In this paper, the properties of S-hyperlattice ideals are explored. It is proved that the direct product of S-hyperlattice ideals of two S-hyperlattices is an S-hyperlattice ideal of their direct product, the homomorphic image and preimage of S-hyperlattice ideal are still S-hyperlattice ideals, the set of all S-hyperlattice ideals of an S-hyperlattice is an algebraic topped meet structure. It is illustrated that for an S-hyperlattice congruence, the congruence class of the smallest element zero is generally not an S-hyperlattice ideal, and a sufficient condition for it to be an S-hyperlattice ideal is given. While for an S-hyperlattice ideal, a maximal S-hyperlattice congruence with it as a congruence class is constructed.

A concept of relatively maximal ideals of co-residuated lattices is introduced, and the properties and the relationships among relatively maximal ideals, prime ideals and ideals are studied. A sufficient condition of the ideal lattice of a co-residuated lattice to be a spatial frame is obtained.

In the context of Abelian categories, the introduction of n-super finitely presented objects is employed to characterize n-weak injective objects, thereby some fundamental properties of n-weak injective objects are established. The n-weak injective dimensions of objects in an Abelian category is defined, and the connections between the n-weak injective dimensions of three objects in a short exact sequence and the cotorsion theory are discussed.

This paper determines the graphs with maximum and minimum distance spectral radii in the complement graphs of all clique trees, and provides the graphs with maximum and minimum distance spectral radii in the complement graphs of all graphs with cut vertices.

The E-total colorings of complete bipartite graphs K1,n, K2,n and K3,n which vertex-distinguished by multiple sets are discussed by using the method of proof by contradiction and the pre-assignment of chromatic sets, the vertex-distinguishing E-total chromatic numbers of these graphs are determined.

Given a graph G, the weighted Szeged index of a graph G, denoted by S_w(G)=∑_uv=e∈E(G)(d_G(u)+d_G(v))n^G_u(e)n^G_v(e), where d_G(u)is the degree of u in G. For edge uv=e∈E(G), n_u(e)represents the number of vertices closer to vertex u than vertex v in graph G. Some graph transformations are given, by using these graph transformations, the upper bound and lower bound of weighted Szeged index of blossomed stars and the upper bound of weighted Szeged index of trees with given diameters are obtained, and the corresponding extreme value graphs are characterized.

An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most one. The graph G is said to be equitably k-colorable if G has an equitable k-coloring. The maximum average degree is the maximum value of average degree of all nonempty subgraphs of G, denoted by mad(G). In this paper, we utilizes the method of weight transfer to prove that a graph G with mad(G)≤(13)/4 is equitably k-colorable for k≥max{Δ(G),6}, where Δ(G)is the maximum degree of G.

In this paper, we describe the degeneracy of the product graphs by the degeneracy of their factor graphs, combined with conclusions on the linear arboricity of degeneracy graphs, and give the degeneracy conditions for Cartesian product graphs, some direct product graphs and strong product graphs to satisfy the linear arboricity conjecture. Then we prove that the lexicographic product graph of two 1-degenerate graphs satisfies the linear arboricity conjecture and determine its linear arboricity in most cases.

The initial boundary value problem of a class of fourth order hyperbolic equation with viscoelastic term is studied. The local existence of the weak solution is obtained by using Galerkin method. Then the global existence of the weak solution is proved using the potential well method, and the decay estimate of the global solution is given.

A Frenet type Darboux curve with singular points is defined in the three-dimensional Euclidean space, and we give a necessary and sufficient condition for a curve to be a Frenet type Darboux curve. Then we construct the Frenet type Darboux curve through a spherical Legendre curve, and prove that both the framed helix and the framed rectifying curve are Frenet type Darboux curves.

The non-triviality of the product element h1h_n(~overδ)_p∈H^p+2,t(A)is proved by using May spectral sequence and the analysis of degree and differential, where n≥6, the odd prime p≥11, t=q［pⁿ+p4+(p-1)p2+(p-1)p+(p-3)］+p-4, q=2(p-1).

A class of densely defined self-adjoint linear operator a^N is introduced in the functional space L2(M)of normal martingale square-integrable, where a is a positive number, and N is the number operator in L2(M), a^N is called the a-level power-number operator of N. Firstly, the analytical properties of a^N are discussed: a sufficient and necessary condition that a^N is bounded is given, and in this case, {a^N; 0<a≤1} are unit operator on L2(M). Secondly, a^N is compact operator if and only if 0<a<1; the construction and the spectrum of {a^N; a>0} are discussed: {aⁿ; a>0} is the spectrum of a^N and all of its eigenvector forms an orthonormal basis of L2(M), 1 is the unique spectrum of {a^N; a>0} and the vacuum Z_ is the unique common eigenvector of 1. And then the dependence of a^N on a is discussed. Finally, a uniform convergence sequence of a^N for a∈(0,1)and a strong convergence sequence of a^N is constructed when a>1 by means of the quantum Bernoulli noise indexed by Γ.

To simultaneously estimate the number of change points, the location of change points and the model parameters in censored quantile regression model, a linearization technique is employed to obtain estimators for above parameters. This approach overcomes the issues of non-differentiability and non-convexity objective function at the change points. It is capable of capturing the relationship between response and covariate of interest that changes across multiple change points. Furthermore, the proposed estimators strike a balance between flexibility and interpretability, making them become a useful tool for identifying and explaining change points. Simulation studies show that the estimators demonstrate robustness in both homoscedastic and heteroscedastic conditions across various quantile levels. An empirical analysis reveals the existence of two change points and their change point effects.

The paper studies the higher-order differentiability and sensitivity of vector variational inequalities and weak vector variational inequalities. The basic definitions of contingent cones, higher-order tangent sets are introduced. The higher-order differential properties of a class of set-valued maps closely related to vector variational inequalities are studied, and the accurate calculation formula of its higher-order derivatives is obtained. By discussing the relationship between the higher-order derivatives of set-valued mapping and its profile mapping, the higher-order differentiability and sensitivity of vector variational inequalities are obtained.

This paper presents a differential spectral approximation for a fourth-order parabolic problem on a spherical domain based on a dimension reduction scheme. The fourth-order parabolic problem is transformed into an equivalent form in the spherical coordinates. By utilizing the properties of the Laplace-Beltrami operator and the orthogonality of the spherical harmonic functions, the problem is further decomposed into a series of decoupled one-dimensional fourth-order parabolic problems. Based on each one-dimensional fourth-order parabolic problem, its fully discrete scheme is established, and its stability and error estimation of the approximate solution are proved. Some numerical examples are given. The numerical results are shown that the differential spectral approximation algorithm is stable and convergent.