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### Arkadiusz Ploski - One of the best experts on this subject based on the ideXlab platform.

• ##### a bound for the milnor number of plane curve singularities
Open Mathematics, 2014
Abstract:

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

• ##### a bound for the milnor number of plane curve singularities
arXiv: Algebraic Geometry, 2013
Abstract:

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.

### Y Polatoglu - One of the best experts on this subject based on the ideXlab platform.

• ##### bounded log harmonic functions with positive real part
Journal of Mathematical Analysis and Applications, 2013
Co-Authors: Y Polatoglu
Abstract:

Abstract Let H ( D ) be the linear space of all analytic functions defined on the open unit disc D = { z | | z | 1 } , and let B be the set of all functions w ( z ) ∈ H ( D ) such that | w ( z ) | 1 for all z ∈ D . A log-harmonic mapping is a solution of the non-linear elliptic partial differential equation f z ¯ ¯ = w ( z ) ( f ¯ / f ) f z , where w ( z ) is the second dilatation of f and w ( z ) ∈ B . In the present paper we investigate the set of all log-harmonic mappings R defined on the unit disc D which are of the form R = H ( z ) G ( z ) ¯ , where H ( z ) and G ( z ) are in H ( D ) , H ( 0 ) = G ( 0 ) = 1 , and R e ( R ) > 0 for all z ∈ D . The class of such functions is denoted by P L H .

### Arnaud Chéritat - One of the best experts on this subject based on the ideXlab platform.

• ##### Upper bound for the size of quadratic Siegel disks
Inventiones mathematicae, 2004
Co-Authors: Xavier Buff, Arnaud Chéritat
Abstract:

If α is an irrational number, we let { p _ n / q _ n }_ n ≥0, be the approximants given by its continued fraction expansion. The Bruno series B (α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P _α: z ↦ e ^2 i πα z + z ^2 has an indifferent fixed point at the origin. If P _α is linearizable, we let r (α) be the conformal radius of the Siegel disk and we set r (α)=0 otherwise. Yoccoz proved that if B (α)=∞, then r (α)=0 and P _α is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all irrational number α with B (α)

• ##### On the Size of Quadratic Siegel Disks: Part I
arXiv: Dynamical Systems, 2003
Co-Authors: Xavier Buff, Arnaud Chéritat
Abstract:

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all irrational number $\a$ with \$B(\a)

### Mei-chu Chang - One of the best experts on this subject based on the ideXlab platform.

• ##### A sum–product theorem in semi-simple commutative Banach algebras
Journal of Functional Analysis, 2004
Co-Authors: Mei-chu Chang
Abstract:

The following analogue of the Erdos–Szemeredi sum-product theorem is shown. Let A=f1,⋯,fN be a finite set of N arbitrary distinct functions on some set. Then either the sum set fi+fj or the product set fifj has at least N1+c elements, where c>0 is an absolute constant. We use Freiman's lemma and Balog–Szemeredi–Gowers Theorem on graphs and combinatorics. As a corollary, we obtain an Erdos–Szemeredi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂R is a finite set, |A| large enough, then |A+A|+|A.A|>|A|1+c, where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=|H|, v=1N∑x∈Hδx. Then, for all constant c, there is a k=k(c) such that maxz∈Rv(k)(z)

### Pietro Rigo - One of the best experts on this subject based on the ideXlab platform.

• ##### A uniform limit theorem for predictive distributions
Statistics & Probability Letters, 2002
Co-Authors: Patrizia Berti, Pietro Rigo
Abstract:

Abstract Let { F n } be a filtration, {Xn} an adapted sequence of real random variables, and {αn} a predictable sequence of non-negative random variables with α1>0. Set β n = ∑ i=1 n α i and define the random distribution functions F n (t)=(1/β n ) ∑ i=1 n α i I {X i ⩽t} and B n (t)=(1/β n ) ∑ i=1 n α i P(X i ⩽t| F i−1 ) . Under mild assumptions on {αn}, it is shown that sup t |F n (t)−B n (t)|→0 , a.s. on the set {Fn or B n converges uniformly } . Moreover, conditions are given under which Fn converges uniformly with probability 1.