For a famous cubic system given by James and Lloyd, there exist some sufficient conditions such that the system has eight limit cycles. In this paper, we try to derive by computers the necessary and sufficient conditions for this system to have eight limit cycles. In order to find the symbolic real solutions to semi-algebraic systems where polynomials are Lyapunov quantities, we transform the equations into triangular systems by pseudo-division, locate the real solutions of the last equation and verify the inequalities by the Budan-Fourier theorem. The necessary and sufficient conditions for the system to have eight limit cycles are given under a reasonable limitation.

Let $M$ be an $n$-dimensional complete non-compact hypersurface in ${\mathbb R}^{n+1}$ and assume that its mean curvature lies between two positive numbers. Denote by $\Delta$ and $A$ the Laplacian operator and the second fundamental form of $M$, respectively. In this paper, we show that if $3\leq n\leq 5$ and if ${\rm Ind}(\Delta +|A|^2)$ is finite, then $M$ has finitely many ends. We also show that if $2\leq n\leq 5$ and if ${\rm Ind }(\Delta +|A|^2)=0$, then $M$ has only one end.

Let $g:B\to \mathbb C^1$ be a holomorphic map of the unit ball $B$. We study the integral operators $$ T_gf(z)=\int_0^1f(tz)\Re g(tz)\frac{dt}{t}; \ \ L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\qquad z\in B. $$ The boundedness and compactness of the operators $T_g$ and $L_g$ on the Hardy space $H^2$ in the unit ball are discussed in this paper.

We set $D$ to be a simply connected domain and we consider exhaustion function spaces, $X_\infty(D)$ with the projective topology. We show that the natural topology on the topological dual of $X_\infty(D)$, $(X_\infty(D))'$, is the inductive topology. As a main application we assume that $D$ has a Jordan rectifiable boundary $\partial D$, and $M\subset \partial D$ to be an open analytic arc whose Lebesgue measure satisfies $0<m(M)<m(\partial D)$. We prove a result for the dual of ${\mathcal{N}\mathcal{H}}^1_M(D)$, which is the class of holomorphic functions in $D$ which are represented by Carleman formulae on $M\subset \partial{D}$. Furthermore we show that the Cauchy Integral associated to $f \in \near$ is an element of $\near$. Lastly, we solve an extremal problem for the dual of ${\mathcal{N}\mathcal{H}}^1_M(D)$.

Suppose that $\frak A$ is a $C^*$-algebra acting on a Hilbert space $\frak K$, and $\varphi, \psi$ are mappings from $\frak A$ into $B(\frak K)$ which are not assumed to be necessarily linear or continuous. A $(\varphi, \psi)$-derivation is a linear mapping $d: \frak A \to B(\frak K)$ such that $$d(ab)=\varphi(a)d(b)+d(a)\psi(b)\quad (a,b\in \frak A).$$ We prove that if $\varphi$ is a multiplicative (not necessarily linear)\ $*$-mapping, then every $*$-$(\varphi,\varphi)$-derivation is automatically continuous. Using this fact, we show that every $*$-$(\varphi,\psi)$-derivation $d$ from $\frak A$ into $B(\frak K)$ is continuous if and only if the $*$-mappings $\varphi$ and $\psi$ are left and right $d$-continuous, respectively.

Let $f(z,t)$ be a non-normalized subordination chain and assume that $f(\cdot,t)$ is $K$-quasiregular on $B^n$ for $t\in [0,\alpha]$. In this paper we obtain a sufficient condition for $f(\cdot,0)$ to be extended to a quasiconformal homeomorphism of $\overline{\mathbb{R}}^{2n}$ onto $\overline{\mathbb{R}}^{2n}$. Finally we obtain certain applications of this result. One of these applications can be considered the asymptotical case of the $n$-dimensional version of the well known quasiconformal extension result due to Ahlfors and Becker.

We characterize the orbits of the principal congruence subgroup $\Gamma(n)$ of $GL_2(\bbZ)$ acting by fractional linear transformations on the real line. The characterization is in terms of congruence properties $\pmod{n}$ of the numerators and denominators of the convergents of some number in a given orbit.

The improvement of the inverse estimation of approximation theory by trigonometric polynomials in the weighted Lebesgue spaces was obtained and its application in the weighted Smirnov spaces was considered.

We give a multiplicative version of Kostrikin's theorem on $2$-Engel Lie algebras.

Nous donnons une version multiplicative du théorème de Kostrikin sur les algèbres de Lie $2$-Engel.

Let ${\cal S}$ be a left compactly cancellative foundation semigroup with identity $e$ and $M_a({\cal S})$ be its semigroup algebra. In this paper, we give a characterization for the existence of an inner invariant extension of $\delta_e$ from $C_b({\cal S})$ to a mean on $L^\infty({\cal S},M_a({\cal S}))$ in terms of asymptotically central bounded approximate identities in $M_a({\cal S})$. We also consider topological inner invariant means on $L^\infty({\cal S},M_a({\cal S}))$ to study strict inner amenability of $M_a({\cal S})$ and their relation with strict inner amenability of ${\cal S}$.

In this paper, we present a numerical solution of one-dimensional Korteweg-de Vries equation with variant boundary conditions by the Fourier pseudospectral method. Four test problem with known exact solutions were studied to demonstrate the accuracy of the present method. An artificial viscosity was proposed to improve the accuracy of the numerical scheme. The obtained results were compared with the exact solution of each problem and found to be in good agreement with each other.

We show that whenever densely similar operators on a Banach spaces, their approximate point spectra must have non-empty intersection. Also, we introduce the class $\mathcal A$ that consists of those operators for which the Goldberg spectrum coincides with the right essential spectrum. We study spectral properties of quasisimilar operators satisfying Bishop's property $(\beta)$ in the class $\mathcal A$. Finally, as an application to the class $\mathcal N$ that consists of those operators $T$ whose range $R(T)$ is contained in the linear span of finite number of orbits of $T$, we show that any two quasisimilar operators such that are in $\mathcal N$ and satisfying property $(\beta)$ must have the same approximate point spectrum.

Let $K$ be a $p$-adic field and let $f$ be a $K$-analytic function on an open and compact subset of $K^3$. Let $R$ be the valuation ring of $K$ and let $\chi$ be an arbitrary character of $R^{\times}$. Let $Z_{f,\chi}(s)$ be Igusa's $p$-adic zeta function. In this paper, we prove a vanishing result for candidate poles of $Z_{f,\chi}(s)$. This result implies that $Z_{f,\chi}(s)$ has no pole with real part less than $-1$ if $f$ has no point of multiplicity 2.

Let $X$ and $Y$ be linear spaces. It is shown that for a fixed positive integer $n\geq2,$ if a mapping $Q:X\to Y$ satisfies the following functional equation \begin{equation}\label{A} \sum_{i=1}^{n}Q(z-x_{i})=\frac{1}{n}\sum_{\substack{1\le i,j\le n\\ j<i}}Q(x_{i}-x_{j})+nQ\bigg(z-\frac{1}{n}\sum_{i=1}^{n}x_{i}\bigg) \end{equation} for all $z, x_1, \dots, x_n\in X,$ then the mapping $Q: X \to Y$ is a {\it quadratic mapping of Apollonius type} and a quadratic mapping. We moreover prove the Hyers-Ulam stability of the functional equation $(\ref{A})$ in Banach spaces.