Resolvent estimates and smoothing for homogeneous partial differential operators on graded Lie groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|^1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?^N some new classes of partial differential operators are treated.     

Vectorial penalization for generalized functional constrained problems

In this paper we use a double penalization procedure in order to reduce a set-valued optimization problem with functional constraints to an unconstrained one. The penalization results are given in several cases: for weak and strong solutions, in global and local settings, and considering two kinds of epigraphical mappings of the set-valued map that defines the constraints. Then necessary and sufficient conditions are obtained separately in terms of Bouligand derivatives of the objective and constraint mappings.     

On Some Methods to Derive Necessary and Sufficient Optimality Conditions in Vector Optimization

The aim of this paper is to address new approaches, in separate ways, to necessary and, respectively, sufficient optimality conditions in constrained vector optimization. In this respect, for the necessary optimality conditions that we derive, we use a kind of vectorial penalization technique, while for the sufficient optimality conditions we make use of an appropriate scalarization method. In both cases, the approaches couple a basic technique (of penalization or scalarization, respectively) with several results in variational analysis and optimization obtained by the authors in the last years. These combinations allow us to arrive to optimality conditions which are, in terms of assumptions made, new.     

NdHO, a novel oxyhydride

A new metal oxyhydride; neodymium oxyhydride, NdHO, has been synthesized from a reactant mixture of metal hydride (CaH₂ or NdH₃) and neodymium oxide (Nd₂O₃). The unit cell dimensions decrease smoothly in the series from LaHO, CeHO, PrHO to NdHO, in line with the lanthanide contraction. The crystal structure of NdHO is described on the basis of Rietveld refinement on neutron powder diffraction data:

Space group: P4/nmm (no. 129, D_4h⁷).

Axis lengths: a=7.8480(5) Å, c=5.5601(8) Å.

Volume: V=342.46(6) Å³.

The tetragonal structure is derived from the fluorite structure, showing complete ordering of hydride and oxide ions over the anion sublatttice. The formation of NdHO was further substantiated by Raman spectroscopy.     

An axially chiral, electron-deficient borane: synthesis, coordination chemistry, Lewis acidity, and reactivity

An axially chiral dihydroborepine with a binaphthyl backbone and a C(6)F(5) substituent at the boron atom was prepared by transmetalation from the corresponding tin precursor. This novel motif was structurally characterized by X-ray diffraction analysis as its THF and its PhCN Lewis acid/base complex. (1)H NMR measurements at variable temperatures of the former adduct revealed a remarkable dynamic behavior in solution. Several more Lewis pairs with oxygen, nitrogen, carbon, and phosphorus σ-donors were synthesized and analyzed by multinuclear NMR spectroscopy. The determination of the borane's Lewis acidity with the Gutmann-Beckett method attests its substantial Lewis acidity [85% with Et(3) PO as well as 74% with Ph(3) PO relative to the parent B(C(6)F(5))(3)]. Representative examples of Si-H bond activation (carbonyl reduction and dehydrogenative Si-O coupling) are included, demonstrating the chemical stability and the synthetic potential of the new chiral boron-based Lewis acid.     

Isolated singularities of polyharmonic inequalities

We study nonnegative classical solutions u of the polyharmonic inequality

     

Ruled quartic surfaces,models and classification

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling) are based on a result of Rohn concerning curves in \mathbbP¹×\mathbbP¹{\mathbb{P}^1\times \mathbb{P}^1} of bi-degree (2, 2). This is given here a conceptional proof.     

Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

We consider the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton’s laws. The corresponding system is a free boundary problem and the main result of the paper is the existence of weak solutions for this problem.     

Analyticity and Gevrey-Class Regularity for the Second-Grade Fluid Equations

We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions that does not vanish as t → ∞, and which is independent of the Rivlin–Ericksen material parameter α. Applications to the damped incompressible Euler equations are also given.