Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions

This paper is concerned with the existence of monotone positive solution for the following third-order nonlocal boundary value problem $u^{\prime \prime \prime }\left(t\right) +f\left( t,u\left( t\right), u^{\prime}\left( t\right)\right) =0,\, 0<t<1$; $u\left( 0\right) =0,$ $au^{\prime}\left( 0\right)-b u^{\prime\prime}\left( 0\right)=\alpha[u],$ $c u^{\prime}\left( 1\right)+d u^{\prime\prime}\left( 1\right)=\beta[u],$ where $f\in C([0,1]\times R^{+}\times R^{+}, R^{+})$, $\alpha[u]=\int_{_{0}}^{1}u(t)dA(t)$ and $\beta[u]=\int_{_{0}}^{1}u(t)dB(t)$ are linear functionals on $C[0,1]$ given by Riemann-Stieltjes integrals. By applying monotone iterative techniques, we not only obtain the existence of monotone positive solution but also establish an iterative scheme for approximating the solution. An example is also included to illustrate the main results

Recent advances and current issues in single-cell sequencing of tumors

AbstractIntratumoral heterogeneity is a recently recognized but important feature of cancer that underlies the various biocharacteristics of cancer tissues. The advent of next-generation sequencing technologies has facilitated large scale capture of genomic data, while the recent development of single-cell sequencing has allowed for more in-depth studies into the complex molecular mechanisms of intratumoral heterogeneity. In this review, the recent advances and current challenges in single-cell sequencing methodologies are discussed, highlighting the potential power of these data to provide insights into oncological processes, from tumorigenesis through progression to metastasis and therapy resistance

Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model

We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges around the critical point $J_z=(1/2)^+$.Comment: 7 pages, 6 figure

(3S,4R)-3-Ethyl-4-hydr­oxy-3-(3-methoxy­phen­yl)-1-methyl­azepanium (2R,3R)-2,3-bis­(benzo­yloxy)-3-carboxy­propionate

The crystal structure of the title compound, C16H26NO2 +·C18H13O8 −, is stabilized by an extensive network of classical N—H⋯O and O—H⋯O hydrogen bonding. The crystal structure also shows an ammonium-driven diastereo­isomerism