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Background Cutaneous Melanoma (CM) is a malignant disease with increasing incidence and high mortality. Ferroptosis is a new kind of cell death and related to tumor blood and lymphatic metastasis. This study aims at using bioinformatics technology to construct a prognostic signature and identify ferroptosis-related biomarkers to improve the prognosis and treatment of cutaneous melanoma. Methods We used bioinformatics tools to analyze RNA sequencing expression data with clinical information from multiple databases, utilized varieties of statistical methods to construct a ferroptosis-related prognostic signature of cutaneous melanoma and screened out specific genes with independent prognostic ability. Results We obtained 22 ferroptosis-related (P Conclusion The present study suggested a 10-biomarker signature can be clinically used to predict the prognosis of cutaneous melanoma, which was better than conventional factors. CP and ZEB1 were independent prognostic genes and can be applied to guide treatment. In addition, ZEB1 mutation was highly related to overall survival in cutaneous melanoma, while CP may be associated with tumor progression. Our study comprehensively analyzed the relationship between iron metabolism, ferroptosis-related genes, and the prognosis of cutaneous melanoma, provided new insight for molecular mechanisms and treatment of ferroptosis and cutaneous melanoma.

Nitrogen-doped porous carbon materials have been the outstanding candidates in supercapacitor applications due to their low cost, high surface area, abundant pore structure, suitable nitrogen doping content etc. Herein, nitrogen-doped porous carbon materials (NPC) were prepared via simple solid grinding using silica spheres as pore-forming agent and nitrogen-rich aniline as carbon/nitrogen precursor. The silica spheres create crossed macroporous structure that contributes to the transfer and storage of charge. NPC is continuously activated to give NPC-K, (where the K presents potassium hydroxide), further increasing its surface area and pore volume. The resulting NPC-K has hierarchical porous structures, high specific surface area, large pore volume, and suitable nitrogen doping. When used as an electrode material for supercapacitors, it has a good capacitance, rate performance, and cycle stability, demonstrating good electrochemical performance.

A facile and effective co-assembly approach has been developed to controllably fabricate mesoporous carbon materials (MC) with different morphologies. 1-Alkyl-3-methylimidazolium bromide and tetraethyl orthosilicate (TEOS) are applied to assemble with resorcinol/formaldehyde resin (RF) through electrostatic interaction to produce tunable morphologies by simply changing the concentration of the reactants. The increasing in concentration of reactant led to tunable structure from irregular particle aggregates to thin sheet, fluffy spheres and then to hollow olive-like structure, which shows variable adsorption performance for pesticide wastewater treatment. MC with regular spherical morphology, dual-mesoporous structure and high surface area shows higher adsorption capacity than MC with other morphologies, proving its excellent prospects in treating pesticide wastewater.

We study the rate of convergence of $${u^\varepsilon}$$ , as $${\varepsilon \to 0+}$$ , to u in periodic homogenization of Hamilton–Jacobi equations. Here, $${u^\varepsilon}$$ and u are viscosity solutions to the oscillatory Hamilton–Jacobi equation and its effective equation $$\left.\begin{array}{ll}{\rm (C)_\varepsilon}\qquad\begin{cases}u_t^{\varepsilon}+H\left(\frac{x}{\varepsilon}, Du^{\varepsilon}\right) = 0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u^{\varepsilon}(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$ and $$\left.\begin{array}{ll}{\rm (C)} \qquad \begin{cases}u_t+\overline{H} \left(Du\right)=0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$ respectively. We assume that the Hamiltonian H = H(y, p) is coercive and convex in the p variable and is $${\mathbb{Z}^n}$$ -periodic in the y variable, and the initial data g is bounded and Lipschitz continuous. Here, $${\overline{H}}$$ is the effective Hamiltonian. We prove that When n = 2 and H is positively homogeneous in p of some fixed degree $${k \geqq 1}$$ , utilizing the Aubry–Mather theory, we obtain the optimal convergence rate $${O(\varepsilon)}$$ $$|u^{\varepsilon}(x, t)-u(x, t) | \leqq C\varepsilon \quad {{\rm for all}\, (x, t)\in \mathbb{R}^2\times [0, \infty).}$$ Here C depends only on H and $${\|Dg\|_{L^{\infty}(\mathbb{R}^2)}}$$ . When n = 1, the optimal convergence rate $${O(\varepsilon)}$$ is established for any coercive and convex H. The convergence rate turns out to have deep connections with the dynamics of the underlying Hamiltonian system and the shape of the effective Hamiltonian $${\overline{H}}$$ . Some related results and counter-examples are obtained as well.

The excessive use of plastic, especially polystyrene (PS), has caused serious environmental pollution. The efficient utilization of plastics and the conversion of plastics into value-added carbon materials are the concerns of researchers. Herein, we propose novel “pyrolysis–deposition” method to convert one popular plastic substance, PS, into ordered mesoporous carbons (OMCs). During the synthesis process, PS is pyrolyzed into small organic gases under high temperature, which is then adsorbed through capillary adsorption into the mesoporous of SBA-15 in the presence of catalyst. The obtained OMCs have high specific surface area, uniform pore size, and ordered pore structure. The OMCs exhibit specific capacitance of 118 F/g at a current density of 0.2 A/g and electrochemical stability of 87.2% at a current density of 2 A/g after 5000 cycles. The pyrolysis–deposition strategy provides a new idea to convert waste plastics into high-performance carbon materials for electrochemical applications.

Carbon spheres with rich porous structure are regarded as ideal materials for practical supercapacitors because of their excellent thermal stabilities, large surface areas, high electrical conductivities and good cycle stabilities. In this work, a novel dissolution–reassembly method is developed for the fabrication of rich porous carbon spheres (PCS) with high capability for supercapacitor. The resorcinol–formaldehyde resin is firstly synthesized then completely dissolved by acetone into oligomer fragments which further reassemble with F127 to form new structured resin spheres. After carbonization, PCS are obtained. The obtained PCS have regular spherical morphology, rich porous structure, high specific surface area and pore volume. As electrode material for supercapacitor, the PCS exhibit excellent performance with capacitance 240 F g−1 at the current density of 1 A g−1 and outstanding cycling life stability (98.1% after 5000 cycles) at the current density of 5 A g−1, showing the great potential for supercapacitor.

We developed a simple approach to synthesize N-doped mesoporous carbon (NMC) material by using silica assistance. Hexamethylenetetramine is converted into formaldehyde and ammonia under hydrothermal conditions, which served as carbon and nitrogen source at the same time. Carbon precursor was formed by self-polymerization of formaldehyde in alkaline condition and nitrogen was in-situ introduced. Silica hydrolyzed from tetraethyl orthosilicate and carbon precursor form resin–silica composite first, which transformed into NMC after carbonization and removal of silica. As a result, the obtained NMC exhibited rich porous structure and specific surface area (569 m2 g−1). As the electrode material, NMC exhibited a specific capacitance (192 F g−1 at the current density of 0.5 A g−1) and good cycling stability (92.9% even after 5000 cycles at 5 A g−1).

Carbon nanotube (CNT) has been intensively studied as supercapacitor electrode materials due to their high mechanical strength and excellent electrical conductivity. However, CNT with limited accessible surface area has low specific capacitance, which limits its application and development in supercapacitor. Herein, we have designed a thin N-doped porous carbon layer-modified CNT nanocomposite by template-free method. In this process, 3-aminophenol and formaldehyde rapidly self-assemble on the CNT in the presence of cationic surfactant CTAB, forming the composite of 3-aminophenol/formaldehyde (3-AF) resin and CNT. Due to the different polymerization degree of inside 3-AF resin, the soluble 3-AF resin oligomer was removed by acetone to form porous 3-AF resin and CNT@N-doped porous carbon (CNT@N-PC) was obtained by annealing process. The CNT@N-PC exhibited abundant porous structure, high specific surface area of 940 m2 g−1, excellent specific capacitance value of 224 F g−1 at 1 A g−1 and good cycle stability, demonstrating its potential as supercapacitor electrode materials.

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,\alpha}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$., Comment: 15 pages; some typos and imprecise points are corrected

N-doped ordered mesoporous carbon (N-OMC) has been one of the most promising choices as the electrode for supercapacitors due to its large surface area and uniform mesoporous structure. However, there is still a big challenge to prepare N-OMC using a relatively simple method. Here, a straightforward preparation of N-OMC was reported in which the precursor zeoliticimidazolate framework was in situ grown in the SBA-15 template by a fast, solvent-free, and atom economic solid–solid grinding strategy. After pyrolysis and removing of the template, the N-OMC was obtained with ordered mesoporous structure, rich oxygen and nitrogen, and a large specific surface area of 1004 m2/g. As the electrode material for supercapacitors, N-OMC displayed an excellent specific capacitance of 228 F/g at 0.2 A/g and superb charge/discharge cycling stability, which is promising for high-performance energy storage. This solid–solid grinding strategy may offer a low-cost and scalable method to produce high-performance N-OMC for the electrode from the zeoliticimidazolate framework.

With the rise of living standards, non-biodegradable waste, especially waste plastics, has caused serious environmental problems. Herein, we prepare nitrogen doped porous carbon nanosheets (N-PCNs) using magnesium hydroxide [Mg(OH)2] sheets, which are modified by Zn and Co bimetallic zeolitic imidazolate framework nanoparticles as templates and polystyrene (PS) as a carbon precursor. During the high-temperature pyrolysis process, PS pyrolyzes into small organic molecule gases, which can be converted into graphene-like carbon material under the catalyst magnesia (MgO) and Co species. Nitrogen is introduced into the carbon material in situ by the pyrolysis of imidazole ligands, and the evaporation of Zn helps increase the surface area. The obtained N-PCNs with porous structure and large specific surface area can be used as electrode material for supercapacitors, exhibiting excellent capacitance of 149 F g−1 at the current density of 0.5 A g−1, an excellent cycling stability of 97.6% for up to 5000 cycles.

Polystyrene, one of the classical plastic, has caused serious environmental problems due to overuse and inability to recycle effectively. Transforming it into functional carbon materials is one of the effective ways to recycle polystyrene and other waste plastics, which has drawn the attention. In this study, we have developed a facile and efficient method for the preparation of three-dimensional (3D) network structure porous carbon (PC) via the Friedel–Crafts reaction with waste polystyrene serves as carbon source. Notably, the constructed carbonyl (–CO–) cross-linking bridges between the linear polystyrenes provide the resulting hierarchical porous polystyrene with a high cross-linking density and amounts of oxygen atoms to achieve the carbonizability of cross-linking polystyrene framework. Moreover, silica particles created more porous structure for carbon material. The prepared PC showed large specific surface area and 3D porous structure and exhibited good capacitance and electrochemical stability as electrode materials for supercapacitor.

Mesoporous tubular carbon (MTC) with high specific surface area has attracted increasing attention due to their low density, fast ion diffusion and good electrical conductivity for supercapacitors. In this work, MTC was synthesized by using resin as carbon precursor, natural tubular Halloysite and F127 as hard-soft templates respectively. The resulted MTC copied the tubular structure of Halloysite. The addition of F127 not only created more abundant pores, but also greatly improved the surface area. The N2 adsorption experiments showed that the MTC possesses a large specific surface area (1034 m2 g−1) and large pore volumes (2.62 m3 g−1). The TEM of MTC showed an obvious tubular structure with abundant mesoporous structure. As the electrode material, MTC had good electrochemical capacitance (232 F g−1) at current density of 1 A g−1 and excellent cycling stability (95.3% after 5000 cycles) at the current density of 5 A g−1 in 6 M KOH electrolyte. The strategy may offer a low-cost and scalable method to produce high-performance electrode materials by using inexpensive natural product as a template.

Two-dimensional (2D) porous carbon nanosheets attract great attention because of their thin sheet-like morphology, abundant pores and high specific surface area, and their potential applicability in many fields including adsorption, oxygen reduction reaction, organic transistor and energy storage. Herein, a feasible method, named self-templating, to prepare 2D nitrogen-doping hierarchically porous carbon nanosheets (N-HPCNs) with prominent performances as supercapacitor electrode is reported. During the process of preparation, the inexpensive and easily available MgO rods are treated in water to form Mg(OH)2 nanosheets further using as templates and then nitrogen contained resorcinol–formaldehyde resin oligomers as carbon and nitrogen precursor co-condense onto the templates by electrostatic interaction. The obtained N-HPCNs with large specific surface area, hierarchical pores and unique interconnected sheet-like structure are the potential candidates for high energy storage devices. As an active electrode material for electrochemical double-layer capacitors, N-HPCNs exhibit a capacitance of 201 F g−1 at current density of 1 A g−1 and high specific capacitance (78.1% retention of initial capacitance even at 10 A g−1), with excellent cycling life stability (3.5% loss after 5000 cycles).

Fe modified hollow carbon spheres with large cavity and mesoporous shell (Fe-MHCs) were successfully synthesized by a simple pyrolysis and simultaneous deposition method. The organic molecules gases (carbon species) from pyrolysis of polystyrene deposited in the hard template at the catalysis of Fe species existing in the sample during calcination at high temperature. The obtained Fe-MHCs showed uniform spherical morphology with large surface area (924 m2 g−1), mesoporous structure and a certain amount of Fe loaded. The Fe species and the special structure endowed the materials excellent catalytic activity in the oxidation of ethylbenzene to acetophenone. The conversion of 94.5% and the high selectivity to targeted product (97.4%) could be achieved and the acceptable recycling stability was also exhibited.

This paper is the first attempt to systematically study properties of the effective Hamiltonian $$\overline{H}$$ arising in the periodic homogenization of some coercive but nonconvex Hamilton–Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min–max formulas for a class of nonconvex $$\overline{H}$$ . Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of $$\overline{H}$$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

Mesoporous materials with bimodal mesopores show advantages in adsorption, energy storage, and catalysis because such unique structures are beneficial to the mass transfer. Here, we describe the synthesis of bimodal mesoporous carbon nanospheres (BMCSs) by using phenolic resin as carbon precursor, triblock copolymer Pluronic F127 as the soft template, and mesoporous silica spheres as hard templates. The BMCSs with uniform spherical morphology, high specific surface area (1489 m2 g− 1), large pore volume (0.92 cm3 g− 1), and bimodal mesoporous structure (3.8 and 6.8 nm) exhibit promising properties for adsorption of methyl orange (MO). The maximum adsorption capacity of the BMCSs is 5.5 × 102 ± 0.2 × 102 mg g− 1, which is higher than that of many adsorbents reported. The kinetics studies show a better fit of pseudo-second-order model. Meanwhile, fitting equilibrium data show that the Langmuir model is more suitable to describe the MO adsorption than Freundlich model.

We demonstrated a facile method to prepare photoluminescent graphene quantum dots using commercial polyacrylonitrile (PAN) based carbon fibers (CFs) as the raw material by facile chemical oxidation and exfoliation method. The as-prepared GQDs with uniform size exhibit an excitation-independent photoluminescence behavior, which is similar to other semiconductor quantum dots. Moreover, when acting as catalyst the uniform GQDs have better activity for electrochemical oxidation of dopamine (DA) than graphene oxides (GOs). The square wave voltammogram (SWV) peak values of GQDs are in good correspondence with DA concentrations and can act as a sensor of DA.

We present high nitrogen-doped hollow carbon spheres (HNHCSs) via cooperative template-directed coating method. The SiO2 spheres serve as hard template; ionic liquids, such as 1-hexadecyl-3-methylimidazolium bromide (CMMB), serve as cationic surfactant; resorcinol formaldehyde (RF) resins as carbon source; and melamine as nitrogen source. The CMMB used in the coating procedure improves the SiO2 spheres dispersion in the reaction media and introduces nitrogen atom into carbon skeleton. The CMMB, SiO2 spheres, and RF resins self-assemble into inorganic–organic core–shell structures through electrostatic interaction. After being treated with melamine, the nitrogen content of the sample further improved, which can enhance the performance of supercapacitor. Due to high nitrogen content, specific surface area, and unique hollow structure, the HNHCSs exhibit a reversible specific capacitance of 368 F g−1 at 0.2 A g−1 in 6 M KOH, and maintains a good high-class capacitance retention capability (85.1%) at 5.0 A g−1 for up to 5000 cycles. These results show that the HNHCSs with high nitrogen content, specific surface area, and hollow structure represent a promising alternative of efficient electrode material for supercapacitor.

A facile synthesis route to the mesoporous carbon nanospheres (MCNs) with tunable size has been prepared via silica cooperative self-assembly using resorcinol as a polymer precursor, tetraethyl orthosilicate as an inorganic precursor, and hexadecyl trimethyl ammonium bromide and triblock copolymer Pluronic F127 as a co-template. The sizes of carbon nanospheres are uniform and easily controlled in the range of 80–170 nm by varying the content of F127. The MCNs exhibit promising properties for adsorption of bulky dye molecules due to their high specific surface area (1481 m2 g−1), large pore volume (2.55 cm3 g−1), and dual mesoporous texture (4.1 and 24.1 nm). Kinetic and isothermal analysis demonstrates the strong interactions between dye molecules and the MCNs. Furthermore, the regenerated MCNs show quite stable adsorption performance, which can be reused for dye removal. The results demonstrated in this work should give a useful enlightenment for the design of adsorbent to remove organic pollutant in wastewater.

Macro-mesoporous carbon materials and hollow core/mesoporous shell (HCMS) carbon spheres have been prepared through an aqueous route with monodisperse silica spheres as hard template and F127 as soft template. By tuning reaction conditions and the ratio of SiO2/F127, ordered macro-mesoporous carbon materials, disordered macro-mesoporous carbon materials and HCMS carbon spheres were fabricated. All samples possess interconnected macropores and uniform mesopores with a narrow size distribution centred at 6–9 nm. The prepared HCMS carbon spheres show high specific surface area (972 m2 g−1), large pore volume (1.27 cm3 g−1) and thin shell thickness of 5 nm, which exhibit promising properties for supercapacitors with high capacitance due to its unique hollow structures.

Carbon materials have attracted great attention in CO2 capture and energy storage due to their excellent characteristics such as tunable pore structure, modulated surface properties and superior bulk conductivities, etc. Biomass, provided by nature with non-toxic, widespread, abundant, and sustainable advantages, is considered to be a very promising precursor of carbons for the view of economic, environmental, and societal issues. However, the preparation of high-performance biomass-derived carbons is still a big challenge because of the multistep process for their synthesis and subsequent activation. Herein, hierarchically porous structured carbon materials have been prepared by directly carbonizing dried cauliflowers without any addition of agents and activation process, featuring with large specific surface area, hierarchically porous structure and improved pore volume, as well as suitable nitrogen content. Being used as a solid-state CO2 adsorbent, the obtained product exhibited a high CO2 adsorption capacity of 3.1 mmol g−1 under 1 bar and 25 °C and a remarkable reusability of 96.7% retention after 20 adsorption/regeneration cycles. Our study reveals that choosing a good biomass source was significant as the unique structure of precursor endows the carbonized product with abundant pores without the need of any post-treatment. Used as an electrode material in electrochemical capacitor, the non-activated porous carbon displayed a fairly high specific capacitance of 228.9 F g−1 at 0.5 A g−1 and an outstanding stability of 99.2% retention after 5000 cycles at 5 A g−1.

Graphitic carbon spheres (GCS) with uniform particle diameter have been synthesized by K3[Fe(CN)6] catalytic graphitization of resorcinol/formaldehyde (RF) precursors. XRD and Raman spectra reveal that highly graphitic structure is developed by catalytic graphitization at 1000 °C successfully. The catalytic graphitization by K3[Fe(CN)6] maintains the spherical morphology of the RF resins, resulting in GCS-1000 (1000 = catalytic graphitization temperature of 1000 °C) with graphitic spherical morphology (625 nm in diameter). Besides conventionally microporous structure of carbonized RF resins, macropores with the diameter about 100 nm are yielded from the removal of iron particles, leading to hierarchical micro/macroporous structure and high surface area (381 m2 g−1). With a high degree of graphitization and prominent hierarchical porous structure, GCS exhibit a high capacitance of 155 F g−1 and a good electrochemical stability, which make this type of GCS promising candidates as electrode materials for supercapacitors.

Recently, the rapid development of graphene industry in the world, especially in China, provides more opportunities for the further extension of the application field of graphene-based materials. Graphene has also been considered as a promising candidate for use in supercapacitors. Here, nitrogen-doped graphene hollow spheres (NGHS) have been successfully synthesized by using industrialized and pre-processed graphene oxide (GO) as raw material, SiO2 spheres as hard templates, and urea as reducing-doping agents. The results demonstrate that the content and pretreatment of GO sheets have important effect on the uniform spherical morphologies of the obtained samples. Industrialized GO and low-cost urea are used to prepare graphene hollow spheres, which can be a promising route to achieve mass production of NGHS. The obtained NGHS have a cavity of about 270 nm, specific surface area of 402.9 m2 g−1, ultrathin porous shells of 2.8 nm, and nitrogen content of 6.9 at.%. As electrode material for supercapacitors, the NGHS exhibit a specific capacitance of 159 F g−1 at a current density of 1 A g−1 in 6 M KOH aqueous electrolyte. Moreover, the NGHS exhibit superior cycling stability with 99.24% capacitive retention after 5000 charge/discharge cycles at a current density of 5 A g−1.

Porous carbon monoliths have attracted great interest in many fields due to their easy availability, large specific surface area, desirable electronic conductivity, and tunable pore structure. In this work, hierarchical porous nitrogen-doped partial graphitized carbon monoliths (N–MC–Fe) with ordered mesoporous have been successfully synthesized by using resorcinol-formaldehyde as precursors, iron salts as catalyst, and mixed triblock copolymers as templates via a one-step hydrothermal method. In the reactant system, hexamethylenetetramine (HMT) is used as nitrogen source and one of the carbon precursors under hydrothermal conditions instead of using toxic formaldehyde. The N–MC–Fe show hierarchically porous structures, with interconnected macroporous and ordered hexagonally arranged mesoporous. Nitrogen element is in situ doped into carbon through decomposition of HMT. Iron catalyst is helpful to improve the graphitization degree and pore volume of N–MC–Fe. The synthesis strategy is user-friendly, cost-effective, and can be easily scaled up for production. As supercapacitors, the N–MC–Fe show good capacity with high specific capacitance and good electrochemical stability.

We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton–Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed sT (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as sT ≧ O(A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and sT is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), sT is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.

For a bounded domain \({\Omega\subset\mathbb R^n}\) and \({g\in C(\delta\Omega)}\), assume that \({H\in C^2(\mathbb R^n)}\) is convex and coercive, and that \({\{p\in \mathbb R^n\| H(p)={\rm min}_{\mathbb R^n} H\}}\) has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation: $$ \begin{aligned} {\mathcal A} [u]:= H_p(\nabla u)\otimes H_p(\nabla u):\nabla^2 u &=0, \,\, {\rm in}\, \Omega \\ u&=g, \,\,{\rm on}\,\delta\Omega. \end{aligned}$$ For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and that of the Hamilton–Jacobi equation \({H(\nabla u,x)=c}\).

In this paper, we prove that viscosity solutions of Aronsson equations are absolute minimizers in certain L∞ variational problems.

In this paper, we will prove a “tangent line touching” condition for supersolutions of the infinity Laplacian. This is a kind of quantitative estimate for the failure of the strong maximal principle. When n = 2, this also implies the failure of the principle of unique continuation. In addition, we will establish a theorem of continuation.

The objective was to evaluate the clinical value of preventative ileostomy following ultralow anterior rectal resection in decreasing the incidence of anastomotic leakage. For this purpose, 62 cases that had undergone ultralow anterior rectal resection during the period from June 2007 to June 2008 were included in this study. Preventative ileostomy was performed in 36 cases (group A) and 26 cases with no preventative ileostomy performed were included as controls (group B). The incidence rate of anastomotic leakage in both groups was compared. The results show that five cases in group A reported anastomotic leakage while no anastomotic leakage was reported in group B. Therefore, it was concluded that preventative ileostomy could effectively decrease the incidence of anastomotic leakage.

Resveratrol has attracted considerable public and scientific attention because of its beneficial effects on human health. In this study, resveratrol was encapsulated into chitosan microspheres for improving the stability and evaluating the ability of controlled release. Chitosan microspheres were prepared by using an emulsification-crosslinking method, and specifically, a natural extraction genipin was used as crosslinking agent for pursuit better physiological compatibility. Factors which influence on the shape and dispersity of microspheres were investigated to optimize the preparing process of resveratrol loaded genipin-chitosanmicrospheres. The optimum condition was obtained and the prepared resveratrol loaded genipin-chitosanmicrospheres were characterized by many technical instruments. Entrapment efficiency of resveratrol in genipin-chitosanmicrospheres is up to 87.6 %, and loading capacity is 7.9 %. The in vitro release rate in pH 3.6 buffer solution reached to 33.8 % within 72 h, but was lower in pH 7.4 buffer solution.