On the finiteness of certain Rabinowitsch polynomials II

Let m be a positive integer and f_m(x) be a polynomial of the form f_m(x)=x²+x−m. We call a polynomial f_m(x) a Rabinowitsch polynomial if for and consecutive integers is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials f_m(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free.     

Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains

In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: $$\begin{aligned} -\Delta u=u^p \quad \text{ in }\; \Omega _t,\quad u=0 \quad \text{ on }\; \partial \Omega _t. \end{aligned}$$ Here $1<p<\frac{N+2}{N-2}$ when $N\ge 3,\,1<p<\infty $ when $N=2$ and $\Omega _t$ is a tubular domain which expands as $t\rightarrow \infty $ . See (1.6) below for a precise definition of expanding tubular domain. When the section $D$ of $\Omega _t$ is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1–2), 23–55, 2002) and by Ackermann et al. (Milan J Math, 79(1), 221–232, 2011) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all $N\ge 2$ without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.     

Class number 2 problem for certain real quadratic fields of Richaud-Degert type

In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85-104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179-194] to class number 2 problem of real quadratic fields of Richaud-Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n²+1 is a even square free integer.     

Multiplexed evaluation of capture agent binding kinetics using arrays of silicon photonic microring resonators

High affinity capture agents recognizing biomolecular targets are essential in the performance of many proteomic detection methods. Herein, we report the application of a label-free silicon photonic biomolecular analysis platform for simultaneously determining kinetic association and dissociation constants for two representative protein capture agents: a thrombin-binding DNA aptamer and an anti-thrombin monoclonal antibody. The scalability and inherent multiplexing capability of the technology make it an attractive platform for simultaneously evaluating the binding characteristics of multiple capture agents recognizing the same target antigen, and thus a tool complementary to emerging high-throughput capture agent generation strategies.     

Site-selective catalytic surface activation via aerosol nanoparticles for use in metal micropatterning

There is great interest in the fabrication of micro- and nanopatterned metallic structures on substrates for a wide range of electronic, photonic, and magnetic devices. One of the most widely used techniques is the electroless deposition (ELD) of metal, which requires the surface activation of the substrates with a metal catalyst. This paper introduces a method of catalytic surface activation by producing platinum aerosol nanoparticles via spark generation and then thermophoretically depositing the particles onto a flexible polyimide (PI) substrate through the pattern hole of a mask. After annealing, the catalytically activated substrate is placed into a solution for electroless silver deposition. The silver is then formed only on the activated regions of the substrate. Silver line patterns having a width of 18 microm and a height of 1 microm are created with the ability to be effectively reproduced. The average value of the resistivities is approximately 6.8 mu Omega.cm, which is almost comparable to the theoretical resistivity of bulk silver (1.6 mu Omega.cm). Other silver micropatterns containing square dot array, line, line array, Y-branched line, and tapered line using different pattern masks are also demonstrated.     

Determination of relative tensor orientations by γ-encoded chemical shift anisotropy/heteronuclear dipolar coupling 3D NMR spectroscopy in biological solids

In this paper, we present 3D chemical shift anisotropy (CSA)/dipolar coupling correlation experiments, based on γ-encoded R-type symmetry sequences. The γ-encoded correlation spectra are exquisitely sensitive to the relative orientation of the CSA and dipolar tensors and can provide important structural and dynamic information in peptides and proteins. We show that the first-order (m = ±1) and second-order (m = ±2) Hamiltonians in the R-symmetry recoupling sequences give rise to different correlation patterns due to their different dependencies on the crystallite orientation. The relative orientation between CSA and dipolar tensors can be determined by fitting the corresponding correlation patterns. The orientation of (15)N CSA tensor in the quasi-molecular frame is determined by the relative Euler angles, α(NH) and β(NH), when the combined symmetry schemes are applied for orientational studies of (1)H-(15)N dipolar and (15)N CSA tensors. The correlation experiments introduced here work at moderate magic angle spinning frequencies (10-20 kHz) and allow for simultaneous measurement of multiple sites of interest. We studied the orientational sensitivity of γ-encoded symmetry-based recoupling techniques numerically and experimentally. The results are demonstrated on [(15)N]-N-acetyl-valine (NAV) and N-formyl-Met-Leu-Phe (MLF) tripeptide.     

Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Let Ω be a bounded domain in Rn, n?3, with the boundary ∂Ω∈C³. We consider the following singularly perturbed nonlinear elliptic problem on Ω

     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).