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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67 views

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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1answer
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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1answer
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 ...
1
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1answer
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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2answers
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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1answer
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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1answer
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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0answers
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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4answers
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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0answers
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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0answers
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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3answers
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 ...
2
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1answer
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 ...
2
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1answer
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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1answer
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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0answers
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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0answers
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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1answer
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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0answers
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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2answers
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 ...
1
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1answer
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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2answers
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 ...
1
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1answer
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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0answers
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. ...
2
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0answers
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 ...
6
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0answers
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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6answers
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 ...
8
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0answers
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? ...
2
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0answers
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 ...
0
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1answer
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 ...
2
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0answers
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 ...
2
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2answers
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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0answers
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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0answers
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) \...
2
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1answer
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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0answers
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 ...
2
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0answers
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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0answers
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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1answer
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 ...
6
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0answers
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)^...
6
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0answers
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 ...
2
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0answers
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 ...
3
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0answers
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 ...
6
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1answer
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 \...
3
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1answer
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 = (...
7
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0answers
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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0answers
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 ...