Flip-graph moduli spaces of filling surfaces

This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve so we call them filling surfaces. The associated flip-graphs are infinite whenever the mapping class group of the surface (the group of self-homeomorphisms up to isotopy) is infinite, and we can obtain moduli spaces of flip-graphs by considering the flip-graphs up to the action of the mapping class group. This always results in finite graphs and we are interested in their geometry. Our main focus is on the diameter growth of these graphs as n increases. We obtain general estimates that hold for all topological types of filling surface. We find more precise estimates for certain families of filling surfaces and obtain asymptotic growth results for several of them. In particular, we find the exact diameter of modular flip-graphs when the filling surface is a cylinder with a single vertex on the non-privileged boundary curve.Comment: 52 pages, 29 figure

A definable (p,q)(p,q)-theorem for NIP theories

We prove a definable version of Matou\v{s}ek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in the distal case, utilizing the notion of locally compressible types which appeared in the work of the author with Bays and Simon.Comment: Added section 4.1 about VC-densit

Image-based registration methods for quantification and compensation of prostate motion during trans-rectal ultrasound (TRUS)-guided biopsy

Prostate biopsy is the clinical standard for cancer diagnosis and is typically performed under two-dimensional (2D) transrectal ultrasound (TRUS) for needle guidance. Unfortunately, most early stage prostate cancers are not visible on ultrasound and the procedure suffers from high false negative rates due to the lack of visible targets. Fusion of pre-biopsy MRI to 3D TRUS for targeted biopsy could improve cancer detection rates and volume of tumor sampled. In MRI-TRUS fusion biopsy systems, patient or prostate motion during the procedure causes misalignments in the MR targets mapped to the live 2D TRUS images, limiting the targeting accuracy of the biopsy system. In order to sample smallest clinically significant tumours of 0.5 cm3with 95% confidence, the root mean square (RMS) error of the biopsy system needs to be The target misalignments due to intermittent prostate motion during the procedure can be compensated by registering the live 2D TRUS images acquired during the biopsy procedure to the pre-acquired baseline 3D TRUS image. The registration must be performed both accurately and quickly in order to be useful during the clinical procedure. We developed an intensity-based 2D-3D rigid registration algorithm and validated it by calculating the target registration error (TRE) using manually identified fiducials within the prostate. We discuss two different approaches that can be used to improve the robustness of this registration to meet the clinical requirements. Firstly, we evaluated the impact of intra-procedural 3D TRUS imaging on motion compensation accuracy since the limited anatomical context available in live 2D TRUS images could limit the robustness of the 2D-3D registration. The results indicated that TRE improved when intra-procedural 3D TRUS images were used in registration, with larger improvements in the base and apex regions as compared with the mid-gland region. Secondly, we developed and evaluated a registration algorithm whose optimization is based on learned prostate motion characteristics. Compared to our initial approach, the updated optimization improved the robustness during 2D-3D registration by reducing the number of registrations with a TRE \u3e 5 mm from 9.2% to 1.2% with an overall RMS TRE of 2.3 mm. The methods developed in this work were intended to improve the needle targeting accuracy of 3D TRUS-guided biopsy systems. The successful integration of the techniques into current 3D TRUS-guided systems could improve the overall cancer detection rate during the biopsy and help to achieve earlier diagnosis and fewer repeat biopsy procedures in prostate cancer diagnosis

Detailed black hole state counting in loop quantum gravity

We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime