Selection of quasi-stationary states in the Navier-Stokes equation on the torus

The two dimensional incompressible Navier-Stokes equation on $D_\delta := [0, 2\pi\delta] \times [0, 2\pi]$ with $\delta \approx 1$, periodic boundary conditions, and viscosity $0 < \nu \ll 1$ is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale $\mathcal{O}(e^{-\nu t})$ and have been shown to play a key role in its long-time evolution. Of particular interest is the role that $\delta$ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if $\delta \neq 1$, while the dipole will be selected if $\delta = 1$. Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of $\delta$ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all $\delta \approx 1$, there is an initial time period in which the high modes decay at the rapid rate $\mathcal{O}(e^{-t/\nu})$, while the low modes evolve at the slower $\mathcal{O}(e^{-\nu t})$ rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.Comment: 29 pages, 4 figure

Secukinumab in the treatment of psoriasis: patient selection and perspectives.

Secukinumab is a human monoclonal antibody targeting IL-17A that has been approved for three indications: moderate-to-severe plaque psoriasis, psoriatic arthritis, and ankylosing spondylitis. In Phase III clinical trials for each of these three indications, secukinumab has proven to be both highly efficacious and well-tolerated. However, several biologic medications are currently approved for the treatment of moderate-to-severe plaque psoriasis, and many demonstrate excellent efficacy and safety. Due to this wide selection, it is often unclear how to choose biologics for specific patients. Important considerations in biologic selection include clinical efficacy, safety, cost, convenience, onset of action, and management of comorbid disease. This article aims to outline the key considerations in patient selection for the treatment of plaque psoriasis with secukinumab

Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.Comment: 23 pages, 27 figure

Theoretical Comparison of the Excited Electronic States of the Linear Uranyl (UO\u3csub\u3e2\u3c/sub\u3e\u3csup\u3e2+\u3c/sup\u3e) and Tetrahedral Uranate (UO\u3csub\u3e4\u3c/sub\u3e\u3csup\u3e2-\u3c/sup\u3e) Ions Using Relativistic Computational Methods

This thesis examines the ground and excited electronic states of the uranyl (UO2+) and uranate (UO42-) ions using Hartree-Fock self-consistent field (HF SCF), multi-configuration self-consistent field (MCSCF) and multi-reference single and double excitation configuration interaction (MR- CISD) methods. The MR-CISD SD calculation included spin-orbit operators. Molecular geometries were obtained from self-consistent field (SCF ) second-order perturbation theory (MP2), and density functional theory (DFT) geometry optimizations using the NWChem 4.01 massively parallel ab initio software package. COLUMBUS version 5.8 was used to perform in-depth analysis on the HF SCF MCSCF and MR-CISD potential energy surfaces. Excited state calculations for the uranyl ion were performed using both a large- and small-core relativistic effective core potential (RECP) in order to calibrate the method. This calibration included comparison to previous theoretical and experimental work on the uranyl ion. Uranate excited states were performed using the small-core RECP as well as the methodology developed using the uranyl ion

A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions

An accurate and efficient hybrid Density Functional Theory (DFT)/Multireference Configuration Interaction (MRCI) model for computing electronic excitation energies in heavy element atoms and molecules was developed. This model incorporated relativistic effects essential for accurate qualitative and quantitative spectroscopic predictions on heavy elements, while simultaneously removing spin-multiplicity limitations inherent in the original model on which it is based. This model was used to successfully compute ground and low-lying electronic states for atoms in the first two rows of the period table, which were used for calibration. Once calibrated, calculations on carbon monoxide, bromine fluoride, the bromine atom, uranium +4 and +5 ions and the uranyl (UO22+) ion showed the model achieved reductions in relative error with respect to Time Dependent Density Functional Theory (TDDFT) of 11-42%, with a corresponding reduction in computational effort in terms of MRCI expansion sizes of a factor of 25-64

The effects of external focus of attention exercise rehabilitation on dual task walking in Parkinson\u27s disease

Parkinson’s disease impairs control of well-learned movements, and therefore, individuals with Parkinson’s disease are forced to walk with greater conscious control. This causes difficulties while walking and completing a secondary task simultaneously (dual tasking), in that distractions from conscious control of walking increase the risk of falls and injury. Although, attention-based exercise may be a potential avenue to decrease the demands associated with walking in Parkinson’s disease. For example, an external focus of attention (on manipulated objects) has been found to recruit the networks that are important for walking with little conscious control (automatic control networks). In contrast, an internal focus (on limb movements) has been found to recruit conscious control networks. Unexpectedly, an external focus of attention (compared to an internal) has been found to improve postural stability in Parkinson’s disease (compared to internal) despite that this attentional focus recruits damaged automatic control circuits. Therefore, promoting an external focus during goal-based exercise may improve upon automatic control functioning, and therefore, improve the ability to dual task and reduce the severity of symptoms. A parallel group, single blind, randomized controlled trial was conducted. Thirty-nine participants with Parkinson’s disease were randomized to one of two exercise groups: 1) External focus of attention (focus on movement of coloured labels attached to limbs, n=19) or 2) Internal focus of attention (focus on movement of limbs, n=20). Both exercise groups completed 33 one-hour goal-based exercise sessions over 11 weeks. Eleven participants were assigned to a non-exercise control group. Walking ability (single and dual tasking) and symptom severity (Unified Parkinson’s disease Rating Scale motor subsection [UPDRS-III] ON and OFF dopamine medications) were assessed before and after (pre/post) the completion of the program, and 8 weeks after exercise cessation (washout). As a result of the intervention, walking ability while completing a secondary task became significantly worse in the Internal focus of attention exercise group, while dual tasking ability did not change in the external group. Symptom severity significantly improved in only the External group from pre to post. From pre to washout, dual task walking ability and motor symptom severity improved in both exercise groups. However, the Internal group had increased errors on the dual task, whereas the External group did not, indicating that improvements were only demonstrated in the External group. Thus, External focus of attention exercise may provide benefits that establish a foundation for improvements to dual task walking ability in Parkinson’s disease, whereas adopting an Internal focus of attention during goal-based exercise appeared to increase reliance on conscious control of movement, hindering dual tasking ability. Additionally, after the exercise program, greater improvements to symptom severity were found after externally focused exercise compared to internally focused. Together, these findings indicate that focusing externally on the manipulation of coloured labels while exercising provides greater rehabilitation effectiveness in Parkinson’s disease compared to focusing internally on limb movements

Profile of tildrakizumab-asmn in the treatment of moderate-to-severe plaque psoriasis: evidence to date.

Plaque psoriasis is an immune-mediated skin disease that affects roughly 3% of adults in the United States. Advances over the past 20 years in understanding the immune-mediated pathophysiology of psoriasis have led to the development of targeted biologic therapies for this condition. Currently, biologic medications approved for the treatment of plaque psoriasis include tumor necrosis factor α inhibitors, interleukin (IL)-17 or IL-17 receptor inhibitors, IL-12/23 inhibitors, and IL-23 inhibitors. Tildrakizumab-asmn is a monoclonal antibody that targets the p19 subunit of IL-23 and is approved for use in adult patients with moderate-to-severe plaque psoriasis who are candidates for systemic therapy or phototherapy. This article reviews the current pharmacologic, efficacy, and safety data on tildrakizumab-asmn

Tropical Geometry and the Motivic Nearby Fiber

We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic nearby fiber." This invariant specializes in the schon case to the Hodge-Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge-Deligne polynomial in the cases of schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.Comment: 27 pages. Compositio Mathematica, to appea