A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure $R$ and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:$$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}\<+\infty \mathrm{forsome}0\<{\alpha }^{*}\<+\infty .$$In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend...

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure R and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result,...

In the paper existence and uniqueness results for the linear differential system on the interval [0,1] $A_1{\left(A_{0}x\right)}^{\text{'}}-A_2^{\text{'}}x=f^{\text{'}}$ with distributional coefficients and solutions from the space of regulated functions are obtained.

Given the probability measure $\nu$ over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points in $\Omega$ in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Given the probability measure ν over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.