# All Questions

132,032
questions

**2**

votes

**0**answers

139 views

### Interpolation inequality involving negative Sobolev space

$\newcommand\norm[1]{\left\|#1\right\|}\newcommand\inner[2]{\langle #1,#2\rangle}$
Let $u\in \dot{H}^1(\mathbb{R}^n)$ for $n\geq 3$ where $\dot{H}^{1}$ denotes the homogeneous Sobolev space that is ...

**1**

vote

**2**answers

216 views

### Dimensions of $\frak{sl}_n$-representations

The dimension of any irreducible $\frak{sl}_n$-representation $V$ is clearly equal to the dimension of its dual representation $V^*$. Does the dimension of an irreducible $\frak{sl}_n$-representation ...

**1**

vote

**0**answers

58 views

### Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?

Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...

**1**

vote

**0**answers

114 views

### Conformal changes of metric and Ricci curvature

Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...

**2**

votes

**1**answer

115 views

### Entire reflection symmetric function which is near $i$ when $\textrm{Im }z$ is big

Is there an analytic entire function $f:\mathbb{C} \rightarrow \mathbb{C}$ such that
$f(z)=\overline{f(\overline{z})},$
For every $\varepsilon>0$ there is a $\delta >0$ such that if $\textrm{Im ...

**0**

votes

**1**answer

80 views

### Nonlinear differential equations with zero initial conditions

Suppose that we have $n$ differential equations of the form:
$\dot{x}_i(t) = f_i(g(x_1(t)), \ldots, g(x_n(t))) \qquad \qquad$ ($i=1,\ldots,n$).
where $f_i$ are linear functions, and $g$ is an ...

**3**

votes

**1**answer

120 views

### Lower Gaussian estimates for Dirichlet heat kernel on manifolds

Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental ...

**2**

votes

**0**answers

36 views

### Formulas to determine the value of graph energy with addition or deletion of edges

If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...

**0**

votes

**1**answer

122 views

### What to call a function that is negative on a set

Let $Y$ be a nonempty region in $\mathbb{R}^n$. I am designing an algorithm which given a point $x_0$ outside $Y$ in a finite number of steps lead to a point $x_n∈ Y$. The way I do it is that I have a ...

**0**

votes

**0**answers

23 views

### Understanding non-convex subgradients and normal cones

I think I have a very good understanding of subgradients of convex functions and normal cones to convex sets. On the other hand, I have a lot of difficulties understanding them in the non-convex setup....

**1**

vote

**1**answer

183 views

### localizing subcategories of a nice triangulated category

Suppose that $D(A)$ is the derived category of of a ring A. Let $b\in D(A)$ be a compact object and $B$ the localizing subcategory generated by b (having arbitrary coproduct).
Does the inclusion ...

**3**

votes

**0**answers

99 views

### Clarification about extensions of Ornstein-Uhlenbeck operator

I am reading stuffs regarding the Ornstein-Uhlenbeck operator and its various extensions to $L^p(\gamma)$, with $p \in (1,+\infty)$ and with $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$. ...

**0**

votes

**1**answer

163 views

### Associativity rule for integration against fractional Brownian motion

In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...

**2**

votes

**0**answers

37 views

### Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares?

Suppose that I have a square and positive definite for every evaluation $x\in\mathbb{R}^{n}$ symmetric matrix $M(x)\in(\mathbb{R}[x])^{s\times s}.$
Does there always exist a polynomial $p(x)\in\...

**0**

votes

**0**answers

30 views

### The set of bounded lipschitz functions over a compact is barrelled but not a neighborhood of zero?

I recently learned that Banach spaces are barrelled, i.e any convex, balanced, absorbing and closed subset is a neighborhood of zero (wikipedia). I'm having trouble understanding why the following ...

**2**

votes

**1**answer

182 views

### Uniqueness of the direct sum of $C^*$ algebras as quotient of free products

Suppose that you have $A, B$ two unital $C^*$ algebras and let $A \ast B$ the reduced free product (I think that it is the reduced amalgamated product over the common $*$-subalgebra $\mathbb{C} 1$) ...

**1**

vote

**1**answer

57 views

### Confidence interval for the difference of lognormally distributed random variables

I have two lognormally distributed random variables $Y_i=e^{X_i}$ where $X_i \sim \mathcal{N}\big(\mu_i, \: \sigma_i^2 \big)$ for $i=1,2$, and $X_1$ and $X_2$ are correlated by $\rho_{12}$. Now, Let $...

**0**

votes

**0**answers

98 views

### classification for some groups

Let $G$ be a finite group. Suppose that $G$ acts on a set, say $X$, transitively such that for every $x\in X$, $G_x^g=G_x$ or $G_x^g\cap G_x=\{1\}$. Could you please tell me if there is a ...

**5**

votes

**2**answers

237 views

### First time appearance of Lie crossed module (crossed module of Lie groups) in literature

Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time?
I see many papers "recall" the definition of the Lie crossed module but, I ...

**6**

votes

**1**answer

102 views

### Tameable hypergraphs

Let $H=(V,E)$ be a hypergraph. We say that $I\subseteq V$ is an independent set if $e\not\subseteq I$ for all $e\in E$.
We say that $H$ is tameable if every independent set is contained in a maximal ...

**0**

votes

**0**answers

79 views

### Linear independence of algebraic integers of equal norm, part II

In a previous question, I asked whether for a given degree $n$ number field $K$ whether there exist algebraic integers $\alpha_1, \cdots, \alpha_n$, pairwise non-associates and having equal norm ...

**9**

votes

**1**answer

266 views

### Origin and context of adjunctions inducing equivalences between full subcategories

The following is well-known.
Theorem. Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{...

**0**

votes

**0**answers

25 views

### Permutation realised as Kronecker product of unitary matrices

Under what circumstances can a permutation of a vector $\pmb{v}$ be achieved with the Kronecker product $U \otimes U$ of two unitary matrices? In other words,
$P \, \pmb{v} = \left( U \otimes U \right)...

**11**

votes

**1**answer

280 views

### Translation lengths in CAT(0) spaces

Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...

**0**

votes

**0**answers

78 views

### Sheafification in an arbitrary category

Let $\mathcal{F}$ be a presheaf valued in an arbitrary category $\mathcal{C}$ on a topological space $X$, with $\mathcal{C} $ has limits(or, $\mathcal{C}$ has equalizers so that a sheaf valued in $\...

**0**

votes

**1**answer

101 views

### Physical significance of a fractional operator

Is there any physical significance of the operator $(-\Delta)^s\pm \Delta$ when $0<s<1.$ I would like to know if there is any real life applications other than in pure mathematics.

**2**

votes

**3**answers

173 views

### Help with a limit involving incomplete beta integral

In trying to prove that the limit of a certain function approaches 1 as the positive integer parameter $n$ approaches infinity, I have ended up with the following intermediate expressions:
$$f(n)=2^{...

**2**

votes

**0**answers

42 views

### The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively.
$$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$
$$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$
In the ...

**25**

votes

**0**answers

944 views

### Spectral sequences as deformation theory

I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...

**5**

votes

**3**answers

440 views

### Linear independence of algebraic integers of equal norm

Let $K$ be a number field with $[K:\mathbb{Q}]=n$ with $n \geq 2$ and let $\mathcal{O}_K$ be its ring of integers. Suppose that $\alpha_1, \cdots, \alpha_n \in \mathcal{O}_K$ are distinct algebraic ...

**5**

votes

**0**answers

87 views

### Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects

In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In ...

**7**

votes

**0**answers

193 views

### Determinacy of symmetric games

Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...

**3**

votes

**0**answers

47 views

### Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...

**4**

votes

**0**answers

361 views

### Why $K_5$ and $K_{3,3}$?

Most people will have already guessed that this is about Kuratowski's theorem.
The theorem states that every non-planar graph must contain a complete graph $K_5$ with five vertices or a complete ...

**11**

votes

**1**answer

667 views

### Where should I learn about the p-adic L-functions of elliptic curves?

Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ...

**3**

votes

**1**answer

117 views

### Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...

**2**

votes

**0**answers

78 views

### symmetry for a pair of statistics on partitions

Let $\lambda\vdash n$ denote a partition $\lambda$ of $n$ and let $\square\in\lambda$ denote a box $\square$ in the Young diagram of $\lambda$.
QUESTION. Can you list a pair of (distinct) statistics $...

**6**

votes

**1**answer

490 views

### If $M\otimes_S T$ is an $A$-module, is $M$ an $A$-module?

Let $\mathbb{C}$ be the field of the complex numbers. Let $R=\mathbb{C}[x]$, $T=\mathbb{C}\langle x\rangle$ be the ring of entire series with convergence radius at least $1$, and let $S=\mathbb{C}\...

**3**

votes

**1**answer

194 views

### Local existence of flat metrics with degenerate singular values

It has been proved that,
If $\lambda_1,\,\lambda_2,\cdots,\lambda_n$ are real analytic functions from $\mathbb{R}^n$ to $\mathbb{R}$, such that $\lambda_i(0)\neq \lambda_j(0)$ for $i\neq j$, then ...

**5**

votes

**1**answer

179 views

### Randomized version of Turán's theorem

Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...

**0**

votes

**4**answers

391 views

### Confining a polytope to one side of an affine hyperplane

Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
This answer on math.stackexchange.com claims the ...

**7**

votes

**1**answer

162 views

### Hybrid online/ in-person workshops, conferences, and summer schools

I am writing to ask if people in the community can post any noteworthy experiences they have had with hybrid online/in-person workshops. Does anyone have experience with these, as either organizer or ...

**0**

votes

**0**answers

131 views

### Holomorphic automorphic/cusp forms on real Lie groups

An automorphic form on a real Lie group $G$ for a discrete subgroup $\Gamma$ is a function $f:G\to\mathbb{C}$ with some properties (see Borel’s definition in Proceedings of Symposia in PURE ...

**4**

votes

**1**answer

127 views

### Solve differential system of equations

Consider the following system:
$$
\begin{cases}
x_1 + 3 x_3 = 4a, \\
f(x_1) + 3 f(x_3) = 8 f(a), \\
f'(x_1) = 3 f'(x_3).
\end{cases}
$$
I want to find all functions (or at least learn some properties ...

**1**

vote

**1**answer

177 views

### The Schoenflies Theorem on two dimensional surfaces

Let $S$ be a surface and $U$ an open connected subset of $S$. If the frontier of $U$ in $S$ is a two sided circle $C$, then the closure of $U$ in $S$ is a surface whose boundary is $C$. I would like ...

**7**

votes

**1**answer

260 views

### What are some compact Hessian manifolds?

In case this is too general, here is a more specific question.
Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)?
Background
A Hessian manifold is a Riemannian ...

**4**

votes

**0**answers

190 views

### Motivic Galois correspondence

Is there a Galois correspondence in motivic Galois theory ? If so, is there a mathematical work on this correspondence that i can find on the net ?
Thanks in advance for your help.

**1**

vote

**0**answers

33 views

### Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...

**6**

votes

**0**answers

126 views

### Expressing a polynomial as the determinant of a matrix of linear forms

I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...

**13**

votes

**1**answer

266 views

### Question about definition of Hopf algebra in Hatcher

I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283):
Definition: A Hopf algebra is a graded ...