Mean Rational Approximation for Some Compact Planar Subsets

In 1991, J. Thomson obtained celebrated structural results for $P^t(\mu).$ Later, J. Brennan (2008) generalized Thomson's theorem to $R^t(K,\mu)$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. The results indicate that if $R^t(K,\mu)$ is pure, then $R^t(K,\mu) \cap L^\infty (\mu)$ is the "same as" the algebra of bounded analytic functions on \mbox{abpe}(R^t(K, \mu)), the set of analytic bounded point evaluations. We show that if the diameters of the components of $\mathbb C\setminus K$ are allowed to tend to zero, then even though \text{int}(K) = \mbox{abpe}(R^t(K, \mu)) and $K =\overline {\text{int}(K)},$ the algebra $R^t(K,\mu) \cap L^\infty (\mu)$ may "be equal to" a proper sub-algebra of bounded analytic functions on $\text{int}(K),$ where functions in the sub-algebra are "continuous" on certain portions of the inner boundary of $K.$Comment: arXiv admin note: text overlap with arXiv:1904.0644

Directed Ramsey number for trees

In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $\overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Ramsey number $\overleftrightarrow{R}(H,k)$ of $H$, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order $n$. Here we show that $\overleftrightarrow{R}(T,k) \le c_k|T|^{k-1}$ for any oriented tree $T$, which is again tight up to a constant factor, and it generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined the $2$-colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure

Mean Rational Approximation for Compact Subsets with Thin Boundaries

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(\mu),$ the closed subspace of $L^t(\mu)$ spanned by the analytic polynomials, when 1 \le t < \i. In 2008, J. Brennan \cite{b08} generalized Thomson's theorem to $R^t(K, \mu),$ the closed subspace of $L^t(\mu)$ spanned by the rational functions with poles off a compact subset $K$ containing the support of $\mu,$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. We obtain a necessary and sufficient condition for $R^t(K, \mu)$ to ensure such a decomposition theorem holdsComment: arXiv admin note: text overlap with arXiv:1904.06446, arXiv:2212.0539

The Commutant of Multiplication by z on the Closure of Rational Functions in Lt(μ)L^t(\mu)

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$ on $K,$ and 1 \le t < \i, let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of $\text{Rat}(K)$ in $L^t(\mu).$ For a bounded Borel subset $\mathcal D\subset \mathbb C,$ let \area_{\mathcal D} denote the area (Lebesgue) measure restricted to $\mathcal D$ and let H^\i (\mathcal D) be the weak-star closed sub-algebra of L^\i(\area_{\mathcal D}) spanned by $f,$ bounded and analytic on $\mathbb C\setminus E_f$ for some compact subset $E_f \subset \mathbb C\setminus \mathcal D.$ We show that if $R^t(K, \mu)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\mathcal R \subset K$ whose closure contains the support of $\mu$ and there exists an isometric isomorphism and a weak-star homeomorphism $\rho$ from $R^t(K, \mu) \cap L^\infty(\mu)$ onto $H^\infty(\mathcal R)$ such that $\rho(r) = r$ for all $r\in\text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081

Precise tail asymptotics of fixed points of the smoothing transform with general weights

We consider solutions of the stochastic equation $R=_d\sum_{i=1}^NA_iR_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent both from $A_i$'s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}|A_1|^{\alpha}=1/N$ and $\mathbb{E}|A_1|^{\alpha}\log|A_1|>0$, the limit $\lim_{t\to\infty}t^{\alpha}\mathbb{P}[|R|>t]=K$ exists. In the present paper, we prove positivity of $K$.Comment: Published at http://dx.doi.org/10.3150/13-BEJ576 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The short-time behavior of kinetic spherical model with long-ranged interactions

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m_{0} quenched from a very high temperature to T < T_{c} is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order m_{r} ~ -t^{-k} is found in the intermediate time-regime. The short-time scaling relation of small m_{0} are generalized to an arbitrary m_{0} and all the time larger than t_{mic}. The characteristic functions $\phi (b,m_{0})$ for the scaling of m_{0} and $\epsilon (b,T')$ for T'=T/T_{c} are obtained. The crossover between scaling regimes is discussed in detail.Comment: 22 pages, 3 figure