Newest Questions
159,021 questions
1
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135
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What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
- 1,037
2
votes
0
answers
91
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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
8
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0
answers
344
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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 ...
- 19k
-4
votes
0
answers
24
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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
7
votes
1
answer
210
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Size doubling amoeba
Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
- 6,285
3
votes
0
answers
42
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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
-3
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28
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Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
- 1
7
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146
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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
-4
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212
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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 ...
3
votes
3
answers
399
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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, $v(t,r)$ ...
- 870
1
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0
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25
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On $N$-partition of some common subsets $\Omega\subset\mathbb R^d$
Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.
In this post we set $d=...
- 1,399
1
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0
answers
129
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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
3
votes
0
answers
50
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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
3
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1
answer
128
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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
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0
answers
133
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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
-5
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answers
82
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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 ...
9
votes
1
answer
326
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An elementary proof of the equivalence of the Bol and Moufang identities
By a well-known result of Bol (1937) and Bruck (1946), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 42k
0
votes
0
answers
53
views
Special determinant formula
Consider two column vectors $\textbf{a}$ and $\textbf{b}$ of length $k$ and $m$ respectively, $km$ variables denoted $y_{i,j}$ (i=1 to k, j=1 to m), and a quadratic form $\textbf{y}^{T}\mathbb{M}\...
- 419
-1
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0
answers
45
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Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
4
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1
answer
82
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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
16
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3
answers
1k
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Is there a natural topology for sets of topological spaces?
The Gromov–Hausdorff metric makes a set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
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1
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104
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when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
0
votes
0
answers
85
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Is there a symbolic computation program that can deal with differential forms, Stokes theorems, the Hodge * operator, etc
I am looking at a messy series of computations of integrals of differential forms on manifolds with boundary, involving repeated application of Stokes' theorem, and also involving the Hodge * operator....
- 1
30
votes
1
answer
3k
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Closed formula for the factorial over naturals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on natural numbers, powers of natural numbers, and fixed natural numbers?
The same question over the ...
- 19k
-1
votes
0
answers
23
views
A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 49
1
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0
answers
31
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Primary invariants on MAGMA for a graded ring
I have asked this question on mathstacks, but a collegue of mine recommended me to post it here.
I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
- 11
0
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0
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119
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A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$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}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
3
votes
0
answers
140
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A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\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
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votes
0
answers
61
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Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
2
votes
1
answer
143
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 985
1
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0
answers
63
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Construction of the smallest nucleus above a prenucleus: what does this proof tell us?
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is implicit:
...
- 32.5k
5
votes
1
answer
238
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Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 53
3
votes
0
answers
89
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+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
3
votes
0
answers
60
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What upper bounds are known, for the number of divisors of Mersenne numbers?
Short version. What upper bounds are known, for the number of divisors of Mersenne numbers?
Long version.
Studying the structure of the factors of $M_n = 2^n - 1$ appears to be an active and difficult ...
- 1,444
2
votes
0
answers
89
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
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0
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0
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38
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Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
1
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0
answers
59
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Minimum number of sets for a union-closed sets conjecture counterexample
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 1,000
0
votes
1
answer
71
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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
2
votes
0
answers
52
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Can we bound the degree of a one dimensional smooth compact leaf of a holomorphic foliation in terms of its genus?
Let $X$ be a smooth projective variety over the complex numbers with a fixed ample line bundle $H$. Suppose that $\cal F$ is a foliation in curves over $X$ (which may be singular).
Can you find a ...
- 388
1
vote
0
answers
49
views
Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,207
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
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.
...
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-5
votes
0
answers
45
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New Framework for Handling Undefined Mathematical Operations: Seeking Comments and Suggestions [closed]
The framework addresses undefined mathematical operations like zero raised to the first power (0^1) and division by zero (1/0) by introducing definitions that include infinity as the inverse of zero. ...
- 1
13
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
-5
votes
0
answers
82
views
Research without context is no research? [closed]
I was wondering what is research in mathematics. Can it be considered research only the research in open problems or it can be considered research also the finding of new formulas, new classes of ...
4
votes
0
answers
187
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
5
votes
3
answers
340
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
0
votes
0
answers
116
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
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
0
votes
1
answer
77
views
Newton method for polynomials with random starting points
I know that this question exists, but unfortunately it doesn't cover my issue sufficiently.
Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
- 1,171