# All Questions

**41**

votes

**15**answers

5k views

### How does the work of a pure mathematician impact society? [closed]

First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.
I am in a really bad position, because ...

**62**

votes

**17**answers

25k views

### Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...

**3**

votes

**2**answers

405 views

### 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 ...

**7**

votes

**3**answers

737 views

### Conjectured integral for Catalan's constant

Numerical evidence suggests:
$$ \int_0^{\frac12}\int_0^{\frac12}\frac{1}{1-x^2-y^2} dy \, dx= \frac{G}{3}\qquad (1)$$
Couldn't find the indefinite integral, though maple simplifies (1) to
$$ \int ...

**7**

votes

**2**answers

135 views

### Image of poset with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = ...

**5**

votes

**1**answer

116 views

### Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and ...

**4**

votes

**1**answer

80 views

### Order-preserving images of $(\mathcal{P}(\kappa),\subseteq)$

Is there a cardinal $\kappa \neq \emptyset$ and a connected poset $P$ of cardinality $\leq \kappa$ such that there is no surjective order-preserving map from $(\mathcal{P}(\kappa),\subseteq)$ onto ...

**32**

votes

**12**answers

10k views

### Is pi a good random number generator?

Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, ...

**65**

votes

**11**answers

3k views

### Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...

**12**

votes

**3**answers

527 views

### Generalization's of Greene's Theorem for the Robinson-Schensted correspondence

One important property of the Robinson-Schensted correspondence (RS) is that the longest increasing subsequence of the permutation $\sigma$ is $\lambda_1$, the first entry of the shape ...

**25**

votes

**12**answers

9k views

### why is it so cool to square numbers? (in terms of finding the standard deviation)

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...

**16**

votes

**3**answers

527 views

### Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...

**27**

votes

**5**answers

6k views

### Computing the Galois group of a polynomial

Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of ...

**7**

votes

**3**answers

562 views

### A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...

**7**

votes

**2**answers

512 views

### Ergodic theory and dynamical systems books references

I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks).
I am asking here for an advice of a book (or maybe a set of papers) to look at ...

**10**

votes

**2**answers

688 views

### An integrality question about expressing an integer as a product of numbers below $n$

Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod_{j=1}^{n} j^{x_j}
$$
where $x_1$, $\ldots$, ...

**30**

votes

**6**answers

5k views

### Book on mathematical “rigorous” String Theory?

I've been looking high and low for a mathematical Book on String Theory. The only Book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only ...

**30**

votes

**8**answers

1k views

### Natural examples of sequences of adjoint functors

I am looking for examples of sequences of adjoint functors. That are (possibly bounded) sequences
$$(...,F_{-1}, F_{0}, F_1, F_2,...)$$
such that each $F_n$ is left adjoint to $F_{n+1}$. We call such ...

**42**

votes

**1**answer

3k views

### Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...

**21**

votes

**5**answers

2k views

### Generating finite simple groups with $2$ elements

Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is ...

**24**

votes

**1**answer

2k views

### An etale version of the van Kampen theorem

Let $V$ be a smooth connected algebraic variety over an algebraically closed field $k$. Let $W_1, W_2$ be closed subvarieties of $V$ of positive codimension whose intersection $W_1 \cap W_2$ has ...

**4**

votes

**3**answers

2k views

### Number of Normal subgroups In a p-Group

Dear all,
Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n $ ($ n$ is some natural number ? ) .
Is there anyway ...

**22**

votes

**4**answers

2k views

### Is there a universal countable group? (a countable group containing every countable group as a subgroup)

This recent MO
question,
answered now several times over, inquired whether an
infinite group can contain every finite group as a
subgroup. The answer is yes by a variety of means.
So let us raise the ...

**22**

votes

**2**answers

1k views

### The difference between a handle decomposition and a CW decomposition

Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it ...

**3**

votes

**3**answers

704 views

### Computing the Euler characteristic of the complex projective plane using differential topology

I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for ...

**3**

votes

**2**answers

209 views

### Left determined model structure on delta-generated topological spaces

Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). ...

**8**

votes

**6**answers

8k views

### Reading materials for mathematical logic

Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject?

**3**

votes

**1**answer

489 views

### A chain homotopy that does not arise from a homotopy of spaces?

Algebraic topologists like to cook up algebraic invariants on topological spaces in order to answer questions, so they are often concerned with how strong those invariants are. Currently, I am ...

**9**

votes

**0**answers

176 views

### Is this (funny) combinatorial optimization problem NP-hard ? (cutting numbers and placing them in urns)

The parameters of the problem are $m$ numbers which are integers (these numbers are denoted $b_i$), $n$ urns and in each urn, we can place $C$ numbers. We assume $nC \geq m$ so that the problem is ...

**14**

votes

**2**answers

1k views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

**0**

votes

**0**answers

11 views

### Distance minimization and submodular functions

I have a question about square distance minimization and submodular functions. Suppose I have a random variable $X = (X_1,...,X_n)$ over $\mathbb{R}^n$. Let $x$ be a realization of this random ...

**6**

votes

**0**answers

160 views

### A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...

**4**

votes

**1**answer

108 views

### Stinespring's dilation without $C^{\ast}$-algebras

Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?
I will now state the version of Stinespring's dilation ...

**0**

votes

**0**answers

22 views

### exponential growth of an ordered structure (like a dcpo) [on hold]

Here is a paper that relates hyperbolic spacetimes to a special type of Domain (dcpo) called an interval domain. Inflation is a well understood aspect of the history of our spacetime and can be ...

**8**

votes

**0**answers

94 views

### When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor?

When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor? Is there a general approach for any number of variables, aka when is the variety defined by the ...

**7**

votes

**2**answers

186 views

### Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...

**29**

votes

**19**answers

9k views

### Math paper authors' order

It seems in writing math papers collaborators put their names in the alphabetical order of their last name. Is this a universal accepted norm? I could not find a place putting this down formally.

**3**

votes

**0**answers

230 views

### Bounding the degrees in a Bézout relation for integer polynomials

Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that
$$
A \cdot U + B \cdot V=1.
$$
...

**10**

votes

**6**answers

908 views

### Sum of $n$ vectors in $(\mathbb Z/n)^k$

Let $n,k$ be positive integers. What is the smallest value of $N$ such that for any $N$ vectors (may be repeated) in $(\mathbb Z/(n))^k$, one can pick $n$ vectors whose sum is $0$?
My guess is ...

**0**

votes

**1**answer

108 views

### Concerning Jump process (Lévy process)

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A ...

**7**

votes

**2**answers

2k views

### About the definition of Borel and Radon measures

I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...

**1**

vote

**0**answers

39 views

### A slight generalization of Mehta's integral.

I am trying to find the value of following integral
$$\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\prod_{i=1}^ne^{-\frac{t_i^2}{2}+\alpha_i t_i}\prod_{1\le i<j\le ...

**2**

votes

**1**answer

49 views

### Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...

**5**

votes

**6**answers

1k views

### Exceptional Lie algebras

I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8.
Can anybody explain to me what prevents us from constructing e(9) from e(8)? One can use the e(8) lattice vectors and try ...

**0**

votes

**0**answers

58 views

### Complement and fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...

**10**

votes

**1**answer

455 views

### is there a general statement about structures on spheres relating to division algebras?

It is classical to take a division algebra over $\mathbb{R}$ and defining an H-space structure on the unit spheres by restricting and normalizing.
There are commutative division algebras of dimension ...

**0**

votes

**3**answers

381 views

### Cohomology of complexes

I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*) $?

**26**

votes

**1**answer

791 views

### Understanding “infinite” relations in groups

Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ ...

**5**

votes

**0**answers

62 views

+100

### Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science.
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...

**4**

votes

**0**answers

328 views

### Complexity of edge coloring in planar graphs?

I asked this on StackExchange TCS but did not get a satisfactory answer:
Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". However, 3-edge coloring of ...