In the present paper, we define the minimum degree of polynomials. By using the minimum degree of polynomials and Zariski topology, we give a complete description of the images of polynomials with zero constant term on strictly upper triangular matrix algebras over an algebraically closed field.

The study of t-designs with certain highly symmetric properties and the classification of their automorphism groups is one of the important topics of algebraic combinatorics. Hitherto the classification of flag-transitive Steiner t-designs has been completely settled by Huber. Nevertheless, it is interesting that the results of t-designs will get richer when G-flag-transitivity are weakened to G-block-transitivity and the research work becomes more challenging as well. Depending on the classification of finite simple groups and 2-transitive permutation groups, we show the classification of block-transitive Steiner t-designs admitting a block-transitive automorphism group with sporadic socle, and draw a conclusion as follows: Let D=(P,E) be a non-trivial Steiner t-design and G\le \mathrm{A}\mathrm{u}\mathrm{t}(D) be point-primitive. If the socle of G is a sporadic simple group, then G acts block-transitively on D if and only if G acts flag-transitively on D, and D is one of the five known flag-transitive Steiner t-designs.

In this paper, we consider the boundary value problems of third-order nonlinear ordinary differential equation where \text{?}f:[\mathrm{0}，\mathrm{1}]\times {\mathbf{R}}^{\mathrm{3}}\to \mathbf{R}\text{?}satisfies Carathéodory conditions. Under some suitable growth conditions on f, we show that the above problem has at least one solution. The proof of the main results is based on Leray-Schauder fixed point theorem.

We consider the following class of p-Laplacian problem where 1<p<N,a,b are positive parameters,q\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}（（\mathrm{1},+\mathrm{\infty })，\text{?}[\mathrm{0},+\mathrm{\infty }))，\text{?}f\in C([\mathrm{0},+\mathrm{\infty }),[\mathrm{0}，+\mathrm{\infty })). By applying the fixed point theorem in cones, the method of upper and lower solutions and topological degree theory, we obtain the existence and multiplicity of positive solutions for the above p-Laplacian problem. -div(| ?u | p-2?u)=q(| x |)f(u), | x |>1,x∈RN,u(x)=b, | x |=1,u(x)→a, | x |→+∞, （P）

A class of conical degenerate elliptic equations with potential functions is studied. The existence of the ground state solution for the equation is established by studying the constrained minimization problem. The blow-up behavior is analyzed as well. Specifically, we prove that there exists at least one minimizer for the constraint minimization problem under certain conditions, but the minimizer does not exist when these conditions are unsatisfied. The blow-up behavior of the minimizer is analyzed in detail when the parameter approaches the critical value.

An inertial viscous iterative algorithm is constructed for the common elements of variational inequality problems and fixed point problems. Under appropriate assumptions, it is proved that the iterative sequence generated by the constructed algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequalities and the common fixed point set of a finite family of semi-contractive mappings by using demi-closed at zero,projection operator and other analysis techniques. Numerical experiments illustrate the effectiveness of the algorithm. The study of this paper improves and extends some recent relative results.

In order to analyze the mean random chord length of convex sets under different kinds of random processes, we take circles, equilateral triangles, rectangles, and squares as examples. Their mean values are obtained using definition method. Then the extreme values of the mean chord length of convex sets are discussed by means of the chord power integrals and their inequalities. Furthermore, some inequalities about these mean values are established, and two conjectures are proposed.

In this paper, we introduce the concepts of G-non recurrent point and G-average shadowing property. Then, we study the topological structure of G-non recurrent point in the inverse limit space under group action and the dynamical properties of G-average shadowing property in metric G-space. The results are as follows: (1) The self-map f has G-non recurrent point if and only if the shift map \sigma has \bar{G}-non recurrent point; (2) If f^k has G-average shadowing property, then f has G-average shadowing property. These results generalize the conclusions of non recurrent point set of shift mapping in the inverse limit space and average shadowing property of iterative mapping in metric space.

This paper studies the existence and uniqueness of mild solution to stochastic Cahn-Hilliard equations, driven by fractional-colored noise, which is fractional in time and colored in space, with spatial kernel \mathrm{f}. A local solution is found by truncating drift term and applying variable substitution to stochastic integral. We prove the tightness of truncated solution by estimating Green function. Finally, a weak convergence of local solution is explored to verify the existence and uniqueness for mild solution of original equation. Coefficient conditions related to Hurst exponent H is then revealed. Furthermore, regularity of the skeleton are checked by applying Cauchy-Schwarz, Burkholder's inequalities and estimating Green function. It makes use of Gronwall's lemma and Girsanov's theorem to reduce large deviation form. We obtain Freidlin-Wentzell inequality in a special space, in which extension of Garsia's lemma plays an important role. The large deviation principle with a small perturbation can then be established.

An exact algorithm is presented for a parallel machine scheduling problem with additional resources constraints. Specifically, the jobs need be processed on parallel machines, and the processing of each job requires a unit of some additional resources in addition to machine resource. The variety and amount of additional resources are limited, our goal is to make the optimal schedule of the problem so that the completion time of the final completed job is minimized. This problem originates from the field of earth observation satellite data download, and has been widely used in intelligent manufacturing and information processing. The integer programming model, the lower bound of the optimal solution and the branch and bound algorithm of the problem are given. An effective branching strategy is proposed to avoid repeated branching and a bound method is designed to improve the convergence rate of the algorithm. The correctness and the effectiveness of the algorithm under different parameter configuration are verified by a large number of numerical simulation experiments and a small-sized instance example.

Based on the equivalent integral-differential form of the equation considered, an alternate directional implicit (ADI) numerical scheme for two-dimensional nonlinear time-fractional wave equations with initial singularity was developed by combining the convolution quadrature formula with the Crank-Nicolson technique. Theoretical derivation shows that the scheme has at least order γ accuracy in time direction and order 4 accuracy in space direction. Finally, a numerical example is demonstrated to verify the conclusion.

The spread and outbreak of computer viruses in the information age have had a huge impact on social production and people's lives, and it is particularly important to effectively prevent and suppress the spread of computer viruses. In this paper, a computer virus propagation model with immune function is established making use of the optimal control method. It obtains the basic reproduction number R_{\mathrm{0}} based on the method of the next generation matrix; It builds the Liapunov function, and proves the condition of the global stability of the disease-free equilibrium point when R_{\mathrm{0}}\le \mathrm{1}; The optimal solution is obtained based on Pontryagin's principle of maximum value. Numerical simulation results show that the optimal control model can detect and deal with infected computers while improving the installation and protection rate of antivirus software, and the control target time is the shortest and the control cost is the lowest.

The catalysts PtxTi/MCF and Pt/yTi-MCF were obtained by adding Ti element to mesocellular silica foam (MCF) support in two different ways. The catalytic activities of these catalysts for the complete oxidation of benzene, n-hexane and ethyl acetate were studied. The effects of different catalyst preparation methods, Ti addition amount and calcination temperature were investigated in detail. In addition, the structure and physicochemical properties of some of the support and catalyst were characterized by nitrogen isothermal adsorption-desorption, Raman spectrum analysis, small and wide angle XRD. The results show that, Pt8Ti/MCF has good catalytic activity and stability for the complete oxidation of benzene, n-hexane, and especially ethyl acetate. It was prepared simply by impregnating the mixture solution of platinum acetylacetone and titanium acetylacetone and then calcination, which exhibits a good application prospect in future.

To improve the catalytic ability of nano zero-valent iron in a Fenton like reaction system and reduce the agglomeration of nano zero-valent iron materials, the liquid phase reduction method was used to prepare biochar loaded nano zero valent iron materials (nZVI@BC). The material was characterized and the methylene blue was degraded to explore the material properties, the optimal degradation conditions and the action mechanism. The results showed that nano zero-valent iron had been successfully and evenly distributed on the surface of biochar. The prepared material was of high purity and good stability. In the degradation optimization experiment, the optimal reaction condition for degradation of 100 mg·L^-1 methylene blue solution was 25 ℃, the initial pH=3, the dosage of nZVI@BC is 30 mg·L^-1 and the dosage of H₂O₂ is 4 mmol·L^-1. Under this condition, methylene blue solution can be almost completely degraded in 10 minutes. It had found that hydroxyl radicals played a major role in the degradation process. The chromogenic functional groups first broke, followed by the destruction of the aromatic ring structure, and ultimately completed degradation. And the material had a good recycling performance. After three cycles, the removal rate of methylene blue was still above 85%. In actual printing and dyeing wastewater treatment experiments, nZVI@BC had a good wastewater treatment effect. When the dosage of H₂O₂ was 0.9 mmol·L^-1, the dosage of nZVI@BC was 60 mg·L^-1 and the pH≤ 4.5, the COD_cr decreased to below 50 mg·L^-1, up to the standards for reuse water.

Advanced oxidative processes of persulfate (PS-AOPs) have promising applications in antibiotic wastewater treatment due to their strong oxidation capacity and good selectivity. In this study, Sargassum fusiforme was selected as a raw biomass, and ZnCl₂ carbonation-pyrolysis was adopted to prepare zinc chloride modified sheep sorrel biochar which was used as a catalyst to catalyze persulfate (PDS) to degrade the norfloxacin (NOR) in the water. The results showed that different ratios of ZnCl₂ doping would significantly affect the performance of biochar, and the catalytic efficiency of the prepared biochar was optimal when the ratio of algal powder to ZnCl₂ was 1∶1. Moreover, SEM, BET, XRD characterizations were employed to investigate the physical and chemical properties of YQZn in order to further explore the modification mechanism of ZnCl₂. The catalytic efficiency result of YQZn under different conditions showed that the biochar had the best NOR removal rate of 95.09% under the conditions of PDS concentration was 1 mmol·L^-1, modified biochar addition was 0.3 g·L^-1, pH=9 and NOR concentration was 10 mg·L^-1. \mathrm{S}{\mathrm{O}}_{\mathrm{4}}^{·-}, HO· radicals, and ¹{\mathrm{O}}_{\mathrm{2}} played a leading role in activating PDS in the system by free radical quenching. To solve the difficulties in biochar powder recovery YQZn geopolymer composite catalyst (GB) was prepared by using biochar loaded on porous geopolymer. The results showed that GB had excellent degradation activity and reuse performance, which successfully realized the loading and recycling of biochar, and thus had great potential for continuous treatment of norfloxacin in water.

With the impact of the epidemic and the deepening of population aging in China, the distribution and quality of medical resources have become important factors affecting housing prices, gradually generating the capitalization effects of medical resources. In this study, the differences in the resident's preference for the infectious hospital and general 3A hospitals in Harbin were explored in depth through a questionnaire survey and a comparative analysis of their capitalization effects. Furthermore, the social heterogeneity of the capitalization effects of medical resources and the homogeneity of the two kinds of medical resources were analyzed based on quantile regression models and interaction effects tests. The results show that (1) the infectious hospital depresses the prices of nearby housings, and general 3A hospitals increase the prices of nearby housings. The capitalization effect of both medical resources gradually decreases with increasing distance. (2) Medium-priced housings are more sensitive to the proximity of the infectious hospital, and the capitalization effect of general 3A hospitals gradually increases as the price of housings increases. (3) There is an interaction between the capitalization effects of the two kinds of medical resources, and the proximity of general 3A hospitals enhances the NIMBY (not in my backyard) effect of infectious hospitals.