In this paper, we discuss the conditions for a center for the generalized Liénard system \[ \frac {{\rm d}x}{{\rm d}t}=\varphi (y)-F(x), \qquad \frac {{\rm d}y}{{\rm d}t}=-g(x), \] or \[ \frac {{\rm d}x}{{\rm d}t}=\psi (y), \qquad \frac {{\rm dy}}{{\rm d}t}= -f(x)h(y)-g(x),
\] with $f(x)$, $g(x)$, $\varphi (y)$, $\psi (y)$, $h(y)\: \mathbb R\rightarrow \mathbb R$, $F(x)=\int _0^xf(x)\mathrm{d}x$, and $xg(x)>0$ for $x\ne 0$. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].
In this work, we study the properties of central paths, defined with res\-pect to a large class of penalty and barrier functions, for convex semidefinite programs. The type of programs studied here is characterized by the minimization of a smooth and convex objective function subject to a linear matrix inequality constraint. So, it is a particular case of convex programming with conic constraints. The studied class of functions consists of spectrally defined functions induced by penalty or barrier maps defined over the real nonnegative numbers. We prove the convergence of the (primal, dual and primal-dual) central path toward a (primal, dual, primal-dual, respectively) solution of our problem. Finally, we prove the global existence of Cauchy trajectories in our context and we recall its relation with primal central path when linear semidefinite programs are considered. Some illustrative examples are shown at the end of this paper.
Crack propagation in anisotropic materials is a persistent problem. A general concept to predict crack growth is the energy principle: A crack can only grow, if energy is released. We study the change of potential energy caused by a propagating crack in a fully three-dimensional solid consisting of an anisotropic material. Based on methods of asymptotic analysis (method of matched asymptotic expansions) we give a formula for the decrease in potential energy if a smooth inner crack grows along a small crack extension.
We characterize Banach lattices E and F on which the adjoint of each operator from E into F which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if E and F are two Banach lattices then each order DunfordPettis and weak Dunford-Pettis operator T from E into F has an adjoint Dunford-Pettis operator T ′ from F ′ into E ′ if, and only if, the norm of E ′ is order continuous or F ′ has the Schur property. As a consequence we show that, if E and F are two Banach lattices such that E or F has the Dunford-Pettis property, then each order Dunford-Pettis operator T from E into F has an adjoint T ′ : F ′ → E ′ which is Dunford-Pettis if, and only if, the norm of E ′ is order continuous or F ′ has the Schur property.
Let $X$ denote a specific space of the class of $X_{\alpha ,p}$ Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily $\ell _p$ Banach spaces. We show that for $p>1$ the Banach space $X$ contains asymptotically isometric copies of $\ell _{p}$. It is known that any member of the class is a dual space. We show that the predual of $X$ contains isometric copies of $\ell _q$ where $\frac{1}{p}+\frac{1}{q}=1$. For $p=1$ it is known that the predual of the Banach space $X$ contains asymptotically isometric copies of $c_0$. Here we give a direct proof of the known result that $X$ contains asymptotically isometric copies of $\ell _1$.
Policy makers and researchers require raw data collected from agencies and companies for their analysis. Nevertheless, any transmission of data to third parties should satisfy some privacy requirements in order to avoid the disclosure of sensitive information. The areas of privacy preserving data mining and statistical disclosure control develop mechanisms for ensuring data privacy. Masking methods are one of such mechanisms. With them, third parties can do computations with a limited risk of disclosure. Disclosure risk and information loss measures have been developed in order to evaluate in which extent data is protected and in which extent data is perturbated. Most of the information loss measures currently existing in the literature are general purpose ones (i. e., not oriented to a particular application). In this work we develop cluster specific information loss measures (for fuzzy clustering). For this purpose we study how to compare the results of fuzzy clustering. I. e., how to compare fuzzy clusters.
Let $E=\bigcup _{n=1}^{\infty }I_{n}$ be the union of infinitely many disjoint closed intervals where $I_{n}=[a_{n}$, $b_{n}]$, $0<a_{1}<b_{1}<a_{2}<b_{2}<\dots <b_{n}<\dots $, $\lim _{n\rightarrow \infty }b_{n}=\infty .$ Let $\alpha (t)$ be a nonnegative function and $\{\lambda _{n}\}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\{t^{\lambda _{n}}\log ^{m_{n}}t\}$ in $C_{0}(E)$ is obtained where $C_{0}(E)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f(t){\rm e}^{-\alpha (t)}$ vanishing at infinity.