All Questions
153,398
questions
1
vote
1
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155
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Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?
Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...
- 413
3
votes
0
answers
124
views
multiplicity of components in resolution of surface singularity
Let $X$ be a smooth compact algebraic surface, and $C=\bigcup E_i$ be a connected divisor where the $E_i$ are integral and the matrix $(E_i\cdot E_j)$ is negative definite. In the analytic category, ...
- 735
13
votes
3
answers
3k
views
Differentiability of Eigenvalues - Perturbation Theory
first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
- 138
2
votes
2
answers
448
views
These polynomials are always either even or odd [duplicate]
The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by
$$Ef(x)=f(x+1) \qquad \text{and} \qquad \...
- 41.8k
5
votes
2
answers
482
views
Gaussians at lattice points
Let $\epsilon > 0$. I would like to know if there exists $c < \infty$ such that for all $d \in \mathbb{N}$ the following holds. If $x \in \mathbb{R}^d$ let $N_x$ be the standard Gaussian ...
- 1,689
2
votes
1
answer
205
views
Flatness of Fano Contractions
In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the ...
user69183
1
vote
0
answers
274
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If $R$ is UFD , then does $R \cong R[X,Y]$ imply $R \cong R[X]$?
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$.
My questions are: Is it possible ...
user111524
3
votes
0
answers
195
views
Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
- 1,949
8
votes
1
answer
231
views
Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?
I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
- 2,031
1
vote
0
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134
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On the solution of Laplace equation with mixed boundary condition
Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system
$$
\begin{...
- 363
2
votes
1
answer
151
views
A particular separation example
Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...
- 3,972
6
votes
0
answers
239
views
Dowker and neighborhood complexes: reference wanted
Let $R$ be a 0-1 matrix whose rows or columns are maximal.
Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)?
From 0-1 matrix corresponding to an abstract simplicial ...
- 15.3k
2
votes
2
answers
413
views
Euclidean model structure on multipointed $d$-spaces
I use the notation of this question. A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $...
- 3,759
1
vote
1
answer
271
views
Variation of global sections of line bundles
The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
- 1,949
6
votes
3
answers
1k
views
Where to find the results of Onishchik?
I would like to have a good reference where the results in
"Inclusion relations between transitive compact transformation groups" https://mathscinet.ams.org/mathscinet-getitem?mr=27:3740
can be ...
- 73
3
votes
0
answers
610
views
Diagonal elements of Hermitian matrices with given eigenvalues
Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
- 31
1
vote
0
answers
162
views
Integral model of etale covering
Let $K$ be a $p$-adic field whose residue field $k$ is algebraically closed. Let $X$ be a hyperbolic curve over $K$, what I mean by a curve is a smooth geometrically connected scheme of dimension one ...
- 1,378
3
votes
1
answer
498
views
Is the linear span of irrep matrices a complete matrix basis?
Let $G$ be a finite or compact group and $\rho: G \to \mathrm{U}(d)$ a $d$-dimensional unitary representation of $G$. If $\rho$ is irreducible then the following seems to be true:
$$
\mathrm{span}_\...
- 489
9
votes
0
answers
182
views
$p$-groups and the arithmetic of $p$
I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
- 5,200
3
votes
1
answer
216
views
Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$
Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...
- 13.6k
9
votes
0
answers
285
views
Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
- 1,534
-2
votes
2
answers
432
views
Can there be an upper bound on definability of cardinal numbers in ZF? [closed]
Is there a known result to the effect that it cannot be the case that for some natural $n$, there is a formula of length $n$ such that all cardinals can be defined by a formula whose length is shorter ...
- 9,266
1
vote
1
answer
410
views
Non-invertible Karp reduction
Karp (many-one) reducibility between $NP$-complete problems $A$ and $B$ is defined as a polynomial-time computable function $f$ such that $a \in A$ if and only if $f(a) \in B$. Berman-Hartmanis ...
- 2,951
5
votes
1
answer
277
views
Existence of geodesic convex functions
By a result of Shing-Tung Yau [1974, Mathematische Annalen 207: 269-270], there are no non-trivial continuous geodesic convex functions on complete manifolds with finite volume.
What happened if we ...
- 1,991
5
votes
1
answer
281
views
Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
- 309
2
votes
0
answers
180
views
Well-definedness of Guillemin and Pollack's $df_x$
In Guillemin and Pollack, given a smooth map $f:X\to Y$ on smooth manifolds, parametrizations $\phi:U\to X$ and $\psi:V\to Y$ in the commutative diagram
$$\begin{array}{ll}
X & \overset{f}{\to} &...
- 151
2
votes
1
answer
352
views
Daniell integral vs. Borel measure
Let $X$ be a locally compact, Hausdorff, topological space and denote by $\mathcal B_+(X)$ the collection of all Borel-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals).
...
- 2,233
1
vote
0
answers
86
views
An equality of inner products of holomorphic curves
The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The ...
- 91
4
votes
1
answer
374
views
Gaps between roots of consecutive Hermite polynomials
Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials.
It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple ...
- 1,771
3
votes
0
answers
84
views
Hunting for holomorphic functions part 2: this time with non-local boundary condition
This is a follow-up to Looking for holomorphic function on a sector with specified boundary behavior. There I was looking for a holomorphic function on a sector with real boundary condition on one ...
- 1,569
2
votes
0
answers
231
views
Is there any generalization of Weil conjecture for non-smooth variety?
Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
- 806
6
votes
2
answers
321
views
Cocycle description of gerbes
I am trying to understand cocycle description of gerbes as in https://arxiv.org/pdf/math/0611317.pdf.
Let $\mathcal{P}$ be a gerbe on a topological space $X$ i.e., $\mathcal{P}$ is a stack over ...
- 5,931
4
votes
1
answer
410
views
Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
user111524
0
votes
1
answer
163
views
Unique continuation for the wave equation
Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation
$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$
where $f$ and $1-c$ ...
1
vote
1
answer
388
views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
- 19
2
votes
1
answer
149
views
Boundedness of Dirac deltas
Suppose that $X$ is a metric space and let $C_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C_k(X)$ is a topological vector space. Let ...
- 25
9
votes
1
answer
737
views
Symmetric Powers for Lie Algebras
Let $V_\lambda$ be the irreducible representation of $sl_{n}(\mathfrak{C})$ with highest weight $\lambda$. There are well known formulas for the decomposition of $V_\lambda^{\otimes^k}= V_\lambda\...
- 93
5
votes
0
answers
239
views
On the Choice Content of Carathéodory's Conformal Mapping Theorem
The Schoenflies theorem, as a variant of the well-known Jordan curve theorem, states that the interior and the exterior planar regions determined by a simple closed curve (aka Jordan curve) in $\...
2
votes
0
answers
96
views
An extremal sum for hypergraph degrees
Consider a rank-$r$ hypergraph $H = (V,E)$. I would a lower-bound of the following form:
$$
\sum_{e \in E} \frac{ \sum_{v \in e} \text{deg}(v) }{\max_{v \in e} \text{deg}(v)} \geq c \sum_{v \in V} \...
- 3,407
1
vote
2
answers
604
views
Is Weierstrass function nowhere pointwise Lipschitz?
Consider the classical Weierstrass function
$$
W(x)=\sum_{n=1}^\infty \frac{e^{i2^nx}}{2^n}.
$$
It is a well-known result that this function is nowhere differentiable (Hardy, TAMS 1916, Thm 1.31). In ...
- 126
4
votes
1
answer
202
views
If $T_1T_2 = T_2T_1$, why $r(T_1 + T_2) \leq r(T_1) + r(T_2)$?
Let $T_1$ and $T_2$ be two bounded linear operators in a complex banach space $X$.
If $T_1T_2 = T_2T_1$, I want to know how to show that
$$
r(T_1+T_2) \leq r(T_1) + r(T_2),
$$
where $r(A)$ ...
- 1,154
8
votes
0
answers
455
views
Lefschetz pencils and perverse sheaves
I have been reading BBD and Geordie Williamson's “An illustrated guide to perverse sheaves”. The latter has been tremendously helpful to get some intuition for the former.
Let $K$ be some field, and ...
- 5,444
5
votes
0
answers
128
views
Do non-constant maps specialize to non-constant maps?
Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.
Is the ...
- 51
1
vote
0
answers
343
views
approaching the border between absolute convergence and divergence of series
Let us consider absolute convergent series $\ell^{1^+}$ ordered under eventual dominance (mod finite) $<^*$. T. Bartoszynski proved that unbounded number ${\frak b}(\ell^{1^+}, <^*)$ equals ...
- 88
3
votes
1
answer
203
views
Discrete spectrum of Schrodinger operator
Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...
- 1,905
14
votes
0
answers
478
views
If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 103k
8
votes
2
answers
708
views
Lower bound on the entries of the Perron vector
Let $A$ be a matrix that satisfies all the conditions of Perron- Frobenius theorem. From the theorem it is known that the entries of the eigenvector corresponding to the largest eigenvalue will be ...
- 183
1
vote
1
answer
215
views
Connected Hausdorff spaces with different cardinalities of open sets
Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| ...
- 45.5k
1
vote
1
answer
188
views
Approximation of constructible abelian étale sheaves
Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$.
Suppose $X = \varprojlim X_j$, ...
user124171
3
votes
1
answer
188
views
Projective embeddings and quasi-compactness
Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding.
Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...
user124171