Let $G$ be a simple graph with vertex set $V$, such that for any two vertices $u,v\in V$, we have at least $k$ edge-disjoint paths of length $2$ (i.e., formed by $2$ edges) connecting $u$ with $v$. Let $n=|V|$ be the total number of vertices of $G$. Question: What is the minimum value of $k$, ex...

Consider a 4-regular graph with $N^2$ vertices, which can be interpreted as a $N\times N$ lattice with periodic boundary conditions so that every vertex has degree 4. For an unweighted and undirected graph, the Laplacian matrix can be written as $\mathbf{L}=4\mathbf{I}-\mathbf{A}$. Where $\mathbf...

Let $W$ be a standard Brownian motion, and $\mathcal F_t$ its natural filtration. Let $X$ be an $\mathcal F_t$-predictable process. Question: Fix $b > a > 0$. Is it true that for all sequences $t_n \downarrow 0$ of positive numbers, we have $$\underset{n \to \infty}{\lim} \int_{a}^b \mathbb E[X_s...

Consider a SDE $$dX_t = b(t,X_t)dt + f\big(a(t,X_t)\big)dW_t,\quad \quad\quad\quad\quad\quad\quad\quad\quad(\ast)$$ where $(W_t)_{t\ge 0}$ is a Brownian motion and $$f(z):={\bf 1}_{\{z>0\}} +\frac{1}{2}{\bf 1}_{\{z\le 0\}}.$$ Assume $b, a: \mathbb R_+\times\mathbb R \to\mathbb R$ are both bounded...

I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof of theorem 2.2, that is the index formula, the author claims that we can assume that $\Psi(s,t)$...

Let $X$ and $Y$ be Polish metric spaces with $X$ compact. Let $C^{\alpha}(X,Y)$ denote the set of $\alpha$-Hölder functions from $X$ to $Y$ (for $\alpha \in (0,1]$) and let $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with compact-open topology. Under what conditions is the se...

Is there an example of a birational morphism of smooth complex projective varieties $f\colon X\to Y$, that cannot be factored into a chain $X\to X_1\to\cdots\to X_n\to Y$ of blow-down along smooth centers? (By weak factorization theorem, we know in general that $f$ can be factorized into a zig-za...

Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class. Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, consider the operator given by the kernel $T(i,j) K(i,j) $ for some numbers $K(i,j)$ such that $\sup_{i,...

Is there a name for an adjunction between two categories such that i) the unit of the adjunction is a natural isomorphism, ii) the counit of the adjunction is a natural isomorphism?