On the Stationary Navier–Stokes Problem in $$\mathbb {R}^3$$: An Approach in Weighted Sobolev Spaces

In this paper, we study the steady-state Navier–Stokes equations in \(\mathbb {R}^3\). First, we establish the existence of very weak solution in \(\varvec{L}^p(\mathbb {R}^3)\) with \(3/2< p < +\infty \) under smallness conditions on the data. A uniqueness result is also given in case the data belong to \(\mathbb {L}^r(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) with \(3/2<r<3\). We also discuss the case where the data are not necessarily small. In particular, these results enhance those obtained by Bjorland et al. (Commun Partial Differ Equ 26:216–246, 2011), and are in agreement with those obtained by Kim and Kozono (J Math Anal Appl 395(2):486–495, 2012). Second, we prove a result of existence and uniqueness of weak solution in the weighted Sobolev space \(\varvec{W}_0^{1,p}(\mathbb {R}^3)\cap \varvec{W}_0^{1,\,3/2}(\mathbb {R}^3)\) in case of small external forces given by \(\mathrm{div}\mathbb {F}\) with \(\mathbb {F} \in \mathbb {L}^p(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) and \(1<p<3\).     

Gaussian Analytic Functions in the Polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of \(\mathbb {C}^n\). Following the scheme previously used in the unit ball, we first study the asymptotics of fluctuations of linear statistics as the directional intensities \(L_j\), \(j=1,\dots ,n\) tend to \(\infty \). Then, we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact Kähler manifolds.     

Bimetallic nanoclusters with strong red fluorescence for sensitive detection of hypochlorite in tap water

Gold-silver nanoclusters (Au-AgNCs) were synthesized by simultaneous chemical reduction of Au(III) and Ag(I) ions in one pot, using bovine serum albumin as both a template and a reductant. The Au-AgNCs have an average size of 2.4 nm and display strong red fluorescence (with an emission peak at 610 nm on excitation at 360 nm). The fluorescence quantum yield can reach 18.6%. Fluorescence is strongly quenched by hypochlorite, while other common anions have minor (or no) effects on fluorescence. Based on these findings, a fluorometric method was developed for the determination of hypochlorite. The method has a linear response in the 0.7 to 15 μM concentration range, with a limit of detection as low as 80 nM. It was successfully applied to the determination of hypochlorite in (spiked) tap water.

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Graphical abstract Gold-silver nanoclusters with strong red fluorescence were synthesized by simultaneous chemical reduction of Au(III) and Ag(I) ions in one pot, and a sensitive and selective method for the detection of hypochlorite was developed based on the quenching of the fluorescence of the nanoclusters.

     

Microsphere based continuous-flow immunoassay in a microfluidic device for determination of clinically relevant insulin levels

This paper describes a miniaturized microsphere-based immunoassay integrated into a microfluidic device for rapid quantitation of insulin. Analysis of bionic pancreas studies have revealed that the rates of absorption of insulin analogs vary from patient to patient, and even within patients on different occasions. Thus development of an approach to monitor insulin continuously allows the pharmacokinetic characteristics of insulin analogs to be determined in real-time. The authors have developed a microsphere-based continuous flow assay in a microfluidic chip that allows for the detection of insulin within seconds with high sensitivity and specificity. The method was applied to near real-time monitoring of clinical samples. Calibration plot were established for different insulin analogs such as insulin aspart (Novolog), insulin lispro (Humalog), and regular human insulin (RHI) and the insulin detection limit was 0.26 ng.mL^?1 (44 pM). This sensitivity allows to detect the fasting insulin levels of T1D patients, which are reported in the range of 50–180 pM (0.3–1 ng.mL^?1), after treatment with subcutaneous insulin administration. This fast approach was also applied to sera collected in intervals from T1D patients after a bolus of insulin aspart delivery. The insulin profile obtained by this method is similar to the basal and peak insulin levels as determined using the standard non-continuous ELISA reference method. In our perception, this assay will improve healthcare by personalizing diagnostics for better clinical outcome and provide real-time feedback on sensing and actuation.

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Graphical abstract Schematic illustration of the microfluidic microsphere based Microfluidic Lab-On-Chip device for near real-time insulin monitoring.