Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms _R and _D on functions f: (0, 1)→ such that to each bounded domain Ω in ⁿ, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality _R(u*(|Ω| t))C_D(|^mu|* (|Ω| t)), uC^m₀(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which _D need not be rearrangement-invariant, _R(u*(|Ω| t))C_D((d/dt) ∫_{{x ⁿ : |u(x)|>u*(|Ω| t)}} |(u)(x)| dx), uC¹₀(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that _R cannot be replaced by an essentially larger quasinorm and _D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.     

Electrochemical DNA biosensor using a disposable electrochemical printed (DEP) chip for the detection of SNPs from unpurified PCR amplicons

In this study, we are reporting for the first time the elucidation of single nucleotide polymorphisms (SNPs) of clinically important alleles from consenting human subjects using a disposable electrochemical printed (DEP) chip in connection with differential pulse voltammetry (DPV) and a redox active molecule Hoechst 33258 [H33258, 2'-(4-hydroxyphenyl)-5-(4-methyl-1-piperazinyl)-2,5'-bi(1H-benzimidazole)]. Post-PCR products were analyzed directly without any purification process. The aggregation of the DNA-H33258 complex causes a significant drop in the peak current intensity of H33258 oxidation. The phenomenon of DNA aggregation induced by H33258 in addition to changes in anodic current peak are used to detect SNPs. Since laborious probe immobilization was not required, our biosensor offers several benefits due to its simplicity and rapid response as a promising device for genetic analysis.     

An electrochemical on-field sensor system for the detection of compost maturity

A maturity sensor system was developed, based on the combination of three electrically measured parameters, pH, NH₄⁺ concentration, and phosphatase activity in the water extracts of compost samples. One of these parameters, the apparent phosphatase activity in crude test solutions was determined using screen-printed carbon strips (SPCSs) coated with α-naphthyl phosphate (α-NP) in Nafion film. The phosphatase activity was monitored in connection with differential pulse voltammetry (DPV) with an aliquot (30 μL) of the test solution on SPCS. The phosphatase activity sensor was validated using alkaline phosphatase (ALP) in Tris-HCl buffer (pH 8.0) and acid phosphatase (ACP) in citric acid buffer (pH 5.0). The activity of the spiked enzymes in the water extract of the compost sample could be confirmed with the change of corresponding oxidation peak current signal of the product, α-naphthol. The water extracts of compost samples (n = 24) collected in various composting days were applied to our compost maturity sensor system, and the conventional germination tests. Using multiple regression analysis, the germination index (GI) was expressed by the multi-linear regression equation consisting of pH, NH₄⁺ concentration, and the phosphatase activity. The calculated GI from the regression equation had a good correlation with the measured GI of the corresponding composts (r = 0.873). As a result, we have determined an equation for the determination of the compost stability using our portable sensor system rapidly at the composting site.     

Boundary trace inequalities and rearrangements

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived. This research was partially supported by the Italian research project “Geometric properties of solutions to variational problems” of GNAMPA (INdAM) 2006, by the research project MSM 0021620839 of the Czech Ministry of Education, by grants 201/03/0935, 201/05/2033 and 201/07/0388 of the Grant Agency of the Czech Republic and by the Nečas Center for Mathematical Modeling project no. LC06052 financed by the Czech Ministry of Education.     

Non-resonant microwave absorption studies of superconducting MgB2 and MgB2 + MgO

Non-resonant microwave absorption (NRMA) studies of superconducting MgB₂ and a sample containing ∼10% by weight of MgO in MgB₂ are reported. The NRMA results indicate near absence of intergranular weak links in the pure MgB₂ sample. A linear temperature dependence of the lower critical field H _c1 is observed indicating a non-s wave superconductivity. However, the phase reversal of the NRMA signal which could suggest d wave symmetry is also not observed. In the MgB₂ + MgO sample, much larger low field dependent absorption is observed indicating the presence of intergranular weak links. The hysteretic behavior of NRMA is compared and contrasted in the two samples. In the pure MgB₂ sample, a large hysteresis is observed between the forward and the reverse scans of the magnetic field indicating strong pinning of flux lines. This hysteresis saturates a few degrees below T _c while in the MgB₂ + MgO sample, a much slower increase of hysteresis with decreasing temperature is observed, a signature of weaker pinning.     

Optical trapping for the characterization of amyloid-beta aggregation kinetics

Alzheimer's disease (AD) is marked by the accumulation of neuronal plaques from insoluble amyloid-beta (Aβ) peptides. Growing evidence for the role of Aβ oligomers in neuronal cell cytotoxicity and pathogenesis has prompted the development of novel techniques to better understand the early stages of aggregation. Near infrared (NIR) optical trapping was applied to characterize the early stages of Aβ aggregation in the presence of a β-sheet intercalating dye, Congo Red (CR), as the fluorescent marker. The integration of fluorescence analysis with NIR optical trapping has provided a new outlook into the first two hours of Aβ aggregation.     

Sensitive detection of cold cesium molecules formed on Feshbach resonances

We observe the dynamic formation of quasibound Cs2 molecules near Feshbach resonances in a cold sample of atomic cesium. Using an external probe beam, more than 15 weakly coupled molecular states are detected with high sensitivity, whose collisional formation cross sections are as small as sigma=2 x 10(-16) cm(2). By modeling the molecule formation and dissociation processes with rate equations, we conclude that at an atomic density of 10(13) cm(-3) and temperature of 5 microK, more than 5(1)x10(5) Cs2 molecules in a single rovibrational state coexist with 10(8) Cs atoms in our trap.     

Coherent Higgs and W pair production

Cross sections for the coherent electromagnetic production of the intermediate vector bosons and the Higgs boson in ultrarelativistic heavy-ion collisions are calculated from standard electroweak theory. The distributions of the resulting decay products are also considered. Particular attention is given to the energy régime of the Superconducting Supercollider and the CERN Large Hadron Collider.     

Fluctuations of fractional charge in soliton anti-soliton systems

The total charge in a soliton-anti-soliton system has integer eigenvalues. However, when soliton-induced charge fractionization occurs, it is possible to make a Bogoliubov transformation on the eigenstates of the total charge, so that in the limit of infinite soliton anti-soliton separation., the transformed states become eigenstates of a charge operator suitably localized at the soliton. In the limit, the localized charge operator has fractional eigenvalues, without fluctuations.