On a rigidity result of singularities

The aim of this note is to prove, in the spirit of a rigidity result for isolated singularities of Schlessinger see Schlessinger (Invent Math 14:17–26, 1971) or also Kleiman and Landolfi (Compositio Math 23:407–434, 1971), a variant of a rigidity criterion for arbitrary singularities (Theorem 2.1 below). The proof of this result does not use Schlessinger’s Deformation Theory [Schlessinger (Trans Am Math Soc 130:208–222, 1968) and Schlessinger (Invent Math 14:17–26, 1971)]. Instead it makes use of Local Grothendieck-Lefschetz Theory, see (Grothendieck 1968, Éxposé 9, Proposition 1.4, page 106) and a Lemma of Zariski, see (Zariski, Am J Math 87:507–536, 1965, Lemma 4, page 526). I hope that this proof, although works only in characteristic zero, might also have some interest in its own.     

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.     

Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

Synthesis and Antimicrobial Activity of Some New N‐substituted‐5‐arylidene‐thiazolidine‐2,4‐diones

A total of 17 new N‐substituted derivatives ( 2b , 2c , 2d , 2e , 2f , 2g , 2h , 2i , 2j , 2k and 3b , 3c , 3d , 3e , 3f , 3g , 3h ) of 5‐((2‐phenylthiazol‐4‐yl)methylene) thiazolidine‐2,4‐dione ( 2a ) and 5‐(2,6‐dichloro‐ benzylidene)thiazolidine‐2,4‐dione ( 3a ) were synthesized. The structural elucidation of the newly synthesized compounds was based on elemental analysis and spectroscopic data (MS, ¹H NMR, ¹³C NMR), and their antimicrobial activities were assessed in vitro against several strains of Gram‐positive and Gram‐negative bacteria and one fungal strain (Candida albicans) as growth inhibition diameter. Some of them showed modest to good antibacterial activity against Gram‐negative Escherichia coli and Salmonella typhimurium and Gram‐positive Staphylococcus aureus, Bacillus cereus, and Enterococcus fecalis bacterial strains, whereas almost all the compounds were inactive against Listeria monocytogenes. All of the synthesized compounds showed moderate to very good activity against C. albicans.     

Characterisation of nickel–zinc ferrite/silica nanocomposites with low ferrite concentration obtained by an improved modified sol–gel method

The paper presents a study regarding the structure, morphology and magnetic behaviour of x% (Ni_0.65Zn_0.35Fe₂O₄)/(100 − x)% SiO₂ ferrimagnetic nanocomposites for low Ni–Zn ferrite concentration (x = 5, 10, 15, 20 and 30 mass percent) obtained by an improved modified sol–gel method. The obtained gels and nanocomposites have been characterized by fast Fourier transform-infrared (FT-IR) spectrometry, X-ray diffraction (XRD), transmission electron microscopy (TEM) and magnetic measurements (MM). The addition of a supplementary quantity of diol in the synthesis, corresponding to a molar ratio EG : TEOS = 1:1, and the control of the thermal treatment applied to the precursor xerogels tetraethylortosilicate (TEOS)–metal nitrates (MN)–ethylene glycol (EG) leads to fine (~2–9 nm), almost spherical Ni–Zn ferrite nanoparticles homogenously dispersed inside the amorphous SiO₂ matrix. TEM images reveal the fine nature and the narrow size distribution of the ferrite nanoparticles. Nanoparticles diameter increases with the ferrite concentration and with the annealing temperature. For all concentrations of ferrite in SiO₂ and all annealing temperature, we have obtained Ni_0.65Zn_0.35Fe₂O₄ ferrite as single phase (proven by XRD) in the amorphous silica matrix, only after a pre-treatment of synthesized gels, at 573 K, for 3 h. The magnetic behaviour of ferrite nanoparticles in quasi-static magnetic fields is very particular, depending on the annealing temperature and the ferrite content in silica matrix. We have obtained superparamagnetic behaviour for the nanocomposites, for a concentration of 30% ferrite in SiO₂ at high annealing temperature, of 1,273 K.     

Biotechnological Production of 20-alpha-Dihydrodydrogesterone at Pilot Scale

The human sex hormone progesterone plays an essential and complex role in a number of physiological processes. Progesterone deficiency is associated with menstrual disorders and infertility as well as premature birth and abortion. For progesterone replacement therapy, the synthetic progestogen dydrogesterone is commonly used. In the body, this drug is metabolized to 20α-dihydrodydrogesterone (20α-DHD), which also shows extensive pharmacological effects and hence could act as a therapeutic agent itself. In this study, we describe an efficient biotechnological production procedure for 20α-DHD that employs the stereo- and regioselective reduction of dydrogesterone in a whole-cell biotransformation process based on recombinant fission yeast cells expressing the human enzyme AKR1C1 (20α-hydroxysteroid dehydrogenase, 20α-HSD). In a fed-batch fermentation at pilot scale (70 L) with a genetically improved production strain and under optimized reaction conditions, an average 20α-DHD production rate of 190 μM day⁻¹ was determined for a total biotransformation time of 136 h. Combined with an effective and reliable downstream processing, a continuous production rate of 12.3 ± 1.4 g 20α-DHD per week and fermenter was achieved. We thus established an AKR-dependent whole-cell biotransformation process that can also be used for the production of other AKR1C1 substrates (as exemplarily shown by the production of 20α-dihydroprogesterone in gram scale) and is in principle suited for the production of further human AKR metabolites at industrial scale.     

Existence results for proper efficient solutions of vector equilibrium problems and applications

In this paper, we present sufficient conditions for the existence of Henig efficient solutions, superefficient solutions and Henig globally efficient solutions of a vector equilibrium problem in topological vector spaces, using a well-known separation theorem in infinite dimensional spaces. As an application, using a scalarization technique, existence results for proper efficient solutions of generalized vector variational inequalities are given.     

Turn-on fluorescence in tetraphenylethylene-based metal-organic frameworks: an alternative to aggregation-induced emission

Coordinative immobilization of functionalized tetraphenylethylene within rigid porous metal-organic frameworks (MOFs) turns on fluorescence in the typically non-emissive tetraphenylethylene core. The matrix coordination-induced emission effect (MCIE) is complementary to aggregation-induced emission. Despite the large interchromophore distances imposed by coordination to metal ions, a carboxylate analogue of tetraphenylethylene anchored by Zn(2+) and Cd(2+) ions inside MOFs shows fluorescence lifetimes in line with those of close-packed molecular aggregates. Turn-on fluorescence by coordinative ligation in a porous matrix is a powerful approach that may lead to new materials made from chromophores with molecular rotors. The potential utility of MCIE toward building new sensing materials is demonstrated by tuning the fluorescence response of the porous MOFs as a function of adsorbed small analytes.     

Reductive electrosynthesis of crystalline metal-organic frameworks

Electroreduction of oxoanions affords hydroxide equivalents that induce selective deposition of crystalline metal-organic frameworks (MOFs) on conductive surfaces. The method is illustrated by cathodic electrodeposition of Zn(4)O(BDC)(3) (MOF-5; BDC = 1,4-benzenedicarboxylate), which is deposited at room temperature in only 15 min under cathodic potential. Although many crystalline phases are known in the Zn(2+)/BDC(2-) system, MOF-5 is the only observed crystalline MOF phase under these conditions. This fast and mild method of synthesizing MOFs is amenable to direct surface functionalization and could impact applications requiring conformal coatings of microporous MOFs, such as gas separation membranes and electrochemical sensors.