Dehn filling trivialization on a knot group: separation and realization

Let $K$ be a non-trivial knot in $S^3$ with the exterior $E(K)$. For a slope $r \in \mathbb{Q}$, let $K(r)$ be the result of $r$--Dehn filling of $E(K)$. To each element $g$ of the knot group $G(K)$ assign $\mathcal{S}_K(g)$ as the set of slopes $r$ such that $g$ becomes the trivial element in $\pi_1(K(r))$. The purpose of this article is to prove somewhat surprising flexibilities -- a separation property and a realization property -- of the set $\mathcal{S}_K(g)$, which are refinements of the Property P in the context of Dehn filling trivialization. We construct infinitely many, mutually non-conjugate elements $g$ (in the commutator subgroup) of $G(K)$ such that $\mathcal{S}_K(g)$ is the empty set, namely, elements of $G(K)$ that survive all the Dehn fillings of $K$ whenever $K$ has no cyclic surgery. Then we prove the Separation Theorem that can be seen as a Dehn filling analogue of various separability properties of 3-manifold groups: for every non-torus knot $K$ and any disjoint finite sets $\mathcal{R}$ and $\mathcal{S}$ of slopes, there exists an element $g$ of $G(K)$ such that $\mathcal{S}_K(g)$ contains $\mathcal{R}$, but does not contain any slopes in $\mathcal{S}$ whenever $\mathcal{S}$ contains no Seifert surgery slopes. We develop this to establish the Realization Theorem asserting that for any hyperbolic knot $K$ without torsion surgery slope, every finite set of slopes whose complement does not contain Seifert surgery slopes can be realized as the set $\mathcal{S}_K(g)$ for infinitely many, mutually non-conjugate elements $g \in G(K)$. We also provide some examples showing that the Separation Theorem and the Realization Theorem do not hold unconditionally.Comment: 59 pages, 2 figure

Stochastic model-based assessment of power systems subject to extreme wind power fluctuation

Extreme outliers of wind power fluctuation are a source of severe damage to power systems. In our previous work, we proposed a modelling framework, verified its usefulness via real data, and developed a model-based evaluation method of the impact of such extreme outliers. However, it has been a drawback that the obtained estimates of frequency fluctuation of power systems are sometimes excessively conservative for their practical use. To overcome this weakness, theory and methods for tightening the fluctuation estimates are investigated in this paper. This is done by applying a robust performance analysis method of a Lur'e system to the error analysis of stochastic linearization. The usefulness of our proposed method is shown through a load frequency control model

Ablation for idiopathic left ventricular tachycardia in a patient with double outlet right ventricle who underwent Fontan operation: a case report

Background The incidence of ventricular tachycardia (VT) in patients following Fontan operation is reported as 3.5%. Furthermore, in patients with repaired double outlet right ventricle (DORV), scar-related VT and outflow tract VT have been reported; however, Purkinje-related VT has not previously been reported. In this report, we present the case of idiopathic left VT (ILVT) in a patient with DORV who underwent Fontan operation. Case summary A 31-year-old man was diagnosed as having DORV with complete atrioventricular defect at birth. When he was 17 years old, he underwent surgical repair, including extracardiac Fontan operation and common atrioventricular valve replacement. Five years later, VT was detected. Since some medications were ineffective in suppressing VT, he was referred to our hospital for definitive treatment. Ventricular tachycardia was induced by atrial and ventricular programmed electrical stimulations. The mechanism of the VT was determined to be re-entry. The earliest activation site was located at the mid-inferior septum of the hypoplastic left ventricle, in which Purkinje potentials were observed before the local ventricular electrogram. Radiofrequency catheter ablation (RFCA) was performed at this site to eliminate VT. Discussion Most VTs originate from surgical scars in patients with congenital heart disease. Catheter ablation was feasible in scar-related VT. To the best of our knowledge, this is the first report of ILVT treated successfully with RFCA in a DORV patient who had undergone Fontan operation