All Questions
152,900
questions
17
votes
2
answers
945
views
Isoperimetric-like inequality for non-connected sets
The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
3
votes
1
answer
478
views
Find recurrence in Pascal-like triangle of polynomials
Consider an infinite, upper-triangular Toepliz-matrix, with first row $x_1,x_2,\dots,x_n.$
Then there is a sequence of determinants obtained from the $m \times m-$ sub-matrices
with upper left corner ...
1
vote
0
answers
1k
views
Is there an upper bound to strongly connected components
Hey,
given $G = (V,E)$ a directed Graph. Is there any way to calculate the maximum number of strongly connected components $k$, only based on $n$ nodes and $e$ edges?
5
votes
1
answer
191
views
Expressing a element of a Matrix subgroup in terms of subgroup generators
I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
7
votes
2
answers
572
views
Pairs of Permutations up to Simultaneous Conjugation
The conjugacy classes of $S_n$ are the cycle types since if $\tau = (\dots)(\dots)\dots(\dots)$, the conjugation $\tau \mapsto \sigma \tau \sigma^{-1}$ permutes the labels in the cycles of $\tau$.
...
0
votes
1
answer
258
views
Number of points on a complex sphere with pairwise inner product restriction
Considered the following inner products:
$(1)$ $\langle x,y \rangle = \sum_{t=1}^{n}x_{t}y_{t}$
$(2)$ $\langle x,y \rangle_{c} = \sum_{t=1}^{n}x_{t}\bar{y}_{t}$
consider the following surfaces:
$\...
4
votes
4
answers
606
views
A question about the additive group of a finitely generated integral domain
Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I ...
2
votes
1
answer
256
views
A Volterra-type equation
Consider the following integral equation
$\phi(x) = f(x) + \frac{1}{x}\int_0^x N(x,y)\phi(y)\;dy$,
where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation ...
2
votes
1
answer
663
views
Different cuspidal automorphic representations with same representations at infinity
Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$).
Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$.
$$K=\Pi_{v<\infty}K_v$$
$K_v$ is $...
3
votes
2
answers
730
views
What's known about complete split primes in Q(E[p])?
Let E be an elliptic curve over $\mathbf{Q}$, and p be a prime of good reduction for E such that the Galois representation $\bar\rho_p$ of $\mathbf{Q}$ on the p-torsion of E surjects onto Aut(E[p]). ...
12
votes
1
answer
516
views
Are there nonobvious cases where equations have finitely many algebraic integer solutions?
Let $X$ be a scheme of finite type over $\mathbb{Z}$. Let $R$ be the ring of algebraic integers. My intuition is that $X(R)$ is practically always infinite.
More specifically, suppose that $X$ is ...
0
votes
2
answers
526
views
Grobner fan linked to associahedra?
In "Computing Grobner Fans" by Fukuda/Jensen/Thomas on page 2210 in Table 1 are the numbers (1,20,120,300,330,132) for some statistics on Grobner fans for Grass(2,5). This is a vector found in A126216,...
1
vote
0
answers
162
views
CH for tilings of the plane
Given any set of tiles (jordan curves) that can tile the plane, how to prove that the number of possible tilings using tiles from this set is either in bijection with the real numbers or a (possibly ...
71
votes
3
answers
9k
views
What exactly is the relation between string theory and conformal field theory?
Maybe it would be helpful for me to summarize the little bit I
think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and
an operator
$$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\...
11
votes
1
answer
459
views
Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?
Background
I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea ...
7
votes
3
answers
954
views
Simple basis for Barnes-Wall lattices in dimension `$2^n$`
I'm searching for a "simple" description of the basis of the Barnes-Wall lattices
in (real) dimension $2^n$, if possible in a basis of minimal vectors, so that I can
do some calculations.
Can ...
2
votes
2
answers
912
views
Question about Godel's 2nd Theorem
Let Con(PA) be the sentence of arithmetic which translates as "Peano Arithmetic is consistent." Then according to Godel's 2nd incompleteness theorem, assuming PA is consistent then PA can neither ...
1
vote
1
answer
353
views
multigrid boundary conditions and variable operator
Hi,
I have two question about solving an elliptic equation using multigrids.
First, in case of Neumann boundary conditions : Does the error equation, which is solved on the coarse grid, have the ...
2
votes
0
answers
396
views
Do inverse images respect flabby sheaves?
Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
5
votes
0
answers
500
views
Categorifying idempotent relations
Generalizing partial orders: A relation $R$ is transitive if $R \circ R \subseteq R$ and interpolative if $R \subseteq R \circ R$. It is idempotent if $R \circ R = R$. Interpolativeness means that ...
0
votes
0
answers
263
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
9
votes
1
answer
843
views
$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars
Hi,
I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem:
Definition:
Given some class of formulas ...
20
votes
2
answers
3k
views
D-modules and Algebraic Solutions of PDEs
I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...
14
votes
0
answers
463
views
Cohomological interpretations of quadratic form invariants over rings?
The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
2
votes
0
answers
168
views
Bundles of (co)chain complexes
A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic (co)...
8
votes
1
answer
1k
views
Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds
The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over ...
5
votes
5
answers
826
views
For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
3
votes
1
answer
490
views
Are Weierstrass points algebraic
Let $X$ be a compact connected Riemann surface of genus $g>0$.
Suppose that $X$ can be defined over a number field (as an algebraic curve). Then, is it clear that each Weierstrass point of $X$ is ...
4
votes
1
answer
424
views
Inverse of Kleisli star, or "extension operator"
While thinking about monads in the theory of denotational semantics, I have made an observation about the Kleisli category that I would like to check
Suppose $F : \mathcal D \to \mathcal C$, $G : \...
12
votes
1
answer
3k
views
Is the degree of a finite morphism stable by base change
Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes.
Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$?
If ...
1
vote
0
answers
230
views
Lower bound for intersection number
The base scheme is an algebraically closed field.
Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
0
votes
2
answers
807
views
Structural definition of "product" in set theory
At first sight there is no abstract (= structural) definition of "product" in set theory. E.g. the Cartesian product of sets $A$ and $B$ is defined as the set of all ordered pairs $(x,y)$, $x \in A$, $...
10
votes
1
answer
1k
views
Use of games to approximate solutions to Partial Differential Equations
Hi there,
Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "...
8
votes
2
answers
1k
views
Is the free product of arbitrarily many copies of `${\mathbb{Z}}$` and `${\mathbb{Z}}/2$` residually nilpotent?
A group is residually nilpotent if the intersection of the terms in its lower central series is the trivial group.
Is the free product of arbitarily (possibly infinite) many abelian groups residually ...
7
votes
1
answer
800
views
sheaf valued functors: how much can you prove about them just using category theory
This question is related to my earlier question. I have learnt a lot since I asked that question and I think I can phrase my thoughts more clearly now.
To someone who is comfortable with category ...
2
votes
1
answer
571
views
Moschovakis Coding Lemma
I trying to study the Coding Lemma (in descriptive Set Theory) and there is a small point in the proof that I don't understand. Let me first recall the version I'm studying ( there are different ...
13
votes
3
answers
1k
views
Random Reidemeister moves to unknot
Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing ...
2
votes
1
answer
277
views
Tree graph restructuring.
I have a graph $G(V,E)$ and a tree $T(V',E')$ where $|V|=|V'|$ and $T$ is isomorphic to a subgraph of $G$. In other words I found a spanning tree of $G$ and made one of its nodes act as the root.
I ...
15
votes
0
answers
633
views
Are "fpqc algebraic spaces" algebraic spaces?
Suppose $F:Sch^\text{op}\to Set$ is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must $F$ be an algebraic space? That is, must $F$ have an ...
3
votes
1
answer
300
views
Hermit H-machines
I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H.
Is there a ...
4
votes
2
answers
2k
views
Ample divisors on blown-up projective space
Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...
19
votes
2
answers
2k
views
Area of distance sphere in manifold with Ricci $\ge 0$.
Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is true:...
2
votes
0
answers
259
views
Generic Sets and Category Theory
Hi,
I am interested in categorical versions of forcing techniques, and I was wondering if there is anyway that topological/set-theoretical concepts such as filters, density and generic sets can be ...
1
vote
0
answers
222
views
(Non-)Surjectivity of the Maslov index
Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
8
votes
1
answer
841
views
Symmetrization map for universal enveloppings. What are Harish-Chandra images of symmetrization (poisson-center of S(gl_n) ) ?
For any Lie algebra symmetrization maps Poisson center of S(g) to center of U(g).
Consider g=gl_n, the Poisson center of S(gl_n) is isomorphic to algebra of symmetric polynomials in eigenvalues.
...
5
votes
1
answer
609
views
Characterizing invertible nonnegative matrices with bounded sums
Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
0
votes
2
answers
425
views
Formula with prime-density 1 in the integers [closed]
The Hermite-grand-conjecture implies that f(k)=(2^(2^5^11^(7k+1))+1)/3 is prime for all natural numbers $k$.
Is there any explicit formula that has so far been proven to produce primes for all ...
13
votes
3
answers
1k
views
Schemes over ℤ with a “graded existence over 𝔽₁”
Let $X$ be a (separated, and with whatever other tameness conditions are appropriate) scheme over the integers $\mathbb{Z}$. (If you don't like schemes much, imagine that $X$ is described by ...
3
votes
1
answer
462
views
Base change for the Gauss-Manin sheaf
I want to see the following thing:
$\ \ $If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable ...
3
votes
1
answer
517
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...