On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations

The putative singular set S in space-time of a suitable weak solution u of the 3D Navier–Stokes equations has box-counting dimension no greater than 5/3. This allows one to prove that almost all trajectories avoid S. Moreover, for each point x that does not belong to S, one can find a neighbourhood U of x such that the function u is continuous on U and space derivatives of u are bounded on every compact subset of U. It follows that almost all Lagrangian trajectories corresponding to u are C^{1} functions of time (Robinson & Sadowski, Nonlinearity 2009). We recall the main idea of the proof, give examples that clarify in what sense the uniqueness of trajectories is considered, and make some comments on how this result might be improved

Learning the game: Breakdowns, breakthroughs and player strategies

Digital games are rich learning environments that require players to engage with challenging situations in order to progress. Recent research indicates that game-play involves overcoming breakdowns and achieving breakthroughs in relation to player action, understanding and involvement. In particular, breakthroughs involve moments of insight where learning occurs which, in turn, can help increase involvement. However, little is known about how players actually achieve breakthroughs. We applied the breakdown/breakthrough “lens” to explore how players attempt to achieve breakthroughs in relation to two single player games. We identified a finite number of strategies that illustrate how players learn in games. These strategies are considered in relation to producing playable and engaging games

A field programmable gate array based Langmuir probe system for measurement of plasma parameters at 500 kHz in a high-power impulse magnetron sputtering plasma.

By utilizing Field Programmable Gate Arrays in a configuration similar to that of the Mirror Langmuir Probe, it is possible to bias a single probe at three precise voltages in sequence. These voltages can be dynamically adjusted in real-time based on the measured plasma electron temperature to ensure the transition region is always sampled. The first results have been obtained by employing this method and have generated real-time outputs of electron temperature, ion saturation current, and floating potential on a low temperature pulsed-DC magnetron at 500 kHz. These results are in good agreement with the analysis of a conventionally swept Langmuir probe. This probe is designed with the intention of being implemented on MAST-U to aid in the study of exhaust physics and enable further investigation into filamentary behavior. [Abstract copyright: © 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

The HR image slicer for GNIRS at Gemini North: optical design and performance

GNIRS (Gemini Near-InfraRed Spectrograph) is a multi-function spectrograph at Gemini North telescope offering four observational modes in the spectral range of 0.8 to 5.4 µm. It provides 2-pixel spectral resolutions from 1,200 up to 18,0000 and has single disperser and cross-disperser modes yielding simultaneous spectral bandwidths from 40 nm to 1,650 nm. GNIRS presented three existing modes: long-slit (50-100" slit), cross-dispersed (5-7" slit) and low resolution (LR) Integral Field Unit (IFU) (3.15" x 4.80") and it is now being upgraded with a fourth mode allowing high resolution (HR) IFU spectroscopy using an image slicer optimised for fully adaptively corrected images over a field of view of 2.25 arcsec2 (1.80" x 1.25") covered by 25 slices of 410 µm width offering a spatial sampling of 0.05 x 0.05 arscec2 with a diffraction limited optical quality. The proposed layout meets specifications and some challenging design constraints: it shall be contained within the same envelope defined by the LR image slicer (0.1 x 0.2 x 0.1 m3 ), the input and output focal-ratios of both image slicers shall be the same and at exact positions but providing different anamorphic magnifications and preserving the optical quality. The length of the generated slit will be similar to the length of the slit in long-slit mode to maximise detector use and avoid vignetting. This communication presents the optical design and performance of the high resolution image slicer compliant with all specifications and constraints and it shows some design adaptations adopted in order to facilitate its manufacturing in metal at Durham University

A storm in a shelf sea: Variation in phosphorus distribution and organic matter stoichiometry

Organic matter (OM) plays an important role in productive shelf seas and their contribution to global carbon (C) and nutrient cycles. We investigated the impact of storm mixing on OM dynamics in the seasonally stratified Celtic Sea. After the storm, OM production was decoupled from consumption in the euphotic layer. Over the 15 day study, dissolved OM (DOM) became phosphorus (P) rich relative to C, whereas particulate OM (POM) became P-deplete relative to C. Upward diapycnal phosphate fluxes were accompanied by reciprocal downward mixing of dissolved organic P (DOP) and particulate P (PPhos). Transfer of DOP and PPhos below the thermocline accounts for 22% and 26%, respectively, of the upward phosphate flux. Given the changes in stoichiometry of POM and DOM after the storm, the form in which OM is transferred below the thermocline has important implications for the efficiency of elemental transfer, impacting C cycling and storage in the ocean

Optical spectroscopic studies of light-harvesting by pigment-reconstituted peridinin-chlorophyll-proteins at cryogenic temperatures

Low temperature, steady-state, optical spectroscopic methods were used to study the spectral features of peridinin-chlorophyll-protein (PCP) complexes in which recombinant apoprotein has been refolded in the presence of peridinin and either chlorophyll a (Chl a), chlorophyll b (Chl b), chlorophyll d (Chl d), 3-acetyl-chlorophyll a (3-acetyl-Chl a) or bacteriochlorophyll a (BChl a). Absorption spectra taken at 10 K provide better resolution of the spectroscopic bands than seen at room temperature and reveal specific pigment–protein interactions responsible for the positions of the Q(y) bands of the chlorophylls. The study reveals that the functional groups attached to Ring I of the two protein-bound chlorophylls modulate the Q(y) and Soret transition energies. Fluorescence excitation spectra were used to compute energy transfer efficiencies of the various complexes at room temperature and these were correlated with previously reported ultrafast, time-resolved optical spectroscopic dynamics data. The results illustrate the robust nature and value of the PCP complex, which maintains a high efficiency of antenna function even in the presence of non-native chlorophyll species, as an effective tool for elucidating the molecular details of photosynthetic light-harvesting

Towards standard setting for patient-reported outcomes in the NHS homeopathic hospitals

We report findings from a pilot data collection study within a programme of quality assurance, improvement and development across all five homeopathic hospitals in the UK National Health Service (NHS).<p></p> <b>Aims</b> (1) To pilot the collection of clinical data in the homeopathic hospital outpatient setting, recording patient-reported outcome since first appointment; (2) to sample the range of medical complaints that secondary-care doctors treat using homeopathy, and thus identify the nature and complexity of complaints most frequently treated nationally; (3) to present a cross section of outcome scores by appointment number, including that for the most frequently treated medical complaints; (4) to explore approaches to standard setting for homeopathic practice outcome in patients treated at the homeopathic hospitals.<p></p> <b>Methods</b> A total of 51 medical practitioners took part in data collection over a 4-week period. Consecutive patient appointments were recorded under the headings: (1) date of first appointment in the current series; (2) appointment number; (3) age of patient; (4) sex of patient; (5) main medical complaint being treated; (6) whether other main medical complaint(s); (7) patient-reported change in health, using Outcome Related to Impact on Daily Living (ORIDL) and its derivative, the ORIDL Profile Score (ORIDL-PS; range, –4 to +4, where a score ≤−2 or ≥+2 indicates an effect on the quality of a patient's daily life); (8) receipt of other complementary medicine for their main medical complaint.<p></p> <b>Results</b> The distribution of patient age was bimodal: main peak, 49 years; secondary peak, 6 years. Male:female ratio was 1:3.5. Data were recorded on a total of 1797 individual patients: 195 first appointments, 1602 follow-ups (FUs). Size of clinical service and proportion of patients who attended more than six visits varied between hospitals. A total of 235 different medical complaints were reported. The 30 most commonly treated complaints were (in decreasing order of frequency): eczema; chronic fatigue syndrome (CFS); menopausal disorder; osteoarthritis; depression; breast cancer; rheumatoid arthritis; asthma; anxiety; irritable bowel syndrome; multiple sclerosis; psoriasis; allergy (unspecified); fibromyalgia; migraine; premenstrual syndrome; chronic rhinitis; headache; vitiligo; seasonal allergic rhinitis; chronic intractable pain; insomnia; ulcerative colitis; acne; psoriatic arthropathy; urticaria; ovarian cancer; attention-deficit hyperactivity disorder (ADHD); epilepsy; sinusitis. The proportion of patients with important co-morbidity was higher in those seen after visit 6 (56.9%) compared with those seen up to and including that point (40.7%; P < 0.001). The proportion of FU patients reporting ORIDL-PS ≥ +2 (improvement affecting daily living) increased overall with appointment number: 34.5% of patients at visit 2 and 59.3% of patients at visit 6, for example. Amongst the four most frequently treated complaints, the proportion of patients that reported ORIDL-PS ≥ +2 at visit numbers greater than 6 varied between 59.3% (CFS) and 73.3% (menopausal disorder).<p></p> <b>Conclusions</b> We have successfully piloted a process of national clinical data collection using patient-reported outcome in homeopathic hospital outpatients, identifying a wide range and complexity of medical complaints treated in that setting. After a series of homeopathy appointments, a high proportion of patients, often representing “effectiveness gaps” for conventional medical treatment, reported improvement in health affecting their daily living. These pilot findings are informing our developing programme of standard setting for homeopathic care in the hospital outpatient context