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1. CMB 2013 (vol 56 pp. 673)

Ayadi, K.; Hbaib, M.; Mahjoub, F.
 Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type $$\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}$$ where $(A, B, C)\in (\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$. In particular, we will give, under some conditions on the polynomials $A$, $B$ and $C$, well approximated elements satisfying this equation. Keywords:diophantine approximation, formal power series, continued fractionCategories:11J61, 11J70

2. CMB 2011 (vol 56 pp. 337)

Fan, Qingzhai
 Certain Properties of $K_0$-monoids Preserved by Tracial Approximation We show that the following $K_0$-monoid properties of $C^*$-algebras in the class $\Omega$ are inherited by simple unital $C^*$-algebras in the class $TA\Omega$: (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative. Keywords:$C^*$-algebra, tracial approximation, $K_0$-monoidCategories:46L05, 46L80, 46L35

3. CMB 2011 (vol 55 pp. 752)

Hickel, M.; Rond, G.
 Approximation of Holomorphic Solutions of a System of Real Analytic Equations We prove the existence of an approximation function for holomorphic solutions of a system of real analytic equations. For this we use ultraproducts and Weierstrass systems introduced by J. Denef and L. Lipshitz. We also prove a version of the PÅoski smoothing theorem in this case. Keywords:Artin approximation, real analytic equationsCategories:13B40, 13L05, 14F12

4. CMB 2011 (vol 55 pp. 762)

Li, Hanfeng
 Smooth Approximation of Lipschitz Projections We show that any Lipschitz projection-valued function $p$ on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions $q$ with Lipschitz constant close to that of $p$. This answers a question of Rieffel. Keywords:approximation, Lipschitz constant, projectionCategory:19K14

5. CMB 2011 (vol 54 pp. 566)

Zhou, Xiang-Jun; Shi, Lei; Zhou, Ding-Xuan
 Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter. Keywords:multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence ratesCategories:68T05, 62J02

6. CMB 2010 (vol 53 pp. 614)

Böröczky, Károly J.; Schneider, Rolf
 The Mean Width of Circumscribed Random Polytopes For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained. Keywords:random polytope, mean width, approximationCategories:52A22, 60D05, 52A27

7. CMB 2009 (vol 53 pp. 11)

Burke, Maxim R.
 Approximation and Interpolation by Entire Functions of Several Variables Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen. Keywords:entire function, complex approximation, interpolation, several complex variablesCategory:32A15

8. CMB 2009 (vol 52 pp. 28)

Choi, Changsun; Kim, Ju Myung; Lee, Keun Young
 Right and Left Weak Approximation Properties in Banach Spaces New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed. Keywords:approximation property, quasi approximation property, weak approximation propertyCategories:46B28, 46B10

9. CMB 2008 (vol 51 pp. 372)

Ezquerro, J. A.; Hernández, M. A.
 Picard's Iterations for Integral Equations of Mixed Hammerstein Type A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed. Keywords:nonlinear equations in Banach spaces, successive approximations, semilocal convergence theorem, Picard's iteration, Hammerstein integral equationsCategories:45G10, 47H99, 65J15

10. CMB 2007 (vol 50 pp. 434)

Õzarslan, M. Ali; Duman, Oktay
 MKZ Type Operators Providing a Better Estimation on $[1/2,1)$ In the present paper, we introduce a modification of the Meyer-K\"{o}nig and Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and $f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r-$th order generalization of our operators and study their approximation properties. Keywords:Meyer-KÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionalsCategories:41A25, 41A36

11. CMB 2006 (vol 49 pp. 237)

Gauthier, P. M.; Zeron, E. S.
 Approximation by Rational Mappings, via Homotopy Theory Continuous mappings defined from compact subsets $K$ of complex Euclidean space $\cc^n$ into complex projective space $\pp^m$ are approximated by rational mappings. The fundamental tool employed is homotopy theory. Keywords:Rational approximation, homotopy type, null-homotopicCategories:32E30, 32C18

12. CMB 2002 (vol 45 pp. 80)

Gauthier, P. M.; Zeron, E. S.
 Approximation On Arcs and Dendrites Going to Infinity in $\C^n$ On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions. Keywords:tangential approximation, CarlemanCategories:32E30, 32E25