# All Questions

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88 views

### Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy?
...

**1**

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6 views

### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...

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6 views

### Number of non-isomorphic models

I had this question up on Math stackexchange: http://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763 . While it was answered partially there, I'm posting here ...

**0**

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**0**answers

17 views

### What is the significance of the eigendecomposition of a Cayley table?

Treating the Cayley table of a group $G$ as a matrix $M_g$, one notices interesting things about its eigendecomposition.
For instance, for the symmetric groups $\{S_n\}$, the rank of the Cayley table ...

**1**

vote

**1**answer

233 views

### About Abhyankar's conjecture

I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book Rigid Geometry and Its Applications. It claims that all quasi $p$-group $G$ ...

**2**

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**0**answers

21 views

### Expected absolute value of the determinant of an $n$ by $n$ Toeplitz $(0,1)$ matrix

If $A$ is chosen uniformly over all $n$ by $n$ $(0,1)$-Toeplitz matrices, what is the expected absolute value of the determinant?

**0**

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**0**answers

12 views

### Variational Properties of the Perelman Functional

After reading a bit more about Perelman's entropy and gradient solitons, I came up with a hunch, which I must test. Non-singular solitons can be regarded as critical points of Perelman's entropy, or ...

**7**

votes

**2**answers

102 views

### Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...

**3**

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**0**answers

26 views

### Does hypoellipticity imply the existence of a parametrix?

Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...

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**0**answers

101 views

### Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...

**6**

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**0**answers

111 views

### What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?

Over the years, advances in machine learning has allowed us to communicate and interact, using the same natural language, more and more semantically with computers, e.g. Google, Siri, Watson, etc. On ...

**13**

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**2**answers

487 views

### Counter examples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...

**1**

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**0**answers

86 views

### The representation theory for the fake Heisenberg groups over non-perfect local field

Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
...

**13**

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**1**answer

224 views

### Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem ...

**0**

votes

**1**answer

173 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

**3**

votes

**3**answers

103 views

### Integrals involving the Tricomi hypergeometric function

I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...

**6**

votes

**1**answer

122 views

### How small can the Mumford-Tate group of hypersurface be?

Is there some way of giving a lower bound on the dimension of the Mumford-Tate group of a hypersurface? Let's say it's of general type, say, of degree $10$ inside $ \mathbb{P}^3$.
(Edited from here ...

**1**

vote

**1**answer

100 views

### Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those ...

**5**

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**0**answers

143 views

### Log smooth models for abelian varieties

Let $K$ be a field which is complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some ...

**5**

votes

**1**answer

40 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...

**4**

votes

**0**answers

47 views

### Expected size of determinant of $AA^T$ for non-square random Toeplitz $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ Toeplitz (0,1)-matrices, what is the expected size of the value of the determinant of $AA^T$? We can assume $m \leq n$ and all ...

**18**

votes

**2**answers

438 views

### Why is every variety of bands determined by a single identity?

A band is a semigroup where every element is idempotent: $a a = a$. A collection of bands is a variety if it is closed under taking subobjects (subsemigroups), quotient objects (images of ...

**95**

votes

**107**answers

32k views

### Most memorable titles [closed]

Apparently, for a large number of readers, the choice whether they select to read a paper or not is often strongly influenced by the title.
I was wondering if the MO-users would be willing to share ...

**1**

vote

**1**answer

123 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...

**83**

votes

**90**answers

11k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**5**

votes

**0**answers

83 views

### Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...

**23**

votes

**1**answer

790 views

### Homotopy Type Theory: What is it?

My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following:
There are three directions:
...

**27**

votes

**23**answers

7k views

### Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but
I have been curious ever since
Suvrit's popular October 2010
Most memorable titles MO question
if there were any ...

**133**

votes

**16**answers

54k views

### What's a mathematician to do? [closed]

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ...

**2**

votes

**2**answers

339 views

+50

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**3**

votes

**1**answer

123 views

### Is the following a sufficient condition for being a primal algebra?

I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...

**14**

votes

**18**answers

22k views

### A good book of functional analysis

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the ...

**12**

votes

**1**answer

238 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

**1**

vote

**1**answer

141 views

### Does the associated Lie algebra determine a group?

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...

**10**

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**1**answer

191 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

**13**

votes

**2**answers

1k views

### Symplectic formulation of statistical physics

Does there exists a symplectic formulation of statistical physics?
I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated ...

**35**

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**2**answers

2k views

### Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples.
Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, ...

**5**

votes

**1**answer

619 views

### What is the geometric interpretation of this quantity?

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic
structure on the tangent bundle, $(TM, ...

**32**

votes

**3**answers

3k views

### What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...

**11**

votes

**4**answers

1k views

### Weil conjecture for algebraic surfaces

Deligne's proof of the Weil conjecture is difficult.
On the other hand, there are some "simpler" proofs of the Weil conjecture in the case of algebraic curves.
For instance, in GTM52, one see it ...

**25**

votes

**8**answers

3k views

### Geometric intuition for limits

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects ...

**10**

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**2**answers

1k views

### Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant:
1) Is eta a topological invariant ...

**3**

votes

**3**answers

308 views

### Generators of the kernel of multiplication of a ring

Is the following true?
Let $R$ be a commutative ring and $S$ be an $R$-algebra. Let $\mu: S\otimes_R S\to S$ be the multiplication.
Let $I$ be the kernel of $\mu$.
Let $J\subset I/I^2$ be the ...

**8**

votes

**5**answers

1k views

### What are the zero entropy invariant measures for an Anosov geodesic flow?

Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed ...

**5**

votes

**2**answers

262 views

### How strict can I be in the definition of “2-group”?

Recall that a group is an associative, unital monoid $G$ such that the map $(p_1,m) : G \times G \to G\times G$ is an isomorphism of sets. Here $p_1$ is the first projection and $m$ is the ...

**0**

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**0**answers

2 views

### Are these two $q$-continued fractions equivalent?

In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24,
$$U(q) = ...

**0**

votes

**0**answers

15 views

### How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$

I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset ...

**0**

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**0**answers

28 views

### Elementary bound on operator norm on symmetric tensors: reference request

My education didn't really cover Tensors very well, so I'm getting stumped by a quite elementary question.
Let $T_k$ be a type k symmetric tensor. We can define the "operator norm" (or the induced ...

**1**

vote

**1**answer

52 views

### Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...

**-4**

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**0**answers

29 views

### linear algebra, existense of polynomials [on hold]

True or False? How to prove it?
There do not exist polynomials p(t) and q(t), and scalars a, b, c, d, e, f , for which the following equations hold for all t:
(Hint: use Theorem 9.)
ap(t) + bq(t) = 1 ...