p-adic MRA and GMRA are important tools for constructing wavelet frames in L²(R₊). That a nested subspace sequence in L²(R₊) has trivial intersection and L²(R₊) union is a fundamental requirement for it to form a p-adic MRA and GMRA. This paper addresses the intersection and union of p-adic dilates of a singly generated p-adic shift-invariant subspace. We prove that, for a singly generated p-adic shift-invariant subspace, the intersection of its p-adic dilates is 0, and the union of its p-adic dilates is a Walsh p-adic reducing subspace of L²(R₊) if the generator φ is Walsh p-adic refinable in addition. In particular, the dilates form a p-adic GMRA for L²(R₊) if and only if ∪_j∈Zp^jsupp(Fφ)=R₊, where F is the Walsh p-adic Fourier transform on L²(R₊). It is worth noticing that our results are similar to the case of usual L²(R), while their proofs are nontrivial. It is because the p-adic addition ⊕ on R₊ is different from the usual addition + on R.

We prove that every nonlinear mixed Lie triple derivable mapping from any factor von Neumann algebra A into itself is an additive *-derivation, with the classical derivable mapping and matrix block.

We consider the asymptotic behavior of non-autonomous stochastic fractional wave equations on an unbounded domain Rⁿ. We firstly transform the equation into a random equation whose solutions generate a random one system. Then we establish the asymptotical compactness of the system by the splitting technique. Finally the existence of random attractors is proved.

We first discuss two special quasi-cyclic modules over principal ideal domains and then investigate the structures of vector spaces of finite A-width. We show that the poset of A-invariant subspaces of a vector space of finite A-width must satisfy the minimal condition, and give a sufficient and necessary condition for a vector space (as a F[λ]-module) to be a quasi-cyclic module.

Let λ₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all negative, and let λ₁/λ₂ be irrational and algebraic. Let V be a well-spaced sequence and δ > 0. We prove that for any given ε > 0, the number of v satisfying v ∈ V and v ≤ X for which|λ₁p₁²+λ₂p₂²+λ₃p₃³+λ₄p₄³-v|<v^-δ has no solution in primes p₁, p₂, p₃, p₄ does not exceed O(X^7/8+2δ+ε). This gives an improvement of an earlier result.

We explicitly compute the first and second cohomology groups of the Super Schrödinger algebra J (1/1) with coefficients in the trivial module and the finitedimentional irreducible modules. We also show that the first and second cohomology groups of J (1/1) with coefficients in the universal enveloping algebras U(J (1/1)) (under the adjoint action) are infinite dimensional.

Let H₄ be the Sweedler's 4-dimensional Hopf algebra. In this paper by the definition and property of Rota-Baxter operator, we establish the system of quadratic equations of the matrix elements of Rota-Baxter operators of H₄ with weight λ for a given base. By solving the system of the equations with weight λ=0 and λ=1 the matrix forms of Rota-Baxter operators are given.Keywords Sweedler's algebra; weight; Rota-Baxter algebra

We propose a joint modeling approach for the analysis of recurrent event data with a terminal event, where an additive-multiplicative rates model is specified for the recurrent event process, the Cox hazards frailty model is specified for the terminal event, and the shared frailty is used to account for the association between the two processes. An estimating equation approach is developed for estimating the model parameters. The asymptotic properties of the proposed estimators are established. Simulation studies are constructed to examine performances of the proposed estimators under finite samples. Finally, we use the proposed method to analyze a medical cost study of chronic heart failure patients.Keywords additive-multiplicative rates model; estimating equation; frailty; recurrent event; terminal event

Partially linear additive panel data models with strong interpretability and flexibility have been widely used in a variety of research fields. Considered a fixed effects partially linear additive panel data model with correlation structure within subjects, we derived the estimators by using penalized quadratic inference functions method under the basis of combining exponential spline function and LSDV method; the asymptotic normality of parametric estimators and convergence of nonparametric estimators were proved under suitable regular conditions; Monte Carlo simulations show that our estimates have good performances in small sample cases; meanwhile, the estimation techniques were used to analyse a real data set.

In this paper, using theory of value distribution, we investigate whether a class of order n complex differential-difference equation
ω⁽ⁿ⁾(z)² +[αω(z + c) - βω(z)]²=1
and a class of systems of order n complex differential-difference equations 

have entire solutions with finite order or not. Our results extend and improve the results due to Gao Lingyun and Liu Kai et al.

In this paper, we study the analytical solutions for the extended generalized two-component Dullin-Gottwald-Holm shallow water system. By the resulting of Emden equation, we investigate the global existence and finite-time blowup phenomena. Furthermore, the perturbation method and characteristic method are used to construct two types of exact solutions for this system.

Let A be an abelian category and X a subcategory of A. Sather-Wagstaff, Sharif and White introduced the Gorenstein subcategory G (X). Denote by PP the class of pure-projective R-modules and by P the class of projective R-modules. We show that there are some rings such that G (P) ⊆ G(PP), which gives a negative answer to the Question that whether G (W) is contained in G (X) provided that W is a subcategory of X. In addition, we give some characterizations of when G (P) ⊆ G (PP) and G (PP) ⊆ G (P) hold.

We investigate a scalarizing function's continuity and convexity under K-conditions. Utilizing that function, we convert set-valued optimization problems into equilibrium problems, and then study existence of efficient solutions of set-valued optimization problems with constraints, the upper semicontinuity and lower semicontinuity of strongly approximate solution mappings to the parametric set-valued optimization problems. As compared with relevant literatures, our methods are new, our conclusions and conditions are more general.

We study univalent functions f for which log f' belongs to the analytic Morrey spaces. We establish some new characterizations of the analytic Morrey domains.

We first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As applications in Finsler geometry, we prove that a Finsler manifold is a Berwald space if it is a Landsberg space and satisfies one of the following conditions: closed Cartan-type form, vanishing S curvature or vanishing mean Berwald curvature.

An extension of algebras is a homomorphism of algebras preserving identities. The relative global dimension of an extension f:B→A is defined to be the supremum of relative projective dimensions of all A-modules. We give a necessary and sufficient condition for an extension of algebras to have finite relative global dimension. As an application, we give a new short proof for a result in Hochschild [Relative homological algebra,Trans. Am. Math. Soc., 1956, 82: 246–269].

We are concerned with the existence of the periodic solutions for a class of weakly coupled systems. Under some sub-linear conditions about time-mapping, we prove the existence of at least one harnomic solution by applying the Poincaré–Bohl theorem and infinite periodic solutions with period 2mπ to equations by applying the higher dimensional version of the Poincaré–Birkhoff theorem, where m ∈ Z and m>1.

A finite group G is called a CN-group if every subgroup H of G is C-normal in G. In this paper, we will give first a complete classification of the 2-generator CN-p-groups. Then by applying the structure of the 2-generator CN-p-groups, we obtain the following results: If p is a odd prime, then G is a CN-p-group if and only if Φ(G)≤ Z(G). If p = 2, then G is a CN-p-group if and only if for any given a ∈ G and for any g ∈ Φ(G), we have g^a=g or g^a=g^-1. We also get some criteria of CN-p-groups in terms of direct product of CN-p-groups.

We use the analysis method, the arithmetical properties of Dedekind sums and first kind Chebyshev polynomials to study the asymptotic estimation problem of one kind hybrid mean value involving the Dedekind sums and the first kind Chebyshev polynomials. At last, we obtain a sharp asymptotic formula for it.

The main purpose of this paper is using the analytic method and the properties of classical Gauss sums and trigonometric sums to study the computational problem of one kind hybrid power mean of two different Gauss sums, and give an exact computational formula for it.

Recently Ding has constructed certain cyclic codes by using new cyclotomy (V₀,V₁) and studied the properties. In this paper we construct new binary sequences of order two and length pq by using the cyclotomy (V₀,V₁), and calculate the autocorrelation values, linear complexity and minimal polynomials.

Let φ(n),S(n) be the Euler function and the Smarandache function of the positive integer n, respectively. Based on elementary methods and techniques, according to the algorithm formula of the Smarandache function, all solutions of the equation φ(p^αm)=S(p^αk) are given, where p is a prime, α and m are both positive integers, and gcd(m,p)=1. And then we get the solutions of the equation φ(n)=S(n^k). Furthermore, all positive integers n satisfying the condition S(n)|σ(n) are determined. At last, basing on the Mobius transformation inversion theorem, we prove that the equation φ(n)=Σ_d|n S(d) has only two solutions, namely, n=2⁵ and n=3×2⁵.

The main purpose of this paper is using the analytic method and the properties of trigonometric sums to study the computational problem of one kind hybrid power mean of two different Gauss sums, and give an exact computational formula for it. As an application of our result, we give an exact formula for the number of solutions of one kind diagonal congruence equation mod p, where p is an odd prime.

The main purpose of this paper is using the elementary methods and the properties of the trigonometric sums to study the computational problem of one kind fourth power mean of the two-term exponential sums, and give a precise computational formula for it.

This paper establishes the boundedness of the commutator M_b generated by the Marcinkiewicz integral M and the regularized bounded mean oscillation space with the discrete coefficient RBMO(μ) over non-homogeneous metric measure space. Under the assumption that the dominating function λ satisfies the ∈-weak reverse doubling condition, when p ∈ (1,∞), the authors prove that the Mb is bounded on the Lebesgue space L^p(μ). Furthermore, the boundedness of the Mb on the Morrey space is also obtained.

In this work we extend the quantitative and sharp weighted bounds for the A_p theorem to the q-variation of ω-Calderón–Zygmund operators. These results make use of the new sparse dominating techniques given recently by Lerner to control the q-variation. Compared with the work of Hytönen etc., which also involved the sharp weighted estimates of q-variations,ω in our case only satisfies the Dini condition, and related cut-off is sharp.

We investigate the ergodic property of Markov chains in general state space, an exponentially ergodic conclusion Markov chains, adding conditions π(f^p)<∞,p＞1. By using the coupling method, there exists the full absorption set, such that the Markov chain is f-exponentially ergodic on it.

In this article, some properties of measurable operators associated with a von Neumann algebra are considered. The concept of step operator is defined and it is proved that any positive measurable operator can be strongly approximated by some step operators on its domain, which means that any positive measurable operator can be strongly approximated by some projections on its domain. In addition, the measurability of composition operator of measurable operator and bounded operator is discussed.

Isometry is a significant subject in the study of the structure of space. In this paper, we will introduce a special F-space,b⁽²⁾ space, and give the representation theorem for the onto isometric mapping on the unit spheres of the b⁽²⁾ spaces, then we solved the Tingley's problem on b⁽²⁾ space.

Let K/F_q be a global function field over the finite field F_q,and l be a prime number different from the characteristic of K. Denote by ζ_l a primitive l-th root of unity in a fixed algebraic closure of K. For two given elements a,b ∈ K^*-(K^*)^l, we study in this paper the properties of radical extensions K(^l√a) and K(^l√a,^l√b) of K. By the Kummer theory, we give a necessary and sufficient condition for K(^l√a)/K and K(^l√a,^l√b)/K being not geometric extensions. Suppose that a,b ∈ K^* - (K^*)^l are l-independent. For a prime divisor P of K and the corresponding discrete valuation ring O_P, a necessary and sufficient condition for a,b,generating cyclic group (O_P/P)^* is presented by the properties of the above two function fields extensions. With the help of results obtained, the Dirichlet density of M_a,b, which is the set of prime divisors of K such that cyclic group (O_P/P)^* can be generated by a,b, is given explicitly in this paper.

We prove that the global Ding projective dimension and global Ding injective dimension coincide for any ring. We investigate the relationship between singularity categories and stable categories with respect to Ding modules, and characterize Gorenstein (regular) rings and the finiteness of left global dimension of rings in terms of singularity categories and Ding modules.

We discuss some algebraic properties of Toeplitz operators with a class of radial and quasi-homogeneous symbols on the Fock space of the complex plane. We give necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator, and study the zero-product problem of several such Toeplitz operators. Furthermore, the corresponding commuting problem of Toeplitz operators with quasi-homogeneous symbols is studied.

We develop c-supplemented subalgebras, E-algebras and Frattini theory of Lie algebras for Lie-Rinehart algebras, obtain its some important properties and give a necessary conditions for solvable Lie-Rinehart algebras. Moreover, we obtain a necessary and sufficient conditions for E-Lie-Rinehart algebras and c-supplemented Lie-Rinehart algebras, respectively.

On the basis of the Prüfer-Baer theorem of the bounded module over a principal ideal domain, this paper study several basic problems about the algebraic linear transformation of some vector space (infinite dimensional). Let V be a vector space (infinite dimensional) over a field F, A be an algebraic linear transformation of V:
(1) Suppose any linear transformation commuting with A commutes also with a linear transformation B, then B=f(A), where f is a polynomial over F.
(2) There exists a basis for V such that the matrix of A relative to this basis has the rational canonical form (classical canonical form). Moreover the classical canonical form becomes the Jordan canonical form when F is algebraic closed.
(3) There exists the Jordan-Chevalley decomposition of A when F is algebraically closed.
This result prevails for the perfect field in general. These results extend some theorems of finite dimensional vector spaces to infinite dimensional vector spaces.

We study the properties of the category of the Yetter-Drinfeld modules over a weak Hopf algebra, and give sufficient condition for the Yetter-Drinfeld category to be semisimple.

We establish the fundamental W^1,p estimate for the weak solution of a system in a bounded domain Ω in R³. The system is related to the steady-state of Maxwell's equations for the magnetic field. The inverse of the principle coefficient matrix is assumed to be in the VMO space. We transform the system to scalar elliptic equations by using the properties of curl and divergence of vector fields in R³. By the regularity theory of elliptic equations, we get the W^1,p estimate for 1 < p < ∞.

Let dwI denote the class of #-injective complexes of left R-modules (i.e., complexes of injective left R-modules). We prove that over left noetherian rings R, the pair (^⊥(dwI), dwI) is a perfect injective cotorsion pair. In particular, we get that every complex of left R-modules has a #-injective envelope. As an application, we prove that over left noetherian rings R, every complex of left R-modules has a special E_tac (I)-preenvelope, where E_tac (I) is the class of complete acyclic complexes of injective left R-modules.

We study the conformal transformations between two (α, β)-metrics. We prove that, if F is a locally dually flat regular (α, β)-metric and is conformally related to F, that is, F=e^σ(x)F, then F is also a locally dually flat (α, β)-metric if and only if the conformal transformation is a homothety. Further, in the case with singularity, we prove that any conformal transformation between two locally dually flat general Kropina metrics must be a homothety.

We consider two Toeplitz operators T_u and T_v on the generalized Fock space over the complex plane C. Let's assume that u is a radial function and the two operators commute. Under certain growth condition at infinity of u and v, we prove that v must be a radial function as well. Finally, we also construct a S_p class of Toeplitz operators on the generalized Fock space with symbols which are essentially unbounded on any point of the complex plane C.

Fixed point theorems for several cyclic contractive mappings are established in dislocated quasi-b-metric spaces. The results obtained in this paper improve and unify some previous results in the existing literature. Moreover, several nontrivial examples are given to illustrate the superiority of the main results.