Uniqueness of Radial Solutions for the Fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (?Δ)^s with s ? (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (?Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

Low-Temperature High-Resolution Solid-State (cryoMAS) NMR of Han Purple BaCuSi2O6

We present low-temperature (low-T, sub-liquid-N₂) high-speed high-resolution ²⁹Si solid-state (cryoMAS) nuclear magnetic resonance studies on a model 2D-BEC quantum magnet BaCuSi₂O₆, known also as Han Purple. We observe broadened ²⁹Si lines below the well-established 100 K structural phase transition confirming the existence of inhomogeneities at low temperatures. Interestingly, the low-T spectra of \({\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}\) closely resemble those of the novel compound Ba₂CoSi₂O₆Cl₂ taken at room temperature. This suggests that the Co compound features structural modulations or inhomogeneities already at room temperature. The low-T crystal structure and magnetism of \({\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}\) are more complex than previously believed, and deserve further investigation.     

[μ‐(Me3SiCH2Sb)5–Sb1,Sb3–{W(CO)5}2] und [{(Me3Si)2CHSb}3Fe(CO)4] – zwei cyclische Komplexe mit Antimon‐Liganden

Syntheses and Crystal Structures of [μ‐(Me₃SiCH₂Sb)₅–Sb¹,Sb³–{W(CO)₅}₂] and [{(Me₃Si)₂CHSb}₃Fe(CO)₄] – Two Cyclic Complexes with Antimony Ligands cyclo‐(Me₃SiCH₂Sb)₅ reacts with [(THF)W(CO)₅] (THF = tetrahydrofuran) to form cyclo‐[μ‐(Me₃SiCH₂Sb)₅–Sb¹,Sb³–{W(CO)₅}₂] ( 1 ). The heterocycle cyclo‐ [{(Me₃Si)₂CHSb}₃Fe(CO)₄] ( 2 ) is formed by an insertion reaction of cyclo‐[(Me₃Si)₂CHSb]₃ and [Fe₂(CO)₉]. The crystal structures of 1 and 2 are reported.     

α-Trifluoroacetyl-δ-valerolactone: synthesis, acyl-lactone rearrangement and unexpected easy decarboxylation of the rearranged product

Claisen condensation of δ-valerolactone and ethyl trifluoroacetate leads to α-trifluoroacetyl-δ-valerolactone, existing completely in CDCl₃ solution as an enol. Upon treatment of this cyclic 1,3-ketoester with aqueous HCl, an acyl-lactone rearrangement takes place. The structure of the resulting pyran was confirmed by an X-ray analysis. The ability of the rearrangement product to lose its carboxylic group under unexpected mild conditions was used for the synthesis of the corresponding 3-unsubstituted cyclic hemi-ketal.     

Non-Oxido-Vanadium(IV) Metalloradical Complexes with Bidentate 1,2-Dithienylethene Ligands: Observation of Reversible Cyclization of the Ligand Scaffold in Solution

Derivatives of 1,2-dithienylethene (DTE) have superb photochromic properties due to an efficient reversible photocyclization reaction of their hexatriene structure and, thus, have application potential in materials for optoelectronics and (multi-responsive) molecular switches. Transition-metal complexes bearing switchable DTE motifs commonly incorporate their coordination site rather distant from the hexatriene system. In this work the redox active ligand 1,2-bis(2,5-dimethylthiophen-3-yl)ethane-1,2-dione is described, which reacts with [V(TMEDA)₂Cl₂] to give a rare non-oxido vanadium(IV) species 3(M,M/P,P) . This blue complex has two bidentate en-diolato ligands which chelate the V^IV center and give rise to two five-membered metallacycles with the adjacent hexatriene DTE backbone bearing axial chirality. Upon irradiation with UVA light or prolonged heating in solution, the blue compound 3(M,M/P,P) converts into the purple atropisomer 4(para,M/para,P) . Both complexes were isolated and structurally characterized by single-crystal X-ray diffraction analysis (using lab source and synchrotron radiation). The antiparallel configuration (M or P helicity) present in both 3(M,M/P,P) and 4(para,M/para,P) is a prerequisite for (reversible) 6π cyclization reactions. A CW EPR spectroscopic study reveals the metalloradical character for 3(M,M/P,P) and 4(para,M/para,P) and indicates dynamic reversible cyclization of the DTE backbone in complex 3(M,M/P,P) at ambient temperature in solution.     

An effective synthetic route to ortho-difluoromethyl arylphosphosphonates: studies on the reactivity of phosphorus- and fluorine-containing functions

Herein, we report a new and convenient methodology for the synthesis of ortho-XCF₂ arylphosphonates via Diels-Alder reaction of selected 1,3-butadienes with XCF₂-≡-P(O)(OEt)₂, followed by the aromatization of the cyclic vinylphosphonates obtained using the KMnO₄/Al₂O₃ system. The reactivity of ortho-XCF₂ arylphosphonates was then examined to give the respective dichlorides that were converted to the corresponding phosphonic acids, phosphine oxides or a carboxylic acid (upon hydrolysis of the CF₂Br group). When ortho-XCF₂ arylphosphonates (X=Br) were treated with Zn/CuBr and an electrophile, the dimeric product ArCFCFAr was isolated only. The lithiation of the CF₂H group (X=H) allowed however to obtain products of nucleophilic substitution with various electrophiles.     

How Many Molecules are Required to Obtain a Steady Faradaic Current from Mediated Electron Transfer at a Single Nanoparticle on a Supporting Surface?

We investigate the chronoamperometric noise characteristics of electron‐transfer reactions occurring on single nanoparticles (NPs) and assemblies of well‐separated NPs on a supporting surface. To this end, we combine a formerly described expression for the steady‐state current of a single particle with the shot‐noise model and derive an expression for the signal‐to‐noise ratio as a function of bulk concentration and particle radius. Our findings are supported by random‐walk simulations, which closely match the analytical results.