All Questions
153,402
questions
1
vote
0
answers
141
views
Estimating the sum of Dirichlet character $\sum_{0 \leq x < q} \chi(F(x))$ where $F(x)$ is a polynomial
Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not:
$$
\sum_{0 ...
- 3,595
6
votes
1
answer
825
views
Rice's theorem in type theory
From the formula
$$\forall f\colon A\to A\,\exists x\colon A\,(f(x)=x)$$
we can get the scheme
$$\forall x\colon A\,(\phi(x)\vee\neg\phi(x))\Rightarrow\forall x\colon A\,\phi(x)\vee\forall x\colon A\,\...
3
votes
0
answers
134
views
Moving between first and second order models using recursion
It seems that there are times when parts of the second order model of a certain structure can be determined to a significant degree using the first order model of the structure and recursion. For ...
- 9,009
4
votes
1
answer
604
views
Fixed point of a group action
Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...
- 43.7k
11
votes
1
answer
629
views
Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex
Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$.
(Terminology: By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is ...
- 2,074
2
votes
0
answers
1k
views
Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?
In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality:
Let $\Omega\...
- 1,207
2
votes
1
answer
269
views
Mapping between Notations
As in my other question, it is assumed that the (total) function describing a given notation is denoted as $address:p\rightarrow \Bbb{N}$ and assumed to be bijective.
Suppose we are given two ...
- 861
5
votes
1
answer
226
views
A Schur-like product theorem on groups
Let $G$ be a finite group, and consider the composition $X * Y$ on $\mathbb{C}G$ defined by $$(\sum_g \alpha_g u_g) * (\sum_g \beta_g u_g) = \sum_g \alpha_g \beta_g u_g.$$ This composition can be ...
- 25.9k
17
votes
0
answers
635
views
Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
- 7,549
4
votes
1
answer
716
views
Weight of Weil numbers in the residue field of $\overline{\mathbb{Q}}_l$
Let $l$ be a prime number and $q$ be a power of a prime number different from $l$. Recall that a Weil number (relative to $q$) is an element $x$ of $\overline{\mathbb{Q}_l}$ that is an algebraic ...
- 2,640
13
votes
1
answer
592
views
About primitively recursively recognizable ordinals
Preliminary: I believe the notion of primitive recursive functions on ordinals is standard and unproblematic (the main difference with the finite case is that one needs to introduce a $\sup$ or $\...
- 30.2k
2
votes
1
answer
161
views
Maximum number of points in convex position on a grid made by two sets of concentric circles
Here is a similar but more difficult problem than the one I asked already:
Consider two points $p_1$ and $p_2$ in the euclidean plane and a set of $n$ concentric circles around $p_1$ and a set of $m$ ...
- 51
7
votes
1
answer
214
views
Are free ultrafilters as posets product-irreducible?
Let $\kappa\geq \aleph_0$ be a cardinal, and suppose that ${\cal U}$ is a non-principal ultrafilter on $\kappa$. We regard ${\cal U}$ as a poset $({\cal U}, \subseteq)$.
Suppose that there are posets ...
- 45.5k
0
votes
1
answer
122
views
Maximum number of points in convex position on a grid
Guys this problem really bothers me (I don`t know how to prove it) please help:
What is the maximum number of points in convex position on a $n\times m$ grid?
(My guess would be $2*(m+n)-4$.)
- 51
10
votes
1
answer
1k
views
K-theory of an elliptic curve
Given an elliptic curve $E$ over $\mathbb{Q}$, I have read somewhere (But I can't remember exactly where) that the Beilinson conjecture asserts that: The rank of the albelian group $K_{2}(E)$ (the ...
- 223
14
votes
2
answers
895
views
Terminology: Lost in translation or multiple-meanings
I was reading Uniformization of Riemann Surfaces by Henri Paul de Saint Gervais (not a real person, but a group of French mathematicians), and the translator kindly points out that the name of "the ...
8
votes
3
answers
445
views
How to get this integral's asymptotics?
Consider the following integral
$$
\int_0^{\infty}\frac{e^{-x}-1}{x^{2+\frac{A}{\log b-5/6}}}\frac{1}{\log(b/x)-i\pi/2}\,dx
$$
where $A>0$ and $b>0$. I am interested in the small $b$ asymptotics ...
- 247
4
votes
2
answers
230
views
Number of non-equivalent graph embeddings
Given a graph $G$, there is a minimal integer $g$ associated with it which captures the minimum genus a surface needs to have so that $G$ embeds in the surface without edge crossings.
Is there a way ...
- 13.7k
38
votes
4
answers
30k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 499
1
vote
0
answers
135
views
Example of the Main Theorem of Complex Multiplication [closed]
I am trying to understand the main theorem ov CM for elliptic curves. I work with the version stated in the second chapter of Silvermans "Advanced Topics in the Arithmetics of Elliptic Curves".
I ...
- 41
0
votes
1
answer
367
views
How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]
Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...
- 6,001
5
votes
2
answers
362
views
Are continuous rational functions arc-analytic?
Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...
- 953
4
votes
0
answers
170
views
Kuranishi family and smoothing of Calabi-Yau n-fold
Consider $X$ be a Calabi-Yau n-fold with at
most one ordinary double point singularity. Suppose $X$ is smoothable. Then it is known that the Kuranishi family of $X$ is a smoothing of $X$.
Now, ...
- 81
3
votes
0
answers
206
views
Hook-content polynomial
Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
- 685
22
votes
2
answers
1k
views
Why doesn't this construction of the tangent space work for non-Riemannian metric manifolds?
In the 1957 paper, On the differentiability of isometries, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1).
...
- 2,622
1
vote
0
answers
109
views
generalized sieve implementations
Suppose I have a multiplicative function $f$ and I want to compute it for all integers from $1$ to $N$ (or maybe just for a long sequence of consecutive integers). It is clear that this is far faster ...
- 95.6k
8
votes
0
answers
92
views
Is there a quaternionic analogue of Kodaira's embedding theorem?
Let $M$ be a $4m$-dimensional Quaternion-Kähler manifold of positive scalar curvature. Does there exist an $n$ large enough, so that $M$ can be embedded inside $\mathbb{H}P^n$ via a quaternionic ...
- 5,011
5
votes
0
answers
185
views
How to count Isomorphism Types of arbitrary structures?
For all relational signatures $\sigma$ and nonnegative integers $n$, I want to count the number of isomorphism types of structures of order $n$ of the signature $\sigma$.
What I mean by structure is ...
- 381
20
votes
5
answers
933
views
If a $\otimes$-idempotent object has a dual, must it be self-dual?
Let $C$ be a symmetric monoidal category.
Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...
- 61.6k
8
votes
0
answers
1k
views
Galois descent for schemes over fields
Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ ...
- 953
13
votes
0
answers
2k
views
Identifying poisoned wines, with a twist
(This is a joint musing with Andrew Gordon and Wyatt Mackey)
There is a classic, elementary riddle, discussed before on MO and math.SE: suppose you have 1000 bottles of wine, and one is poisoned. The ...
- 303
8
votes
2
answers
589
views
Does the clique-coclique bound hold for all walk-regular graphs?
The clique-coclique bound is said to hold for a simple graph $G$ on $n$ vertices if $\lvert \omega(G) \rvert \lvert \alpha(G) \lvert \leq n$, letting $\omega(G)$ and $\alpha(G)$ denote its clique and ...
- 155
0
votes
0
answers
162
views
Bounding exponential sum of the form $\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n (x_n) e(a F(\mathbf{x})/q)$
I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and
I would greatly appreciate any suggestions on how ...
- 3,595
6
votes
0
answers
101
views
Squaring operation in KO theory
There's an operation $\Lambda^2: KO^0(X) \to KO^0(X)$ such that $\Lambda^2(x+y)-\Lambda^2(x)-\Lambda^2(y)=xy$, which comes from the antisymmetric power. Similarly, there's a $\Lambda^2: KO^4(X) \to KO^...
- 6,063
20
votes
0
answers
623
views
Is $\sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an ...
user14319
6
votes
0
answers
77
views
Delaying the first zero of a trigonometric series
Let $a_1,\ldots,a_n \in [0,1], \omega_1,\ldots,\omega_n \in (0,+\infty)$, and define, for every $t \ge 0$,
$$
f(t) := \sum_{i = 1}^n a_i \sin(\omega_i t).
$$
I'm interested in trying to optimize the ...
- 562
4
votes
0
answers
215
views
What is Grothendieck and Hartshorne's local duality?
I am looking for a simple and brief answer to the following question:
What is Grothendieck and Hartshorne's local duality and what is its relationship with dualizing modules?
How do (semi-)dualizing ...
- 191
2
votes
0
answers
83
views
What are the Cartan geometries modeled on $\mathbb{H}P^m$?
I am not an expert on Cartan Geometry (in fact, I have just read and understood the definition, at a basic level). I have the following questions:
1) Can someone please describe what are the Cartan ...
- 5,011
3
votes
1
answer
323
views
At what point does exponential integral coincide with exponential?
I'm looking for the solution to the equation $E_1(x)=e^{-x}$, where $E_1(x)$ is the exponential integral function. The solution is approximately 0.434818204, but is this constant well known/studied?
...
- 1,304
2
votes
1
answer
128
views
circular convolution over Invariant subspace
Consider a set of distinct vectors {$a,b,c,d,...$} in a subspace $C$ of $R^n$.
$C = \pi (C)$ where $\pi : R^n \rightarrow R^n$ defined by $\pi (x_1,x_2,...,x_n)=(x_2,...,x_n,x_1)$
How can I prove ...
- 53
0
votes
2
answers
199
views
Is this equality between an integral and a series wrong?
In this paper (Maroun's PhD dissertation, 2013) at page 46 the following formula is given (apparently without a reference):
$$\int_0^{\infty } e^{i a x^s+i b x^p} \, dx=\sum _{n=0}^{\infty } \frac{\...
- 9,312
8
votes
1
answer
243
views
Pairs of roots of unity whose real part satisfies a polynomial identity
Some motivation: The matrix $M$ is Butson Hadamard if the entries of $M$ are $k^{\textrm{th}}$ roots of unity (for some $k$), and distinct pairs of rows are orthogonal under the usual Hermitian inner ...
- 1,210
1
vote
1
answer
116
views
Conformal prediction for the case of single tailed events
I'll start with a motivating example and only then proceed to the question.
Consider a list of total packages of milk that were purchased on 9 consecutive days on a given store,
$z_1,\ldots,z_9 = 1,...
- 153
1
vote
0
answers
131
views
A mixed Vandermonde-Wronskian matrix
I am trying to prove that a matrix of the following form is generically nonsingular:
$A= \begin{bmatrix}
1&1&1&1 \\
f_1 & f_2 & f_3 &f_4 \\
(f_1 -\frac{d}{dt}).f_1& (...
1
vote
1
answer
212
views
Is there any 1-cusp 3-manifold without Lens space Dehn filling?
For a knot complement on 3-sphere, there's at least one Lens-space Dehn filling (1/0-slope). Is it also true for any 1-cusp 3-manifold?
- 115
10
votes
1
answer
355
views
Wiener's axiomatization of the group law based on division
Gian-Carlo Rota wrote that [*]:
Wiener axiomatized the group law by taking $xy^{-1}$ as the basic operation, and his axiomatization is quite different from any of the other axiom systems for groups....
- 3,353
21
votes
2
answers
2k
views
Infinite dimensional symplectic geometry
Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
- 483
2
votes
1
answer
128
views
Is the Lascar group over $A$ trivial when $T=T^{eq}$ and $A = acl(A)$?
Let $T$ be a first-order theory which eliminates imaginaries, and let $A$ be an algebraically closed set in a model of $T$. Let $Gal_L(T[A])$ be the Lascar group of the theory $T[A]$, which is $T$ ...
- 61.6k
5
votes
2
answers
839
views
Representation theory over any field
I understand that representation theory of complex reductive groups is essentially combinatorial. By general principles, I imagine Galois theory then determines the theory over any field. For example, ...
- 51
5
votes
2
answers
679
views
What condition makes unitary reductive group unramified?
I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...
- 1,719