Trending questions
159,029 questions
2
votes
0
answers
85
views
Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
- 719
1
vote
0
answers
96
views
Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
- 111
2
votes
0
answers
34
views
Maximum number of connected components of surfaces in three dimensions, what is known?
Part of Hilbert's 16th problem is:
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the ...
- 21
3
votes
3
answers
403
views
Huygens' principle or finite speed of propagation?
I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.
For context, $v(t,r)$ ...
- 890
-3
votes
0
answers
73
views
Is this a conclusion group for a new fundamental geometry problem? [closed]
Let σ(n) represent all possible values of the types of different lengths of segments connected to each other among n points in the definition,such as in the plane,σ(3)=(1,2,3),σ(3)min=1,in the ...
5
votes
1
answer
238
views
Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 53
3
votes
1
answer
139
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+50
Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 380
2
votes
0
answers
93
views
Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?
Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
- 249
6
votes
1
answer
648
views
Is decomposability of polynomials over a field an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 310
0
votes
0
answers
133
views
A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
2
votes
0
answers
89
views
Non-functoriality of transition maps between $\kappa$-condensed sets
I am currently following Scholze's lecture notes (https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) on condensed mathematics. We would like to define condensed sets as sheaves on the site $\...
- 121
5
votes
3
answers
343
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
1
vote
0
answers
136
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
2
votes
1
answer
149
views
Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 985
4
votes
0
answers
92
views
Transferring $A_\infty$-structure from a module to its homology
Given an $A_\infty$-module $M$, which is a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_k$ with morphisms $m_n^M\colon A^{\otimes(n-1)}\otimes M\rightarrow M$ of degree 2-n satisfying the $A_\infty$ ...
- 213
0
votes
1
answer
69
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
8
votes
0
answers
363
views
Closed formula for the factorial over reals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on real numbers, powers of real numbers, and fixed real numbers?
Similar question have been asked ...
- 19.1k
3
votes
0
answers
92
views
+100
Can I get a spherical coordinate from a real cocycle?
The Setting
I am currently working on a project in Topological Data Analysis (TDA), where I aim to construct a density-robust spherical coordinate associated with a dataset $X$, sampled from a ...
- 71
-1
votes
0
answers
58
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Solving special multivariable limits by Euclidean geometry
General Problem: Inspect function $L(a)$ for $a\in \mathbb{R}^{n}$, given that:
$$L(a)=\lim_{x\to 0_{+}}\frac{x^{a}}{F(x)}$$
Notation legends:
$x=(x_1,\ldots,x_n),\ a=(a_1,...,a_n),\ x^a=x_{1}^{a_1}\...
- 231
3
votes
0
answers
50
views
Martingale problem for the Wiener process
Consider $\Omega \triangleq \mathbf{C}([0,T];\mathbb{R})$, $\mathbf{F} \triangleq \mathbf{B}(\mathbf{C}[0,T];\mathbb{R})$ (Borelian $\sigma$-algebra) and $\mathbf{F}_t \triangleq \sigma \left \{ W_s, \...
- 395
0
votes
0
answers
91
views
Integral form of linking number
I am reading the paper "Gapless Floquet topology" by Cardoso et al and the following section got me confused.
I understand that now $F$ lives in a space where a 2 dimensional subspace is ...
- 101
4
votes
1
answer
170
views
Diamonds at $\omega_2$ under PFA
Let $\lambda$ be a cardinal, and $S\subset \lambda^+$ be stationary. A $\diamondsuit^+(S)$-sequence is a sequence $\langle \mathcal{A}_\alpha\mid \alpha\in S\rangle$ such that each $\mathcal{A}_\alpha\...
- 61
7
votes
0
answers
148
views
Cardinal characteristics and $\mathfrak{c} < \aleph_\omega$
Let $\mathsf{R}$ denote some finitely many relations about finitely many cardinal characteristics (e.g. $\mathfrak{a} \leq \mathfrak{s}$, $\mathfrak{a} < \mathfrak{d} = \mathfrak{r}$, $\mathfrak{b} ...
- 1,442
1
vote
1
answer
89
views
Simple modules of the Weyl algebra
Let $\mathbb{F}$ be a field and W be the Weyl algebra, as the algebra over $\mathbb{F}$ generated by $a,b$ with relation $ab-ba=1$.
The description of simple modules over the Weyl algebra over ...
- 467
-4
votes
0
answers
212
views
Can a mathematics research paper have just propositions & corollaries? [closed]
I am writing a research paper, in which I am proving some properties of new convolution operation $\star$ for some transform, like linearity, associativity, commutativity,distributivity, shift ...
4
votes
1
answer
275
views
Maximum density of sum-free sets with respect to Knuth's "addition"
A subset $S\subseteq\mathbb{N}$ is said to be sum-free if whenever $s,t\in S$, then $s+t\notin S$. For instance the set of odd numbers is sum-free and has (lower and upper) asymptotic density 1/2.
...
- 48.9k
8
votes
1
answer
849
views
What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
- 35.5k
-3
votes
1
answer
104
views
when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
1
vote
0
answers
111
views
Zariski Connectedness Theorem: From Analytic & Topological Viewpoint
Let $p:Y \to X$ be a proper, flat (see later why) surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ ...
- 6,038
3
votes
0
answers
140
views
A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$
where
$$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
- 531
3
votes
1
answer
316
views
Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
- 4,418
4
votes
1
answer
89
views
Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
- 1,124
8
votes
1
answer
230
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
6
votes
1
answer
2k
views
Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
- 95
17
votes
1
answer
1k
views
Are integers conservatively embedded in the field of complex numbers?
I am looking for a reference to the fact that $\mathbb{Z}$ is conservatively embedded into the field $\mathbb{C}$ of complex numbers, that is anything in $\mathbb{Z}$ which is definable in $(\mathbb{C}...
- 301
2
votes
0
answers
134
views
Effective Bombieri-Lang conjecture
The Bombieri-Lang conjecture is the following well-known conjecture:
Let $X$ be a projective variety defined over a number field $K$. Suppose that $X$ is general type. Then $X(K)$, the set of $K$-...
- 26.9k
3
votes
1
answer
142
views
Whitney stratifications of hypersurfaces
Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$
Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a ...
- 55
0
votes
0
answers
32
views
question about some algebraic simplifications performed as we solve differential equations with Laplace transform
I am trying to follow this discussion of Laplace transforms on youtube:
https://www.youtube.com/watch?v=ofvkZXgbIxE&t=610s
The relevant portion is 10 minutes in to the video.
There is some algebra ...
- 101
-5
votes
0
answers
83
views
Every smooth function contains a bijection [closed]
Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
1
vote
0
answers
31
views
Primary invariants on MAGMA for a graded ring
I have asked this question on mathstacks, but a collegue of mine recommended me to post it here.
I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
- 11
-4
votes
0
answers
25
views
Instrumental Variable model prove inequality holds [closed]
very stuck on this proof for my homework, professor didn't really teach this concept as it was supposed to be "self-learning", so I'm not really sure where to start.
homework problem
3
votes
0
answers
42
views
When does a lax monad morphism induce a functor between categories of algebras that preserves reflexive coequalisers?
Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is ...
- 10.7k
4
votes
0
answers
190
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 150
0
votes
1
answer
71
views
Formula for $P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \mathbb{N}_+} \left( \prod_{i=1}^m k_i^{a_i} \right) $
Let $\mathbb{N}_+ = \{ 1, 2, \dots\} $. For a given sequence of elements $\{a_i \}_{1 \leq i \leq m} $in $ \mathbb{N}_+ $, we define
\begin{equation}
P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \...
- 89
1
vote
1
answer
85
views
Question on gamma matrices
Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
- 1,171
7
votes
1
answer
203
views
Is there a more natural way to define the Young symmetrizer and the Specht module?
It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.
For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
3
votes
1
answer
88
views
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
- 833
5
votes
0
answers
122
views
+50
Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 269
3
votes
1
answer
158
views
How to maximize the variance of a subset of integers?
$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
- 41
2
votes
1
answer
111
views
Second order differential equation with non constant coefficient
Is it possible to solve the differential equation for $y(t)$ the following ODE?
$$
y^{\prime \prime}(t)+ \frac{f^{\prime}(t)}{2 f(t)} y^{\prime}(t) + k^{2} y(t) = 0
$$
It can also be rewritten as
$$
\...
- 45