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2
votes
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
3answers
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
0answers
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
0answers
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
3answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
4answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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 ...

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