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

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4
votes
1answer
189 views

Mazur's Galois Deformations paper for non-residually irreducible case

In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know ...
1
vote
0answers
43 views

References for a minor variant of the Rayleigh quotient

I believe this variant of the Rayleigh quotient inequality must be well known but I could not find references for it online. It's proof is straightforward. Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
0
votes
0answers
65 views

Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
4
votes
0answers
76 views

Kruskal-Katona for homocyclic groups?

I need a version of the Kruskal-Katona theorem (better still, of the Lovasz "approximate" version thereof) for the elementary abelian / homocyclic groups, in the following spirit: What is the ...
4
votes
0answers
166 views

Lie group structure over diffeomorphisms group

Let $M$ be a smooth manifold. Is it true that for every subgroup $G$ of $diff(M)$ there is at most one Lie group structure on $G$ such that the natural left $G$-action on $M$ is smooth? Edit: by "Lie ...
6
votes
1answer
270 views

Encyclopedia of properties of nonnegative matrices

I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $A_{i,j}\geq 0$ for all $i,j$. Chapter 8 in ...
8
votes
1answer
146 views

Is the modularisation of a unitary fusion category always unitary?

Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
2
votes
0answers
134 views

On the use of the term “field of sets” in Maharam's papers

I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for ...
13
votes
1answer
207 views

Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the ...
6
votes
1answer
172 views

Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?

A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on ...
3
votes
0answers
113 views

Hamiltonian on the torus

In discrete models like Ising we have Hamiltonians of the form $$H(\sigma)=\frac{1}{N}\sum_{i=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$$ where $\sigma_{i}=\pm 1$ , $J_{ij}$ interaction coefficients and N ...
0
votes
1answer
73 views

Boundary behaviour of a second order pde with characteristics

Good morning everybody. My question is inspired from the following fact: Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial_x^2 ...
5
votes
1answer
270 views

reference request for mod p and p-adic K-theory

Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book ...
4
votes
1answer
144 views

Reference to a variant of Abel's summation formula

Edit. A stronger version of the formula is true (details follow). Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that ...
3
votes
1answer
104 views

Category enriched over a monoidal 2-category

Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by ...
0
votes
0answers
58 views

Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE $$\begin{cases}\mathbf{u}'(t)\in-\partial ...
3
votes
0answers
57 views

A measurable implicit function with differentiated arguments

I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers. The setup is as follows. Let ...
9
votes
0answers
172 views

Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence

This is related to this question (edit: now answered). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ is some ...
7
votes
2answers
310 views

Request for classical articles in representation theory

I am planning in running a Ph.D. student seminar next year on representation theory in the spirit of MIT Kan's Seminar where students give lectures on classical articles on representation theory that ...
4
votes
1answer
152 views

Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?

The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix $$ \left\|\frac{\partial^2u}{\partial ...
5
votes
2answers
237 views

Why is the number of irreducible components upper semicontinuous in nice situations?

Suppose $f: X \rightarrow Y$ is a flat projective morphism of finite type schemes over an algebraically closed field so that the fibers over the closed points of Y are (geometrically) reduced. Why is ...
4
votes
0answers
88 views

Where do I read about semi-algebraic/analytic sets?

What's a good introduction to semi-algebraic/semi-analytic sets? I'm coming from algebraic geometry, so ideally I'd prefer material that has a similar point of view and highlights analogies. I've ...
1
vote
2answers
146 views

Elementary question: Cubic 4-fold and rational quartic scroll

Forgive me to ask an elementary question, because I really need the answer to this today (I already asked this in Stackexchange.) Let $S$ be the rational quartic scroll in $\mathbb{P}^5$ ($S$ is the ...
1
vote
1answer
102 views

example of an $\ell_1$-saturated Banach space without an unconditional basis

Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but ...
11
votes
3answers
676 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
59
votes
15answers
6k views

Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
5
votes
0answers
63 views

Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims

I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...
2
votes
0answers
57 views

Reference request on a notion of independence for families of [real-valued] functions

This is basically another reference request. Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that ...
3
votes
1answer
250 views

Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations

In Diophantine equations over the twentieth century: a (very) brief overview , p. 5 Problem Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine ...
2
votes
0answers
209 views

Looking for a reference in commutative algebra

I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...
2
votes
0answers
60 views

How sensitive are the n-th step transition probabilities of the simple random walk to a small perturbation of an infinite graph?

Suppose that $G$ is an infinite, locally finite, connected graph. Fix a vertex $o$ in the graph and for each $n$ and $x$ let $p(n,o,x)$ be the probability that a simple random walk (at each step a ...
18
votes
1answer
489 views

Why can the general quintic be transformed to $w^5-5\beta w^3+10\beta^2w-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$ This form is well-known for its connection to the symmetries of the ...
7
votes
1answer
266 views

What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...
2
votes
0answers
48 views

Reference for branching processes

A popular model of a continuous time branching process was introduced around 1970, which is now called the Crump-Mode-Jagers branching process, was introduced here: A General Age-Dependent Branching ...
3
votes
0answers
132 views

Almost primes in short intervals

Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by ...
1
vote
1answer
142 views

A boundary of the second fundamental theorem of calculus

Let's say that a set $X\subseteq [0,1]$ has Property Q if the following holds: For every continuous $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ and derivative existing and bounded by 1 on $[0,1]\setminus X$, ...
-1
votes
1answer
79 views

Group associated to the monoid $({\cal P}(X\times X), \circ)$

Consider the set ${\cal P}(X\times X)$. It can be endowed with a binary operation $\circ$ where $$A\circ B = \{(a,b)\in X\times X:\exists z\in X((a,z)\in A\land (z,b)\in B)\}.$$ Note that ...
5
votes
2answers
238 views

3-sphere bundles over 4-sphere bound smooth disc bundles

I saw in the answer of this post: Is it true that all sphere bundles are boundaries of disk bundles? that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is ...
1
vote
1answer
126 views

Linear Schrödinger equation on $\mathbb{H}^{d}$

Consider the linear Schrödinger equation $i\partial_t u = -\Delta u$, where $\Delta$ is the Laplacian on the hyperbolic space $\mathbb{H}^d$. What are the admissible pairs $(p, q)$ such that we have ...
3
votes
0answers
95 views

$\deg \mathcal{F}=\sum_{i=1}^r \textrm{rank}(\mathcal{F}|_{X_i}) \cdot \deg X_i$

Let $K$ be a field. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^n_K$ whose scheme theoretic support is a reduced, closed subscheme $X \subseteq \mathbb{P}^n_K$ of dimension $k$. Let $X_1, ...
1
vote
0answers
161 views

Why the Castelnuovo exact sequence is named after Castelnuovo

I have seen variations of the following exact sequence referred throughout the literature as the Castelnuovo sequence: $$0\longrightarrow \mathscr I_{X:H}(-d)\longrightarrow \mathscr ...
10
votes
3answers
563 views

Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$

Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
4
votes
0answers
114 views

Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
2
votes
0answers
116 views

Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...
4
votes
1answer
209 views

Open questions on (finite) tensor categories

I would like to know about problems on (finite) tensor categories. I have read Etingof´s notes from his course at MIT. I have a question: There exists any reference where I can find an open problem ...
-1
votes
1answer
134 views

separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
11
votes
0answers
370 views

Goodwillie's notes from MSRI Lecture Series

Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
2
votes
0answers
196 views

Is Carlos Simpson's Descent available online?

I am not sure whether this question is suitable for MO. Is the paper "Descent" by Carlos Simpson in the book "Alexandre Grothendieck: A Mathematical Portrait" page 83-142 (or a similar version of that ...
2
votes
0answers
51 views

Geometric unfolding of a difference equation

Does anyone know a link to the (superb) slides of a talk given by Zeeman in 1996, I think (with same title as this question)? It used to be at www.math.utsa.edu/ecz/gu.html .
2
votes
0answers
20 views

Fractional parts of two multiples [duplicate]

There is a theorem (I can't remember its name) saying that for any irrational number $x$ and any $0<a<b<1$, there exists a positive integer $n$ such that $\{nx\}\in (a,b)$, where $\{\cdot\}$ ...