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For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed a …

This is a continuation of Should coffee machines be deconcentrated? Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the peopl …

We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the populatio …

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is symm …

Let $\Phi$ be the CDF of a standard Gaussian distribution, i.e. $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-y^2/2}dy,\quad \forall~ x\in \mathbb R.$$ Denote by $\Phi^{-1}$ its inverse functi …

$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\mathb …

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is symm …

For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the in …

Let $L:[0,1]\times \mathbb R^m\times \mathbb R^n\to \mathbb R$ be defined by $$L(\lambda, x,y):=\sum_{1\le i\le m}\alpha_i x_i + \sum_{1\le j\le n}\beta_j y_j -\sum_{1\le i\le m, 1\le j\le n} p_{i,j}\ …

Let $\mu$ be a measure on some measurable space $(\Omega, \mathcal F)$ such that $$\mu(B)\in \{0,1\},\quad \forall B\in \mathcal F.$$ Can we show that $\mu$ must be a Dirac measure (under suitable con …

Let $C_p\equiv C_p(\mathbb R_+,\mathbb R_+)$ be the space of right-continuous piecewise constant functions $f: \mathbb R_+\to \mathbb R_+$, i.e. $f\in C_p$ iff $$f(t)=\sum_{k=1}^n {\mathbf 1}_{[t_{k-1 …

Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems : $$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma …

Let $d,n\ge 1$ be fixed integers. Given some compact subset $E\subset \mathbb R^d$, consider the function $f: E^n\ni (x_1,\ldots, x_n) \longrightarrow f(x_1,\ldots, x_n)\in \mathbb R$ defined by $$f( …

Let $X=(X_t:t\ge 0)$ be a stochastic process (martingale in general) starting at $X_0=0$. For $T>0$ and $a<b$, let $U^T_{a,b}(X)$ be the number of upcrossings of $X$ across the interval $[a,b]$ over $ …

Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows: $$ 1-t= \int\limits_0^\infty p(x)\Phi\left(\frac …