Landau’s theorem for log-p-harmonic mappings

The main aim of this paper is to prove the existence of Landau-Bloch constant for log-p-harmonic mappings.     

A Liouville type theorem for polyharmonic elliptic systems

In this paper, we consider the polyharmonic system m(−Δ)U=Vq,m(−Δ)V=Up in RN, for m>1, N>2m, with p?1, q?1, but not both equal to 1, where m(−Δ) is the polyharmonic operator. Set α=2m(q+1)/(pq−1), β=2m(p+1)/(pq−1), for α,β∈[(N−2)/2,N−2m), we prove the nonexistence of positive solutions.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

某类双调和映射的Landau型定理

令f(z)=h(z)+g(z)为开单位圆盘U内的调和映射.本文首先建立了有界调和映射的精确系数估计,其次得到了满足有界性条件|h(z)|+|g(z)|≤M的正规化调和映射的精确系数估计.作为其应用,得到了双调和映射的一些Landau型定理.这些结果改进和推广了早期作者的相关结果.     

On lengths,areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.     

Landau’s theorem for functions with logharmonic Laplacian

In this paper, we show the existence of Landau constant for functions with logharmonic Laplacian of the form F(z) = ∣z∣²L(z) + K(z), ∣z∣ < 1, where L is logharmonic and K is harmonic. Moreover, the problem of minimizing the area is solved     

Fractional Noether’s theorem in the Riesz-Caputo sense

We prove a Noether’s theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.     

Application of Rouche’s theorem for MP filter design

The method for the minimum-phase (MP) finite impulse response (FIR) filter design, based on Rouche’s theorem from complex analysis is presented here. The filter is designed directly from a given specification. The method uses the cosine filters and the sharpening technique resulting in a multiplierless filter.     

A degeneracy theorem for meromorphic mappings with truncated multiplicities

In this article, we prove a degeneracy theorem for three linearly non-degenerate meromorphic mappings from C n into P N (C), sharing 2N + 2 hyperplanes in general position, counted with multiplicities truncated by 2.     

A general Isserlis theorem for mixed-Gaussian random variables

This work generalizes a widely used result derived by L. Isserlis for the expectations of products of jointly Gaussian random variables by extending it to include mixed-Gaussian random variables.     

Coincidence theorem for admissible set-valued mappings and its applications in FC-spaces

A new coincidence theorem for admissible set-valued mappings is proved in FC-spaces with a more general convexity structure. As applications, an abstract variational inequality, a KKM type theorem and a fixed point theorem are obtained. Our results generalize and improve the corresponding results in the literature.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

A uniqueness theorem for meromorphic mappings in several complex variables

In this paper, we give a uniqueness theorem for meromorphic mappings from ?n into ?N(?) with rank≥μregardless of multiplicities.     

A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed   relative residual error tolerance converges Q$Q$-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance.     

A generalization of Sperner’s theorem and an application to graph orientations

A generalization of Sperner’s theorem is established: For a multifamily M={Y₁,…,Y_p} of subsets of {1,…,n} in which the repetition of subsets is allowed, a sharp lower bound for the number φ(M) of ordered pairs (i,j) satisfying i≠j and Y_i⊆Y_j is determined. As an application, the minimum average distance of orientations of complete bipartite graphs is determined.     

On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes

Let {X(t),t≥0}$\{X\left(t\right),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004)  [23], which we refer to as Piterbarg’s max-discretisation theorem gives the joint asymptotic behaviour (T→∞$T\to \infty$) of the continuous time maximum M(T)=_maxt∈[0,T]X(t)$M\left(T\right)={max}_{t\in [0,T]}X\left(t\right)$, and the maximum M^δ(T)=_maxt∈R(δ)X(t)$M^{\delta }\left(T\right)={max}_{t\in \mathfrak{R}\left(\delta \right)}X\left(t\right)$, with R(δ)⊂[0,T]$\mathfrak{R}\left(\delta \right)\subset [0,T]$ a uniform grid of points of distance δ=δ(T)$\delta =\delta \left(T\right)$. Under some asymptotic restrictions on the correlation function Piterbarg’s max-discretisation theorem shows that for the limit result it is important to know the speed δ(T)$\delta \left(T\right)$ approaches 0 as T→∞$T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.     

Positive entire solutions for a class of singular nonlinear polyharmonic equations on

In the paper, the existence of positive radially symmetric entire solutions for a class of singular nonlinear polyharmonic equations on is proved. Some properties are obtained. The results of this paper are generalizations of those in Refs. [Math. Z. 67 (1957) 32; Arch. Math. 9 (1958) 308; Proc. R. Soc. Edinburg 82A (1979) 189; Hiroshima Math. J. 17 (1989) 13].     

The computational content of Walras’ existence theorem

In [Y. Tanaka, Undecidability of the Uzawa equivalence theorem and LLPO, Appl. Math. Comput. 201 (2008) 378-383] Yasuhito Tanaka showed that Walras’ existence theorem implies the nonconstructive lesser limited principle of omniscience (LLPO); it follows that Walras’ existence theorem does not admit a constructive proof. We give a constructive proof of an approximate version of Walras’ existence theorem from which the full theorem can be recovered with an application of LLPO. We then push Uzawa’s equivalence theorem to the level of approximate solutions, before considering economies with at most one equilibrium.     

On the existence of radial positive entire solutions for polyharmonic systems

Two-dimensional nonlinear, polyharmonic systems of the type