ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR

In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.

Weakly R-KKM mappings-intersection theorems and minimax inequalities in topological spaces

In this paper, we introduce the concepts of weakly R-KKM mappings, R- convex and R-β-quasiconvex in general topological spaces without any convex structure. Relating to these, we obtain an extension to general topological spaces of Fan＇s matching theorem, namely that Lemma 1.2 in this paper. On this basis, two intersection theorems are proved in topological spaces. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved in topological spaces. Our results generalize and improve the corresponding results in the literature.

ON DEGENERATE WEAKLY (K_1,K_2)-QUASIREGULAR MAPPINGS

The authors first give the definition of degenerate weakly （K1,K2）-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.

Weak WT_2-class of differential forms and weakly A-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.

Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property

In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

HARMONIC MAPPINGS OF THE HEXAGASKET TO THE CIRCLE

Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].

FIXED POINT INDEX THEORY FOR WEAKLY INWARD MAPPINGS

In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes ns to deal with these kinds of mappings more easily. As obvious applications, some results in [3],[5],[7],[9],[10] are deepened and extended.

FIXED POINTS FOR WEAKLY INWARD MAPPINGS IN BANACH SPACES (Ⅱ)

S. Hu and Y. Sun[1] defined the fixed point index for weakly inward mappings, investigated its properties and studied fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we further investigate boundary conditions, under which the fixed point index for i（A, Ω, p） is equal to nonzero, where i（A, Ω, p） is the completely continuous and weakly inward mapping. Correspondingly, we can obtain many new fixed point theorems of the completely continuous and weakly inward mapping, which generalize some famous theorems such as Rothe＇s theorem, Altman＇s theorem, Petryshyn＇s theorem etc. in the case of weakly inward mappings. In addition, our conclusions extend the famous fixed point theorem of cone expansion and compression to the case of weakly inward mappings. Moreover, the main results contain and generalize the corresponding results in the recent work[2].

Weak harmonic signal detection method in chaotic interference based on extended Kalman filter

The traditional methods of weak harmonic signal detection under strong chaotic interference often suffer from high computational complexity and poor performance. In this paper, an Extended Kalman Filter (EKF) based detection method is proposed for the detection of weak harmonic signal. The EKF method avoids matrix inversion by iterating measurement equation and state equation, which simultaneously improves the robustness and reduces the complexity. Compared with the existing detection methods, the proposed method has the following advantages: 1) it has better performance than the neural network method;2) it has similar performance with the optimal filtering method, but with lower computational complexity;3) it is more robust compared with the optimal filtering method.

UNIQUENESS FOR SOLUTIONS OF NONHOMOGENEOUS A -HARMONIC EQUATIONS WITH VERY WEAK BOUNDARY VALUES

A uniqueness result for nonhomogeneous quasilinear elliptic partial differential equations with very weak boundary values was proved.

Heat flow of harmonic maps from noncompact manifolds

The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.

Harmonicity Spectrum

Perceiving harmonic information (especially weak harmonic information) in time series has important scientific and engineering significance. Fourier spectrum and time-frequency spectrum are commonly used tools for perceiving harmonic information, but they are often ineffective in perceiving weak harmonic signals because they are based on energy or amplitude analysis. Based on the theory of Normal time-frequency transform (NTFT) and complex correlation coefficient, a new type of spectrum, the Harmonicity Spectrum (HS), is developed to perceive harmonic information in time series. HS is based on the degree of signal harmony rather than energy or amplitude analysis, and can therefore perceive very weak harmonic information in signals sensitively. Simulation examples show that HS can detect harmonic information that cannot be detected by Fourier spectrum or time-frequency spectrum. Acoustic data analysis shows that HS has better resolution than traditional LOFAR spectrum.

ODEs IN BANACH SPACES AND FIXED POINTS OF WEAKLY INWARD MAPS

This paper is an attempt to investigate systematically fixed points of weakly inward maps by using some basic results from differential equations in Banach spaces. By investigating the Poincare operators for such differential equations, we establish a fixed point index theory for two classes of weakly inward maps.

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

In this paper, we explore the linear combinations of right half-plane mappings and vertical strip mappings. We demonstrate that the combinations of these harmonic mappings are convex in the vertical direction provided they are locally univalent and sense-preserving. Furthermore, we extend this analysis to a more general case by setting specific conditions. Additionally, we take some common parameters such as as the dilatation of these harmonic mappings, and prove the sufficient conditions that their combinations are locally univalent and convex in the vertical direction. Several examples are constructed by the Mathematica software to demonstrate our main results.

Common Fixed Point Theorems for Compatible and Weakly Compatible Maps in G-Metric Spaces

In this paper first we prove common fixed point theorems for compatible and weakly compatible maps. Secondly, we prove common fixed point theorems for weakly compatible maps along with property (E.A.) and (CLRg) property respectively.

ON VERY WEAK SOLUTIONS OF A-HARMONICEQUATION WITH VERY WEAK BOUNDARYVALUES

In this paper,the following result is given by using Hodge decomposition: There exists r(0) = r(0)(n,p,a,b), such that if u is an element of W-loc(1,r)(Omega) is a very weak solution of (1.1),with C max{1,p - 1} < r < p and u is an element of W-0(1,r)(Omega;partial derivativeOmega\E) where E subset of partial derivativeOmega is a closed set and small in an appropriate capacity sense, then u = 0, a.e. in Omega provided that r(0) < r < p.

f-Harmonic maps of doubly warped product manifolds

In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×（μ,λ） N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.

EXISTENCE AND NONUNIQUENESS FOR THE HEAT FLOW OF INHOMOGENEOUS HARMONIC MAP EQUATION

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On the Relativistic Harmonic Oscillator

The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) &#8594;1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.

Some results on P-harmonic maps and exponentially harmonic maps between Finsler manifolds

This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].