The authors consider the classes of the univalent functions denoted by SH(β), SP and SP(α, β). On these classes, the univalence conditions for a general integral operator are studied.
Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. Let h(z) ≢ 0 and ∞ be a meromorphic function on D. Assume that the following two conditions hold for every f ∈ F: $\begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $ Then F is normal on D.
The authors mainly concern the set U f of c ∈ ℂ such that the power deformation $z(\frac{{f(z)}}{z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a 2 z 2 + … in the unit disk. It is shown that U f is a compact, polynomially convex subset of the complex plane ℂ unless f is the identity function. In particular, the interior of U f is simply connected. This fact enables us to apply various versions of the λ-lemma for the holomorphic family $z(\frac{{f(z)}}{z})^c $ of injections parametrized over the interior of U f. The necessary or sufficient conditions for U f to contain 0 or 1 as an interior point are also given.
A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate solutions obtained by the numerical algorithm developed by using nonpolynomial quintic spline functions are better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.
The authors give an upper bound of the essential norm of a composition operator on H 2(B n), which involves the counting function in the higher dimensional value distribution theory defined by S. S. Chern. A criterion is also given to assure that the composition operator on H 2( Bn) is bounded or compact.
The smoothness of the solutions to the full Landau equation for Fermi-Dirac particles is investigated. It is shown that the classical solutions near equilibrium to the Landau-Fermi-Dirac equation have a regularizing effects in all variables (time, space and velocity), that is, they become immediately smooth with respect to all variables.
Let f(z) be a holomorphic Hecke eigencuspform of weight k for the full modular group. Let λ f(n) be the nth normalized Fourier coefficient of f(z). Suppose that L(sym2 f, s) is the symmetric square L-function associated with f(z), and $\lambda _{sym^2 f} (n)$ (n) denotes the nth coefficient L(sym2 f, s). In this paper, it is proved that $\sum\limits_{n \leqslant x} {\lambda _{sym^2 f}^4 (n)} = xP2(\log x) + O(x^{\frac{{79}}{{81}} + \varepsilon } ),$, where P 2(t) is a polynomial in t of degree 2. Similarly, it is obtained that $\sum\limits_{n \leqslant x} {\lambda _f^4 (n^2 )} = x\tilde P2(\log x) + O(x^{\frac{{79}}{{81}} + \varepsilon } ),$, where $\tilde P_2 (t)$ is a polynomial in t of degree 2.
The authors extend the notion of statistical structure from Riemannian geometry to the general framework of path spaces endowed with a nonlinear connection and a generalized metric. Two particular cases of statistical data are defined. The existence and uniqueness of a nonlinear connection corresponding to these classes is proved. Two Koszul tensors are introduced in accordance with the Riemannian approach. As applications, the authors treat the Finslerian (α, β)-metrics and the Beil metrics used in relativity and field theories while the support Riemannian metric is the Fisher-Rao metric of a statistical model.
The authors prove that an operator with the cellular indecomposable property has no singular points in the semi-Fredholm domain, by applying the 4 × 4 matrix model of semi-Fredholm operators due to Fang in 2004. This result fills a gap in the result of Olin and Thomson in 1984.
The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.
Consider a sequence of i.i.d. positive random variables with the underlying distribution in the domain of attraction of a stable distribution with an exponent in (1, 2]. A universal result in the almost sure limit theorem for products of partial sums is established. Our results significantly generalize and improve those on the almost sure central limit theory previously obtained by Gonchigdanzan and Rempale and by Gonchigdanzan. In a sense, our results reach the optimal form.
The author studies the L 2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λ i ≥ 0 and c 0, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.
The empirical likelihood approach is suggested to the pretest-posttest trial based on the constrains, which we construct to summarize all the given information. The author obtains a log-empirical likelihood ratio test statistic that has a standard chi-squared limiting distribution. Thus, in making inferences, there is no need to estimate variance explicitly, and inferential procedures are easier to implement. Simulation results show that the approach of this paper is more efficient compared with ANCOVA II due to the sufficient and appropriate use of information.
Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ψ ∈ L 2(ℝ d), such that the set $\left\{ {\left| {\det A} \right|^{\frac{n}{2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\}$ forms a Parseval frame for L 2(ℝ d). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of dψ⌃ is an A-dilation Parseval frame wavelet whenever ψ is an A-dilation Parseval frame wavelet, where ψ⌃ denotes the Fourier transform of ψ. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L 2(ℝ d) is discussed.