Bohr's equivalence relation in the space of Besicovitch almost periodic functions

Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, $B(\mathbb{R},\mathbb{C})$, defined in terms of polynomial approximations. From this, we show that in an important subspace $B^2(\mathbb{R},\mathbb{C})\subset B(\mathbb{R},\mathbb{C})$, where Parseval's equality and Riesz-Fischer theorem holds, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.Comment: Because of a mistake detected in one of the references, the equivalence relation which is inspired by that of Bohr is revised to adapt correctly the situation in the general case. arXiv admin note: text overlap with arXiv:1801.0803

A generalization of Bohr's Equivalence Theorem

Based on a generalization of Bohr's equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr's equivalence theorem extended to the case of these functions.Comment: Because of a mistake detected in one of the references, the previous version of this paper has been modified by the authors to restrict the scope of its application to the case of existence of an integral basi

Measuring cortical connectivity in Alzheimer's disease as a brain neural network pathology: Toward clinical applications

Objectives: The objective was to review the literature on diffusion tensor imaging as well as resting-state functional magnetic resonance imaging and electroencephalography (EEG) to unveil neuroanatomical and neurophysiological substrates of Alzheimer’s disease (AD) as a brain neural network pathology affecting structural and functional cortical connectivity underlying human cognition. Methods: We reviewed papers registered in PubMed and other scientific repositories on the use of these techniques in amnesic mild cognitive impairment (MCI) and clinically mild AD dementia patients compared to cognitively intact elderly individuals (Controls). Results: Hundreds of peer-reviewed (cross-sectional and longitudinal) papers have shown in patients with MCI and mild AD compared to Controls (1) impairment of callosal (splenium), thalamic, and anterior–posterior white matter bundles; (2) reduced correlation of resting state blood oxygen level-dependent activity across several intrinsic brain circuits including default mode and attention-related networks; and (3) abnormal power and functional coupling of resting state cortical EEG rhythms. Clinical applications of these measures are still limited. Conclusions: Structural and functional (in vivo) cortical connectivity measures represent a reliable marker of cerebral reserve capacity and should be used to predict and monitor the evolution of AD and its relative impact on cognitive domains in pre-clinical, prodromal, and dementia stages of AD. (JINS, 2016, 22, 138–163

Cohen strongly p-summing holomorphic mappings on Banach spaces

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and analyse their linearizations from $\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space $\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and endowed with a natural norm $d_p^{\mathcal{H}^\infty}$, we show that it is a regular Banach ideal of bounded holomorphic mappings generated by composition with the ideal of strongly $p$-summing linear operators. Moreover, we identify the space $(\mathcal{D}_p^{\mathcal{H}^\infty}(U,F^*),d_p^{\mathcal{H}^\infty})$ with the dual of the completion of tensor product space $\mathcal{G}^\infty(U)\otimes F$ endowed with the Chevet--Saphar norm $g_p$

Deuterium and 15^{15}N fractionation in N2_2H+^+ during the formation of a Sun-like star

Although chemical models predict that the deuterium fractionation in N$_2$H$^+$ is a good evolutionary tracer in the star formation process, the fractionation of nitrogen is still a poorly understood process. Recent models have questioned the similar evolutionary trend expected for the two fractionation mechanisms in N$_2$H$^+$, based on a classical scenario in which ion-neutral reactions occurring in cold gas should have caused an enhancement of the abundance of N$_2$D$^+$, $^{15}$NNH$^+$, and N$^{15}$NH$^+$. In the framework of the ASAI IRAM-30m large program, we have investigated the fractionation of deuterium and $^{15}$N in N$_2$H$^+$ in the best known representatives of the different evolutionary stages of the Sun-like star formation process. The goal is to ultimately confirm (or deny) the classical "ion-neutral reactions" scenario that predicts a similar trend for D and $^{15}$N fractionation. We do not find any evolutionary trend of the $^{14}$N/$^{15}$N ratio from both the $^{15}$NNH$^+$ and N$^{15}$NH$^+$ isotopologues. Therefore, our findings confirm that, during the formation of a Sun-like star, the core evolution is irrelevant in the fractionation of $^{15}$N. The independence of the $^{14}$N/$^{15}$N ratio with time, found also in high-mass star-forming cores, indicates that the enrichment in $^{15}$N revealed in comets and protoplanetary disks is unlikely to happen at core scales. Nevertheless, we have firmly confirmed the evolutionary trend expected for the H/D ratio, with the N$_2$H$^+$/N$_2$D$^+$ ratio decreasing before the pre-stellar core phase, and increasing monotonically during the protostellar phase. We have also confirmed clearly that the two fractionation mechanisms are not related.Comment: 9 pages, 2 figures, accepted for publication in MNRA