All Questions
17,266
questions
7
votes
1
answer
931
views
Decomposition of induced representations in S_n
Let C be a cyclic subgroup of S_n.
How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components?
Is there are a way to know which ...
7
votes
1
answer
588
views
Failure of Shoenfield's Absoluteness
Shoenfield's absoluteness states that if $M \subseteq N$ are models of $ZF$ and $M \supseteq \omega_1^N$, then every $\Sigma^1_2$ formula with parameters in $M$ is absolute between $M$ and $N$. In ...
7
votes
0
answers
997
views
Isolated singularities and tangent cones
Assume that I have an affine hypersurface $X =V(f)\subset \mathbb{C}^4$ of degree $d$ with an isolated singularity of multiplicity $m$ at the origin $o=(0,0,0,0)$. Let $$f:=f_m + f_{m+1}+ \cdots +f_d$$...
7
votes
2
answers
914
views
Principal ideal ring, does there exist an invertible matrix such that certain matrix is upper triangular?
I asked here on Math Stack Exchange the following question.
Let $R$ be a principal ideal ring. If $A$ is any $p \times q$ matrix over $R$, then does there exist an invertible matrix $U$ in $\text{M}...
7
votes
1
answer
471
views
Abandoned notions in mathematics? [duplicate]
I'm looking for examples of abandoned or demised notions/concepts in mathematics, preferably (but not necessarily) after the age of foundations. To be clear: I'm not looking for abandoned ideas or ...
7
votes
1
answer
3k
views
Why we study Geometric invariant theory?
I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...
7
votes
1
answer
353
views
Singular Fisher information matrix and existence of unbiased estimators
I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
7
votes
1
answer
412
views
How big can a wedge of 2-forms be?
The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$
for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on
$\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
7
votes
4
answers
2k
views
Visualizing functions with a number of independent variables
I need to graph real valued functions (for exposition and analysis).
The issue is: there are more independent variables so that the conventional graphing methods can't be used, and furthermore I don't ...
7
votes
2
answers
1k
views
A curious martingale
Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely?
Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
7
votes
0
answers
480
views
"Nontrivial" singular points on the eigencurve?
Let $\mathscr{C}$ be the Coleman-Mazur-Buzzard eigencurve of some fixed tame level $N$. Are there any known examples of a singular point $x\in \mathscr{C}$ which lies in a unique irreducible ...
7
votes
2
answers
828
views
what is the number of paths returning to 0 on the hexagonal lattice
I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice.
I can find plenty on references on self avoiding paths, but I am looking ...
7
votes
1
answer
365
views
Selberg Zeta Function and Fenchel-Nielsen Coordinates
According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...
7
votes
1
answer
291
views
Separating unit disks by lines
Given $n\ge 2$. For a real $d>2$, consider a constellation $C$ of $2n$ disks of radius $1$ in the plane such that $h(C)$, the minimal distance between any two of their centers, is equal to $d$. Let ...
7
votes
0
answers
850
views
Optimal Gear Trains
Suppose you need to slow down a turning motor so that a gear turns at
an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and
$b$ are natural numbers. For example, this set of ...
7
votes
2
answers
302
views
Homotopicity of two certain sections of frame bundle of $GL(n,\mathbb{R})$
Edit: According to comment of Prof. GoodWillie we revise the question.
Put $M=GL(n,\mathbb{R})$.
We identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$:
The identification is based on the ...
7
votes
1
answer
1k
views
Quick reference for general Weyl's inequality in number theory
I would like a reference for the result here. Having that $t$ there makes me happy. I would prefer not to have to, in my paper, run through and (not trivially but not too greatly) alter the proof of ...
7
votes
1
answer
418
views
Proof of global Peano existence theorem in ZF?
By global Peano existence theorem I mean the existence of a maximal interval of solution of a first order ODE $x'=f(x,t)$ with continuous $f$.
The proofs of the global Peano Theorem found in the ...
7
votes
1
answer
3k
views
Integer solutions of x^n + y^n = z^{n-1}
This is related to another question
I am interested in the non-trivial integer solutions of
$$ x^n + y^n = z^{n-1} $$
for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$.
There are ...
7
votes
2
answers
781
views
measurable linear functionals are also continuous on separable Banach spaces?
It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.
7
votes
1
answer
762
views
Can $x^4+y^4+1$ be a perfect power?
Recall that a perfect power has the form $x^m$ with $m,x\in\{2,3,\ldots\}$. Motivated by Fermat's result that the equation $x^4+y^4=z^2$ has no positive integer solution, here I ask the following ...
7
votes
0
answers
428
views
Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?
I am sure this is written down somewhere but cannot find it.
Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...
7
votes
2
answers
406
views
Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and $\phi_{2}:\mathfrak{sl}_2(\mathbb{C})\rightarrow\...
7
votes
2
answers
311
views
Perfect matchings of a regular, uniform, partite hypergraph
This is in relation to the question here. What, if any, are the known conditions for the existence of a perfect matching for a $r$-regular, $r$-uniform, $r$-partite hypergraph. I specifically ...
7
votes
1
answer
557
views
Zeros of a combination of exponentials
Is there any known result about the necessary and sufficient conditions for the existence of zeros for a function $f(x)=\sum_{n=1}^{N} a_n e^{b_n x}$, where $a_n,b_n \in \mathbb{R}\, \forall n=1,2,\...
7
votes
3
answers
510
views
Trace of a nonlinear matrix equation (cont'd)
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
7
votes
2
answers
240
views
Double dual of free $\mathbb{Z}_{(p)}$-modules
For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D_{(p)}A=\text{Hom}(A,\mathbb{Z}_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $...
7
votes
1
answer
2k
views
The fibre product of two quotient stacks
My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times_S Y/G \times H]$? And if yes, do you have any reference for it?
Thank you.
7
votes
2
answers
748
views
Mass of spinor genus, positive integral quadratic forms
There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some ...
7
votes
1
answer
192
views
Trisecting $3$-fold sumsets, II: is the middle part ever thin?
This is a refined version of the question I asked yesterday.
Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[...
7
votes
2
answers
800
views
On the consistency of the definition of the conductor for automorphic forms
Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
By its ...
7
votes
3
answers
2k
views
Changing coordinates so that one Riemannian metric matches another, up to second derivatives
Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ ...
7
votes
1
answer
299
views
Large gaps between consecutive irreducible polynomials with small heights
For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then $N+2,\...
7
votes
3
answers
2k
views
Grover's Quantum Search Algorithm
I am confused about an extremely basic point concerning Grover's quantum search algorithm; my confusion suggests to me that maybe I've missed the entire point.
My understanding of the algorithm is ...
7
votes
1
answer
724
views
von dyck groups and solvability
A von Dyck group is a group with presentation $< a,b | a^m=b^n=(ab)^p=1 >$ with m,n,p natural numbers. Is it known which of these groups are solvable and which are not? Is there a reference ...
7
votes
1
answer
596
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
7
votes
1
answer
365
views
Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...
7
votes
1
answer
477
views
Moments of a random variable and of its conditional expectation
Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...
7
votes
2
answers
1k
views
Strata of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...
7
votes
2
answers
1k
views
Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
7
votes
2
answers
746
views
Probabilistic Interpretation of First Dirichlet Eigenvalue?
The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to
$$
-\Delta\psi = \...
7
votes
1
answer
675
views
Sources of Theorem drafts by the original author
When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always ...
7
votes
2
answers
514
views
The kernel of all invariant means
Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in ...
7
votes
2
answers
592
views
Will (general points + small number of arbitrary points) impose independent condtions on plane curves?
It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
7
votes
0
answers
191
views
longest path transversals
In a connected graph any two longest paths intersect at a common vertex. It is open whether any three longest paths in a connected graph intersect at a common vertex. For a connected graph $G$, let ...
7
votes
1
answer
406
views
Open cell decomposition after applying a Weyl group element
Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For Zariski-...
7
votes
1
answer
371
views
Optimum Tournament Strategy
Consider a symmetric N-player game in which all players partition one total unit of
energy among individual games. The probability of winning each game is simply proportional to the spent energy (...
7
votes
2
answers
2k
views
Arbitrary union of meagre open sets
Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you ...
7
votes
1
answer
189
views
$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?
Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
7
votes
1
answer
307
views
Is there a model-independent characterization of the gaunt strict $n$-categories amongst the weak $(\infty,n)$-categories?
Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. ...