A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness

In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^{d} \subset\mathbb{R}^{d+1}$ . The reproducing kernel is given by an integral representation using the truncated power function $(\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}$ supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β=1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003). We show that the reproducing kernel is a sum of the Euclidean distance ∥x?y∥ of the arguments of the kernel raised to the power of 2β?1 and an adjustment in the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β?1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For $\beta\in\mathbb{N}$ , the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Stolarsky’s invariance principle states that the sum of all mutual distances among N points plus a certain multiple of the spherical cap $\mathbb{L}_{2}$ -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap $\mathbb{L}_{2}$ -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over $\mathbb{S}^{d}$ of smoothness (d+1)/2 provided with the reproducing kernel 1?C _d ∥x?y∥ for some constant C _d . Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap $\mathbb{L}_{2}$ -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over $\mathbb{S}^{d}$ of arbitrary smoothness s=β?1/2+d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0,1] ^s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).     

Desorption of C60 upon thermal decomposition of cesium C58 fullerides

A monodispersed fullerene material comprising exclusively C(58) cages was doped with Cs to generate Cs(x)C(58) films of various compositions. The resulting modified properties have been studied using a variety of surface analysis methods with emphasis on thermal desorption and ultraviolet photoelectron spectroscopies. Cs doping raises the thermal stability of C(58) films which are characterized by quasi-covalent cage-cage bonds between annelated pentagon sites. Desorption mass spectra show emission of significant amounts of C(60) at elevated temperatures implying that Cs doping can activate C(58)→C(60) conversion in the condensed phase. In the case of saturated Cs(x)C(58) films, up to 4.5% of the initially deposited C(58) can be desorbed as C(60). From the spectroscopic data, we infer that Cs insertion and transport into the interstitial sites of the C(58) solid is accompanied by spontaneous electron transfer to the electronegative fullerene framework-leading to a weakening of intercage carbon-carbon bonds. At the same time, the overall cohesion of the solid film is enhanced by the formation of multiple ionic Cs(+) (β)C(58) (-) (δ) interactions. Near 800 K, Cs(+) activates∕catalyzes concerted disproportionation reactions resulting in the transfer of C(2) from C(58) (-) (δ) to neighbouring cages to yield C(60) (and C(56)). Heating Cs(x)C(58) films to beyond this temperature range yields a (high temperature) stable reaction product with a significantly modified UP spectrum and a finite density of states at the Fermi level.     

Synthesis and biological evaluation of 1,4-naphthoquinones and quinoline-5,8-diones as antimalarial and schistosomicidal agents

Improving the solubility of polysubstituted 1,4-naphthoquinone derivatives was achieved by introducing nitrogen in two different positions of the naphthoquinone core, at C-5 and at C-8 of menadione through a two-step, straightforward synthesis based on the regioselective hetero-Diels-Alder reaction. The antimalarial and the antischistosomal activities of these polysubstituted aza-1,4-naphthoquinone derivatives were evaluated and led to the selection of distinct compounds for antimalarial versus antischistosomal action. The Ag(II)-assisted oxidative radical decarboxylation of the phenyl acetic acids using AgNO(3) and ammonium peroxodisulfate was modified to generate the 3-picolinyl-menadione with improved pharmacokinetic parameters, high antimalarial effects and capacity to inhibit the formation of β-hematin.     

Nanophotonic lab-on-a-chip platforms including novel bimodal interferometers, microfluidics and grating couplers

One of the main limitations for achieving truly lab-on-a-chip (LOC) devices for point-of-care diagnosis is the incorporation of the "on-chip" detection. Indeed, most of the state-of-the-art LOC devices usually require complex read-out instrumentation, losing the main advantages of portability and simplicity. In this context, we present our last advances towards the achievement of a portable and label-free LOC platform with highly sensitive "on-chip" detection by using nanophotonic biosensors. Bimodal waveguide interferometers fabricated by standard silicon processes have been integrated with sub-micronic grating couplers for efficient light in-coupling, showing a phase resolution of 6.6 × 10(-4)× 2π rad and a limit of detection of 3.3 × 10(-7) refractive index unit (RIU) in bulk. A 3D network of SU-8 polymer microfluidics monolithically assembled at the wafer-level was included, ensuring perfect sealing and compact packaging. To overcome some of the drawbacks inherent to interferometric read-outs, a novel all-optical wavelength modulation system has been implemented, providing a linear response and a direct read-out of the phase variation. Sensitivity, specificity and reproducibility of the wavelength modulated BiMW sensor has been demonstrated through the label-free immunodetection of the human hormone hTSH at picomolar level using a reliable biofunctionalization process.     

Total synthesis of branimycin: an evolutionary approach

The first total synthesis of the macrolactone antibiotic branimycin (4) has been described. The key disconnection leads to a cis-dehydrodecalone core and a polyketide side chain which are connected via organometallic addition. The dehydrodecalone core was targeted via altogether five different approaches featuring various kinds of chiral elements and ring-closing methodology. In the end the most successful method starting from diepoxynaphthalene 109 was chosen to carry on with the synthesis. Thus the oxygen functions and carbon appendages were introduced via organometallic desymmetrization reactions to generate epoxy ketone 107, to which vinyl iodide 11 was added after conversion into the organolithium species. The synthesis was completed by introducing the ester side chain via Michael addition and subsequent macrolactonization.     

A non-photochemical approach to the bicyclo[3.2.0]heptane core of bielschowskysin

An asymmetric synthesis of the tricyclic core (-)-1 of the marine diterpene bielschowskysin is described. In particular, a methodology was developed to introduce the crucial quaternary center at C-12.     

Self condensation of enamines mediated by acetylation. A novel approach to 1-(azol-5-yl)-(1E,3Z)-butadiene-4-N,N-dimethylamines

Novel self-condensation of 3-(azol-5-yl)-1,1-dimethylenamines has been found to form new C-C bonds leading to 2,4-(1,2,3-triazole-1,2,3-thiadiazole-3-phenylisothiazole)-(1E,3Z)-5-yl-butadiene-1-amines. The discovered reaction represents a new example of C-H functionalization in unsaturated systems and can serve an efficient synthetic approach to rational design of new 2,4-(diazole-5-yl)-dieneamines.     

Compact complex surfaces with geometric structures related to split quaternions

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler, are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that a compact 4-manifold carries a para-hyperkähler structure iff it has a metric of split signature together with two parallel, null, orthogonal, pointwise linearly independent vector fields. Every compact complex surface admitting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite-dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperkähler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type S⁰$S^0$ and provide a list of compact complex surfaces which could carry para-hypercomplex structures.