and its dependence on $d$

It's well known that the structure of the maximal order of $\mathbb{Q}[\sqrt{d}]$ depends on $d$ modulo $4$: (assuming $d$ is squarefree), the maximal order is $\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\...

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**3**answers

346 views

### Oldest abstract algebra book with exercises?

Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of ...

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114 views

### English translation of “Une inégalité pour martingales à indices multiples et ses applications”

Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples
et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...

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**1**answer

57 views

### There is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon

Conjecture 1: With $n\ge 5$, given general n-polygon, there is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon (with one and only one ...

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63 views

### Origins of the ``baby Freiman'' theorem

It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) ...

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**1**answer

78 views

### Literature on the polynomials and equations, in structures with zero-divisors

I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.
For example, there is literature ...

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25 views

### Gelfand-Kirillov dimension for non-associative algebras

Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $...

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53 views

### When Riemannian manifold with boundary is Alexandrov space?

I am looking for a proof or, better, a reference to a proof of the following known fact.
Let $(M,g)$ be a smooth Riemannian manifold with boundary. Assume the sectional curvature of $M$ is at least $\...

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**1**answer

149 views

### Reference to a covering theorem by Ketonen

I am reading a paper by Goldberg, and he uses a theorem by Ketonen, which is highlighted in red below:
Do you know where can I find the theorem and it's proof?
Link to the article: https://arxiv.org/...

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69 views

### arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field

A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety.
I am interested in the arithmetic analogue, a 2-dimensional ...

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279 views

### Fundamental group of a compact branched cover

My problem originates from the following classical result, proved, as far as I know, by Grauert and Remmert:
Theorem. Let $Y$ be a compact complex manifold, $B \subset Y$ be a connected submanifold ...

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133 views

### conditions on a morphism $f:X\rightarrow Y$ to ensure $X$ is reduced, given $Y$ is reduced?

Let $X,Y$ be finite type projective schemes over $\mathbb{C}$, and $f:X\rightarrow Y$ be a surjective morphism (but not an isomorphism). Suppose it is known that $Y$ is reduced, and the fibers of $f$ ...

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181 views

### Looking for references for NBG theory meant for the working mathematician (not for someone interested at Foundations of Mathematics)

As most mathematicians, I've always used sets as the main tool for doing mathematics. My knowledge of the subject is limited to what I've learned from Halmos's "Naive Set Theory" and, for ...

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84 views

### Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary

I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.
I know that there are several generalizations of this theorem, ...

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46 views

### Equivalence relation induced by Kolmogorov quotients

Recall: given a (possibly non-$T_0$) topological space $X$, its Kolmogorov quotient $KX$ is the $T_0$ topological space formed by $X/\sim$ where $x\sim y$ if they are topologically indistinguishable. ...

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113 views

### Resolution of pairs in characteristic p

Let $R$ be a complete DVR of characteristic $p$, say $R=\mathbb{F}_p[[t]]$, and $X$ be a reduced scheme of finite type over $R$. Let also $X_s$ denote the special fiber of $X$. If I understand ...

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80 views

### Shortest path on Riemannian manifold with boundary

Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume
$$II_{\partial M}(...

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**6**answers

631 views

### Reference for topological graph theory (research / problem-oriented)

I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...

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144 views

### Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...

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201 views

### Roots of determinant of matrix with polynomial entries — a generalization

For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...

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**1**answer

77 views

### References for irrational random walks

I am interested in the symmetric random walk on $\mathbb{R}$ which increments have the discrete law $$\mu=\sum_{i=1}^q p_i (\delta_{\omega_i}+\delta_{-\omega_i})$$
where the $p_i$ sum to $1/2$ and the ...

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46 views

### Is there an analog of polarization for skew-symmetric forms?

This question might be too lightweight here but on math.SE it did not receive any feedback since May 2, so...
Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of ...

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**2**answers

478 views

### What are your opinions on Zeidler's QFT books? [closed]

I am interested in mathematically rigorous treatment of quantum field theory, constructive QFT in particular.
I have read 'QFT, A Tourist Guide for Mathematicians' and am going to read "Quantum ...

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96 views

### Viewing limit as a map

Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which
$$
f_{\infty}(x) = \lim\limits_{n \...

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27 views

### Rate of convergence for point processes in Skorokhod J1 topology

Skorohod J1 Topology space $D[0,1]$ is a metric space, see its definition in https://encyclopediaofmath.org/wiki/Skorokhod_topology
Assume we have a sequence of point processes $(X^n_t: (\Omega, P) \...

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**1**answer

124 views

### Roots of determinant of matrix with polynomial entries

Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...

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41 views

### Trivial extension algebra and finite global dimension

Let $A=kQ/I$ a finite dimensional quiver algebra.
Question: Assume that the trivial extension of $A$ is periodic (or at least that all simple modules are periodic). Does $A$ have finite global ...

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161 views

### On the Chowla and twin prime conjectures

I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...

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61 views

### mean curvature for codimension $>1$?

The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...

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282 views

### Extending a holomorphic vector bundle: a reference request

Let $Y$ be a complex manifold, $X\subset Y$ a compact submanifold, and $E\to X$ a holomorphic vector bundle. Can $E$ be extended
to a bundle over an open neighborhood of $X$ in $Y$? (Four years ...

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182 views

### When $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$?

When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^...

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100 views

### Secondary operations in $H^*P$

I would be very grateful for any answer to this or pointing at a reference.
Let $P$ be the infinite dimensional real projective space and $\Phi_{j,j}$ be the Adams operation based on the Adem relation ...

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26 views

### Involutions on Hopf algebra crossed products

I am interested in a (cocycle) crossed product of a Hopf algebra $H$ with a (twisted) $H$-module algebra $A$, often denoted $H\#_\sigma A$, where $\sigma$ is the associated cocycle. This construction ...

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43 views

### (Big) Category O for rational Cherednik algebras

Let $H_{c}$, for simplicity, be the rational Cherednik algebra with parameter $t=1$, with triangular strucuture $\mathbb{C}[h] \otimes \mathbb{C} W \otimes \mathbb{C}[h^*]$, and $(W,h)$ the defining ...

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166 views

### Name for a class of almost symplectic manifolds

A $2n$-dimensional manifold $M$ is said to be almost symplectic if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \...

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194 views

### Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite

I was told that if $A$ is the subring of $\mathbb{C}[x_1,\ldots, x_n]$ generated by the polynomials $p_1(x_1,\ldots, x_n),\ldots, p_1(x_1,\ldots, x_n)$, then the preimage $p^{-1}(c)$ via the map $p = (...

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77 views

### Literature and history for: lifting matrix units modulo various kinds of ideal

This is not so much a mathematics question as a cross between a "history of mathematics" question and a reference request.
My PhD student has been working on some problems concerning ...

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35 views

### Beginner's book on stochastic prediction theory

I read the papers of Wiener and Masani on stochastic prediction theory. I do not have a statistics background, but for some reason I need to know this subject. My questions are:
Can someone please ...