Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in ℝ d is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of ℤ d ,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)N(d,β), anyx ∈ ℤ d , and anyA ⊂ {1, ,n} d •{y∈ℤ whereν A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ →

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there exists \rho (d,\b)N(d,\b), any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to \infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne

Hypersurface Bohm-Dirac models

We define a class of Lorentz invariant Bohmian quantum models for N entangled but noninteracting Dirac particles. Lorentz invariance is achieved for these models through the incorporation of an additional dynamical space-time structure provided by a foliation of space-time. These models can be regarded as the extension of Bohm's model for N Dirac particles, corresponding to the foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to more general foliations. As with Bohm's model, there exists for these models an equivariant measure on the leaves of the foliation. This makes possible a simple statistical analysis of position correlations analogous to the equilibrium analysis for (the nonrelativistic) Bohmian mechanics.Comment: 17 pages, 3 figures, RevTex. Completely revised versio

The Swiss IMRT dosimetry intercomparison using a thorax phantom

Purpose: In 2008, a national intensity modulated radiation therapy (IMRT) dosimetry intercomparison was carried out for all 23 radiation oncology institutions in Switzerland. It was the aim to check the treatment chain focused on the planning, dose calculation, and irradiation process. Methods: A thorax phantom with inhomogeneities was used, in which thermoluminescence dosimeter (TLD) and ionization chamber measurements were performed. Additionally, absolute dosimetry of the applied beams has been checked. Altogether, 30 plan-measurement combinations have been used in the comparison study. The results have been grouped according to dose calculation algorithms, classified as ``type a`` or ``type b,`` as proposed by Knoos et al. [``Comparison of dose calculation algorithms for treatment planning in external photon beam therapy for clinical situations,`` Phys. Med. Biol. 51, 5785-5807 (2006)]. Results: Absolute dosimetry check under standard conditions: The mean ratio between the dose derived from the single field measurement and the stated dose, calculated with the treatment planning system, was 1.007 +/- 0.010 for the ionization chamber and 1.002 +/- 0.014 (mean +/- standard deviation) for the TLD measurements. IMRT Plan Check: In the lung tissue of the planning target volume, a significantly better agreement between measurements (TLD, ionization chamber) and calculations is shown for type b algorithms than for type a (p > 0.001). In regions outside the lungs, the absolute differences between TLD measured and stated dose values, relative to the prescribed dose, vertical bar(D-m - D-s) / D-prescribed vertical bar, are 1.9 +/- 0.4% and 1.4 +/- 0.3%, respectively. These data show the same degree of accuracy between the two algorithm types if low-density medium is not present. Conclusions: The results demonstrate that the performed intercomparison is feasible and confirm the calculation accuracies of type a and type b algorithms in a water equivalent and low-density environment. It is now planned to offer the intercomparison on a regular basis to all Swiss institutions using IMRT techniques. (C) 2010 American Association of Physicists in Medicine. [DOT: 10.1118/1.3460795