All Questions
153,439
questions
2
votes
1
answer
238
views
Estimating probability that a large sum of i.i.d variables is positive
Let $X$ and $Y$ be i.i.d. random variables with exponential distribution with mean $1$, and let $Z=(X-1)(Y-X)$. Let $Z_1,...,Z_n$ are i.i.d. copies of $Z$, and let $f(n)=P[\sum_{i=1}^n Z_i > 0]$. ...
- 743
0
votes
0
answers
64
views
Sufficient conditions for measurability: a collection of distributions
Let $(\Omega,\Sigma,\mathbb{P})$ be the underlying probability space. Suppose we have a collection of probability density functions $F(\cdot;x)$ parameterized by $x\in[0,1]$.
Let $X$ be a random ...
- 383
2
votes
0
answers
203
views
How to judge the solution process of an SDE to lie on the sphere?
Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...
- 261
9
votes
1
answer
297
views
Reference for Schur multiplier identity
Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}_2(G/H,\mathbb{Z}) \cong \frac{\...
- 93
3
votes
0
answers
176
views
Abelianess of $K(j(E))$
Let $E$ be an elliptic curve with CM by an order in the imaginary quadratic field $K$. Is there some easy way how to prove that the extension $K(j(E))/\mathbb Q$ is abelian?
Update
In general, the ...
- 2,335
2
votes
0
answers
127
views
Is there a Hodge structure on $\text{Hom}(V,W)$?
Let $V, W$ be real (pure) Hodge structures of weight $m, n$. Is there a natural Hodge structure on $\text{Hom}(V,W)$?
As I understand, there is one in the case $V = W$, although the definition I ...
- 4,124
5
votes
0
answers
229
views
The bridge index and crookedness of a knot
I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much ...
- 492
4
votes
1
answer
335
views
$X(\mathbb{Z}/p\mathbb{Z})$ versus $\{X(\mathbb{Z})\pmod{p}\}$
Let $P_1$,...,$P_m$ be polynomials in $n$ variables with coefficients in $\mathbb{Z}$ and consider the set
$$X(\mathbb{Z})=\{(x_1,...,x_n)\in \mathbb{Z}^n \ |\ P_i(x_1,...,x_n)=0 ~ ,\ \forall i \in\{1,...
- 1,479
1
vote
1
answer
131
views
Random optimization problem
Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...
- 1,677
8
votes
2
answers
949
views
History of the kernel of a homomorphism?
This previous question traces the notion of group homomorphism to Jordan (1870) and the term "homomorphic" to Fricke and Klein (1897) and to earlier lectures of Klein:
Whence “homomorphism” and “...
- 3,741
3
votes
1
answer
131
views
Factoring $x^p H(x) + x^q B(x) + T(x)$ over a finite field
$\newcommand\S{\mathcal S}$
Let $p$ be a prime, and suppose that $0.9(p-1)<q<p-1$. Suppose, furthermore, that $H,B,T\in\mathbb F_p[x]$ satisfy $\max\{\deg H,\deg T\}\le q-1$ and $\deg B\le p-1-q$...
- 22.8k
10
votes
0
answers
451
views
Ccc forcings and measurable cardinals
Suppose $\kappa$ is measurable with a witnessing normal ultrafilter $U$ and $P$ is a ccc forcing. Let $\langle p_i: i < \kappa \rangle$ be a sequence of conditions in $P$ such that for every $X \...
- 9,781
3
votes
0
answers
192
views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
user138661
15
votes
2
answers
966
views
Is an HNN extension of a virtually torsion-free group virtually torsion-free?
This is a cross post from Math.StackExchange after 2 weeks without an answer and a bounty being placed on the question.
Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free ...
- 835
10
votes
0
answers
306
views
Understanding a monad from its fixed points
Let $(T, \eta, \mu)$ be monad over $\mathsf{C}$.
And let $\iota : \mathsf{Fix}(T) \hookrightarrow \mathsf{C}$ be the inclusion of the full subcategory of fixed points of $T$. By the universal ...
- 8,186
4
votes
0
answers
239
views
Proper and flat morphism implies finitely presented?
I´ve been reading the Deligne-Mumford construction of the moduli of curves with a given genus and I have some questions about the article http://www.numdam.org/article/PMIHES_1969__36__75_0.pdf
1) ...
- 439
3
votes
1
answer
301
views
Is there a linearly Lindelof space which is not weakly Lindelof?
Recall that a space is:
"Lindelof", if every open cover has a countable subcover.
"Linearly Lindelof", if every open cover which is linearly ordered by $\subseteq$ has a countable subcover.
"weakly ...
- 4,343
6
votes
0
answers
128
views
Q-analogue of an inequality
Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$.
It is not super-difficult to prove the inequality
$$
\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.
$$
This is actually quite a nice inequality that was ...
- 15.6k
6
votes
0
answers
187
views
Computing the difficult integral $\int_0^\infty J_0(x)^4\log(x)dx$
Computing numerically integrals of oscillating functions from $0$ to $\infty$ is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to ...
- 11.8k
0
votes
0
answers
54
views
Do all induced subgraphs of powers of cycles have a perfect matching
Do all independence induced subgraphs of powers of cycles have a distinct 1-factor? By independence induced, I mean those induced subgraphs which are formed by removing a maximal independent set of ...
- 2,027
3
votes
1
answer
218
views
Maximal disjoint collections and matrix rank
First, a combinatorial question fit for an undergrad course. Say I have a collection $\mathcal{C}$ of non-empty subsets of $S=\{1,\dotsc,n\}$ such that every element of $\mathcal{C}$ has at most $k$ ...
- 19.4k
3
votes
1
answer
186
views
Automorphism of ruled surfaces associated to stable vector bundles
Let $X$ be a compact Riemann surface, and let $P \rightarrow X$ be a holomorphic $\mathbb P^1$-bundle over $X$. Then we know that $P$ is of form $\mathbb P(E)$ for some vector bundle $E \rightarrow X$ ...
- 713
4
votes
0
answers
103
views
Borel selections of usco maps on metrizable compacta
The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
- 40.9k
3
votes
1
answer
134
views
The "semi-symmetric" algebra of a vector space
If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
1
vote
1
answer
423
views
Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
$g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
$n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
- 161
2
votes
0
answers
184
views
Stabilizing an open book with Anosov piece
It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
- 1,379
2
votes
1
answer
288
views
Flatness of submodules of free modules
Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
- 5,851
2
votes
1
answer
240
views
Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
- 1,816
6
votes
3
answers
710
views
Want to prove an inequality
I want to show that
$9*\left[\frac{xy}{x+y}-q(1-q)\right]-12*[xy-q(1-q)]+(1-q-x)^{3}+(x+y)^{3}+(q-y)^{3}-1\geq0$ where
$0<q<1$
$0<x<1-q$
$0<y<q$
$(x+y)\left[1+max\{\frac{1-q}{...
- 177
5
votes
2
answers
1k
views
Cardinality of certain subsets in vector spaces over finite fields
Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
- 228
4
votes
0
answers
66
views
Can a nonlinear dynamical system be rewritten in terms of constraints?
My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. ...
- 141
4
votes
1
answer
219
views
specific modules over the Steenrod algebra with one generator
I'd be happy to clarify the following. Consider the module which is a quotient of the Steenrod algebra mod $2$ by the left ideal generated by $\operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{...
- 171
14
votes
1
answer
1k
views
Floor of Riemann zeta function
How to show that $$\left\lfloor\zeta\left(1+\frac{1}{n}\right)\right\rfloor=n$$ for every positive integer $n$?
- 537
3
votes
0
answers
223
views
Locality in Floer theory
There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as '...
- 11
2
votes
1
answer
371
views
Feynman-Kac formula for lattice heat equation with non-diagonal potential
Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\...
- 891
3
votes
0
answers
135
views
Milnor Number of real and imaginary parts of holomorphic germs?
By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor ...
- 73
10
votes
1
answer
318
views
Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.5k
3
votes
1
answer
152
views
A rationality problem about $F$-points in tori
Let $F$ be a finite field, and $T$ be a torus over $F$. Assume that $T_1,T_2$ are two $F$-subtori of $T$, such that $T_1 \times T_2 \to T,(t_1,t_2) \mapsto t_1 t_2$ is surjective with finite kernel $K$...
- 363
4
votes
1
answer
176
views
Uniform boundedness of the number of number fields having fixed discriminant
Let $d$ be an integer. It is a well-known theorem, attributed to Hermite and Minkowski, which asserts that the number of number fields $K$, allowed to have any degree over $\mathbb{Q}$, having ...
- 25.5k
4
votes
1
answer
136
views
Linear combination of coordinates of random unit vector
Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of
$$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$
Does it happen ...
- 1,475
1
vote
0
answers
152
views
Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true
The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$:
Theorem (Hermite-Lindemann-Weierstrass): Let $\...
- 25.5k
2
votes
1
answer
681
views
Motivation and examples of parabolic manifolds
Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...
- 371
5
votes
1
answer
212
views
Zero divisors in compact quantum groups
Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the polynomial Hopf algebra Pol$(\...
- 949
1
vote
0
answers
63
views
Non-Constant Solutions of Nonlinear Elliptic PDE System
Apologies if this is a simple question, but I've left PDE as a field and a friend recently asked me the following question regarding solutions $u$ and $v$ of a system of PDE. Consider
$$ \nabla \cdot (...
- 111
4
votes
1
answer
374
views
Homotopy limit over a diagram of nullhomotopic maps
Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit
$$
\underset{i \in I}{\mathrm{holim}}X(i),
$$
where the maps $X(i) \to X(j)$ are ...
- 751
3
votes
0
answers
348
views
Existence and uniqueness for reaction-diffusion equations
I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$
\begin{align*}
&\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\
& u(0)=u_0\in L_2
\end{align*}
where the ...
- 921
1
vote
2
answers
605
views
A curve is proper iff the space of global sections is finite-dimensional
Let $k$ be a field, $X\rightarrow \mathrm{Spec}\,k$ be a separated morphism of finite type of relative dimension$\leq 1$ (as defined here). Is it true that $f$ is proper iff $f_* \mathcal{O}_X$ is ...
- 11
10
votes
1
answer
233
views
Is there a proof of the Hawking bound for the efficiency of a black holes merger?
Consider two black holes with masses $m_1,m_2$ and zero angular momenta
merging to form a single one with the mass $m$ and the rotation parameter $a=J/m$. Hawking, in "Black Holes in General ...
- 6,861
2
votes
1
answer
182
views
Generalization of du Bois-Reymond Lemma into dimension 2?
The du Bois-Reymond lemma reads as follows:
Let $ f \in L^1 (a,b) $ satisfies
\begin{equation*}
\int^b_a f(t) \varphi'(t) dt =0, \ \ \forall \varphi \in C^{\infty}_0(a,b),
\end{equation*}
...
- 267
1
vote
0
answers
113
views
Some doubt on crossed product von Neumann algebras
There are two definitions in different books. Let $G \curvearrowright M$, then there is the definition of forming Group ring $M[G]$, define product and addition then make its algebra. represent the ...
- 667