Hamiltonian and long paths in bipartite graphs with connectivity

Let G be a graph, $\nu (G)$ the order of G, $\kappa (G)$ the connectivity of G and k a positive integer such that $k\le (\nu (G)?2)/2$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets $V_1$ and $V_2$ is defined to be Hamiltonian-laceable if such that $|V_1|=|V_2|$ and for every pair of vertices $p\in V_1$ and $q\in V_2$, there exists a Hamiltonian path in G with endpoints p and q, or $|V_1|=|V_2|+1$ and for every pair of vertices $p,q\in V_1,p\ne q$, there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition $(X,Y)$. Define $bn(G)$ to be a maximum integer such that $0\le bn(G)<min\{|X|,|Y|\}$ and (1) for each non-empty subset S of X, if $|S|\le |X|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|X|?bn(G)<|S|\le |X|$, then $N(S)=Y$; and (2) for each non-empty subset S of Y, if $|S|\le |Y|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|Y|?bn(G)<|S|\le |Y|$, then $N(S)=X$; and (3) $bn(G)=0$ if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|$ and $bn(G)>0$. In this paper, we show that if $\nu (G)\le 2\kappa (G)+4bn(G)?4$, then G is Hamiltonian-laceable; or if $\nu (G)>6bn(G)?2$, then for every pair of vertices $x\in X$ and $y\in Y$, there is an $(x,y)$-path P in G with $|V(P)|\ge 6bn(G)?2$. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.     

Examples of compactλ-hypersurfaces in Euclidean spaces

In this paper,we first construct compact embeddedλ-hypersurfaces with the topology of torus which are calledλ-torus in Euclidean spaces?n^+1.Then,we give many compact immersedλ-hypersurfaces in Euclidean spaces?n^+1.     

A boundary expansion of solutions to nonlinear singular elliptic equations

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profile of solutions at the boundary.We deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Freestanding Carbon Nanotube Film for Flexible Straplike Lithium/Sulfur Batteries

Flexible lithium/sulfur (Li/S) batteries are promising to meet the emerging power demand for flexible electronic devices. The key challenge for a flexible Li/S battery is to design a cathode with excellent electrochemical performance and mechanical flexibility. In this work, a flexible strap-like Li/S battery based on a S@carbon nanotube/Pt@carbon nanotube hybrid film cathode was designed. It delivers a specific capacity of 1145 mAh g⁻¹ at the first cycle and retains a specific capacity of 822 mAh g⁻¹ after 100 cycles. Moreover, the flexible Li/S battery retains stabile specific capacity and Coulombic efficiency even under severe bending conditions. As a demonstration of practical applications, an LED array is shown stably powered by the flexible Li/S battery under flattened and bent states. We also use the strap-like flexible Li/S battery as a real strap for a watch, which at the same time provides a reliable power supply to the watch.     

A pH-Sensitive Zwitterionic Iron Complex Probe with High Biocompatibility for Tumor-Specific Magnetic Resonance Imaging

Accurate diagnosis of tumor characteristics, including its location and boundary, is of immense value to subsequent therapy. Activatable magnetic resonance imaging (MRI) contrast agents that respond to tumor-specific microenvironments, such as the redox state, pH, and enzyme activity, enable better mapping of tumor tissue. However, the practical application of most reported activatable agents is hampered by problems including potential toxicity, inefficient elimination, and slow activation. In this study, we developed a zwitterionic iron complex (Fe-ZDS) as a positive MRI contrast agent for tumor-specific imaging. Fe-ZDS could dissociate in weakly acidic solution rapidly, accompanied by clear longitudinal relaxivity (r₁) enhancement, which enabled the complex to act as a pH-sensitive contrast agent for tumor-specific MR imaging. In vivo experiments showed that Fe-ZDS rapidly enhanced the tumor-to-normal contrast ratio by >40 %, which assisted in distinguishing the tumor boundary. Furthermore, Fe-ZDS circulated freely in the bloodstream and was excreted relatively safely via kidneys owing to its zwitterionic nature. Therefore, Fe-ZDS is an ideal candidate for a tumor-specific MRI contrast agent and holds considerable potential for clinical translation.     

Binary neutron stars gravitational wave detection based on wavelet packet analysis and convolutional neural networks

This work investigates the detection of binary neutron stars gravitational wave based on convolutional neural network(CNN).To promote the detection performance and efficiency,we proposed a scheme based on wavelet packet(WP)decomposition and CNN.The WP decomposition is a time-frequency method and can enhance the discriminant features between gravitational wave signal and noise before detection.The CNN conducts the gravitational wave detection by learning a function mapping relation from the data under being processed to the space of detection results.This function-mapping-relation style detection scheme can detection efficiency significantly.In this work,instrument effects are con-sidered,and the noise are computed from a power spectral density(PSD)equivalent to the Advanced LIGO design sensitivity.The quantitative evaluations and comparisons with the state-of-art method matched filtering show the excellent performances for BNS gravitational wave detection.On efficiency,the current experiments show that this WP-CNN-based scheme is more than 960 times faster than the matched filtering.     

Sombor index: review of extremal results and bounds

Journal of Mathematical Chemistry - The Sombor index was introduced in 2020, and was soon followed by a remarkably large number of studies, both chemical and mathematical. In this review we collect...