Let G be a generalized extraspecial p-group, where p is a prime. P(G)denotes the group of all power automorphisms of G.(ⅰ)When p is odd, If the exponent of G is p, then P(G)=1. If the exponent of G is p², then P(G)≅Z_p.(ⅱ)When p=2, if G is the central product of n dihedral groups of order 8, then P(G)=1. If G is the central product of n-1 dihedral groups of order 8 and a quaternion group of order 8, then P(G)≅Z₂×Z₂.

Let S be a monoid. The characterization of monoids for which all flat-acts satisfy condition(E)is studied. Under certain conditions, it is proved that all flat S-acts satisfy condition(E)if and only if S is a monoid formed by adding an identity element to a left zero semigroup, or S is a trivial monoid. The existing results are generalized.

Let G be a finite group and n_σ(G) be the number of conjugacy classes of all non σ-subnormal subgroups. Applying the proof method of minimal order counterexamples and some σ-subnormal properties, new conclusions of σ-solvable groups and generalize some previous results are given by comparing quantitative relationships between n_σ(G) and |σ(G)|.

Let a finite group A act on a finite p-group P with |P|>p^e≥p3, where e is a fixed positive integer. In this paper, it is proved that if every non-cyclic and non-maximal class subgroup of order p^e of P is O ^p(A)-invariant, then O ^p(A^A_p-1) acts trivially on P(namely C_P(O ^p(A^A_p-1))=P), where A_p-1 is a formation which consists of all abelian groups with exponent dividing p-1 and A^A_p-1 is the A_p-1-residue of A. Some sufficient conditions for a finite group to be p-supersolvable are also given.

Let P O_n be the semigroup of all order-preserving partial transformation on X_n={1,2,…,n}. For 1≤r≤n, putP On,r={α∈P On:xα=x,∠x∈dom(α)∩{1,2,…,r}}.It is easy to check that the semigroup P On,r is subsemigroup of P O_n. In this paper, we prove that P On,r is generated by idempotents and its idempotent rank is equal to 3n-2r-1.

Let G be a finite group. A subgroup A of G is said to be σ-permutable in G if G possesses a complete Hall σ-set H such that AH ^x=H ^xA for all H∈H and all x∈G. A subgroup H of G is said to be σ-permutably embedded in G if every Hall σ_i-subgroup of H is also a Hall σ_i-subgroup of some σ-permutable subgroup of G. By studying the σ-permutably embedded subgroups of Sylow p-subgroups of G, some new criteria for G to be a p-nilpotent group are given.

The(2+1)-dimensional Boussinesq equation is investigated, which is a multidimensional extension of the typical Boussinesq equation. The soliton solutions of(2+1)-dimensional Boussinesq equation are constructed by using the Hirota bilinear method. The local characteristics of the soliton solutions are also analyzed, and the dynamic behaviors of these soliton solutions are given analytically. Additionally, under the limitation of special parameters, we also obtain the resonance soliton solutions, and due to the resonance collision, the soliton solution presents a "V" shape, which is no longer the traditional cross type. The dynamics of higher-order resonance soliton solutions are more complex and diverse, consisting of the superposition of fundamental resonant solitons.

By using the fixed point theorem, we establish the existence and multiplicity of(positive)solutions for the following discrete four-point boundary value problem with one-dimension Minkowski mean curvature operator{-(Δφ(Δu(k-1)))=f(k,u(k),Δu(k)), k∈［1,N］_Z,u(0)=αu(l₁), u(N+1)=βu(l₂),where f:［1,N］_Z×R×R→R is continuous, α, β∈［0,1)are constants and α≠β, l₁,l₂∈［1,N］_Z, l₁<l₂, φ:(-a,a)→R(0<a<∞)is a monotonic increasing operator and φ(s)=s/((1-s²)^1/2), ［1,N］_Z={1,2,3,…,N}, N≥6.

This paper studies the existence of positive solutions of a second-order Neumann boundary value problem with derivative term{-u″+k²u=f(t,u,u'), 0<t<1,u'(0)=u'(1)=0,where k>0 is a constant, f:［0,1］×［0,∞)×R→［0,∞)is a continuous function. Under superlinearity condition and Nagumo-type condition on f, the problem has at least one positive solution. The proof of the main results is based on fixed-point index.

The long time asymptotic behavior of solutions of a class of two dimensional incompressible magnetohydrodynamic equations with magnetic damping term is discussed. By proving the existence of the bounded absorbing set and using the C-condition method, the existence of the global attractors on the phase space with higher regularity is obtained.

In the first-order logical metric space, three different types of approximate reasoning patterns are defined based on pseudo distance, and the equivalence relationship between the three approximate reasoning patterns is proved. In addition, a new approximate reasoning mode Γ^δα is proposed based on similarity, and the relationship between this reasoning mode and three reasoning modes is studied. Finally, a strong approximate reasoning mode is proposed.

In order to study some new characterizations of orthomodular law in orthogonal modular lattices, we study orthomodular law from an algebraic perspective. By using complement operations, the reciprocal nature of partial addition and subtraction operations,and residuated property of global multiplication and implication, the equivalent characterizations of orthomodular law are given, respectively. These facts reveal the inherent laws of the generation of orthomodular law. By some examples, it is shown that in general the global operations of addition and multiplication on orthogonal modular lattices are non-associative and non-commutative.

Using the compatibility of mutation and Bongartz co-completion in cluster algebras with the corresponding G -systems and the combination results in G -systems, the required projection is constructed, and the exchange graph of any cluster algebra is proved to have the non-leaving-face property.

Twisted generalized Reed-Solomon(TGRS)codes of order three are constructed. The necessary and sufficient conditions characterizing this class of codes as maximum distance separable(MDS)codes are established. A new construction method for MDS codes is provided and the research on MDS-TGRS codes of order one and order two are expanded.

Using the generalized harmonic number and the method of series transformation, some super congruences involving central binomial coefficient, Catalan number and Bernoulli number are established, and some known results are generalized.

Let H be a complex separable Hilbert space, and P and Q be two orthogonal projection operators on H. This article mainly studies the numerical range of the Moore-Penrose inverse operator(P+QP)^† of the combination of P and Q. By using the operator block technique, the geometric characterization of the numerical range of the operator(P+QP)^† is given. That is, the closure of its numerical range is a closed convex hull of some explicit ellipses parameterized by points in the spectrum of operator PQ.

A relationship between parameter estimators of a class of stochastic McKean-Vlasov equations with Lévy noise and their corresponding interacting particle systems is established. The simple mean field limit result is given, that is, the convergence of the solution of the interacting particle system in the sense of L2(Ω). The maximum likelihood estimators of unknown parameters (＾overθ)^N and (-overθ) in the two systems are constructed. We show that when N tends to infinity, (＾overθ)^N converges to (-overθ)in probability.

For formalized fuzzy knowledge space and fuzzy closure space mesh, the concept of distributed serial fuzzy relationship is introduced. The union of a fuzzy approximate space family in which some basic properties given. Conditions for the meshability of a family of fuzzy knowledge spaces are presented. The meshability of a distributed skills function and knowledge structure are obtained.

By using the concept of fuzzy element transfer, a generalized internal P-fuzzy set A^(-overF) contracting inward and a generalized outer P-fuzzy set A^F expanding outward are constructed in the same universe of discourse. The mathematical structure of the generalized P-fuzzy set is given. Based on the generalized P-fuzzy set, the generalized P-fuzzy dependence degree is defined, and its characteristics are analyzed. The relationship between the generalized P-fuzzy set and Zadeh fuzzy set is further discussed, and their relationship theorems are given. Several forms of generated P-set from the generalized P-fuzzy set are obtained. It is concluded that the generalized P-fuzzy set is a generalization of Zadeh fuzzy set and the P-set, and both Zadeh fuzzy set and the P-set are special cases of the generalized P-fuzzy set. Finally, an application of the generalized P-fuzzy set in the venture capital management system is given.

To study the characteristics of atmospheric mercury pollution in coastal Qingdao and evaluate the influences of pollution sources and meteorological conditions, an online measurement of total gaseous mercury(TGM)was conducted on Xiaomai Island in Qingdao during the summer of 2023. A generalized additive model(GAM)was constructed to investigate the linear and nonlinear relationships between TGM and air pollutants as well as meteorological parameters. The results showed the mass concentration of TGM was(2.94±0.75)ng·m^-3, within a range of 1.74-6.49 ng·m^-3, indicating a moderate pollution level in China. TGM mass concentrations were significantly affected by anthropogenic emissions, meteorological conditions, and sea-land breezes. In the univariate GAM model results, CO concentration, relative humidity(RH), and sea-level pressure(P₀)had substantial effects on TGM variations. Specifically, CO and NOx mass concentration exhibited increasing relationships with TGM, indicating that anthropogenic emissions significantly contribute to TGM levels. While RH and P₀ showed decreasing relationships, indicating that favorable meteorological conditions contribute to the dilution or dispersion of TGM. Multivariate GAM model incorporating conventional pollutants and meteorological parameters further improved the fitting results, achieving an explanatory power of 84.6% for TGM variations. The interactions between atmospheric pollutants, WD, RH, and boundary layer height(BLH)significantly influenced TGM variability. This study quantitatively visualizes the complex impacts of various factors and their interactions on TGM variations, revealing the combined effects of pollution transport and meteorological conditions, which is critical for understanding the atmospheric mercury cycle and pollution control in coastal areas.