Lagrange interpolation and quasi-interpolation using trivariate splines on a uniform partition

We develop quasi-interpolation methods and a Lagrange interpolation method for trivariate splines on a regular tetrahedral partition, based on the Bernstein-Bézier representation of polynomials. The partition is based on the bodycentered cubic grid. Our quasi-interpolation operators use quintic C2 splines and are defined by giving explicit formulae for each coefficient. One operator satisfies a certain convexity condition, but has sub-optimal approximation order. A second operator has optimal approximation order, while a third operator interpolates the provided data values. The first two operators are defined by a small set of computation rules which can be applied independently to all tetrahedra of the underlying partition. The interpolating operator is more complex while maintaining the best-possible approximation order for the spline space. It relies on a decomposition of the partition into four classes, for each of which a set of computation rules is provided. Moreover, we develop algorithms that construct blending operators which are based on two quasi-interpolation operators defined for the same spline space, one of which is convex. The resulting blending operator satisfies the convexity condition for a given data set. The local Lagrange interpolation method is based on cubic C1 splines and focuses on low locality. Our method is 2-local, while comparable methods are at least 4-local. We provide numerical tests which confirm the results, and high-quality visualizations of both artificial and real-world data sets

The genetic architecture of the human cerebral cortex

The cerebral cortex underlies our complex cognitive capabilities, yet little is known about the specific genetic loci that influence human cortical structure. To identify genetic variants that affect cortical structure, we conducted a genome-wide association meta-analysis of brain magnetic resonance imaging data from 51,665 individuals. We analyzed the surface area and average thickness of the whole cortex and 34 regions with known functional specializations. We identified 199 significant loci and found significant enrichment for loci influencing total surface area within regulatory elements that are active during prenatal cortical development, supporting the radial unit hypothesis. Loci that affect regional surface area cluster near genes in Wnt signaling pathways, which influence progenitor expansion and areal identity. Variation in cortical structure is genetically correlated with cognitive function, Parkinson's disease, insomnia, depression, neuroticism, and attention deficit hyperactivity disorder

Hard color-singlet exchange in dijet events in proton-proton collisions at root s=13 TeV

Events where the two leading jets are separated by a pseudorapidity interval devoid of particle activity, known as jet-gap-jet events, are studied in proton-proton collisions at root s = 13 TeV. The signature is expected from hard color-singlet exchange. Each of the highest transverse momentum (p(T)) jets must have p(T)(jet) > 40 GeV and pseudorapidity 1.4 0.2 GeV in the interval vertical bar eta vertical bar < 1 between the jets are observed in excess of calculations that assume only color-exchange. The fraction of events produced via color-singlet exchange, f(CSE), is measured as a function of p(T)(jet2), the pseudorapidity difference between the two leading jets, and the azimuthal angular separation between the two leading jets. The fraction f(CSE) has values of 0.4-1.0%. The results are compared with previous measurements and with predictions from perturbative quantum chromodynamics. In addition, the first study of jet-gap-jet events detected in association with an intact proton using a subsample of events with an integrated luminosity of 0.40 pb(-1) is presented. The intact protons are detected with the Roman pot detectors of the TOTEM experiment. The f(CSE) in this sample is 2.91 +/- 0.70(stat)(-1.01)(+1.08)(syst) times larger than that for inclusive dijet production in dijets with similar kinematics.Peer reviewe