A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.Comment: 46 page

Functional Roles of Nova in RNA Metabolism

Nova-1 is a neuron-specific RNA binding protein, distributed both in the nucleus and the cytoplasm of several populations of neurons. The work presented here confirms that Nova-1 binds a sequence of RNA found in an intron of the glycine receptor a2 subunit pre-mRNA. Evidence that Nova-1 acts as a regulator of alternative splicing in transfected cell lines is presented. Furthermore, a direct role of Nova-1 on splicing is demonstrated by establishing an in vitro assay for Nova-l\u27s regulatory role in splicing. Potential functional partners of Nova are suggested by the demonstration of physical interactions between Nova-1 and molecules whose action in the splicing machinery is well described, such as Ul 70K, a protein component of UlsnRNP, and U2AF65. The sedimentation properties of Nova in neuronal cytoplasmic extracts are consistent with the engagement of Nova in heterogeneous structures, probably mRNPs, and with polysomes, suggesting a role for Nova in the regulation of cytoplasmic phenomena of RNA metabolism, such as mRNA localization and translation. These considerations prompted a gene expression screen aimed to identify differences in the pool of mRNAs associated with heavy polysomes in wild type and Nova-1-null mice. Several genes whose mRNAs have been found to undergo changes in abundance specifically in the polysome fraction in the absence of Nova have described functional roles in post-synaptic terminal structure and functions. Nova-1 was also found to be a phospho-protein, and phosphorylation site is in an alternative cassette exon

Space-charge effects in high-energy photoemission

Pump-and-probe photoelectron spectroscopy (PES) with femtosecond pulsed sources opens new perspectives in the investigation of the ultrafast dynamics of physical and chemical processes at the surfaces and interfaces of solids. Nevertheless, for very intense photon pulses a large number of photoelectrons are simultaneously emitted and their mutual Coulomb repulsion is sufficiently strong to significantly modify their trajectory and kinetic energy. This phenomenon, referred as space-charge effect, determines a broadening and shift in energy for the typical PES structures and a dramatic loss of energy resolution. In this article we examine the effects of space charge in PES with a particular focus on time-resolved hard X-ray (~10 keV) experiments. The trajectory of the electrons photoemitted from pure Cu in a hard X-ray PES experiment has been reproduced through $N$-body simulations and the broadening of the photoemission core-level peaks has been monitored as a function of various parameters (photons per pulse, linear dimension of the photon spot, photon energy). The energy broadening results directly proportional to the number $N$ of electrons emitted per pulse (mainly represented by secondary electrons) and inversely proportional to the linear dimension $a$ of the photon spot on the sample surface, in agreement with the literature data about ultraviolet and soft X-ray experiments. The evolution in time of the energy broadening during the flight of the photoelectrons is also studied. Despite its detrimental consequences on the energy spectra, we found that space charge has negligible effects on the momentum distribution of photoelectrons and a momentum broadening is not expected to affect angle-resolved experiments. Strategy to reduce the energy broadening and the feasibility of hard X-ray PES experiments at the new free-electron laser facilities are discussed.Comment: 15 pages, 2 tables, 8 figure

On sets with finite distributional fractional perimeter

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.Comment: 18 page

Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula

Given $\alpha\in[0,1]$ and $p\in[1,+\infty]$, we define the space $\mathcal{DM}^{\alpha,p}(\mathbb R^n)$ of $L^p$ vector fields whose $\alpha$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $\alpha$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.Comment: 22 page

Insight on Hole-Hole Interaction and Magnetic Order from Dichroic Auger-Photoelectron Coincidence Spectra

The absence of sharp structures in the core-valence-valence Auger line shapes of partially filled bands has severely limited the use of electron spectroscopy in magnetic crystals and other correlated materials. Here by a novel interplay of experimental and theoretical techniques we achieve a combined understanding of the Photoelectron, Auger %$M_{23}M_{45}M_{45}$ and Auger-Photoelectron Coincidence Spectra (APECS) of CoO. This is a prototype antiferromagnetic material in which the recently discovered Dichroic Effect in Angle Resolved (DEAR) APECS reveals a complex pattern in the strongly correlated Auger line shape. A calculation of the \textit{unrelaxed} spectral features explains the pattern in detail, labeling the final states by the total spin. The present theoretical analysis shows that the dichroic effect arises from a spin-dependence of the angular distribution of the photoelectron-Auger electron pair detected in coincidence, and from the selective power of the dichroic technique in assigning different weights to the various spin components. Since the spin-dependence of the angular distribution exists in the antiferromagnetic state but vanishes at the N\'eel temperature, the DEAR-APECS technique detects the phase transition from its local effects, thus providing a unique tool to observe and understand magnetic correlations in such circumstances, where the usual methods (neutron diffraction, specific heat measurements) are not applicable.Comment: Accepted by: Physical Review Letter

The fractional variation and the precise representative of BVα,pBV^{\alpha,p} functions

We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space $BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $\alpha\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{\alpha,p}$ function.Comment: 32 page

Relationship between Protein Oxidation Biomarkers and Uterine Health in Dairy Cows during the Postpartum Period

High neutrophil (PMN, Polymorphonuclear neutrophil) counts in the endometrium of cows affected by endometritis, suggests the involvement of oxidative stress (OS) among the causes of impaired fertility. Protein oxidation, in particular, advanced oxidation protein products (AOPP), are OS biomarkers linked to PMN activity. To test this hypothesis, the relationship between protein oxidation and uterus health was studied in thirty-eight dairy cows during the puerperium. The animals were found to be cycling, without any signs of disease and pharmacological treatments. PMN count was performed either through a cytobrush or a uterine horn lavage (UHL). Cows were classified into four groups, based on the uterine ultrasonographic characteristics and the PMN percentage in the uterine horns with a higher percentage of high neutrophil horn (HNH). They were classified as: Healthy (H); Subclinical Endometritis (SCE); Grade 1 Endometritis (EM1); and Grade 2 Endometritis (EM2). AOPP and carbonyls were measured in plasma and UHL. UHL samples underwent Western blot analysis to visualize the carbonyl and dityrosine formation. Plasma AOPP were higher (p < 0.05) in EM2. AOPP and carbonyl group concentrations were higher in the HNH samples (p < 0.05). Protein concentration in the UHL was higher in the EM2 (p < 0.05). Carbonyl and dityrosine formation was more intense in EM1 and EM2. Protein oxidation observed in the EM2 suggests the presence of an inflammatory status in the uterus which, if not adequately hindered, could result in low fertility

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

We continue the study of the space $BV^\alpha(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{\alpha,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $\alpha\in(0,1)$, considered in the previous works arXiv:1809.08575 and arXiv:1910.13419. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{\alpha,p}(\mathbb R^n)=L^{\alpha,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{\alpha,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^\alpha$ strongly converges to the Riesz transform as $\alpha\to0^+$ for $H^1\cap W^{\alpha,1}$ and $S^{\alpha,p}$ functions. We also study the convergence of the $L^1$-norm of the $\alpha$-rescaled fractional gradient of $W^{\alpha,1}$ functions. To achieve the strong limiting behavior of $\nabla^\alpha$ as $\alpha\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.Comment: 43 page