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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 ...
Klim Efremenko's user avatar
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 ...
Ohad Drucker's user avatar
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 $$f:=f_m + f_{m+1}+ \cdots +f_d$$...
Francesco Polizzi's user avatar
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}...
Analysis Student's user avatar
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 ...
qk11's user avatar
  • 505
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 ...
riu_ss's user avatar
  • 87
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 ...
JNL's user avatar
  • 75
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!...
Mikhail Katz's user avatar
  • 15.1k
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 ...
ARi's user avatar
  • 841
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 ...
Nate River's user avatar
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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 ...
David Hansen's user avatar
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 ...
kaleidoscop's user avatar
  • 1,268
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{...
QGravity's user avatar
  • 969
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 ...
Wolfgang's user avatar
  • 13.2k
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 ...
Joseph O'Rourke's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
mathworker21's user avatar
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 ...
Mikhail Katz's user avatar
  • 15.1k
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 ...
joro's user avatar
  • 24.2k
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.
newbie's user avatar
  • 319
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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
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 ...
Wolfgang Loehr's user avatar
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\...
Peter Crooks's user avatar
  • 4,870
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 ...
theGrolarBear's user avatar
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,\...
nicodds's user avatar
  • 73
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) ...
Ludwig's user avatar
  • 2,682
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 $...
Neil Strickland's user avatar
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.
Kimra's user avatar
  • 131
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 ...
Will Jagy's user avatar
  • 25.3k
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 $[...
Seva's user avatar
  • 22.8k
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 ...
Desiderius Severus's user avatar
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 $K$ ...
Tom LaGatta's user avatar
  • 8,372
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,\...
Wolfgang's user avatar
  • 13.2k
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 ...
Steven Landsburg's user avatar
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 ...
dave's user avatar
  • 155
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}(...
ghc1997's user avatar
  • 763
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 ...
Iosif Pinelis's user avatar
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 ...
Hedonist's user avatar
  • 1,269
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 $\mathcal{G}$ ...
Peter Crooks's user avatar
  • 4,870
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 ...
Theo Johnson-Freyd's user avatar
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 = \...
quick_q's user avatar
  • 115
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 ...
ARG's user avatar
  • 4,342
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 ...
Drew's user avatar
  • 1,469
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 ...
David Wood's user avatar
  • 1,243
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-...
Jesko Hüttenhain's user avatar
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 (...
bobuhito's user avatar
  • 1,537
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 ...
Yvoz's user avatar
  • 73
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 ...
Giulia's user avatar
  • 73
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. ...
Tim Campion's user avatar
  • 60.6k

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