We shall deal with the existence and the form of transcendental meromorphic solutions of nonlinear differential equation of the form fn+Qd(z, f)=p1(z)eα1(z)+ p2(z)eα2(z), where n ≥ 4 is an integer, Qd(z, f) is a differential polynomial in f of degree d ≤ n-3 with rational functions as its coefficients, p1, p2 are non-vanishing rational functions and α1, α2 are nonconstant polynomials. By utilizing Nevanlinna value distribution theory, we can derive conditions concerning the terms p1, p2, α1 and α2 that are necessary for the existence and the form of a transcendental meromorphic solution of the equation. In particular, we also consider the existence and the form of transcendental meromorphic solution of the equation when Qd(z, f)=a(z)ff' with n=4, where a(z) is a non-vanishing rational function.
The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of the fourth power mean of the generalized quartic Gauss sums mod p, an odd prime, and give an exact computational formula and asymptotic formula for it according to p ≡ 3 or 1 mod 4.
This paper proposes a new approach to estimate the within-subject covariance of longitudinal rank regression based on the modified Cholesky decomposition matrix. Then new rank-based unbiased estimating functions are developed to improve estimation efficiency in the analysis of unbalanced longitudinal data. Under some regularity conditions, we establish the asymptotically normal distributions of the resulting estimators. Moreover, we propose a robust rank score test for hypothesis on the regression coefficients. Simulation studies and a real data analysis show that the proposed approach yields highly efficient estimators and the proposed rank score test is more powerful than the existing approaches.
We study normality and shared sets of meromorphic functions. For a meromorphic family F, we investigate the relations among numbers of any two functions f, g shared functions, f and its derivative shared functions and multiple values of f. We give a quite general sufficient condition for F to be normal.
We study the commuting Toeplitz operators on the pluriharmonic Hardy space h2(∂Ω). A necessary and sufficient condition is obtained for an analytic Toeplitz operator and a co-analytic Toeplitz operator to be commuting on h2(T2).
Let K/Fq be a global function field over finite field Fq with genus greater than 0. Suppose that Kn:=KFqn is a constant field extension of K with degree n. Together the rational expression for zeta function of K with the properties of constant field extensions, for a specified prime number l, we study in this paper the existence of constant field extension Kn/K with l dividing the order of group Pic0(Kn), which is the group of divisor classes of degree zero of function field Kn.
A damped stochastic viscoelastic wave equation driven by a non-Gaussian Lévy process is studied. Under appropriate conditions, we show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.
Using Nevanlinna theory, we study the uniqueness theorems of L-functions in the (extended) Selberg class and prove that there exist two sets S1 (including one or two elements) and S2 (including three elements) such that f ≡ L if E(Si, f)=E(Si, L) for i=1, 2.
We study the properties and algorithms for the inversion polynomials in the twisted Kazhdan-Lusztig theory. We establish a type of distinguished basis (or D-basis) for Lusztig's dual module M, and show the formulae for the action of Hecke algebra on the basis. In the case of finite Coxeter groups, we get a relation of the structure constants between the Lusztig-Vogan modules.
We give sufficient and necessary conditions for Bergman-type Toeplitz operators on Dirichlet space with bounded harmonic symbols to be a compact operator. And we characterize commuting Bergman-type Toeplitz operators with harmonic symbols on Dirichlet space.
Let p1, p2, p3 ∈ Z\{0, ±1} and e1, e2, e3 be the standard basis of unit column vectors in R3. The self-affine measure μM,D associated with an expanding matrix M=diag[p1, p2, p3] and a digit set D={0, e1, e2, e3} is supported on the spatial Sierpinski gasket T (M, D). It is known that the finiteness or infiniteness of orthogonal exponentials in the corresponding Hilbert space L2(μM,D) has been solved completely. In the finite case, it is conjectured that the cardinality of orthogonal exponentials in L2(μM,D) is at most "4", where the number 4 is the best upper bound. That is, all the four-element orthogonal exponentials are the maximum. In the present paper, we construct a class of the five-element orthogonal exponentials in the Hilbert space L2(μM,D), which shows that the above conjecture is false.
Let R be a prime ring containing a nontrivial idempotent P. Suppose C ∈ R satisfies C=PC. It is shown that an additive map Δ:R → R is derivable at C, that is, Δ(AB)=Δ(A)B + AΔ(B) for every A, B ∈ R with AB=C if and only if there exists a derivation δ:R → R such that Δ(A)=δ(A) + Δ(I)A for all A ∈ R. Similar results are obtained for von Neumann algebras with no central abelian projections. As its application, we obtain that, if nonzero operator C ∈ B(X) such that ran(C) or ker(C) is complementary in X, then an additive map Δ:B(X) → B(X) is derivable at C if and only if it is a derivation. In particular, we show that an additive map from a factor von Neumann algebra into itself is derivable at an arbitrary but fixed nonzero operator if and only if it is a derivation.
In this paper, the central BMO spaces with variable exponent are introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on variable Lebesgue spaces. The boundedness of vector-valued commutators on Herz spaces with variable exponent are also considered.
Due to their potential applications in multiplexing techniques, superframes (also called vector-valued frames) and subspace frames have interested many mathematicians and engineering specialists. A weak bi-frame is a generalization of a bi-frame in a Hilbert space. This paper addresses vector-valued subspace weak Gabor bi-frames (WGBFs) on periodic subsets of the real line, that is, WGBFs for L2(S, CL) with S being periodic subsets of R. Using Zak transform matrix method, we obtain a characterization of WGBFs, which reduces constructing WGBFs to designing Zak transform matrices of finite order; present an example theorem of WGBFs; and derive a density theorem for WGBFs.
We introduce the multilinear Marcinkiewicz integrals with nonsmooth kernel and get their the weighted bounds from Lp1(ω1)×…×Lpm(ωm) to Lp(νω) with p1, …, pm ∈ (1, ∞), 1/p1 + … + 1/pm=1/p and ω=(ω1, …, ωm) a multiple AP weights, where P=(p1, …, pm) and νω=∏k=1m ωkp/pk.
This paper use the weak convergence theorem and probability inequalities of NA sequence, we proved a general result of precise asymptotics in complete moment convergence for NA sequence, improvement and promotion of the exsting results.
We consider the following quasilinear Schrödinger-Poisson system
where f is C1, superlinear and subcritical nonlinearity, V is bounded positive potential. By using the method of perturbation, we prove the system has non-trivial solutions, positive solutions, negative solutions and sign-changing solutions.
Complete moment convergence for product sums of sequence of random variables are discussed by utilizing the method of Wang et al. The sufficient conditions of complete moment convergence for product sums of sequence of random variables are obtained.
