Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local well-posedness is found in $B^\alpha_{p,q}$ for certain values of $p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each $p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition

On the Performance Prediction of BLAS-based Tensor Contractions

Tensor operations are surging as the computational building blocks for a variety of scientific simulations and the development of high-performance kernels for such operations is known to be a challenging task. While for operations on one- and two-dimensional tensors there exist standardized interfaces and highly-optimized libraries (BLAS), for higher dimensional tensors neither standards nor highly-tuned implementations exist yet. In this paper, we consider contractions between two tensors of arbitrary dimensionality and take on the challenge of generating high-performance implementations by resorting to sequences of BLAS kernels. The approach consists in breaking the contraction down into operations that only involve matrices or vectors. Since in general there are many alternative ways of decomposing a contraction, we are able to methodically derive a large family of algorithms. The main contribution of this paper is a systematic methodology to accurately identify the fastest algorithms in the bunch, without executing them. The goal is instead accomplished with the help of a set of cache-aware micro-benchmarks for the underlying BLAS kernels. The predictions we construct from such benchmarks allow us to reliably single out the best-performing algorithms in a tiny fraction of the time taken by the direct execution of the algorithms.Comment: Submitted to PMBS1

Colouring titanium alloys by anodic oxidation

The present study is focused on analyzing the change of colors of anodized titanium and effects of applied electrolytic voltages on chromatics. The titanium specimens were anodize in 20 g/L citric acid and 20 g/L baking soda electrolyte by use of different voltages. The colors of anodize titanium were measured with a spectrophotometer and then evaluated in the CIELAB color space. It is found that different volt produces different colors. Anodizing in the range of 15 V to 150 V produces respectively a wide spectrum of color ranging from brown to fuchsia. It can be concluded that the colors of the anodize titanium are dependent upon the applied voltages

Can Quantum de Sitter Space Have Finite Entropy?

If one tries to view de Sitter as a true (as opposed to a meta-stable) vacuum, there is a tension between the finiteness of its entropy and the infinite-dimensionality of its Hilbert space. We invetsigate the viability of one proposal to reconcile this tension using $q$-deformation. After defining a differential geometry on the quantum de Sitter space, we try to constrain the value of the deformation parameter by imposing the condition that in the undeformed limit, we want the real form of the (inherently complex) quantum group to reduce to the usual SO(4,1) of de Sitter. We find that this forces $q$ to be a real number. Since it is known that quantum groups have finite-dimensional representations only for $q=$ root of unity, this suggests that standard $q$-deformations cannot give rise to finite dimensional Hilbert spaces, ruling out finite entropy for q-deformed de Sitter.Comment: 10 pages, v2: references added, v3: minor corrections, abstract and title made more in-line with the result, v4: published versio

Dark Matter on the Smallest Scales

This work investigates the dark matters structures that form on the smallest cosmological scales. We find that the types and abundances of structures which form at approximately Earth-mass scales are very sensitive to the nature of dark matter. We explore various candidates for dark matter and determine the corresponding properties of small-scale structure. In particular, we discuss possibilities for indirect detection of dark matter through small-scale structure, and comment on the potential of these methods for discriminating between dark matter candidates.Comment: 10 Pages, Proceedings from the Dark Matter working group at the 86th Les Houches Summer School: Particle Physics and Cosmolog

New driver alterations in non-small cell lung cancer. A narrative review

Objective: This review aims to provide an up-to-date snapshot on the state of development of novel biomarker-driven treatments in non-small cell lung cancer (NSCLC). Background: The introduction of immune checkpoint inhibitors and target therapies has revolutionized the natural history of many NSCLCs, allowing for lasting and profound responses. In particular, mutations in the epidermal growth factor receptor (EGFR), rearrangements of the anaplastic lymphoma kinase (ALK), or oncogene c-Ros 1 (ROS1) have marked a paradigm shift in the treatment of NSCLC. Furthermore, new inhibitors for B-Raf proto-oncogene (BRAF), rearranged during transfection (RET), mesenchymal-to-epithelial transition factor (MET), or neurotrophic tyrosine kinase (NTRK) 1–3 have revealed fascinating data, obtaining accelerated approvals from the Food and Drug Administration (FDA) and European Medicines Agency (EMA). Today, the extensive use of next-generation sequencing (NGS) techniques has shown a broad molecular heterogeneity of NSCLC. Many of the mutations identified are considered potential therapeutic targets, and numerous studies are currently evaluating the efficacy of selective inhibitors. Methods: We carried out an extensive review of the literature on PubMed, Web of Science, and Scopus databases and the congress abstracts presented at the American Society of Clinical Oncology (ASCO), European Society for Medical Oncology (ESMO), and World Conference on Lung Cancer (WCLC) in the last 5 years. Our analysis considered works regarding new inhibitors for alterations of Kirsten rat sarcoma viral oncogene homolog (KRAS), PIK3CA, neuregulin-1 (NRG-1), human epidermal growth factor receptor 2 (HER2), fibroblast growth factor receptor (FGFR), genes that have recently become no longer undruggable. Conclusions: Precision oncology is revolutionizing the natural history of NSCLC. Several alterations have been identified as possible treatment targets, and numerous inhibitors show promising results in ongoing clinical trials

Arterial Spin Labeling MRI in Carotid Stenosis: Arterial Transit Artifacts May Predict Symptoms

Background: Stenosis of the internal carotid artery has a higher risk for stroke. Many investigations have focused on structure and plaque composition as signs of plaque vulnerability, but few studies have analyzed hemodynamic changes in the brain as a risk factor. Purpose: To use 3-T MRI methods including contrast material–enhanced MR angiography, carotid plaque imaging, and arterial spin labeling (ASL) to identify imaging parameters that best help distinguish between asymptomatic and symptomatic participants with carotid stenosis. Materials and Methods: Participants with carotid stenosis from two ongoing prospective studies who underwent ASL and carotid plaque imaging with use of 3-T MRI in the same setting from 2014 to 2018 were studied. Participants were assessed clinically for recent symptoms (transient ischemic attack or stroke) and divided equally into symptomatic and nonsymptomatic groups. Reviewers were blinded to the symptomatic status and MRI scans were analyzed for the degree of stenosis, plaque surface structure, presence of intraplaque hemorrhage (IPH), circle of Willis collaterals, and the presence and severity of arterial transit artifacts (ATAs) at ASL imaging. MRI findings were correlated with symptomatic status by using t tests and the Fisher exact test. Results: A total of 44 participants (mean age, 71 years 6 10 [standard deviation]; 31 men) were evaluated. ATAs were seen only in participants with greater than 70% stenosis (16 of 28 patients; P , .001) and were associated with absence of anterior communicating artery (13 of 16 patients; P = .003). There was no association between history of symptoms and degree of stenosis (27 patients with 70% stenosis and 17 patients with ,70%; P = .54), IPH (12 patients with IPH and 32 patients without IPH; P = .31), and plaque surface structure (17 patients with irregular or ulcerated plaque and 27 with smooth plaque; P = .54). Participants with ATAs (n = 16) were more likely to be symptomatic than were those without ATAs (n = 28) (P = .004). Symptomatic status also was associated with the severity of ATAs (P = .002). Conclusion: Arterial transit artifacts were the only factor associated with recent ischemic symptoms in participants with carotid stenosis. The degree of stenosis, plaque ulceration, and intraplaque hemorrhage were not associated with symptomatic statu