The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets S{\mathcal {S}} of the natural numbers such that the Robin inequality holds for all but finitely many n ? S{n \in \mathcal {S}} . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.     

Convection, diffusion and reaction in a surface-based biosensor: modeling of cooperativity and binding site competition on the surface and in the hydrogel

We study theoretically the transport and kinetic processes underlying the operation of a biosensor (particularly the surface plasmon sensor "Biacore") used to study the surface binding kinetics of biomolecules in solution to immobilized receptors. Unlike previous studies, we concentrate mainly on the modeling of system-specific phenomena rather than on the influence of mass transport limitations on the intrinsic kinetic rate constants determined from binding data. In the first problem, the case of two-site binding where each receptor unit on the surface can accommodate two analyte molecules on two different sites is considered. One analyte molecule always binds first to a specific site. Subsequently, the second analyte molecule can bind to the adjacent unoccupied site. In the second problem, two different analytes compete for one binding site on the same surface receptor. Finally, the third problem considers the case of positive cooperativity among bound molecules in the hydrogel using a simple mean-field approach. The transport in both the flow channel and the hydrogel phases of the biosensor is taken into account in this case (with few exceptions, most previous studies assume a simpler model in which the hydrogel is treated as a planar surface with the receptors). We consider simultaneously diffusion and convection through the flow channel together with diffusion and cooperativity binding on the surface and in the hydrogel. In each case, typical results for the concentration contours of the free and bound molecules in the flow channel and hydrogel regions are presented together with the time-dependent association/dissociation curves and reaction rates. For binding site competition, the analysis predicts overshoot phenomena.     

Comparison of analytical and semi-preparative columns for high-performance liquid chromatography-solid-phase extraction-nuclear magnetic resonance

The application of analytical and semi-preparative columns in reversed-phase liquid chromatography-solid-phase extraction-nuclear magnetic resonance (HPLC-SPE-NMR) was compared. The work was aiming at separating a higher sample amount in a single run and in this way to reduce the necessary NMR measurement time of separated compounds. Several parameters for compound separation and trapping procedures were optimised: flow rate of HPLC and make-up water pumps, choice of stationary phase cartridges and drying time. The separation and loadability of nine model compounds on analytical and semi-preparative columns was determined, as well as the focussing capacity of SH-type SPE cartridges. It was found that a semi-preparative column--or multiple peak trapping on analytical columns--gave better results than a standard 4.6mm analytical column for non-polar compounds (e.g. flavonoid aglycones, sesquiterpene lactones, non-polar terpenes, logP>2), but for polar compounds (logP<-2) did not offer any advantage over an analytical column, or was even disadvantageous. For intermediately polar compounds (-2相似文献   

Zeolite framework stabilized copper complex inspired by the 2-His-1-carboxylate facial triad motif yielding oxidation catalysts

The stabilization of a mononuclear copper(II) complex with one MIm2Pr ligand [MIm2Pr = 3,3-bis(1-methylimidazol-2-yl)propionate] in the supercages of zeolite Y was attempted, and the resulting materials were tested for their activity in oxidation catalysis. The preparation procedure yielded initially two species (labeled 1 and 2) within the pore system of the zeolite material, which differ in molecular structure and chemical composition as determined by UV/vis, ESR, IR, and XAFS spectroscopy. In species 1, the copper was found to be five-coordinated, with one MIm2Pr ligand in a facial-type NNO coordination toward copper, the other two coordination sites being occupied by oxygen atoms from either the zeolite framework and/or a water molecule. The total charge of this complex is 1+. In species 2, the copper is surrounded by two MIm2Pr ligands, both in a facial-type coordination mode, identical to the homogeneous Cu(MIm2Pr)2 complex. This neutral species 2 is easily washed out of the zeolite, whereas the mononuclear species 1 remains inside the zeolite material upon washing. The spectroscopic characteristics and activity for 3,5-di-tert-butylcatechol and benzyl alcohol oxidation of species 1 compared closely with that of the zeolite-immobilized Cu(histidine) complexes but differed from that of the homogeneous Cu(MIm2Pr)2 complex. It was therefore found that encapsulation in zeolite offers a route to stabilize a 5-fold-coordinated copper complex with novel catalytic properties. This 1:1 Cu(MIm2Pr) complex is not formed in solution.     

Precision study on capillary electrophoresis methods for metacycline

A CE method for metacycline (MTC) determination was investigated in an inter-laboratory experiment. Many problems were encountered in this study, most of which were related to the transfer of the method to different CE equipment. The reported problems could be classified into different categories: problems related to the precision, to the parameters in the protocol, and to the MTC peak shape. As the peak shape problem was partially responsible for the poor precision, a new CE method was developed in order to obtain a good MTC peak shape on all equipment. The precision of this new method for MTC determination was examined in an intermediate precision study, where the influence of the factors "time" and "equipment" was investigated. Although the new method could be transferred to different instruments, the precision remained poor mainly due to the contributions of the between-replicate and the between-injection variances.     

Dissecting non-ribosomal and polyketide biosynthetic machineries using electrospray ionization Fourier-Transform mass spectrometry

Many virulence factors and bioactive compounds with antifungal, antimicrobial, and antitumor properties are produced via the non-ribosomal peptide synthetase (NRPS) or polyketide synthase(PKS) paradigm. During the biosynthesis of these natural products, substrates, intermediates and side products are covalently tethered to the NRPS or PKS catalyst, introducing mass changes, making these biosynthetic systems ideal candidates for interrogation by large molecule mass spectrometry. This review serves as an introduction into the application of electrospray ionization Fourier-Transform massspectrometry (ESI-FTMS) to investigate NRPS and PKS systems. ESI-FTMS can be used to understand substrate tolerance, timing of covalent linkages, timing of tailoring reactions and the transfer of substrates and biosynthetic intermediates from domain to domain. Therefore we not only highlight key mechanistic insights for thiotemplate systems as found on the enterobactin,yersiniabactin, epothilone, clorobiocin, coumermycin, pyoluteorin, gramicidin, mycosubtilin, C-1027,6-deoxyerythronolide B and FK520 biosynthetic pathways, but we also explain the approaches taken to identify active sites from complex digests and compare the FTMS based assay to traditional assays and other mass spectrometric techniques. Although mass spectrometry was introduced over two decades ago to investigate NRPS and PKS biosynthetic systems, this is the first review devoted to this methodology.     

Generalised expressions for the association and dissociation rate constants of molecules with multiple binding sites

ABSTRACT

Many proteins exhibit multiple binding patches. A patch may harbour a key chemical modification site, but may also simply act as a trap for the binding to another site. Here we consider the scenario in which one molecule (enzyme) binds another molecule (substrate) which contains two sites. We present microscopic expressions for the rate at which the enzyme binds to a particular site on the substrate, both for the scenario in which the enzyme directly binds the site without first visiting the other site, and for the case in which it may visit the other site an arbitrary number of times before binding to the site of interest. We also present the expressions for the corresponding dissociation reactions. These expressions can be used to compute in a single rare-event simulation of the dissociation pathway not only both the intrinsic and effective dissociation rate constants but also both association rate constants.     

BOLD MRI response to hypercapnic hyperoxia in patients with meningiomas: correlation with Gadolinium-DTPA uptake rate

Because meningiomas tend to recur after (partial) surgical resection, radiotherapy is increasingly being applied for the treatment of these tumors. Radiation dose levels are limited, however, to avoid radiation damage to the surrounding normal tissue. The radiosensitivity of tumors can be improved by increasing tumor oxygen levels. The aim of this study was to investigate if breathing a hyperoxic hypercapnic gas mixture could improve the oxygenation of meningiomas. Blood oxygen level-dependent magnetic resonance imaging and dynamic Gadolinium (Gd)-DTPA contrast-enhanced MRI were used to assess changes in tumor blood oxygenation and vascularity, respectively. Ten meningioma patients were each studied twice; without and with breathing a gas mixture consisting of 2% CO(2) and 98% O(2). Values of T(2)* and the Gd-DTPA uptake rate k(ep) were calculated under both conditions. In six tumors a significant increase in the value of T(2)* in the tumor was found, suggesting an improved tumor blood oxygenation, which exceeded the effect in normal brain tissue. Contrarily, two tumors showed a significant T(2)* decrease. The change in T(2)* was found to correlate with both k(ep) and with the change in k(ep). The presence of both vascular effects and oxygenation effects and the heterogeneous response to hypercapnic hyperoxia necessitates individual assessment of the effects of breathing a hyperoxic hypercapnic gas mixture on meningiomas. Thus, the current MRI protocol may assist in radiation treatment selection for patients with meningiomas.     

A comparison study of multishot vs. single-shot DWI-EPI in the neonatal brain: reduced effects of ghosting compared to adults

In the neonatal brain, it is important to use a fast imaging technique to acquire all diffusion weighted images (DWI) for apparent diffusion coefficient (ADC) calculation. Taking into account the occurrence of typical echo planar imaging (EPI) artifacts, we have investigated whether single-shot (SSh) or multishot (MSh) DWI-EPI should be preferred. In 14 neonates, 17 adult patients and 5 adult volunteers, DWIs are obtained both with SSh and MSh EPI. The occurrence of artifacts and their influence on the ADC are explored and further quantified using simulations and phantom studies. Two radiologists scored overall image quality and diagnosability of all images. Single-shot and MSh DWI-EPI scored equally well in neonates with respect to overall image quality and diagnosability. In newborns, more motion artifacts in MSh can be noticed while N/2-ghost artifacts in SSh occur less frequently than in adults. Both N/2-ghost and motion artifacts result in significant ADC abnormalities. There is a serious risk that these artifacts will be mistaken for genuine diffusion abnormalities. N/2-ghost artifacts are hardly noticed in the neonatal brain, which might be due to smaller cerebrospinal fluid (CSF) velocity than in adults. Apparent diffusion coefficient values in MSh are unreliable if motion occurs. We conclude that for ADC calculations in neonates SSh DWI-EPI is more reliable than MSh.