Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms

Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the W1p norm and we consider Lagrange finite elements of arbitrary polynomial order m-1. We establish sharp asymptotic error estimates as N tends to infinity when the optimal anisotropic triangulation is used. A similar problem has been studied earlier, but with the error measured in the Lp norm. The extension of this analysis to the W1p norm is crucial in order to match more closely the needs of numerical PDE analysis, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the Lp error norm. Our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We discuss the extension of our results to finite elements on simplicial partitions of a domain of arbitrary dimension, and we provide with some numerical illustration in two dimensions.Comment: 37 pages, 6 figure

Smooth-AP: Smoothing the Path Towards Large-Scale Image Retrieval

Optimising a ranking-based metric, such as Average Precision (AP), is notoriously challenging due to the fact that it is non-differentiable, and hence cannot be optimised directly using gradient-descent methods. To this end, we introduce an objective that optimises instead a smoothed approximation of AP, coined Smooth-AP. Smooth-AP is a plug-and-play objective function that allows for end-to-end training of deep networks with a simple and elegant implementation. We also present an analysis for why directly optimising the ranking based metric of AP offers benefits over other deep metric learning losses. We apply Smooth-AP to standard retrieval benchmarks: Stanford Online products and VehicleID, and also evaluate on larger-scale datasets: INaturalist for fine-grained category retrieval, and VGGFace2 and IJB-C for face retrieval. In all cases, we improve the performance over the state-of-the-art, especially for larger-scale datasets, thus demonstrating the effectiveness and scalability of Smooth-AP to real-world scenarios.Comment: Accepted at ECCV 202

A Graphene Surface Force Balance

We report a method for transferring graphene, grown by chemical vapor deposition, which produces ultraflat graphene surfaces (root-mean-square roughness of 0.19 nm) free from polymer residues over macroscopic areas (>1 cm2). The critical step in preparing such surfaces involves the use of an intermediate mica template, which itself is atomically smooth. We demonstrate the compatibility of these model surfaces with the surface force balance, opening up the possibility of measuring normal and lateral forces, including friction and adhesion, between two graphene sheets either in contact or across a liquid medium. The conductivity of the graphene surfaces allows forces to be measured while controlling the surface potential. This new apparatus, the graphene surface force balance, is expected to be of importance to the future understanding of graphene in applications from lubrication to electrochemical energy storage systems

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

Description of the two-nucleon system on the basis of the Bargmann representation of the S matrix

For the effective-range function $k\cot \delta$, a pole approximation that involves a small number of parameters is derived on the basis of the Bargmann representation of the $S$ matrix. The parameters of this representation, which have a clear physical meaning, are related to the parameters of the Bargmann $S$ matrix by simple equations. By using a polynomial least-squares fit to the function $k\cot \delta$ at low energies, the triplet low-energy parameters of neutron-proton scattering are obtained for the latest experimental data of Arndt et al. on phase shifts. The results are $a_{t}=5.4030$fm, $r_{t}=1.7494$fm, and $v_{2}=0.163$fm$^{3}$. With allowance for the values found for the low-energy scattering parameters and for the pole parameter, the pole approximation of the function $k\cot \delta$ provides an excellent description of the triplet phase shift for neutron-proton scattering over a wide energy range ($T_{\text{lab}}\lesssim 1000$MeV), substantially improving the description at low energies as well. For the experimental phase shifts of Arndt et al., the triplet shape parameters $v_{n}$ of the effective-range expansion are obtained by using the pole approximation. The description of the phase shift by means of the effective-range expansion featuring values found for the low-energy scattering parameters proves to be fairly accurate over a broad energy region extending to energy values approximately equal to the energy at which this phase shift changes sign, this being indicative of a high accuracy and a considerable value of the effective-range expansion in describing experimental data on nucleon-nucleon scattering. The properties of the deuteron that were calculated by using various approximations of the effective-range function comply well with their experimental values.Comment: 39 pages, 3 figure

Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations

We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known kinetic equations of the Brownian motion theory and contain symmetric fractional derivatives over velocity and space, respectively. With the help of these equations we study analytically the processes of linear relaxation in a force - free case and for linear oscillator. For a weakly damped oscillator we also get kinetic equation for the distribution in slow variables. Linear relaxation processes are also studied numerically by solving corresponding Langevin equations with the source which is a discrete - time approximation to a white Levy noise. Numerical and analytical results agree quantitatively.Comment: 30 pages, LaTeX, 13 figures PostScrip

Entropic Uncertainty Relations in Quantum Physics

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the R\'enyi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentionedComment: 35 pages, review pape

The role of glacier mice in the invertebrate colonisation of glacial surfaces: the moss balls of the Falljökull, Iceland

Glacier surfaces have a surprisingly complex ecology. Cryoconite holes contain diverse invertebrate communities while other invertebrates, such as Collembola often graze on algae and windblown dead organic on the glacier surface. Glacier mice (ovoid unattached moss balls) occur on some glaciers worldwide. Studies of these glacier mice have concentrated on their occurrence and mode of formation. There are no reports of the invertebrate communities. But, such glacier mice may provide a suitable favourable habitat and refuge for a variety of invertebrate groups to colonise the glacier surface. Here we describe the invertebrate fauna of the glacier mice (moss balls) of the Falljökull, Iceland. The glacier mice were composed of Racomitrium sp. and varied in size from 8.0 to 10.0 cm in length. All glacier mice studied contained invertebrates. Two species of Collembola were present. Pseudisotoma sensibilis (Tullberg, 1876) was numerically dominant with between 12 and 73 individuals per glacier mouse while Desoria olivacea (Tullberg, 1871) occurred but in far lower numbers. Tardigrada and Nematoda had mean densities of approximately 200 and 1,000 respectively. No Acari, Arachnida or Enchytraeidae were observed which may be related to the difficulty these groups have in colonizing the glacier mice. We suggest that glacier mice provide an unusual environmentally ameliorated microhabitat for an invertebrate community dwelling on a glacial surface. The glacier mice thereby enable an invertebrate fauna to colonise an otherwise largely inhospitable location with implications for carbon flow in the system

Exotic smooth structures and symplectic forms on closed manifolds

We give a short proof of the (known) result that there are no Kaehler structures on exotic tori. This yields a negative solution to a problem posed by Benson and Gordon. W discuss the symplectic version of the problem and analyze results which yield an evidence for the conjecture that there are no symplectic structures on exotic tori.Comment: AMSLaTeX, 16 pages, a new version. A survey of the symplectic version of the problem is adde

Comparison of crystallization characteristics and mechanical properties of polypropylene processed by ultrasound and conventional micro injection molding

YesUltrasound injection molding has emerged as an alternative production route for the manufacturing of micro-scale polymeric components, where it offers significant benefits over the conventional micro-injection molding process. In this work, the effects of ultrasound melting on the mechanical and morphological properties of micro-polypropylene parts were characterized. The ultrasound injection molding process was experimentally compared to the conventional micro-injection molding process using a novel mold, which allows mounting on both machines and visualization of the melt flow for both molding processes. Direct measurements of the flow front speed and temperature distributions were performed using both conventional and thermal high-speed imaging techniques. The manufacturing of micro-tensile specimens allowed the comparison of the mechanical properties of the parts obtained with the different processes. The results indicated that the ultrasound injection molding process could be an efficient alternative to the conventional process