Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^{2} + y^{2} + 1=0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.

In a more recent paper, the second author has introduced a space|C_{α}|_{k} as the set of all series by absolute summable using Cesàro matrix of order α > -1. In the present paper we extend it to the absolute Nörlund space|N_{p}^{θ}|_{k} taking Nörlund matrix in place of Cesàro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.

Global Existence and Asymptotic Behavior of Solutions to a Free Boundary Problem for the 1D Viscous Radiative and Reactive Gas

In this paper, we study a free boundary problem for the 1D viscous radiative and reactive gas. We prove that for any large initial data, the problem admits a unique global generalized solution. Meanwhile, we obtain the time-asymptotic behavior of the global solutions. Our results improve and generalize the previous work.

K-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces. By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.

Optimal Time Decay of Navier-Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in H^{[N/2]+2}(R^{N}). Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by Danchin. Through our methods, we can get optimal time decay rate with initial data just small in ^{N/2-1,N/2+1} ∩ ^{N/2-1,N/2} and belong to some negative Besov space (need not to be small). Finally, combining the recent results in[25] with our methods, we only need the initial data to be small in homogeneous Besov space ^{N/2-2,N/2} ∩ ^{N/2-1} to get the optimal time decay rate in space L^{2}.

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled "Dunford-Pettis-like properties of continuous vector function spaces". In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map#br#x_{n+1}=r_{n}x_{n}(1-x_{n}), n=0, 1,...,#br#where {r_{n}} is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.

Analysis of a Shil'nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G=(V, E) be a simple graph. A function f:E → {+1, -1} is called a signed cycle domination function (SCDF) of G if ∑_{e∈E(C)}f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ'_{sc} (G)=min{∑_{e∈E}f(e)|f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ'_{sc} (G)=n.

An Affirmative Result of the Open Question on Determining Function Jumps by Spline Wavelets

We study the open question on determination of jumps for functions raised by Shi and Hu in 2009. An affirmative answer is given for the case that spline-wavelet series are used to approximate the functions.