Newest Questions
159,064 questions
2
votes
1
answer
165
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
- 1,043
2
votes
2
answers
823
views
Kolmogorov-Smirnov distance and expectation
Let $P$ and $Q$ be two probability measures over $R^n$, with CDF denoted by $F_P,F_Q$, respectively (that is, $F_P(x)=P(\{x'\in R^n:x'\leq x\})$, where $\leq$ is taken componentwise. The Kolmogorov-...
- 333
2
votes
0
answers
382
views
How to handle a research identity crisis
I have studied applied math and got a PhD (3yrs) in that field with applications in fluid dynamics. Then in my first postdoc (1.5yrs) I did again a postdoc in applied math but studied applications in ...
- 31
0
votes
1
answer
115
views
Summing the max value of the distinct pairs in a multiset
Say one has a multi set of natural numbers, or positive integers if that’s more useful. If $\#A = n$ and one knows $\sum_{a \in A} a$
What is it possible to say about $\sum_{a,a^{‘}\in A, a \neq a^{‘}}...
2
votes
0
answers
63
views
Tightness of the bounding the operator norm of graph by average degree from below
Let $G = (V,E)$ be a simple graph with adjacency matrix $A$. It is well known that the largest eigenvalue
$\lambda_{1}$ of $A$ is contained within the interval $[d, D]$ where $d$ is the average degree ...
- 401
5
votes
0
answers
250
views
Lie algebras, root systems and qubits
This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...
- 5,215
-4
votes
1
answer
605
views
Multiplicative Persistence - Highest persistence found? [closed]
tried to ask on the math reddit but got deleted due to my account being new.
Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
- 21
0
votes
1
answer
451
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 13
1
vote
1
answer
336
views
sum of binomial coefficient approximation by geometric series
I follow a subject almost like this link:
Sum of 'the first k' binomial coefficients for fixed $N$
$$
f(N,k) = \sum^{k}_{i=0} \binom{N}{i} .
$$
Michael Lugo suggest a way with geometric series ...
- 21
2
votes
1
answer
202
views
Upper bound for the rank of a Gram-type matrix
Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
- 93
10
votes
2
answers
388
views
On the intersection of finitely many ultrafilters
I am interested in characterizing those filters that can be written as an intersection of finitely many ultrafilters. I would appreciate any reference on this topic.
3
votes
1
answer
509
views
Existence of a curve of finite length on the image of an embedding which is Sobolev
Suppose that we have an embedding $f:\mathbb{R}^2\to\mathbb{R}^3$ which belongs in the Sobolev space $W^{1,p}_{loc}(\mathbb{R^2},\mathbb{R}^3)$ for some $p>2$. Is it true then that for any two ...
- 81
1
vote
0
answers
129
views
Is set theory interpretable in infinite primitive recursive arithmetic?
In A Formalization of the Theory of Ordinal Numbers, Takeuti interprets $\sf ZFC$ in a first order theory extending first order arithmetic to the infinite ordinal realm, while at the same time ...
- 11.3k
0
votes
0
answers
330
views
Pushforwards in vector bundles over a topological spaces
I have been reading the discussion from Pushforward and pullback..
I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
- 615
21
votes
1
answer
1k
views
Is applying Feit–Thompson’s theorem for the nonexistence of a simple group of order $1004913$ really a circular argument?
In p.212 of Dummit–Foote’s Abstract Algebra, 3rd Edition, an analysis of a hypothetical simple group $G$ of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$ is carried out. The authors write:
We ...
- 321
3
votes
0
answers
177
views
For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
- 5,422
4
votes
0
answers
143
views
Road map for learning about the computational/general theory of modular curves/isogenies of abelian varieties for cryptography
I am a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack ...
- 41
0
votes
1
answer
750
views
A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?
I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...
- 3,084
4
votes
1
answer
489
views
Cohomology of finite symmetric products of manifolds
Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
- 506
0
votes
1
answer
74
views
Non-recursive solution to expected size of set
Let $F(A)$ be a function on an ordered set $A = \{A_1, A_2, \dotsc, A_n\}$ that outputs a set $B$ such that the elements of $B$ are determined as follows.
$A_i\in B$ if $A_i > A_{i-1}, A_{i-2}\...
5
votes
0
answers
234
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
- 1,075
2
votes
0
answers
65
views
Coequalizers and pullbacks in $\infty$-topoi
In an $\infty$-topos, suppose we have two cartesian diagrams of the form
$$
\require{AMScd}
\begin{CD}
\overline{A} @>>> \overline{B} \\
@VVV @VVV \\
A @>>> B .
\end{CD}
$$
Let
$$
\...
- 66
2
votes
1
answer
217
views
Isomorphisms after tensoring with the identity in a monoidal category
Let us take the following assumptions: $\mathscr{M}$ a monoidal category, $X,Y,Z$ three objects in the category, and $f: Y \to Z$ a morphism. If the morphism
$$
\mathrm{id}_X \otimes f: X \otimes Y \...
3
votes
1
answer
550
views
Do CGWH spaces form an exponential ideal in Condensed Sets?
If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
user103549
2
votes
0
answers
93
views
Sobolev inequalities in weighted Sobolev spaces
My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that
$$ \int_\Omega |f|^p (x) w(x)...
- 749
1
vote
0
answers
193
views
Stochastic volatility model question
Let suppose that $S_t$ is a process defined as:
$$ \begin{cases}dS_t = \mu S_t\,dt+m(v_t)\,dW^1_t\\ dv_t = \mu_v(v_t)\,dt + \sigma_v(v_t)\,dW^2_t\end{cases}$$
where the two Brownian motions have ...
- 393
2
votes
1
answer
200
views
Laguerre polynomial and series
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Consider the sum
$$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$
where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$.
I have found this series ...
- 531
0
votes
0
answers
79
views
Projections to orthogonal complements of conditional expectations
For a conditional expectation from a C^* algebra A to a subalgebra B, we can form a positive projection $P:A\to A$ with image equal to $B$. Question: is $Id - P:A\to A$ a positive map?
- 1,143
1
vote
0
answers
162
views
Solution of an equation over free group
Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are ...
- 355
1
vote
1
answer
320
views
I want to ask about two recursive sequences
f(0) = 1
g(0) = 1
f(x+1) = ((f(x))^2 + (g(x))^2)
g(x+1) = f(x)g(x)
Is it possible to find a closed form for these two functions?
- 11
1
vote
1
answer
293
views
Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
- 501
4
votes
1
answer
479
views
Is there an algorithm to generate graphs with given order and diameter?
I saw a question on the nauty emailing list without receiving any response, and it's something I've encountered in my own research as well. I am currently interested in graphs with diameter 3.
I ...
- 1,851
3
votes
1
answer
153
views
Reference request: the free adjunction being free as an $(\infty, 2)$-category?
This question is a particular case of Tim Campion's question.
Let $\mathrm{Adj}$ be the strict $2$-category corepresenting adjunctions, i.e., the free strict $2$-category generated by two objects $x, ...
- 424
4
votes
0
answers
108
views
Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
- 1,075
4
votes
2
answers
262
views
Symmetric monoidal functors from powers of the natural numbers to Set
Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable infinite sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between ...
- 563
4
votes
2
answers
731
views
Obstructions to the existence of a flat connection on a vector bundle
Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
1
vote
0
answers
109
views
Computing Grothendieck group of (unnodal) Enriques surface
Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
- 317
4
votes
2
answers
1k
views
Wasserstein distance between product measures
For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let
$$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$
denote the $p^\text{th}$ Wasserstein distance between $\mu, \nu$, where the ...
- 407
1
vote
0
answers
67
views
Random matrix theory: accounting for mean
Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
5
votes
0
answers
225
views
Quotient of a $F_n$ group which is $F_n$
It is known that quotients of finitely generated groups are finitely generated and that the quotient of a finitely presented group is finitely presented iff the normal subgroup is the normal closure ...
- 911
1
vote
0
answers
122
views
Smooth action on cotangent space of the plane
Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by
$$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$
acts via ...
- 191
6
votes
2
answers
343
views
Is there a notion of a complex/analytic diffeological space?
I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
- 295
1
vote
0
answers
102
views
Burgers' equation with viscosity: modulational analysis and energy estimates for large data
I think my question applies to many PDE that admit stationary solutions or travelling waves. I will simply describe the setting that I am more familiar with, but the technique is standard and applies ...
- 1,075
4
votes
0
answers
168
views
Representation rings of disconnected reductive groups
Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of ...
- 413
4
votes
0
answers
119
views
Is the pushforward of an exponentiable fibration along an exponentiable fibration again exponentiable?
Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold:
The pullback functor $p^*\colon \...
- 9,491
1
vote
0
answers
62
views
Nonintersecting witnesses of connectivity events in graphs
In my research I stumbled across a following result:
Let $G = (V, E)$ be a multigraph with three chosen vertices $a, b, c \in V$. We color its edges into red and blue colors: $E = R \sqcup S$. Events ...
10
votes
2
answers
871
views
When are two semidirect products of two cyclic groups isomorphic
(I have posted this question in Math Stack Exchange, only to have received no answer.)
It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
C_m \rtimes_k C_n ...
- 525
1
vote
1
answer
87
views
Topological modules over a locally compact ring
Let $R$ be a locally compact, separably metrizable ring (commutative with an identity) and let $M$ be a closed submodule of $R \oplus R$. Is the projection of $M$ onto the first coordinate closed?
- 42.8k
0
votes
0
answers
76
views
Numerical method for mixed system of equations and nonlinear inequalities
I am currently encountering challenges in determining the solution method for the following system of equations and inequalities:
$$
\begin{aligned}
&F(x) = 0\\
&G(x) < 0\\
\end{aligned}
$$
...
- 1
0
votes
1
answer
39
views
Is the right-hand term of the autonomous dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, and ...
- 109