Trending questions
159,065 questions
0
votes
0
answers
170
views
Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
2
votes
1
answer
257
views
New (?) Regularization Method for Divergent Series [closed]
Playing with identities ($1$) and ($2$) from this blog post and infinite geometric series, I've noticed the following.
For $x > 1$, the following series is convergent:
$$\sum_{n=0}^{\infty} e^{(2n ...
1
vote
1
answer
69
views
An lower bound for the injective dimension of a module
Let $A$ be a finite dimensional algebra and $M$ a non-injective $A$-module.
Question: Do we have idim $M>$domdim $\tau^{-1}(M)$?
Here idim $N$ denotes the injective dimension of a module $N$ and ...
- 26.5k
7
votes
1
answer
416
views
Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...
- 60.6k
2
votes
0
answers
66
views
Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
0
votes
2
answers
98
views
Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 1
5
votes
1
answer
135
views
Regular elliptic elements are dense in p-adic division algebra
I'm trying to better understand the set $E$ of regular elliptic elements of $D^\times$, where $D$ is a finite dimensional central division algebra over a non-archimedean local field $F$.
For example, ...
- 208
5
votes
0
answers
180
views
Left Adjoint From the Category of Topological Groups to the Category of Condensed Groups
In Scholze's Lecture Notes on Condensed Sets, the author states that (Remark 1.8) the functor that takes a topological group $G$ to its condensation $\underline{G}$ has a left-adjoint, but we do not ...
- 241
3
votes
0
answers
98
views
Square Roots of Non-Negative Even Functions
I'm trying to study properties of maps between quotients of representations of compact Lie groups and I stumbled upon the following problem. Suppose you have a smooth function $f:\mathbb{R}\to\mathbb{...
- 31
-1
votes
1
answer
445
views
A curious Diophantine problem
Let $a, b, c, d$ be positive integers where $\gcd(a, b)=\gcd(c, d)=\gcd(b, d)=\gcd(bd, ad+bc)=1$ and $\min(b, d)>1$. Is it possible to have
$$bd(ad+bc)^{2}\varphi(a)\varphi(c)=ac\varphi(bd)\varphi^{...
- 1,019
6
votes
2
answers
571
views
Problem with definition of tangent vectors as derivations
It is well-known that one can describe tangent vectors to a smooth manifold $M$ as derivations on germs of functions, see this question. However, according to "Riemannsche Geometrie im Großen&...
- 6,813
1
vote
0
answers
192
views
When are the complex points of a scheme an analytic manifold/space
Original Question: Let $X$ be a regular, projective, flat scheme over $\mathbb{Z}$. Let $X(\mathbb{C}$) be the set of complex points of $X$. Why is $X(\mathbb{C}$) a complex analytic manifold? I am ...
0
votes
0
answers
28
views
How to calculate the vertices of a convex polytope (k-DOP)
I am currently reading Christer Ericson's Real-Time Collision Detection Book. The topic I'm particularly interested in, is the chapter about Discrete-orientation Polytopes (k-DOPs). In his words "...
- 1
4
votes
0
answers
180
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
3
votes
0
answers
130
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
1
vote
1
answer
133
views
Can we find a Jouanolou device for $\mathbb{P}^d$ having dimension $<2d$?
Let us work over an algebraically closed field $k$.
A Jouanolou device for a $k$-variety $X$ is an affine space fiberation $f:Y\to X$ such that $Y$ is an affine scheme. (The condition on $f$ means ...
- 2,928
0
votes
0
answers
92
views
Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
4
votes
0
answers
69
views
is a 4-connected planar graph still Hamiltonian after removing an edge?
We know that 4-connected planar graphs are Hamiltonian(by the known Tutte Theorem). Additionally, Thomas and Yu [1] proved that removing two vertices from a 4-connected planar graph still preserves ...
- 1,851
4
votes
1
answer
247
views
Distinct eigenvalues of random matrix over finite field
Let $A$ be a uniformly random matrix in $\mathrm{M}_n(\mathbf{F}_p)$.
It is well known that, as $p$ is fixed and $n$ tends to infinity, $A$ has repeated eigenvalues (over the algebraic closure $\...
- 269
4
votes
1
answer
179
views
Exact forms, gauge transformations, and the Hodge decomposition in non-abelian Gauge theory
I am trying to understand how the Hodge decomposition is affected by gauge transformations in non-abelian in gauge theory (eg $\mathrm{SU}(N)$). In particular, I am searching for a way to generalise ...
- 41
5
votes
1
answer
155
views
Terminology for extremal non-epimorphisms
Is there a special name for a morphism $f : X \to Y$ in a category which doesn't factor through any proper subobject of $Y$? In other words, we have $\mathrm{im}(f)=Y$ for the image of $f$.
This ...
- 63.1k
3
votes
0
answers
140
views
Field of definition of automorphic Galois representation
Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\...
- 190
0
votes
0
answers
128
views
Notion of Kahler differentials for Berkovic spaces
What is, in abstract analytic geometry (I mean, for example, in Berkovic spaces), the approach used for differential forms?
Ordinary Kahler differentials from commutative algebra/algebraic geometry ...
- 101
2
votes
0
answers
125
views
Full level structure Deligne-Rapoport v.s. Katz-Mazur
For modular curves over schemes there are two main references that I use, namely Deligne Rapoport [DR], and Katz-Mazur [KM]. However I recently noticed that there is a difference in conventions in ...
- 2,224
25
votes
1
answer
583
views
Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 587
1
vote
0
answers
59
views
Asymptotic behavior of positive solution to nonlinear scalar field equation
It is well-known that the radial positive solution
$u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
- 705
2
votes
1
answer
93
views
Density of Intersection-Points of "Rational" Lines in the Euclidean Plane
consider the set of lines defined by all pairs of points $\lbrace[(u,v),(u-v,v+u)],\ u,v\in\mathbb{N}\rbrace$ in the euclidean plane
Question:
what is kown about the density of the set of ...
- 13.2k
15
votes
3
answers
3k
views
Finite verification for theorems due to Busy Beaver numbers
I recently learned about the Busy Beaver function, and a formulation of it that essentially tells us if a turing machine of $n$ states takes over $BB(n)$ steps, it will never halt.
One consequence I ...
- 185
5
votes
1
answer
336
views
Counterexample to flat base change for $\mathcal{O}_X$-modules
Consider a cartesian diagram
$$\require{AMScd}
\begin{CD}
X' @>{f'}>> X\\
@V{p'} VV @VV{p} V\\
S' @>{f}>> S
\end{CD}$$
of schemes (or even locally ringed spaces). If $\mathcal{F}$ is ...
- 584
0
votes
1
answer
71
views
Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 269
1
vote
0
answers
45
views
Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
2
votes
1
answer
112
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 6,313
1
vote
0
answers
58
views
Method of characteristics, the characteristic lines follow gradient, is this significant?
The PDE I am working on comes from geology, which I do not have much background on.
Said equation aims to describe describes the erosion by describing it as an advection phenomena: the advection ...
- 11
7
votes
1
answer
292
views
Existence of matrix diagonalizing $x A + y B$ for all $x, y$ and independent of $x, y$
Let $A_1, A_2$ and $B_1, B_2$ be real symmetric matrices.
Suppose $x A_1 + y B_1$ is cospectral with $x A_2 + y B_2$ for all real numbers $x, y$.
Is it true that there exists a fixed orthgonoal matrix ...
- 325
6
votes
0
answers
178
views
Modularity from cubic reciprocity: does it generalize?
Background. Let $a$ be a cubefree integer, $N=3\prod_{p\mid a}p$ and
$$
a_p=\#\{\text{solutions to}\,\, x^3\equiv a\,\,(\text{mod}\,\, p)\}-1.
$$ Let $\zeta$ be a primitive cube root of unity and $A=\...
- 171
1
vote
1
answer
141
views
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
- 69
3
votes
1
answer
280
views
Kleiman criterion for Kähler classes
Demailly and Paun proved the following characterization of nef classes on a compact Kahler manifold:
Theorem 18.13(a). Let $X$ be a compact Kähler manifold. A $(1,1)$-class $\alpha$ on $X$ is nef if ...
- 4,343
0
votes
0
answers
40
views
Optimizing over convex polynomials
I have a minimization problem which reads $\min\limits_P J(P)$, where the minimum is over convex polynomials in $n$ variables, with degree at most $d$, and $J$ is a function taking polynomials as ...
- 101
11
votes
1
answer
872
views
HoTT for the working mathematician (especially the homotopy geometer) - what is the current state?
As a starting point for this question, I want to turn to Mike Shulman's paper "All (∞,1)-toposes have strict univalent universes" (2019). Mike shows that
Every $(∞,1)$-topos can be ...
- 6,962
7
votes
2
answers
257
views
Formulae and integrality for two order four linear recurrences with cubic polynomial coefficients
Why does the following two linear recurrences with cubic polynomial coefficient produce only integers ? What are the explicit formulae for the $A_k$ ?
Sequence 1: $A_0=1,A_1=120,A_2=-9000,A_3=1133760$
...
- 1,362
3
votes
0
answers
164
views
Pro-algebraic fundamental groups
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of ...
- 3,446
1
vote
0
answers
65
views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
2
votes
1
answer
315
views
Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,101
1
vote
0
answers
101
views
Lawvere theory and presentations of groups
In his dissertation on "Functorial semantics of algebraic theories", Lawvere says in his introduction that
"from the category (or more precisely from an underlying-set functor)
we can ...
1
vote
0
answers
70
views
Notion of length in projective space over function field
Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as
$$
d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|...
- 87
2
votes
0
answers
61
views
Iwahori spherical representations of GL(n) with no nonzero fixed vectors under the fixator of a panel of the affine building
Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well ...
- 21
1
vote
0
answers
79
views
Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
52
votes
24
answers
11k
views
Most elementary proof showing that exponential growth wins against polynomial growth
This question is motivated by teaching : I would like to see a completely elementary proof showing for example that for all natural integers $k$ we have eventually $2^n>n^k$.
All proofs I know rely ...
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k