All Questions
152,902
questions
2
votes
1
answer
157
views
Integral over the space of $p$-adic matrices
$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
3
votes
1
answer
136
views
Solving a recursion for polynomials defined by a matrix product
Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix
\begin{align*}
& A = \left(\begin{matrix}
X_1 & \dots & \...
2
votes
0
answers
49
views
Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
14
votes
1
answer
1k
views
Hilbert's sixth problem and QFT description
The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...
1
vote
0
answers
66
views
How to know the character table of the twisted group algebra of the symmetric group $S_4$
Given the character table of its Schur cover group, is there a way to obtain the character table of twisted group algebra from that? I am particularly interested in the symmetric group $S_4$.
1
vote
0
answers
63
views
Fitting a product into the quintuple or Jacobi triple product
The Rogers-Ramanujan functions fit nicely into the QPI or JTP. In fact we have that $$(q^{5};q^{5})_{\infty}(q,q^{4};q^{5})_{\infty}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(5n^{2}-3n)}{2}}$$ and we ...
0
votes
0
answers
31
views
Justification for the "derivative kernel" for a Gaussian Process
In general, it is fairly well established that a method of including derivative information into a a Gaussian Process is to:
Extend the training vector with derivative information $[y, \delta{y}_1, .....
5
votes
0
answers
75
views
Reciprocity for algebra objects in two algebraic categories
I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories.
So, ...
6
votes
0
answers
146
views
Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
5
votes
0
answers
79
views
When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
13
votes
0
answers
438
views
Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
2
votes
0
answers
82
views
Bateman-Horn-type generalization of the Goldbach conjecture
The Bateman-Horn conjecture is a generalization of the twin prime conjecture that roughly states that given a set $S=\{f_1, \dots, f_m\}$ of irreducible polynomials with integer coefficients, there ...
2
votes
0
answers
31
views
0-1 knapsack problem with additional capacity
The 0-1 knapsack problem maximizes the profits of items under a capacity constraint (let's call this capacity $C$).
I am interested in an augmented setting where the algorithm is permitted to use a ...
0
votes
0
answers
66
views
Gibbs Priors form a Martingale
I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
6
votes
0
answers
70
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
1
vote
2
answers
233
views
Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$
Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...
3
votes
0
answers
142
views
Homotopy type theory for semantics
It looks like I have been building up a theory that might require looking closely at Homotopy Type Theory vs. Category Theory with respect to semantics. I am considering two types of semantics that ...
3
votes
1
answer
108
views
Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories
I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...
3
votes
0
answers
73
views
A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
4
votes
0
answers
125
views
On adelic integration
$\DeclareMathOperator\GL{GL}$This question might be beyond naive but I really can't find it nor figure it out. It's related to explicit idelic integration as opposed to just bounding its values.
Let $...
1
vote
0
answers
22
views
Inner product of signatures of piecewise linear paths
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
0
votes
2
answers
126
views
Cauchy's functional multiplicative equation on the unit interval
This question might be trivial, but I didn't find a clean reference and have not attempted to prove it myself yet: Let $f:[0,1]\rightarrow [0,1]$ be a continuous and monotonic function such that $f(0)=...
0
votes
2
answers
57
views
Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix
Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
1
vote
0
answers
49
views
Frobenius acting by autoequivalence on $\text{Isoc}(X/K)$
Let $X_k$ be a smooth quasiprojective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to the fraction field of the Witt ring of $k$, which I denote by $K$.
In various papers I read ...
0
votes
1
answer
90
views
Criterion for Ramification of ray class field $K(\mathfrak{p})$ in $\mathfrak{p}$
The context: We consider ideal theoretic formulation of global class field theory of a number field $K$ and in following all used terminology I'm going to use is adapted from these notes: https://math....
4
votes
0
answers
136
views
Coloured Jones polynomial at 4th root of unity and Arf invariant
Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
9
votes
0
answers
212
views
Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces
Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense?
In other words, when does at set $S\...
1
vote
0
answers
121
views
Can't parse a statement in an article on coalgebras and umbral calculus
This question is cross-posted from MSE.
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
1
vote
0
answers
81
views
Factorial surfaces and smoothness
It is well-known that normal curves are smooth. Moreover, a UFD of Krull dimension one is regular. Is there any higher-dimensional analog?
For example, given a normal projective surface $S$ over $\...
3
votes
1
answer
195
views
A criterion for numerically trivial line bundle
Let $X$ be a projective smooth variety, $D$ a Cartier divisor on $X$. If for any Cartier divisor $B$ on $X$, $\mathcal{O}_X(B)$ and $\mathcal{O}_X(B+D)$ have the same Euler characteristic
$$\chi(X,\...
3
votes
0
answers
70
views
An isomorphic classification of non-associative division octonion algebras
A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...
2
votes
0
answers
53
views
When does an algebraically independent set "satisfy Noether normalization"?
Let $k$ be a field, $A$ a finitely generated $k$-algebra.
By Noether normalization, we know that there exists a finite morphism of $k$-algebras $\varphi : k[x_1, \ldots, x_d] \hookrightarrow A$, with $...
2
votes
0
answers
93
views
Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
1
vote
0
answers
44
views
Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$
Let us assume we have the following extension operator:
$$
\operatorname{ext}_R^\sigma u=
\begin{cases}
u(x) & \text{if }x \in (0,T)\\
u(0) & \text{if }x \in(0,T)^c.
\end{cases}
$$ We ...
1
vote
0
answers
153
views
Are there any relations between perverse t-structure (cohomologies) and standard t-structure (cohomologies)?
I'm reading the Corollary 3.2.3. in Exponential motives by J. Fresan and P. Jossen.
The authors use the following statement in the proof of Corollary 3.2.3: let $C$ be any object in the derived ...
0
votes
1
answer
55
views
Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
12
votes
1
answer
368
views
Partition into antichains
I've read that the following statement is a result of Balcar, but I am unable to find a reference or a proof:
Theorem: If $\kappa\ge \lambda$ are infinite cardinals, then $[\kappa]^{<\lambda}$ can ...
2
votes
1
answer
136
views
Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
0
votes
0
answers
100
views
What are the primitive notions and axioms in model theory? [migrated]
I know every theory has its primitive notions and axioms. Now I am reading Basic Model Theory, and there is no term or sentence referred as to a primitive notion or an axiom. But, I think I know that ...
-2
votes
0
answers
132
views
Elimination over $\mathbb F_p[x,y]$
Let $p$ be a prime. Consider the two independent modular equations:
$$a_1x^2+b_1y^2+c_1xy\equiv d_1\bmod p$$
$$a_2x^2+b_2y^2+c_2xy\equiv d_2\bmod p$$
Is it possible to extract the common roots $(x,y)\...
8
votes
1
answer
335
views
Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?
Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
1
vote
1
answer
173
views
Hilbert scheme of points on an arithmetic surface
$\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Spec{Spec}$Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\Hilb^n(X)$ is regular. My primary ...
1
vote
2
answers
372
views
What is the proper name for this "tersest path" problem in Infinite Craft?
The web game Infinite Craft gives you a starting set of elements $V_0\subset V$ and a mapping $E$ of type $V\times V\rightarrow V$. In fact, $F$ is commutative: $E(v_a,v_b) = E(v_b,v_a)$. So another ...
3
votes
0
answers
47
views
One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
7
votes
1
answer
120
views
Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?
(Originally on MSE.)
Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...
2
votes
1
answer
231
views
If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?
Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...
3
votes
1
answer
184
views
Some questions about induced subgraphs of the directed hypercube graph
Let $Q^n$ be the hypercube graph in $n$ dimensions. Hao Huang famously showed that any induced subgraph on more than $2^{n-1}$ must have maximum degree $ \geq \sqrt{n}$. It is also known that this ...
3
votes
0
answers
69
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
6
votes
1
answer
375
views
Tame-Wild dichotomy; why can't tame algebras be wild?
I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey [D80, CB88].
...
2
votes
0
answers
46
views
Smoothness of the Fréchet Function
Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as
$$
F(x) = \...