Existence of best simultaneous approximations in Lp(S,Σ,X)

Let $(S,\Sigma ,\mu )$ be a complete positive $\sigma$-finite measure space and let $X$ be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in $L_p(S,\Sigma ,X)$. Let ${\Sigma }_0$ be a sub-$\sigma$-algebra of $\Sigma$ and $Y$ a nonempty locally weakly compact convex subset of $X$ such that $\overline{\mathrm{span}Y}$ and its dual have the Radon–Nikodym property. We prove that $L_p(S,{\Sigma }_0,Y)$ is $N$-simultaneous proximinal in $L_p(S,\Sigma ,X)$ (with the additional assumption that $(S,\Sigma ,\mu )$ be finite for the case when $p=1$). Furthermore, for the special case when ${\Sigma }_0=\Sigma$, we show that the assumption that the dual of $\overline{\mathrm{span}Y}$ has the Radon–Nikodym property can be removed.     

The Σ1-provability logic of HA

In this paper we introduce a modal theory ${\mathsf{iH}}_{\sigma }$ which is sound and complete for arithmetical ${\mathrm{\Sigma }}_1$-interpretations in $\mathsf{HA}$, in other words, we will show that ${\mathsf{iH}}_{\sigma }$ is the ${\mathrm{\Sigma }}_1$-provability logic of $\mathsf{HA}$. Moreover we will show that ${\mathsf{iH}}_{\sigma }$ is decidable. As a by-product of these results, we show that $\mathsf{HA}+\square \perp$ has de Jongh property.     

Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process $Z_t=(X_t,Y_t)$ where $X_t$ is the position of a particle and $Y_t$ its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically $\beta$-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density $p$ of $Z$, based on $n$ discrete time observations with time step $\delta$. Two observation schemes are considered: in the first one, $Z$ is the observed process, in the second one, only $X$ is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.     

Maximal determinant over a certain class of matrices and its application to D-optimality of designs

It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of $\alpha {\mathbf{I}}_t-(\mathbf{P}+{\mathbf{P}}^T)$, $\alpha >2$, over the class of t-by-t permutation matrices, and the determinant of $\alpha {\mathbf{I}}_t+\mathbf{P}+{\mathbf{P}}^T$, $\alpha \ge 2.5$, over the class of t-by-t permutation matrices with zero diagonal (derangement matrices). The results are used to characterize D-optimal complete block designs under an interference model.     

The minimum volume of subspace trades

A subspace bitrade of type $T_q(t,k,v)$ is a pair $(T_0,T_1)$ of two disjoint nonempty collections of $k$-dimensional subspaces of a $v$-dimensional space $V$ over the finite field of order $q$ such that every $t$-dimensional subspace of $V$ is covered by the same number of subspaces from $T_0$ and $T_1$. In a previous paper, the minimum cardinality of a subspace $T_q(t,t+1,v)$ bitrade was established. We generalize that result by showing that for admissible $v$, $t$, and $k$, the minimum cardinality of a subspace $T_q(t,k,v)$ bitrade does not depend on $k$. An example of a minimum bitrade is represented using generator matrices in the reduced echelon form. For $t=1$, the uniqueness of a minimum bitrade is proved.