The Collatz tree as a Hilbert hotel:a proof of the 3x + 1 conjecture

The Collatz conjecture maintains that each natural number $n$ is a node in the Collatz tree with a root path to $1$ generated by iterations of the Collatz function, which connects even $n$ to $n/2$ and odd $n$ to $3n+1$. The conjecture holds if a binary subtree $T_{\ge 0}$ exists with as nodes $\textit{all}$ branching numbers constituting the congruence classes ${\left[4,16\right]}_{18}$ with density $2/18$. For each branching number, the $\textit{upward}$ and $\textit{rightward}$ function generate arrows to two branching numbers in the directions $2n$ and $(n-1)/3$. In the automorphism graph of tree $T_{\ge 0}$. The iterates and conjugates of these two functions build outer paths from tree $T_{\ge 0}$ at its root node to two infinite sequences of nodes containing nested subtrees and disjoint cotrees. The nodes and conjugate inner arrows within each subtree and within each cotree correspond one-to-one to the nodes and arrows within tree $T_{\ge 0}$. The cumulative density of the congruence classes of numbers in disjoint cotrees of $T_{\ge 0}$ amounts to the density $2/18$ of all branching numbers, thus proving the conjecture.Comment: Collatz paths between branching classes $[4,16]_18$ and partitioning of the binary tree $T_{ge 0}$ into disjoint cotrees now validate in v5 the density proofs in previous versions (v1 31 Aug 2000

Validation of NODDI estimation of dispersion anisotropy in V1 of the human neocortex

This work validates the estimation of neurite dispersion anisotropy in the brain, using Bingham-NODDI [1], an extension of the diffusion MRI technique called NODDI [2]. The original NODDI provides indices of neurite (axons and dendrites) morphology that are sensitive and specific to microstructural changes [3-7]. Bingham-NODDI additionally allows the estimation of neurite dispersion anisotropy, which can enhance the accuracy of tractography algorithms [8]. The in vivo feasibility of Bingham-NODDI has been evaluated in [1]. The present study validates its indices using high-resolution ex vivo imaging data of the human primary visual cortex (V1), a well characterised region of the neocortex known to include fibres that fan or bend into the cortical layers

Politieke kennis en effecten van nieuws

Welk nieuws doet ertoe en hoeveel nieuws doet ertoe, en in welke mate hangt dit af van de politieke kennis van de ontvanger? Dit artikel beschrijft een longitudinale studie naar de verkiezingsstrijd voor het Nederlandse parlement in 2006

Systemic implications of the bail-in design

The 2007-2008 financial crisis forced governments to choose between the unattractive alternatives of either bailing out a systemically important bank (SIB) or allowing it to fail disruptively. Bail-in has been put forward as an alternative that potentially addresses the too-big-to-fail and contagion risk problems simultaneously. Though its efficacy has been demonstrated for smaller idiosyncratic SIB failures, its ability to maintain stability in cases of large SIB failures and system-wide crises remains untested. This paper’s novelty is to assess the financial-stability implications of bail-in design, explicitly accounting for the multilayered networked nature of the financial system. We present a model of the European financial system that captures all five of the prevailing contagion channels. We demonstrate that it is essential to understand the interaction of multiple contagion mechanisms and that financial institutions other than banks play an important role. Our results indicate that stability hinges on the bank-specific and structural bail-in design. On one hand, a welldesigned bail-in buttresses financial resilience, but on the other hand, an ill-designed bail-in tends to exacerbate financial distress, especially in system-wide crises and when there are large SIB failures. Our analysis suggests that the current bail-in design may be in the region of instability. While policy makers can fix this, the political economy incentives make this unlikely

SWI-informed diffusion tensor tractography

Introduction In diffusion tensor tractography (DTT), white matter structure is inferred in vivo by reconstructing fiber tracts from diffusion weighted images (DWI). Recently [1], white matter structure has also been shown at 7T using susceptibility weighted imaging (SWI) [2]. Most notably, SWI shows excellent contrast between the highly myelinated optic radiation (OR) and the surrounding white matter [3]. Because DTT attempts to reconstruct tracts from voxels orders of magnitude larger than the underlying substrate, it suffers from partial volume effects in voxels that contain multiple or incoherently oriented tracts, resulting in false positive and false negative tracts. Tractography might therefore benefit from the combination of the diffusion tensor with the white matter contrasts in SWI which can be obtained at a much higher resolution. We have adapted a DTT algorithm to include the structure tensor [4] of the SWI magnitude in order to improve tractography in locations where DWI and SWI provide complementary information. Methods DWI (3T; 32-ch coil; twice-refocused spin echo EPI; 61+7 gradient directions; b=1000 s/mm2; TR=8300 ms; TE=95 ms; matrix size=110x110; FOV=220x220 mm; slice thickness=2.0 mm; number of slices=64) and SWI (7T; 8-ch coil; sagittal orientation; TR=36 ms; TE=25 ms; flip angle=15°; matrix size=448x336; FOV=224x168 mm; slice thickness=0.5 mm; number of slices=208; BW=120 Hz/px; acquisition time=20 min) were recorded from a healthy volunteer. The DWI mean b0 image and SWI were bias field corrected and then coregistered with FSL using the normalized mutual information algorithm and weighting volumes. The diffusion tensor and structure tensor fields were reconstructed from the DWI and SWI volumes, respectively. The structure tensor was calculated as the partial derivatives [dxx,dxy,dxz; dyx,dyy,dyz; dzx,dzy,dzz] of the SWI magnitude and every structure tensor component was smoothed (FWHM = 2.5 mm). Tractography was performed using Camino (PICo; 5000 iterations; curvature threshold = 80º; FA threshold = 0.10; step size = 0.50 mm). The structure tensor information was incorporated by requiring that the tracking direction be in the plane orthogonal to the first eigenvector of the structure tensor (ST ϵ1). This plane is assumed to be aligned to the direction of the tract that causes the intensity variation. The tracking direction within this plane is determined as the projection of the diffusion tensor onto the plane. To avoid adapted tracking directions where the structure tensor was non-informative, it was used only if the first eigenvalue of the structure tensor (ST λ1) > 100. For evaluating the performance, seeds were placed in the OR posterior to the point where it merged with the splenium of the corpus callosum (SCC). Waypoints were created anterior to the split in both the OR and SCC. Fractions of streamlines crossing these waypoints were extracted for both DTT and SWI-informed DTT. Results and Discussion Although the main tracts were similar for both DTT and SWI-informed DTT, the algorithms often showed very different branching patterns and more subtle differences in the course of the tracts. Examples are provided in Fig 1&2. In Fig 1 a putatively more accurate tracking of the OR using SWI-informed DTT compared to DTT is shown after seeding in the posterior OR. The seed at the merging of the OR and the SCC resulted in markedly different results for DTT and SWI-informed DTT (Fig 2). Frontal branches emerged for SWI-informed DTT, but not for DTT. A presumably non-veridical split is seen in the SCC for SWI-informed DTT (black arrow), where the structure tensor seems to cause a bias towards the borders of the tract. Fractions of streamlines entering OR/SCC were 0.026 for DTT vs. 0.336 for SWI-informed DTT. Conclusions A modification of a method was proposed to overcome some limitations of diffusion tensor tractography. It was shown that the contrast within the white matter in susceptibility weighted images can provide additional information for tractography algorithms, leading to increased sensitivity at specific locations. To have an unambiguous validation of the findings of SWI-informed DTT, an ex vivo validation of white matter connectivity has to be performed. We have shown that SWI-informed DTT reveals white matter fiber tracts that were not found using standard DTT

Corticosteroid therapy for the management of paradoxical inflammatory reaction in patients with pulmonary tuberculosis

Background Paradoxical reaction after the initiation of tuberculosis treatment is defined as increased inflammation following effective antimycobacterial treatment. This is a phenomenon that can severely complicate a patient's recovery, potentially leading to further morbidity and residual deficits. Paradoxical reaction remains poorly understood regarding its pathophysiology and management. Only a limited number of reports look critically at the available therapeutic options, with evidence of the efficacy of prednisolone therapy being primarily limited to extrapulmonary PR only. Case We describe two HIV negative patients who were admitted to our department with pulmonary tuberculosis, presenting with inflammatory patterns attributable to PR and their response to adjunctive steroid therapy. Discussion and Conclusions The presented cases further highlight the need for immunological studies and randomized trials for corticosteroid therapy are needed to better understand this phenomenon as well as provide an evidence-base for anti-inflammatory treatment. Furthermore, by means of this case series, we are also able to highlight the potential variability in the symptomatology of the lesser known PR phenomenon, in which we observed a hypotensive shock-like syndrome not previously described in literature

Activated Ion Electron Capture Dissociation (AI ECD) of proteins: synchronization of infrared and electron irradiation with ion magnetron motion.

Here, we show that to perform activated ion electron capture dissociation (AI-ECD) in a Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometer equipped with a CO(2) laser, it is necessary to synchronize both infrared irradiation and electron capture dissociation with ion magnetron motion. This requirement is essential for instruments in which the infrared laser is angled off-axis, such as the Thermo Finnigan LTQ FT. Generally, the electron irradiation time required for proteins is much shorter (ms) than that required for peptides (tens of ms), and the modulation of ECD, AI ECD, and infrared multiphoton dissociation (IRMPD) with ion magnetron motion is more pronounced. We have optimized AI ECD for ubiquitin, cytochrome c, and myoglobin; however the results can be extended to other proteins. We demonstrate that pre-ECD and post-ECD activation are physically different and display different kinetics. We also demonstrate how, by use of appropriate AI ECD time sequences and normalization, the kinetics of protein gas-phase refolding can be deconvoluted from the diffusion of the ion cloud and measured on the time scale longer than the period of ion magnetron motion

Diverse politics, diverse news coverage? A longitudinal study of diversity in Dutch political news during two decades of election campaigns

Although diverse political news has been recognized as a requirement for a well-functioning democracy, longitudinal research into this topic is sparse. In this article, we analyse the development of diversity in election coverage in the Netherlands between 1994 and 2012. We distinguish between diversity for party and issue coverage, and look at differences between diversity in newspapers and television news. Results show that news diversity varies over time. Diversity for party types increased over time. We found no clear trend for diversity of issue dimensions. Compared to newspapers, television news is more diverse for party types but less diverse on issue dimensions. The question concerning whether these findings are an indicator of structural bias is discussed

A probabilistic atlas of the cerebellar white matter

Imaging of the cerebellar cortex, deep cerebellar nuclei and their connectivity are gaining attraction, due to the important role the cerebellum plays in cognition and motor control. Atlases of the cerebellar cortex and nuclei are used to locate regions of interest in clinical and neuroscience studies. However, the white matter that connects these relay stations is of at least similar functional importance. Damage to these cerebellar white matter tracts may lead to serious language, cognitive and emotional disturbances, although the pathophysiological mechanism behind it is still debated. Differences in white matter integrity between patients and controls might shed light on structure–function correlations. A probabilistic parcellation atlas of the cerebellar white matter would help these studies by facilitating automatic segmentation of the cerebellar peduncles, the localization of lesions and the comparison of white matter integrity between patients and controls. In this work a digital three-dimensional probabilistic atlas of the cerebellar white matter is presented, based on high quality 3 T, 1.25 mm resolution diffusion MRI data from 90 subjects participating in the Human Connectome Project. The white matter tracts were estimated using probabilistic tractography. Results over 90 subjects were symmetrical and trajectories of superior, middle and inferior cerebellar peduncles resembled the anatomy as known from anatomical studies. This atlas will contribute to a better understanding of cerebellar white matter architecture. It may eventually aid in defining structure–function correlations in patients with cerebellar disorder