% Guide for Authors
% This file may be used as a template for writing a paper for submission to the "U.P.B. Scientific Bulletin"
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\title{Your title}
\author{the first author}
\address{$^1$Lecturer, Dun\u area de Jos University of Gala\c
ti, Romania, e-mail: {\tt gbercu@ugal.ro}}
\author{the second}
\address{$^2$Teacher, Technical College of Nehoiu, Buz\u au County, Romania}
\author{the third}
\address{$^3$Professor, Faculty of Applied Sciences, University ``Politehnica" of Bucharest, Romania}
%\dedicatory{A dedication can be included here}
\begin{document}
\pagestyle{headings}
\maketitle
%%%%%% Abstract
\begin{abstract}
{\it abstract in Romanian}
\mm \mm
{\it \quad\qu abstract in English}
\end{abstract}
%%%%% Keywords
\begin{Keywords}
self-concordant function, ...
\end{Keywords}
%%%%%%% 2000 Mathematics Subject Clasification:
\begin{MSC2000}
53C\,05.
\end{MSC2000}
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\section{Introduction ...s}
According to \cite{jmj} and \cite{ubp1}, we introduce
\begin{definition}\label{d1}
\rm The function $f$ is said to be {\it $k$-self-concordant}, $k\geq
0$, with respect to the Levi-Civita connection $\na$ defined on $M$
if the following condition holds:
$$\left|\na^3f(x)(X_x,X_x,X_x)\right|\leq 2k\left(\na^2f(x)(X_x,X_x)\right)^{3\ov 2},\qu\fo x\in M,~\fo X_x\in T_xM.$$
\end{definition}
\begin{remark}{\rm
Given the metric $g$, and inspired by the linearity of the set of
self-concordant functions \cite{nn}, in our work \cite{bp1}, we used
decomposable functions $f\colon \br_+^n\to\br$, of the form
\begin{equation}\label{e1}
f(x^1,x^2,\ld,x^n)=f_1(x^1)+f_2(x^2)+\cd+f_n(x^n),
\end{equation}
to find a new class of self-concordant functions. Here $f_i\colon
\br_+\to\br$ are differentiable functions}.
\end{remark}
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\section{Section title}
Let be given the Riemannian manifold $M=\br_+^n$, endowed with the
diagonal metric
\begin{equation}\label{e2}
g(x^1, x^2,\ld,x^n)={\rm diag}\left(\di{1\ov g_1^2(x^1)}, ~ \di{1\ov
g_2^2(x^2)},~\cd~ \di{1\ov g_n^2(x^n)}\right),
\end{equation}
\section{Section title}
\begin{remark}{\rm
We emphasize the theories of Nesterov and Nemirovsky \cite{nn} which
use this class of functions for developing linear and convex
quadratic programs with convex quadratic constraints, and Udri\c
ste's works \cite{u1}, \cite{uk} which develop barrier methods for
smooth convex programming on Riemannian manifolds.}\end{remark}
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\section{Conclusions}
We introduced and studied a new class of self-concordant functions,
defined on Riemannian manifolds endowed with metrics of diagonal
type ...
\bigskip
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\begin{thebibliography}{99}
\setlength{\baselineskip}{.45cm}
%author name in italic
%volume number boldface
%standard abreviation is necessary
\bibitem{bp1} {\it G. Bercu and M. Postolache}, Class of
self-concordant functions on Riemannian manifolds, Balkan J. Geom.
Appl., {\bf 14}(2009), No. 2, 13-20.
\bibitem{bcp1} {\it G. Bercu, C. Corcodel and M. Postolache}, On a
study of distinguished structures of Hessian type on
pseudo-Riemannian manifolds, J. Adv. Math. Stud., {\bf 2}(2009), No.
1, 1-16.
\bibitem{hm} {\it V. Helmke and J. B. Moore}, Optimization and Dynamical Systems, Springer-Verlag, London, 1994.
\bibitem{her} {\it D. den Hertog}, Interior Point Approach to Linear, Quadratic and Convex Programming, MAIA 277, Kluwer, 1994.
\bibitem{jmj} {\it D. Jiang, J. B. Moore and H. Ji}, Self-concordant
functions for optimization on smooth manifolds, J. Glob. Optim.,
{\bf 38}(2007), 437-457 ({\tt DOI 10.1007/s10898-006-9095-z}).
\bibitem{nn} {\it Y. Nesterov and A. Nemirovsky}, Interior-point polynomial algorithms in convex programming, Studies in Applied Mathematics (13),
Philadelphia, 1994.
\bibitem{qo} {\it E. A. Quiroz and P. R. Oliveira}, New results on linear optimization through diagonal metrics and Riemannian geometry tools,
Technical Report ES-654/04, PESC COPPE, Federal University of Rio de
Janeiro, 2004.
\bibitem{r2} {\it T. Rapcs\' ak}, Geodesic convexity in nonlinear
optimization, JOTA, {\bf 69}(1991), No. 1, 169-183.
\bibitem{sm} {\it T. Sch\" urmann}, Bias analysis in entropy estimation, J. Phys. A: Math. Gen., {\bf 37}(2004) L295-L301.
\bibitem{ubp1} {\it C. Udri\c ste, G. Bercu and M. Postolache}, $2D$ Hessian Riemannian
manifolds, J. Adv. Math. Stud., {\bf 1}(2008), No. 1-2, 135-142.
\bibitem{u1} {\it C. Udri\c ste}, Optimization methods on Riemannian manifolds, Algebra, Groups and Geometries, {\bf 14}(1997), 339-359.
\bibitem{uk} {\it C.\ Udri\c ste}, Convex Functions and Optimization Methods on Riemannian Manifolds, MAIA 297, Kluwer, 1994.
\end{thebibliography}
\end{document}