In this paper, by Nevanlinna theory of the value distribution of meromophic functions, we investigate the problem of the existence of solutions of some types of systems of complex higher-order q-shift difference equations, and obtain some results. Improvements and extensions of such results are presented in this paper. Some examples show that our results are precise.
In this paper, we prove that the maximal abelian ad-diagonalizable subalgebra of the classical Lie algebras of type A consisiting of n×n matrices over the quaternions always has dimension n-1 and it is conjugated with the algebra of real diagonal matrices of trace zero.
The paper deals with the n-dimensional diffraction problem for quasilinear elliptic system on a bounded domain, where the coefficients of the equations are allowed to be discontinuous on the interfaces and the interfaces are allowed to intersect with the outer boundary of the domain. By constructing an approximation diffraction problems with interfaces which do not intersect with the outer boundary, using various estimates and approximation method, and investigating the regularity of the weak solutions, we get the existence of solutions of the problem. An application is given to the Lotka-Volterra cooperation model with two cooperating species.
In this paper, we consider the following problem -Δu=uN+2/N-2+∈f(x)u, x∈RN, u∈D1,2(RN), u(x)→0, |x|→∞, where ∈ is a positive constant, f(x)∈L∞(RN), N>3. Used perturbation method, we prove that under some conditions on f(x), there exists a ∈0, if 0<ε<ε0, such that there are many solutions for above problem.
In this paper, we investigate the uniqueness of algebroid functions concerning CM shared values, extend a result of Gundersen$^{[10]}$ about the uniqueness of meromorphic functions to algebroid functions.
In this paper, the decay property of solutions for the p-system with general linear dissipation are considered. The main methods the author used are the Fourier analysis method in combination with the fundamental solution used in the Cauchy problem for partial differential equations. Under some suitable conditions on the initial data and the dissipative coefficients, the author gives a detailed description for the decay property of the solutions for the p-system with general linear dissipation. In addition, the discussion can be extended to more general 2×2 hyperbolic equations with the same kind of dissipation.
Let H2 be the Heisenberg group in R5 and D be a bracket generating left invariant distribution with a Lorentzian metric. Let reachable sets I+(0)(resp.J+(0)) be the chronological(resp.causal) future of origin. In this paper, we prove that I+(0) is open and J+(0) is closed for Heisenberg group H2 with sub-Lorentzian metric, which are similar results as in Lorentzian manifolds.
A Hilbert space operator T belongs to k-quasi-*-class A if T*k(|T2|-|T*|2)Tk≥0. The famous Fuglede-Putnam's theorem is as follows: the operator equation AX=XB implies A*X=XB* when A and B are normal operators. In this paper, firstly we prove that if T is a contraction of k-quasi-*-class A operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D=T*k(|T2|-|T*|2)Tk is a strongly stable contraction; secondly we prove that k-quasi-*-class A operators are not supercyclic; at last we show that if X is a Hilbert-Schmidt operator, A and (B*)-1 are k-quasi-*-class A operators such that AX=XB, then A*X=XB*.
In this present paper, we study the long-time behavior for the 2D non-autonomous Navier-Stokes equation with damping that governs the motion of an incompressible fluid. Under suitable assumptions on the non-autonomous external force f(t, x) and initial data uτ, we show the existence of uniform attractors for the 2D non-autonomous Navier-Stokes equation with linear damping by weak continuous method.
In this paper, for isothermal gas, we will think about the existence of classical (Theorem 1.1) and weak (Theorem 1.2) solutions of steady state relativistic Euler equations with spherical symmetry structure, respectively. Theorem 1.1 shows that, by solving two ordinary differential equations and according to the comparison of Mach number M and 1, we establish the existence of classical solutions. Moreover, Theorem 1.2 shows that, under a C2, α, α∈(0,1) steady perturbation of the upstream supersonic flow, we will prove the existence of steady multidimensional transonic shocks (hyperbolic-elliptic shocks) for spherically symmetric relativistic Euler equations.
In this paper, by using the way of weight function and the technique of real analysis, and introducing some parameters and some special functions to jointly characterize the constant factor, a multi-parameters Hilbert-type integral inequality is given. It's equivalent form is considered and their constant factors are proved being the best possible. At the same time, a useful multi-parameters integral inequality with the kernel of hyperbolic cotangent function is obtained.
In this paper, we research the ergodicity of jump processes on general state space, the same conclusion with markov chains of countable states is obtained.
This paper was concerned with the existence of infinitely many weak solutions for the following p&q-Laplacian equation under some appropriate assumptions {-Δpu-Δqu+|u|r-2u=γ|u|s-2u, in Ω, u=0, on ∂Ω.
In this paper, we have given two goals. Firstly we study a single Toeplitz operator Tφ with symbol φ∈L∞,1, which is on the general Dirichlet space Dp (1<p<+∞) of the annulus domain, and further conclude that Tφ is compact if and only if its Berezin transform vanishes at the boundary of the annulus. Secondly we study the Toeplitz operator Tu with symbol $u\in u∈C1(M), which is on classical Dirichlet space D2, and further give a canonical decomposition S=TS+R for some S=∑mi=1∏nj=1Tuij in Toeplitz algebra T and some R in the commutator ideal CT.
This paper addresses the generalization performance of compressed least-square regression learning algorithm. A generalization error bound of this algorithm is established by using the random projection and the theory of the covering number. The obtained results show that the compressed learning can reduce the sample error at the price of increasing the approximation error, but the increment can be controlled. In addition, by using compressed projection, the overfitting problem for the learning algorithm can be overcome to a certain extent.
Let T be a bounded linear operator on a complex Hilbert space H. T is said to be a quasi-k-paranormal operator if ||Tk+2x||||Tx||k≥||T2x||k+1 for all x∈H, where k is a positive integer. Some basic properties of quasi-k-paranormal operators are shown. In particular, quasi-k-paranormal operator is polaroid, as an application, we obtain that Weyl's theorem holds for the class of quasi-k-paranormal operators.
In this paper, we investigate asymptotic properties for realized power variations of a process given by =∫t0ΦsdBHs+ξt, where BH is a fractional Brownian motion with Hurst parameter H∈ (0,1), Φ is a process having finite q-th variation with q<1/(1-H), and ξ is a purely non-Gaussian L\'{e}vy process and is independent of BH. We present some central limit theorems (CLT) for the realized power variations in the situation that the exponent is 1/H. Some limit theorems on the law of large numbers and the CLTs for the realized threshold power variations are obtained as well.
By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta (1978), the result on the existence of a solution u(x) in Lp(Ω) of nonlinear curvature equation with nonlinear Neumann boundary value conditions, where 2N/N+1< p < +∞ and N≥1 denotes the dimension of RN, is studied. The equation discussed in this paper and the methods here are continuation and complement to the previous corresponding results. To obtain the result, some new techniques are used in this paper.
g-frames, as the generalized frames, have many properties similar to those of the frames in Hilbert spaces, but not all the properties are similar. For example, Besselian frames are equivalent to near Riesz bases, but g-Besselian frames are not equivalent to near g-Riesz bases. In this paper, we give an equivalent relation between a g-Besselian frame and a near g-Riesz basis under some conditions, and also have some results on the duality of g-Besselian frames and near g-Riesz bases. Moreover, we discuss the stability of g-Besselian frames and near g-Riesz bases respectively in Hilbert spaces.
In this paper, we consider the initial boundary value problem (IBVP) for one-dimensional compressible bipolar Navier-Stokes-Poisson (BNSP) equations with density-dependent viscosities. First, it is proved that the weak solution for general initial data exists globally in time. Then, it is shown that vacuum state must vanish within finite time. Furthermore, after the vanishing of vacuum state, the global weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time. This extends the previous results for compressible NS [14] to NSP.
In this paper, we will study the continuity of multilinear commutator generated by Marcinkiewicz operator and functions bj on Triebel-Lizorkin space, Hardy space and Herz-Hardy space, where bj belongs to Lipschiz space.
In this paper, we study complete non-compact Riemannian manifolds with radial curvature bounded from below by that of a non-compact model surface of revolution. We find a reasonable condition to ensure that this kind of manifolds are diffeomorphic to a Euclidean space if it contains enough rays starting from the base point.
In this paper, a class of quadratic transformation inequalities for zero-balanced Gaussian hypergeometric function F(a, b; a+b; x) with a, b>0 are presented.
Let k(≥2) be a positive integer, let M be a positive number, let h(z) be a holomorphic function in a domain D, h≠0, and let F be a family of meromorphic functions in D, all of whose zeros have multiplicity at least k+1, and whose poles are multiple. If, for each f ∈F, |f(z)|≥M whenever f k(z)=h(z), then F is normal in D.
The aim of this paper is to investigate the growth of transcendental meromorphic functions with two radially distributed values. The paper is closely related to some previous results due to Hayman et al.
In this paper, we establish the Lp boundedness for a class of rough parametric Marcinkiewicz integral operators associated with polynomial compound mappings, under rather weaker size conditions on the kernels, which essentially improve and generalize certain previous results. As applications, the Lp bounds of the corresponding parametric Marcinkiewicz integral operators related to area integrals and Littlewood-Paley g*λ functions are also obtained.
This paper discusses the problem of determining an unknown source which depends only on one variable for the two dimensional modified Helmholtz equation in a half infinite domain. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the Fourier regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Numerical examples show that our proposed method is effective and stable.
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