Let H be a complex infinite dimensional Hilbert space. B(H) denotes the algebra of all bounded linear operators on H. In this paper, we characterize the operators in B(H) for which f(T) satisfies Weyl's theorem, where f denotes the analytic function on some neighbourhood of the spectrum of T. Also, the relationships between Weyl's theorem for functions of operators and the stability of Weyl's theorem are explored.
We introduce a new algorithm for solving pseudomonotone variational inequality problems and fixed point problems by using the subgradient extragradient method. A weak convergence theorem of proposed algorithm is obtained under some suitable assumptions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.
Let Z and N be the set of all integers and positive integers, respectively. M_{m} (Z) be the set of m×m matrix over Z where m ∈ N. In this paper, by using the result of Fermat's Last Theorem, we show that the following second-order matrix equation has only trivial solutions:X^{n} + Y^{n}=λ^{n}I (λ ∈ Z, λ ≠ 0, X, Y ∈ M_{2}(Z)), where X has an eigenvalue that is a rational number and n ∈ N, n ≥ 3; By using the result of primitive divisors, we show that the second-order matrix equation X^{n} +Y^{n}=(±1)^{n}I (n ∈ N, n ≥ 3, X, Y ∈ M_{2}(Z)) has nontrivial solutions if and only if n=4 or gcd(n,6)=1 and all nontrivial solutions are given; By constructing integer matrix, we show that the following matrix equation has an infinite number of nontrivial solutions:∀_{n} ∈ N, X^{n} + Y^{n}=λ^{n}I (λ ∈ Z, λ ≠ 0, X, Y ∈ M_{n}(Z)); X^{3} + Y^{3}=λ^{3}I (λ ∈ Z, λ ≠ 0, m ∈ N, m ≥ 2, X, Y ∈ M_{m}(Z)).
In this paper, the dominating set of the Bergman space in the unit ball are characterized in terms of the pseudohyperbolic metric ball. Our method is to generalize Luecking's three key lemmas on the unit disc to the unit ball. We then apply those three lemmas to give a complete description of the dominating set of the Bergman space on the unit ball.
The present paper is devoted to the study of the so-called bivariate partial theta function which is first introduced by the author and contains the classical partial theta function as a special case. We focus on its possible product formula, recurrence relation, and series expansion and so on. As main results, we establish a product formula of any two bivariate partial theta functions. It is a generalization of Andrews-Warnaar's product formula for the classical partial theta functions. At the same time, we obtain a second order recurrence relation satisfied by this bivariate partial theta function. Finally, we present two series expansions of the bivariate partial theta function θ(q, x; ab) with respect to {θ(q, axq^{n}; b)|n ≥ 0} and {θ(q, xq^{n}; b)|n ≥ 0}, respectively. As further applications of these results, we also find a product formula of two _{3}φ_{2} series and a ternary representation of the bivariate partial theta function.
Periodicity is one of the most common factor in time series analysis. In the time series analysis of discrete-valued response variables, we use maximum likelihood estimation with penalty to establish a consistent estimator of the unknown period. Given the estimator of the period, we take B-spline to approximate the trend term and the additive function, and at the same time obtain the √n-consistent estimator of the periodic term and the initial estimators of the trend term and the additive function. Then based on the idea of back-fitting, we establish the improved estimators of the trend term and additive function, and the asymptotic normality and efficiency of them are also demonstrated. Simulation experiments and empirical analysis confirm that our proposed method performs well for the finite sample.
We define unstable local entropies for arbitrary Borel probability measures in partially hyperbolic systems. In order to characterize the multifractal spectrum of unstable local entropies, we introduce the concept of unstable (q, μ)-entropy, provide some basic properties of (q, μ)-entropy and establish a relation formula between the Bowen unstable entropy of the multifractal spectrum and the (q, μ)-entropy.
In this paper, the quadratic numerical radius inequalities of off-diagonal block operator matrixwhose entries are bounded operators on the Hilbert space is studied. According to the classical convexity inequalities of non-negative real numbers, the quadratic numerical radius inequalities of A is generalized.
For a graph G=(V (G), E(G)), if a mapping ?:E(G) → {1, 2,..., k} such that ?(e_{1}) =?(e_{2}) for any adjacent edges e_{1}, e_{2}, and there are no bicolored cycles in G, then ? is called an acyclic edge coloring of G. For a list assignment L={L(e)|e ∈ V (E)}, if there exists an acyclic coloring ? such that ?(e) ∈ L(e) for each e ∈ E(G), then ? is called an acyclic L-list coloring of G. If for any L with|L(e)| ≥ k for each e ∈ E(G), there exists an acyclic L-list coloring of G, then we say G is acyclically k-edge choosable. The minimum integer k making G is acyclically k-edge choosable is called the acyclic list chromatic index of G, denoted by a_{l}'(G). In this paper, it is proved that for a connected graph G with maximum degree Δ ≤ 4 and|E(G)| ≤ 2|V(G)|-1, it follows that a_{l}'(G) ≤ 6, which extends the result of Basavaraju and Chandran[J. Graph Theory, 2009, 61(3):192-209].
In this note, it is proved that the Coleman outer automorphism group of a generalized quaternion group is either 1 or an elementary abelian 2-group by using the projection limit property of the group.
We study the following polyharmonic Dirichlet problems in a punctured unit ball where B is the unit ball in R^{N}, ν is the unit outward normal vector of ∂B, N > 2k, k ≥ 2. Under certain assumptions on f, we use the moving plane method to show radial symmetry of any singular positive solution provided that 0 is a nonremovable singularity point. As an application, we can obtain nonexistence of positive solutions for a critical Dirichlet biharmonic problem.
Under the natural assumption of saturation effect, this paper proves the global existence and uniform boundedness of the classical solutions to the 3D initial boundary value problem for a double chemotaxis-Stokes system. Due to the strong nonlinearity in the system, the method developed in this paper can be applied to the related models for the coral spawning, which have attracted much attention recently.
The iterated function system with two-element digit set is the simplest case and the most important case in the study of spectrality or non-spectrality of self- affine measures. The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable. However, the higher dimensional analogue, especially the two-dimensional case has not been solved completely. Also, there is a conjecture to illustrate that in the plane, the remaining cases correspond to nonspectrality of self-affine measures. Motivated by this problem, we provide in this paper some non-spectral conditions for the planar self-affine measures with two-element digit set. Under one of the conditions, we determine the maximal cardinality of orthogonal exponentials. An application of this result and the validity of the conditions are also presented.