1
vote
1answer
43 views

Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...
0
votes
0answers
11 views

Is the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ in $L^\infty(\Omega)$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of $$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$ $$\partial_\nu u = \alpha \quad \text{on ...
5
votes
0answers
39 views

Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is ...
0
votes
0answers
17 views

Cells in affine Weyl groups

This may sound like a very general question, which pretty much reflects my ignorance on the subject. In the case of Weyl groups $W$, there is a notion of left/right/double cells, which is roughly ...
3
votes
1answer
66 views

Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth. We know that ...
7
votes
0answers
54 views

Calculation of the integral related to the gravitational shock wave

The following integral $$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$ can be found in the paper Tevian Dray and Gerard 't Hooft, The ...
4
votes
1answer
157 views

Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ...
1
vote
0answers
38 views

Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$? For examples: are there: Ito ...
14
votes
0answers
474 views
+200

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
1
vote
1answer
74 views

If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space. Usually, we have a ...
64
votes
34answers
11k views

What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein What notions are used but not ...
18
votes
3answers
1k views

What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested). Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]): Let $H$ ...
69
votes
5answers
6k views

What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, ...
6
votes
1answer
361 views

Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid. K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
3
votes
1answer
157 views

Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams. I also found some relation with matroid theory. ...
2
votes
0answers
9 views

Maximal TB number and slice genus relation of a knot in any 3-manifold

Lisca-Stipsicz say that if a knot $K\subset{S^{3}}$ satisfies $g_{s}(K)>{0}$ and $TB(K)=2g_{s}(K)-1$ then $S^{3}_r(K)$ carries positive, tight contact structures for every ...
-2
votes
0answers
23 views

Should I learn information technology? [on hold]

I am 18 y.o., and I want to go to university. I will study Software Engineering (Computer science), but I do not know yet what I want to be exactly. I love math, phisycs, psychology, philosophy, ...
5
votes
0answers
58 views

Stiefel-Whitney classes and counting

We can draw an explicit immersion of a Klein bottle in $\mathbb{R}^3$. Our immersion determines a map from the Klein bottle to $\mathbb{RP}^2$. How do we count the value of $w_1(T(\text{Klein ...
1
vote
1answer
29 views

Continuous non-constant function with infinite intersections with horizontal line on a compact interval?

The title might be misleading, but whether such a function exists is what boggles me about the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all ...
2
votes
0answers
59 views

Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...
3
votes
1answer
42 views

Bounding and dominating numbers ${\frak b}, {\frak d}$ via ultrafilters

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ and suppose that ${\cal U}$ is a free ultrafilter on $\omega$. We write $f \leq_{\cal U} g$ if $$\{n\in\omega: f(n) \leq ...
4
votes
0answers
61 views

Another interpretation of the $16$ dimensional Severi Vairety

I asked about an interpretation of this variety here. There is another one that could be easier. Let $K$ be an algebraically closed field of characteristic $0$. We denote the set of terns of $3\times ...
-3
votes
0answers
37 views

The line graph of a complete graph [on hold]

Show that there exist a $\left\{P_{5},C_{4}\right\}$- decomposition of the graph $L(K_{9})$.
9
votes
4answers
314 views

List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
3
votes
1answer
341 views

Use of infinitude of primes in the Green-Tao theorem [on hold]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
6
votes
2answers
170 views

Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...
0
votes
0answers
12 views

The effective strategy for choosing epsilon_init of random initialization in neural networks [on hold]

gays. i meet some problem in a description about choosing epsilon_init for random initialization in neural networks. enter image description here i don't know why the epsilon_init is related to the ...
1
vote
0answers
18 views

Wave-like equation with 1st order time derivative and non-constant coefficients

We start with the following recurrence relation for complex coefficients $c_{n,m}$: $$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m-2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$ where ...
0
votes
1answer
35 views

CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
1
vote
1answer
83 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let ...
3
votes
1answer
100 views

Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
1
vote
0answers
34 views

Markov Chains and Simple Machine Learning [on hold]

Suppose I have a large training set consisting of many strings of symbols. $TS = \{Str_0, Str_1, ..., Str_n\}$ $Str_i = \{Sym_0 ... Sym_{len}\}$ These strings of symbols are each generated by the ...
2
votes
0answers
42 views

End points of continua

Whyburn (1942) defined an end point x of a continuum X to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines end point locally. ...
1
vote
0answers
18 views

Frobenius series of Fuchsian PDEs

I'm interested in the analyticity of Frobenius-like series solutions to a PDE in $z=(z_1,\ldots ,z_N)\in\mathbb{C}^N$ with regular singular behavior at $z_\alpha=0$ for all $\alpha=1,\ldots, N$. For ...
0
votes
0answers
48 views

canonical decomposition of a linear map [on hold]

I have a linear map $A : {\mathbb F}^k \to {\mathbb F}^{k+m}$ where ${\mathbb F}$ is some field; (you can impose restrictions on ${\mathbb F}$ if it helps). I'd like to decompose $A$ as a product of ...
12
votes
0answers
200 views

Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
2
votes
1answer
112 views

Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...
4
votes
0answers
26 views

Carving a rectilinear polygon

In this question, carving a polygon $P$ means removing an axis-parallel rectangle adjacent to the boundary of $P$. Carving $P$ might break it into two or more polygons. You are given a square $P$. ...
6
votes
0answers
81 views

Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
2
votes
1answer
92 views

Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to ...
2
votes
0answers
199 views

Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...
0
votes
1answer
250 views

If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with quasi-Euler ...
1
vote
1answer
655 views

Hard maths on viXra? [closed]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...
0
votes
1answer
48 views

Do tori in a symplectic group always have invariant maximal isotropic subspaces?

$\newcommand{\mbf}{\mathbf}$ Hi all, I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help. For ...
533
votes
209answers
135k views

Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
148
votes
67answers
49k views

Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
6
votes
4answers
887 views

Topological structure of SO(n) as a product

I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product $$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$ Allen Hatcher [1, p. 293 f.] ...
5
votes
0answers
107 views

Differential operators acting on the Schwartz space

I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome! Let $D$ be a linear differential operator with ...
28
votes
3answers
2k views

Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow ...
45
votes
34answers
6k views

Are there any books that take a 'theorems as problems' approach?

Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof ...

15 30 50 per page