Small Disk Research Articles

Impact of a small disk on a sessile water drop

We address the details of events that follow the impact of a small solid disk on a water drop sitting on another disk of the same diameter. We experimentally demonstrate that fast squeezing of a low-viscosity liquid drop between two approaching disks leads to the formation of complex, radially expanding liquid structures (splashes) outside of the disks. The spatial and temporal dynamics of these splashes are tracked via high-speed video recording and flash photography. We analyze the mechanisms that control the shapes and breakup processes of these structures and derive a mathematical model for their behavior using simple physical arguments. Our investigation indicates that liquid structure formation is the result of a rapid increase in the velocity of liquid ejection from the gap between disks with time.

A Redshifted Inner Disk Atmosphere and Transient Absorbers in the Ultracompact Neutron Star X-Ray Binary 4U 1916–053

Abstract The very small accretion disks in ultracompact X-ray binaries are special laboratories in which to study disk accretion and outflows. We report on three sets of new (250 ks total) and archival (50 ks) Chandra/HETG observations of the “dipping” neutron star X-ray binary 4U 1916–053, which has an orbital period of P ≃ 50 minutes. We find that the bulk of the absorption in all three spectra originates in a disk atmosphere that is redshifted by v ≃ 220–290 km s−1, corresponding to the gravitational redshift at a radius of R ∼ 1200 GM/c 2. This shift is present in the strongest, most highly ionized lines (Si xiv and Fe xxvi), with a significance of 5σ. Absorption lines observed during dipping events (typically associated with the outermost disk) instead display no velocity shifts and serve as a local standard of rest, suggesting that the redshift is intrinsic to an inner disk atmosphere and not due to radial motion in the galaxy or a kick. In two spectra, there is also evidence of a more strongly redshifted component that would correspond to a disk atmosphere at R ∼ 70 GM/c 2; this component is significant at the 3σ level. Finally, in one spectrum, we find evidence of a disk wind with a blueshift of . If real, this wind would require magnetic driving.

Lie-Schwinger block-diagonalization and gapped quantum chains: Analyticity of the ground-state energy

We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have proven that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant; see [5]. The main result of this paper is that, under the same hypotheses, the ground-state energy is analytic for values of the coupling constant belonging to a sufficiently small disk in the complex plane centred at the origin, uniformly in the length of the chain. Furthermore, assuming that the interaction potentials are invariant under translations, we prove that, in the thermodynamic limit, the energy per site is analytic for values of the coupling constant in the same disk. In our proof we use a new method introduced in [7], which is based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. Indeed, to do this we prove a rather strong result concerning complex Hamiltonians corresponding to complex values of the coupling constant.

Indigenous Disk Beads in the Southern Southwest: Contemporary, Ethnographic, Ethnohistorical, and Archaeological Evidence

Within the Phoenix Basin in southern Arizona, disk beads have long been highly valued. Remarkably, the Akimel O’Odham (i.e., Pima) and Pee Posh (i.e., Maricopa) still place great importance on them today. Similar beads were formed from shell, stone, and clay. The effort necessary to manufacture beads from these constituents varies, but beads made from different materials are often macroscopically indistinguishable. Furthermore, some raw materials had to be imported, while other constituents were locally available. These factors should have affected the value of different bead types, and ethnographic evidence shows that indigenous people have clearly defined preferences for attributes, including color and material. In particular, beads made from shell and turquoise are recognized as being most valuable. This paper presents archaeological, ethnohistorical, ethnographic, and contemporary data that show that small disk beads were a form of wealth that was employed in trade transactions.

Neck Injury-Motor Vehicle Accident

HISTORY: A 57-year-old male was involved in a motorcycle accident, landing on his right side. Injuries included a flaccid right arm and hand, laceration of his right thigh, and a fracture of the 3rd metatarsal of his right foot. Patient’s chief complaint was pain in right shoulder and neck and an inability to complete ADLs. PHYSICAL EXAMINATION: Patient demonstrated flaccid right upper extremity with very little finger movement. Patient was able to achieve some finger flexion of all 5 digits and thumb opposition to index finger, although could not achieve pad to pad contact. All PROM of joints was preserved. Sensation was grossly intact. Patient tolerated initial examination well with mild discomfort during objective measures. No red flags arose during medical screening/systems review. DIFFERENTIAL DIAGNOSIS: 1. Cerebrovascular accident 2. Brachial plexus stretch injury 3. Brachial plexus avulsion injury TESTS AND RESULTS: MRI of cervical spine without contrast – small disc herniation, MRI of brachial plexus with and without contrast - edema involving right scalene and adjacent fascial planes extending into brachial plexus as well as into right C6/C7 and C7/T1 neural foramen to confirm nerve root sleeve tears, CTA of neck – negative for artery involvement, EMG study showed that the brachial plexus was not avulsed, but not currently sending EMG signals to hand. FINAL/WORKING DIAGNOSIS: Brachial plexus stretch injury TREATMENT AND OUTCOMES: 1. Therapeutic exercises (ROM, strength, endurance, stability), 2. Therapeutic activity (work specific, ADL specific), 3. Neuromuscular rehabilitation (muscle re-education, sequencing, coordination, neurodynamics), 4. Patient education (home exercise program, home safety). Patient has been responding well to a treatment plan combining aspects of the above mentioned interventions. Has been showing greater activation of right hand and forearm muscles with ability to complete weak grip at this stage. Right upper extremity remains supported in sling.

Evolution of α Centauri b’s protoplanetary disc

ABSTRACT With hydrodynamical simulations we examine the evolution of a protoplanetary disc around α Centauri B including the effect of the eccentric orbit binary companion α Centauri A. The initially circular orbit disc undergoes two types of eccentricity growth. First, the eccentricity oscillates on the orbital period of the binary, Porb, due to the eccentricity of the binary orbit. Secondly, for a sufficiently small disc aspect ratio, the disc undergoes global forced eccentricity oscillations on a time-scale of around $20\, P_{\rm orb}$. These oscillations damp out through viscous dissipation leaving a quasi-steady eccentricity profile for the disc that oscillates only on the binary orbital period. The time-averaged global eccentricity is in the range 0.05–0.1, with no precession in the steady state. The periastrons of the gas particles are aligned to one another. The higher the disc viscosity, the higher the disc eccentricity. With N-body simulations we examine the evolution of a disc of planetesimals that forms with the orbital properties of the quasi-steady protoplanetary disc. We find that the average magnitude of the eccentricity of particles increases and their periastrons become misaligned to each other once they decouple from the gas disc. The low planetesimal collision velocity required for planet formation suggests that for planet formation to have occurred in a disc of planetesimals formed from a protoplanetary disc around α Centauri B, said disc’s viscosity must be have been small and planet formation must have occurred at orbital radii smaller than about $2.5\, \rm au$. Planet formation may be easier with the presence of gas.

Fully Endoscopic Transforaminal Lumbar Discectomy for Upward Migration of Upper Lumbar Disc Herniation: Clinical and Radiological Outcomes and Technical Considerations.

Microdiscectomy for the upward migration of upper lumbar herniated discs has a high risk of isthmus and facet injury. Fully endoscopic transforaminal discectomy can preserve normal bony structures during discectomy. The purpose of this study was to assess the clinical and radiological outcomes of fully endoscopic transforaminal discectomy for upward migrated upper lumbar herniated discs. All patients had upward migrated disc herniation from L1–L2 to L3–L4 levels and were treated using fully endoscopic transforaminal discectomy under local anesthesia. All enrolled patients were monitored for more than 12 months. Clinical outcomes were assessed using the Oswestry Disability Index (ODI) and visual analog scale (VAS) of pain. Surgery-related complications were analyzed. In addition, radiological outcomes were investigated using postoperative magnetic resonance imaging (MRI) and lumbar dynamic X-ray. Twenty-eight patients were enrolled in this study. ODI and VAS significantly decreased after endoscopic transforaminal discectomy. Migrated ruptured disc particles were completely removed and confirmed on postoperative MRI in 26 of the 28 patients. Even though small remnant disc particles were detected in two patients, symptoms improved after endoscopic transforaminal discectomy. Early recurrence of herniated disc occurred at the operated segment in one patient. There were no significant complications associated with fully endoscopic transforaminal discectomy. Three patients experienced a postoperative transient tingling sensation and numbness of the leg. Fully endoscopic transforaminal lumbar discectomy may be an effective and alternative treatment option for upward migrated disc herniation in the upper lumbar area. In addition, fully endoscopic transforaminal lumbar discectomy may prevent complications associated with general endotracheal anesthesia and injuries of the isthmus and the facet joint.

Hints of a Population of Solar System Analog Planets from ALMA

Abstract The recent Atacama Large Millimeter/submillimeter Array (ALMA) Disk Substructures at High Angular Resolution Project (DSHARP) survey provided illuminating results on the diversity of substructures in planet-forming disks. These substructures trace pebble-sized grains accumulated at local pressure maxima, possibly due to planet–disk interactions or other planet formation processes. DSHARP sources are heavily biased to large and massive disks that only represent the high (dust flux) tail end of the disk population. Thus it is unclear whether similar substructures and corresponding physical processes also occur in the majority of disks that are fainter and more compact. Here we explore the presence and characteristics of features in a compact disk around GQ Lup A, the effective radius of which is 1.5–10 times smaller than those of DSHARP disks. We present our analysis of ALMA 1.3 mm continuum observations of the GQ Lup system. By fitting visibility profiles of the continuum emission, we find substructures including a gap at ∼10 au. The compact disk around GQ Lup exhibits similar substructures to those in the DSHARP sample, suggesting that mechanisms of trapping pebble-sized grains are at work in small disks as well. Characteristics of the feature at ∼10 au, if due to a hidden planet, are evidence of planet formation at Saturnian distances. Our results hint at a rich world of substructures to be identified within the common population of compact disks, and subsequently a population of solar system analogs within these disks. Such study is critical to understanding the formation mechanisms and planet populations in the majority of protoplanetary disks.

Angular momentum profiles of Class 0 protostellar envelopes

Context. Understanding the initial properties of star forming material and how they affect the star formation process is a key question. The infalling gas must redistribute most of its initial angular momentum inherited from prestellar cores before reaching the central stellar embryo. Disk formation has been naturally considered as a possible solution to this “angular momentum problem”. However, how the initial angular momentum of protostellar cores is distributed and evolves during the main accretion phase and the beginning of disk formation has largely remained unconstrained up to now. Aims. In the framework of the IRAM CALYPSO survey, we obtained observations of the dense gas kinematics that we used to quantify the amount and distribution of specific angular momentum at all scales in collapsing-rotating Class 0 protostellar envelopes. Methods. We used the high dynamic range C18O (2−1) and N2H+ (1−0) datasets to produce centroid velocity maps and probe the rotational motions in the sample of 12 envelopes from scales ~50 to ~5000 au. Results. We identify differential rotation motions at scales ≲1600 au in 11 out of the 12 protostellar envelopes of our sample by measuring the velocity gradient along the equatorial axis, which we fit with a power-law model v ∝ rα. This suggests that coherent motions dominate the kinematics in the inner protostellar envelopes. The radial distributions of specific angular momentum in the CALYPSO sample suggest the following two distinct regimes within protostellar envelopes: the specific angular momentum decreases as j ∝ r1.6±0.2 down to ~1600 au and then tends to become relatively constant around ~6 × 10−4 km s−1 pc down to ~50 au. Conclusions. The values of specific angular momentum measured in the inner Class 0 envelopes suggest that material directly involved in the star formation process (<1600 au) has a specific angular momentum on the same order of magnitude as what is inferred in small T-Tauri disks. Thus, disk formation appears to be a direct consequence of angular momentum conservation during the collapse. Our analysis reveals a dispersion of the directions of velocity gradients at envelope scales >1600 au, suggesting that these gradients may not be directly related to rotational motions of the envelopes. We conclude that the specific angular momentum observed at these scales could find its origin in other mechanisms, such as core-forming motions (infall, turbulence), or trace an imprint of the initial conditions for the formation of protostellar cores.

A novel numerical radius upper bounds for 2 × 2 operator matrices

ABSTRACT In this paper, we establish some numerical radius inequalities for bounded linear operator defined on a complex Hilbert space. As a natural application, the existence of the all polynomial zeros is identified in a specific small disk. Moreover, we provide a refinement of an earlier numerical radius inequality due to Herzallah et al. [Numerical radius inequalities for certain operator matrices. Integr Equ Oper Theory. 2011;71:129–147] and a generalization of Shebrawi's inequality [Numerical radius inequalities for certain operator matrices II. Linear Algebra Appl. 2017;523:1–12].

Electrodialytically treated MSWI fly ash use in clay bricks

Fly ash from municipal solid waste incineration (MSWI) is classified as hazardous waste, due to high heavy metal and salt content. Thus, beneficial use is restricted, and the fly ash hazardousness should be reduced before testing reuse options. Electrodialytic treatment can remove heavy metals and soluble salts from an MSWI fly ash suspension by applying an electric current to the suspension and thus be used to decontaminate the MSWI fly ash. In Greenland, MSWI fly ash is stored at uncontrolled disposal sites, and a more sustainable solution for handling fly ash is needed. At the same time, most construction materials are imported from Europe to Greenland, and increased use of local materials would greatly benefit the circular economy in the area. In this study, it was investigated if local Greenlandic resources could have potential in brickmaking. Two different clays; a Danish clay (used commercially in brick production) and one Greenlandic (not used commercially) and raw and electrodialytically treated MSWI fly ash from Sisimiut, Greenland were used. Small clay discs with 0, 10, 20 and 30% substitution of clay by MSWI fly ash were fired at 1000 °C for 24 h. Substituting clay with fly ash generally resulted in lower technical properties of the clay discs. Nevertheless, from this initial screening, the clay discs with electrodialytically treated fly ash and Greenlandic clay showed the lowest porosity and water absorption, which could be beneficial for use as construction material in cold climates.

The thermal-radiative wind in low-mass X-ray binary H1743−322 – II. Iron line predictions from Monte Carlo radiation transfer

ABSTRACT We show the best current simulations of the absorption and emission features predicted from thermal-radiative winds produced from X-ray illumination of the outer accretion disc in binary systems. We use the density and velocity structure derived from a radiation hydrodynamic code as input to a Monte Carlo radiation transport calculation. The initial conditions are matched to those of the black hole binary system H1743−322 in its soft, disc dominated state, where wind features are seen in Chandra grating data. Our simulation fits well to the observed line profile, showing that these physical wind models can be the origin of the absorption features seen, rather than requiring a magnetically driven wind. We show how the velocity structure is the key observable discriminator between magnetic and thermal winds. Magnetic winds are faster at smaller radii, whereas thermal winds transition to a static atmosphere at smaller radii. New data from XRISM (due for launch 2022 January) will give an unprecedented view of the physics of the wind launch and acceleration processes, but the existence of static atmospheres in small disc systems already rules out magnetic winds which assume self-similar magnetic fields from the entire disc as the origin of the absorption features seen.

Cold gas and dust: Hunting spiral-like structures in early-type galaxies

Context. Observations of neutral hydrogen (H I) and molecular gas show that 50% of all nearby early-type galaxies contain some cold gas. Molecular gas is always found in small gas discs in the central region of the galaxy, while neutral hydrogen is often distributed in a low-column density disc or ring typically extending well beyond the stellar body. Dust is frequently found in early-type galaxies as well. Aims. The goal of our study is to understand the link between dust and cold gas in nearby early-type galaxies as a function of H I content. Methods. We analyse deep optical g − r images obtained with the MegaCam camera at the Canada-France-Hawaii Telescope for a sample of 21 H I-rich and 41 H I-poor early-type galaxies. Results. We find that all H I-rich galaxies contain dust seen as absorption. Moreover, in 57% of these H I-rich galaxies, the dust is distributed in a large-scale spiral pattern. Although the dust detection rate is relatively high in the H I-poor galaxies (∼59%), most of these systems exhibit simpler dust morphologies without any evidence of spiral structures. We find that the H I-rich galaxies possess more complex dust morphology extending to almost two times larger radii than H I-poor objects. We measured the dust content of the galaxies from the optical colour excess and find that H I-rich galaxies contain six times more dust (in mass) than H I-poor ones. In order to maintain the dust structures in the galaxies, continuous gas accretion is needed, and the substantial H I gas reservoirs in the outer regions of early-type galaxies can satisfy this need for a long time. We find that there is a good correspondence between the observed masses of the gas and dust, and it is also clear that dust is present in regions further than 3 Reff. Conclusions. Our findings indicate an essential relation between the presence of cold gas and dust in early-type galaxies and offer a way to study the interstellar medium in more detail than what is possible with H I observations.

Lack of the Histone Methyltransferase Gene Ash2 Results in the Loss of Citrinin Production in Monascus purpureus

Absent, small, or homeotic discs 2 (Ash2), a histone H3K4 methyltransferase complex, has been implicated in the control of hyphal development and secondary metabolism in many kinds of filamentous fungi. We constructed an Ash2 deletion mutant (ΔAsh2) by using an Agrobacterium-mediated gene knockout method to investigate the function of the Ash2 gene in the mold Monascus purpureus. Lack of the Ash2 gene resulted in the formation of a lower colony phenotype with fluffy aerial hyphae that autolyzed as the colony grew on potato dextrose agar at 30°C. The production of pigments and the number of conidia were significantly lower in the ΔAsh2 than in the wild type. Citrinin production by the ΔAsh2 was not detected during 15 days of fermentation. Relative expression levels of secondary metabolite regulatory genes PigR and CTNR, secondary metabolite synthesizing genes PKSPT and CTN, key genes of mitogen-activated protein kinase pathway Spk1 and its downstream gene mam2, the conidium development control gene BrlA, and global regulatory genes LaeA and VeA were detected by the quantitative real-time PCR. These results indicate that the Ash2 gene is involved in conidial germination, pigment production, and citrinin production and plays a key role in development and secondary metabolism in M. purpureus.

Abstract 2: Heat Generating Nanocomposite To Reduce Infection

Purpose: To evaluate the combination of mild hyperthermia plus antibiotics for reducing Stapylococcus aureus biofilms associated with silicone materials. Methods: The additive effect of non-ablative hyperthermia with antibiotics was explored in a mouse model. Small disks of silicone with [4,4-bis(2-ethylhexyl)cyclopenta[2,1-b;3,4-b’]dithiophene-2,6-diyl-alt-2,1,3-benzoselenadiazole-4,7-diyl] (PCPDTBSe) nanoparticles were inoculated with Xen29 S. aureus biofilms. These disks were implanted in the flanks of hairless, immunocompetent Crl:SKH1-hrBR mice, with each mouse receiving with 1 silicone disk on the left flank and 1 nanocomposite disk on the right flank. After 24 hours, the mice were imaged using IVIS to evaluate the infection using the bioluminescent signature from the Xen 29. Mice were treated with either 200µL 2mg/mL Gentamicin orIP Saline, with or without laser (3W, CW, 800nm, 25s) applied directly over the flank 1 hour afterinjection. Mice were imaged using IVIS 24 hours after treatment. The disks and surrounding infected tissue were extracted and homogenized. The homogenate was diluted and plated in triplicate to calculate the CFU /g per disk. The change in IVIS luminescence before and after treatment was compared. A total of 14 mice were implanted and treated. 2 mice received treatment of IP Gentamicin only, 4 mice received IP saline and laser therapy, and 8 mice received IP Gentamicin and laser therapy. Results: Only mice treated with Gentamicin and laser therapy demonstrated a decrease in luminescence on IVIS after treatment, representing a decrease in bacterial burden. These mice also demonstrated the greatest percent decrease (-65.8%) of CFU/g on the nanocomposite disk relative to the silicone disk. This was also the only treatment group which saw a significant decrease in bacterial burden on the nanocomposite disk relative to the silicone disk. Conclusion: Mice treated with the combined Gentamicin and laser therapy had the largest percent reduction in biofilm burden on the BSe-Si disk relative to the Si disk. This supports the hypothesis that joint heat and antibiotic therapy can be effective at reducing biofilms in vivo. Ultimately, this technology may have clinical utility towards treating biofilms on indwelling medical devices, mitigating a major burden of disease.

Wear Analysis of Compression Ignition Engine using Mesua ferea linn Methyl Eseter

As the standard of living is increasing day by day, the demand for energy consumption is also increasing due to the increasing population and expanding urbanization.At the same time,the whole world is aware about the depletion of petroleum reserves and its impact on environment pollution through exhaust emissions which has become a major threat to the whole world, so to deal with this there is an urgent need of suitable alternative fuels for use in diesel engine. Biodiesel is a suitable alternative fuel which is generally produced from renewable biological resources like vegetable oil and animal fats. It is biodegradable , non toxic, has lower emissions profile and at the same time environment friendly. In this study, biodiesel is prepared using mesua ferea linn(MFL)seed oil by transterification process. Two similar engines were operated for long turm endurance test using optimum biodiesel blend and petroleum-diesel oil, respectively. Atomic absorption spectroscopy (AAS) was used to check the wear debris concentration in lubricating oil samples which were drawn from both the engines after a fixed interval of time. Biodiesel operated engine showed lower wear debris concentration as compared to diesel operated engine. Scanning electron microscopy( SEM) showed lower damage of small disk of piston ring when operated with 20% biodiesel fuel, which revealed the better lubricity of biodiesel fuel. Also, carbon deposition of various parts like fuel injector, piston head were found to be lower in biodiesel used engine than that of diesel used engine.

A Model for ERD2 Function in Higher Plants.

ER lumenal proteins have a K(H)DEL motif at their C-terminus. This is recognized by the ERD2 receptor (KDEL receptor in animals), which localizes to the Golgi apparatus and serves to capture escaped ER lumenal proteins. ERD2-ligand complexes are then transported back to the ER via COPI coated vesicles. The neutral pH of the ER causes the ligands to dissociate with the receptor being returned to the Golgi. According to this generally accepted scenario, ERD2 cycles between the ER and the Golgi, although it has been found to have a predominant Golgi localization. In this short article, we present a model for the functioning of ERD2 receptors in higher plants that explains why it is difficult to detect fluorescently tagged ERD2 proteins in the ER. The model assumes that the residence time for ERD2 in the ER is very brief and restricted to a specific domain of the ER. This is the small disc of ER immediately subjacent to the first cis-cisterna of the Golgi stack, representing specialized ER export and import sites and therefore constituting part of what is known as the “secretory unit”, a mobile aggregate of ER domain plus Golgi stack. ERD2 molecules in the ER domain of the secretory unit may be small in number, transient and optically difficult to differentiate from the larger population of ERD2 molecules in the overlying Golgi stack in the confocal microscope.

A comparison of SMA (styrene maleic acid) and DIBMA (di-isobutylene maleic acid) for membrane protein purification

The use of styrene maleic acid co-polymer (SMA) for membrane protein extraction and purification has grown in recent years. SMA inserts in the membrane and assembles into small discs of bilayer encircled by polymer, termed SMA lipid particles (SMALPs). This allows purification of membrane proteins whilst maintaining their lipid bilayer environment. SMALPs offer several improvements over conventional detergent approaches, however there are limitations, most notably a sensitivity to low pH and divalent cations. Recently it was shown that the aliphatic diisobutylene-maleic acid (DIBMA) copolymer, was also able to directly solubilise membranes forming DIBMALPs (DIBMA lipid particles), and that this polymer overcame some of the limitations of SMA. In this study the ability of DIBMA to solubilise and purify functional membrane proteins has been compared to SMA. It was found that DIBMA is able to solubilise several different membrane proteins from different expression systems, however for some proteins it gives a lower yield and lower degree of purity than SMA. DIBMA extracted G protein-coupled receptors retain ligand- and G protein-binding. DIBMALPS are larger than SMALPs and display a decreased sensitivity to magnesium. However the stability of DIBMALPs appears to be lower than SMALPs. The lower purity and lower stability are likely linked to the larger size of the DIBMALP particle. However, this also offers a potentially less rigid lipid environment which may be more amenable to protein dynamics. Therefore the optimal choice of polymer will depend on which features of a protein are to be investigated.

Fundamental tones of clamped plates in nonpositively curved spaces

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M,g) is universally bounded from below by (n−1)416κ4 whenever the κ-Cartan-Hadamard conjecture holds on (M,g), e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in (M,g) of volume v>0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature −κ2 provided that v≤cn/κn with c2≈21.031 and c3≈1.721, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. K≡κ=0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low-dimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on (M,g)) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

Different modes of star formation – II. Gas accretion phase of initially subcritical star-forming clouds

ABSTRACT The accretion phase of star formation is investigated in magnetically dominated clouds that have an initial subcritical mass-to-flux ratio. We employ non-ideal magnetohydrodynamic simulations that include ambipolar diffusion and ohmic dissipation. During the early prestellar phase, the mass-to-flux ratio rises towards the critical value for collapse, and during this time the angular momentum of the cloud core is reduced significantly by magnetic braking. Once a protostar is formed in the core, the accretion phase is characterized by the presence of a small amount of angular momentum but a large amount of magnetic flux in the near-protostellar environment. The low angular momentum leads to a very small (or even non-existent) disc and weak outflow, while the large magnetic flux can lead to an interchange instability that rapidly removes flux from the central region. The effective magnetic braking in the early collapse phase can even lead to a counterrotating disc and outflow, in which the rotation direction of the disc and outflow is opposite to that of the infalling envelope. The solutions with a counterrotating disc, tiny disc, or non-existent disc (direct collapse) are unique outcomes that are realized in collapse from magnetically dominated clouds with an initial subcritical mass-to-flux ratio.