41
votes
15answers
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
17answers
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
2answers
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
3answers
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
2answers
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
1answer
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
1answer
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
12answers
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
11answers
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
3answers
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
12answers
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
3answers
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
5answers
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
3answers
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
2answers
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
2answers
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
6answers
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
8answers
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
1answer
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
5answers
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
1answer
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
3answers
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
4answers
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
2answers
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
3answers
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
2answers
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
6answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
19answers
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
0answers
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
6answers
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
1answer
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
2answers
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
0answers
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
1answer
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
6answers
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
0answers
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
1answer
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
3answers
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
1answer
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
0answers
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
0answers
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 ...

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