Disk Plane Research Articles

Essential Norm of Sum of Generalized Weighted Composition Operators between Weighted Spaces of Analytic Functions

Let D be the open unit disc in the complex plane C , H ( D ) be the space of analytic functions on D and S ( D ) be the set of analytic self-maps on D . Let Ψ = ( ψ j ) j = 0 k be such that ψ j ∈ H ( D ) and φ ∈ S ( D ) . We obtain estimates of the essential norms of the following sum of generalized weighted composition operators T Ψ , φ k f = ∑ j = 0 k ψ j · f ( j ) ° φ = ∑ j = 0 k D ψ j , φ j f , f ∈ H ( D ) , between weighted Banach spaces of analytic functions H v ∞ ( H v 0 ) and H w ∞ ( H w 0 ) , which unifies the study of products of composition operators, multiplication operators and differentiation operators. As an application, we give the estimates of the essential norm of the operators D ψ , φ n : B v → B w , B v → H w ∞ and T μ 1 , μ 2 , φ n : B v → H w ∞ . Also, we give the new characterizations of the estimates of the essential norms of the operators T Ψ , φ k and D ψ j , φ j .

Numerical investigation on aerodynamic noise of rigid coaxial rotor in forward flight with lift-offset

A computational fluid dynamics (CFD) solver based on Reynolds-averaged Navier–Stokes (RANS) equations and high-efficiency trim model is used to simulate the unsteady aerodynamics of coaxial rotor. Farassat 1 A equations are adopted for predicting rotor far-field aerodynamic noise. Forward flight cases of a two-bladed rigid coaxial rotor in different advance ratios and lift-offsets (LOS) are conducted. Sound pressure histories of different observation points and noise radiation maps are analyzed. Results indicate that, the intensity of rotor thickness noise moves towards the advancing side in the rotor disk plane, with the increase of advance ratio. Due to the superposition effect, the thickness noise of coaxial rotor is symmetrical, which is different with the single rotor. At low advance ratio, loading noise of the rigid coaxial rotor is enhanced near the blade-crossing azimuths caused by the unsteady interaction of the twin rotors. The enhancement turns weak with the increase of advance ratio. At high advance ratio, increasing LOS tends to enhance the rotor-self BVI noise, while weakening the inter-rotor interaction noise. At certain flight states, appropriate LOS can reduce the overall noise radiation intensity of rigid coaxial rotor.

Assessing the robustness of the Galactic rotation curve inferred from the Jeans equations using Gaia DR3 and cosmological simulations

Context. Several authors have recently applied Jeans modelling to Gaia-based datasets to infer the circular velocity curve for the Milky Way. These works have consistently found evidence for a continuous decline in the rotation curve beyond ~15 kpc, which may indicate the existence of a light dark matter (DM) halo. Aims. Using a large sample of Gaia DR3 data, we aim to derive the rotation curve of the Milky Way using the Jeans equations, and to quantify the role of systematic effects, both in the data and those inherent to the Jeans methodology under the assumptions of axisym-metry and time independence. Methods. We used data from the Gaia DR3 radial velocity spectrometer sample, supplemented with distances inferred through Bayesian frameworks, to determine the radial variation of the second moments of the velocity distribution for stars close to the Galactic plane. We used these profiles to determine the rotation curve using the Jeans equations under the assumption of axisym-metry and explored how they vary with azimuth and position above and below the plane of the Galactic disc. We applied the same methodology to an N-body simulation of a Milky Way-like galaxy impacted by a satellite akin the Sagittarius dwarf, and to the Auriga suite of cosmological simulations. Results. The circular velocity curve we infer for the Milky Way is consistent with previous findings out to ~15 kpc, where our statistics are robust. Due to the larger number of stars in our sample, we are able to reveal evidence of disequilibrium and deviations from axisymmetry closer in. For example, we find that the second moment of VR flattens out at R ≳ 12.5 kpc, and that the second moment of Vϕ is different above and below the plane for R ≳ 11 kpc. Our exploration of the simulations indicates that these features are typical of galaxies that have been perturbed by external satellites. From the simulations, we also estimate that the difference between the true circular velocity curve and that inferred from Jeans equations can be as high as 15%, but that it is likely of the order of 10% for the Milky Way. This is higher than the systematic uncertainties associated with the observations or those linked to most modelling assumptions when using the Jeans equations. However, if the density of the tracer population were truncated at large radii instead of being exponential as often assumed, this could lead to the erroneous conclusion of a steeply declining rotation curve. Conclusions. We find that steady-state axisymmetric Jeans modelling becomes less robust at large radii, indicating that particular caution must be exercised when interpreting the rotation curve inferred in those regions. A more careful and sophisticated approach may be necessary for precision measurements of the DM content of our Galaxy.

Complex Bipolar Multi-Fuzzy Sets

Convey human being's information to a mathematical formula and vice versa by a proper tool, is considered a first step to solve the problem of assigning a membership degree (suitable value) of an object to a set by decision-makers. Different information with its periodic circumstances highlighted the need for an extra variable to be built into the mathematical tools. This extra variable adds a significant role to represent a special type of information that has the same data but with a different meaning in different time/levels/phases. However, the idea of extending the range of membership degree to unit disk in the complex plane solved this problem. In this research, we choose bipolar multi-fuzzy information to be generalized to the complex realm using the Cartesian form "x+iy". Therefore, the formal definition of complex bipolar multi-fuzzy sets (CBMFS) is introduced with a range of membership lies in the unit square. The advantage of CBMFS is that the real and imaginary parts of CBMFSs can represent bipolar multi-fuzzy information. Also, some basic operations and numerical examples on CBMFS are presented and studied its properties. Moreover, the relations of CBMFS with each of bipolar fuzzy sets (BFS) and complex bipolar fuzzy sets (CBFS), respectively, are illustrated. Finally, two types of distances and ∂-equality under CBMFS are presented. Also, some illustrative examples and properties of complex bipolar multi-fuzzy distance and ∂-equality are obtained.

Development of a new Bridgman-type high-pressure cell by using built-in gasket up to 9.4GPa and evaluation of deformation.

We have developed a built-in gasket for the Bridgman-type opposed-anvil high-pressure cell, featuring a PTFE (Teflon) capsule of ϕ 2.0 (1.5) × 2.5 mm3, filled with a liquid pressure-transmitting medium. This gasket, comprising a stainless-steel plane disk, a stainless-steel support ring, and pyrophyllite support gaskets, has enhanced the sample space height, allowed for precise adjustment of the anvil top area, and facilitated easy electrical insulation of lead wires. We calibrated the pressure by detecting phase transitions in Bi and Sn through resistivity measurements, achieving nearly hydrostatic pressure up to 9.4GPa with this cell. Our analysis of the deformation of the gasket components under force has provided guidelines for effective pressurization.

Association between condylar surface computed tomography values in the coronal plane and temporomandibular joint disc position in jaw deformity patients: a retrospective study.

This study aimed to examine the relationship between computed tomography (CT) values of the condylar surface in the coronal plane, condylar morphology and the disc positions in the temporomandibular joint (TMJs) in patients with deformities before and after orthognathic surgery. MATERIALS AND METHODS: The maximum CT values (pixel values) on the condylar surface, condylar thickness, condylar width, condylar angle, condylar height, and joint space were measured. TMJ disc position was classified into five types (anterior type, fully covered type, posterior type, anterior disc displacement with [AW] and without reduction [AWO]), using magnetic resonance imaging (MRI), before and 1 year after surgery. Statistical analysis was performed to assess differences between groups and between the pre- and postoperative values. RESULTS: We enrolled 142 TMJs from 71 female patients diagnosed with jaw deformities. Disc position classification did not change after surgery in all TMJ. Significant differences were observed between anterior displacement (AW and AWO) and other types (fully covered type and posterior type) in the lateral CT value and condylar angle before and 1 year after surgery (P < 0.05). CONCLUSIONS: This study demonstrated a strong relationship between condylar surface CT values in the coronal plane and condylar angle with TMJ disc position classification.

The Discovery of Polarized Water Vapor Megamaser Emission in a Molecular Accretion Disk

For the first time in an extragalactic source, we detect linearly polarized H2O maser emission associated with the molecular accretion disk of NGC 1068. The position angles of the electric polarization vectors are perpendicular to the axes of filamentary structures in the molecular accretion disk. The inferred magnetic field threading the molecular disk must lie within ∼35° of the sky plane. The orientation of the magnetic fields relative to the disk plane implies that the maser region is unstable to hydromagnetically powered outflow; we speculate that the maser region may be the source of the larger-scale molecular outflow found in Atacama Large Millimeter/submillimeter Array studies. The new very long baseline interferometry observations also reveal a compact radio continuum source, NGC 1068*, aligned with the near-systemic maser spots. The molecular accretion disk must be viewed nearly edge on, and the revised central mass is M = (16.6 ± 0.1) × 106 M ☉.

A class of Hausdorff–Berezin operators on the unit ball

Abstract The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

Morphology of Molecular Clouds at Kiloparsec Scale in the Milky Way: Shear-induced Alignment and Vertical Confinement

The shape of the cold interstellar molecular gas is determined by several processes, including self-gravity, tidal force, turbulence, magnetic field, and galactic shear. Based on the 3D dust extinction map derived by Vergely et al., we identify a sample of 550 molecular clouds within 3 kpc of the solar vicinity in the Galactic disk. Our sample contains clouds whose size ranges from parsec to kiloparsec, which enables us to study the effect of Galactic-scale processes, such as shear, on cloud evolution. We find that our sample clouds follow a power-law mass–size relation of M∝32.00Rmax1.77 , M∝20.59RS2.04 , and M∝14.41RV2.29 , where Rmax is the major axis-based cloud radius, R S is the area-based radius, and R V is the volume-based radius, respectively. These clouds have a mean constant surface density of ∼7 M ⊙ pc−2 and follow a volume density–size relation of ρ∝2.60Rmax−0.55 . As cloud size increases, their shapes gradually transition from ellipsoidal to disk-like to bar-like structures. Large clouds tend to have a pitch angle of 28°−45°, where the angle is measured concerning the Galactic tangential direction. These giant clouds also tend to stay parallel to the Galactic disk plane and are confined within the Galactic molecular gas disk. Our results show that large molecular clouds in the Milky Way can be shaped by Galactic shear and confined in the vertical direction by gravity.

Multiplication operators on the weighted Sobolev disk algebra

Let ⅅ be the unit disk in the complex plane ℂ. For α > −1, the weighted Sobolev disk algebra SA(ⅅ, dAα ) consists of all analytic functions in the weighted Sobolev space W 2,2(ⅅ, dAα ). In this paper, we prove that the multiplication operator is similar to on SA(ⅅ, dAα ) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(ⅅ, dAα ) belongs to the commutant of by using the matrix representation of P. In addition, we compute the exact norm of Mz on SA(ⅅ, dAα ). Finally, we prove that on the unweighted Sobolev disk algebra SA(ⅅ) the restrictions of to different invariant subspaces zkSA(ⅅ) (k ≥ 1) are not unitarily equivalent, and the restrictions of (n ≥ 2) to different invariant subspaces Sj (0 ≤ j < n) are also not unitarily equivalent.

Subgroups and Homomorphism Structures of Complex Pythagorean Fuzzy Sets

This research introduces the notion of complex Pythagorean fuzzy subgroup (CPFSG). Both complex fuzzy subgroup (CFSG) and complex intuitionistic fuzzy subgroup (CIFSG) have significance in assigning membership grades in the unit disk in the complex plane. CFSG has a limitation solved by CIFSG, while CIFSG deals with a limited range of values. The important novelty of the CPFSG lies in its ability to solve the above limitations simultaneously and gets a wider range of values to be engaged in CPFSG. This work has introduced and investigated CPFSG as a new algebraic structure via the conditions that the sum of the square membership and non-membership lies on the unit interval for both the amplitude term and phase term. The result as any CIFSG is CPFSG but the convers is not true has been proved. Complex Pythagorean fuzzy coset has been defined and complex Pythagorean fuzzy normal subgroup (CPFNSG) and their algebraic characteristic has been demonstrated. Homomorphism on the CPFSG is shown. Some results as the inverse image of CPFSG and CPFNSG under isomorphism function are also a CPFSG and CPFNSG, respectively.

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

Piezo-viscous micropolar fluid flow between two parallel annular disks

A theoretical analysis of the piezo-viscous dependence of micropolar fluid flow between parallel annular disks has been conducted. It is assumed that viscosity changes exponentially with pressure. Pressure is analytically approximated using a small perturbation method with the viscosity coefficient as the perturbation parameter. Squeeze film time, pressure, and load carrying capacity are calculated and graphically demonstrated across various viscosity–pressure characteristics. The study examines how an increase in cross-viscosity correlates with notable growth in pressure, response time, and bearing load. According to the results, micropolar fluid offers an increase in the load-bearing capacity and therefore lengthens the response time to prevent the contact of parallel annular disks. These observations are useful for bearings, seals, and lubrication systems, where controlling viscosity and pressure is crucial for smooth operation and productivity. As the limiting case of micro-polar fluid when the coupling number tends to zero or the couple stress parameter approaches infinity, without the annular ring, the present derivation agrees well with the Newtonian viscous case of parallel planar squeezing disks described by Bird et al. [Transport Phenomena (John Wiley &amp; Sons, 2006)].

Magnetic field and kinetic helicity evolution in simulations of interacting disk galaxies

Context. There are indications that the magnetic field evolution in galaxies is influenced by tidal interactions and mergers between galaxies. Aims. We carried out a parameter study of interacting disk galaxies with impact parameters ranging from central collisions to weakly interacting scenarios. The orientations of the disks were also varied. In particular, we investigated how magnetic field amplification depends on these parameters. Methods. We used magnetohydrodynamics for gas disks in combination with live dark-matter halos in adaptive mesh refinement simulations. The disks were initialized using a setup for isolated disks in hydrostatic equilibrium. Since we focused on the impact of tidal forces on magnetic field evolution, adiabatic physics was applied. Small-scale filtering of the velocity and magnetic field allowed us to estimate the turbulent electromotive force (EMF) and kinetic helicity. Results. Time series of the average magnetic field in central and outer disk regions show pronounced peaks during close encounters and mergers. This agrees with observed magnetic fields at different interaction stages. The central field strength exceeds 10 μG (corresponding to an amplification factor of 2–3) for small impact parameters. As the disks are increasingly disrupted and turbulence is produced by tidal forces, the small-scale EMF reaches a significant fraction of the total EMF. The small-scale kinetic helicity is initially antisymmetric across the disk plane. Though its evolution is sensitive to both the impact parameter and inclinations of the rotation axes with respect to the relative motion of the disks, antisymmetry is generally broken through interactions and the merger remnant loses most of the initial helicity. The EMF and the magnetic field also decay rapidly after coalescence. Conclusions. The strong amplification during close encounters of the interacting galaxies is mostly driven by helical flows and a mean-field dynamo. The small-scale dynamo contributes significantly in post-interaction phases. However, the amplification of the magnetic field cannot be sustained.

Selection of third-party logistics in supply chain finance under probabilistic complex hesitant fuzzy sets and distance measures

Financial institutions attach great importance to the multiple attribute decision-making (MADM) problems of selecting an effective third-party logistics (3PL) service provider in supply chain finance. A suitable 3PL service provider can not only assist financial institutions in carrying out supply chain finance activity but can also replace financial institutions in supervising the operation of a target financing supply chain, therefore minimizing financial institutions' operating risks. However, there is no sufficient study on the selection of a 3PL service provider. Hence, the objective of this paper is to establish an innovative concept of the “probabilistic complex hesitant fuzzy set” (PCHFS), which is a hybrid structure of the complex hesitant fuzzy set and the probabilistic fuzzy set to manage complex information in real-decision theory. This paper examines the periodicity of probabilistic fuzzy information and extends the range from [0,1] to the unit disc in the complex plane to provide more ability to describe the full meaning of the information in a mathematical model. Based on the internal structure of the set and to find the degree of discrimination between the pairs of PCHFSs, the generalized distance measures and modified generalized distance measures are defined. Several properties and their relationships between them are derived in detail. Also, several cases of the proposed measures are exposed, which reduces them to the existing studies. Furthermore, based on these proposed measures, a multiple-attribute decision-making (MADM) approach is established in an uncertain environment, and several numerical examples are given to examine the feasibility and validity of the explored measures. Finally, the modified and parameterized distance measures based on PCHFSs are verified by comparing them with some existing measures.

Evidence for Primordial Alignment: Insights from Stellar Obliquity Measurements for Compact Sub-Saturn Systems

Despite decades of effort, the mechanisms by which the spin axis of a star and the orbital axes of its planets become misaligned remain elusive. In particular, it is of great interest whether the large spin–orbit misalignments observed are driven primarily by high-eccentricity migration—expected to have occurred for short-period, isolated planets—or reflect a more universal process that operates across systems with a variety of present-day architectures. Compact multiplanet systems offer a unique opportunity to differentiate between these competing hypotheses, as their tightly packed configurations preclude violent dynamical histories, including high-eccentricity migration, allowing them to trace the primordial disk plane. In this context, we report measurements of the sky-projected stellar obliquity (λ) via the Rossiter–McLaughlin effect for two sub-Saturns in multiple-transiting systems: TOI-5126 b (λ = 1 ± 48°) and TOI-5398 b ( λ=−8.1−6.3+5.3° ). Both are spin–orbit aligned, joining a fast-growing group of just three other compact sub-Saturn systems, all of which exhibit spin–orbit alignment. In aggregate with archival data, our results strongly suggest that sub-Saturn systems are primordially aligned and become misaligned largely in the postdisk phase, as appears to be the case increasingly for other exoplanet populations.

A dusty magnetospheric stream explaining the light curves of the dipper objects: Finding a new inclination threshold to produce dippers

Context. The so-called “dippers” are young stellar objects that exhibit dimming episodes in their optical light curves. The common interpretation for the occurrence of these dips is that dusty regions periodically or quasi-periodically cross the line of sight toward the object. Aims. We develop a model where we assume that these regions are located at the intersection of the magnetospheric stream with the disk. The stream is fed by gas and dust coming from the disk. As the material follows the magnetic field lines above the disk plane, it forms an opaque screen that partially blocks the stellar emission. The amount of extinction caused by the material crossing the line of sight depends on the abundance and location of the dust along the stream, which depends on the degree of dust evaporation due to the heating by the star. Methods. We run hydrodynamical simulations of dusty accretion streams to produce synthetic dipper light curves for a sample of low-mass young stars still accreting from their disk according to evolutionary models. We compare the distribution of the light curve amplitudes between the synthetic sample and observed samples of dippers from various star-forming regions. Results. Dust evaporation along the accretion column drives the distribution of photometric amplitudes. Our results suggest that most of the observed dippers correspond to systems seen at high inclination. However, dust survival within accretion columns may also produce dippers at lower inclination, down to about 45°. We find that the dust temperature arising from stellar irradiation should be increased by a factor 1.6 to find consistency between the fraction of dippers our model predicts in star-forming regions and the observed fraction of 20–30%. Conclusions. Transient dust survival in accretion columns appear as an alternative (or complementary) mechanism to inner disk warp occultation in order to account for low-inclination dippers in star-forming regions.

The Funk–Finsler Structure on the Unit Disc in the Hyperbolic Plane

The Funk–Finsler Structure on the Unit Disc in the Hyperbolic Plane

High-fidelity computational study of aerodynamic noise of side-by-side rotor in full configuration

This paper presents a numerical study of side-by-side rotor aircraft in full configuration during hover using high-fidelity computational fluid dynamics (CFD) and aeroacoustic simulations. CFD simulations are performed using HPCMP CREATE™-AV Helios, while acoustic calculations are conducted with PSU-WOPWOP. Overset structured grids are applied using the SA-DES turbulence model along with adaptive mesh refinement in the near- and off-body meshes. Four overlap cases at a specific collective pitch angle of 8° are considered. Results reveal a substantial influence of the fuselage on rotor performance, with improvement in the figure of merit being observed for each overlap configuration due to a partial ground effect. There is considerable increase in blade sectional thrust, pressure fluctuations, and noise levels seen in the 0% overlap configuration compared to the 25% overlap configuration. This increase is primarily due to the absence of rotor overlap, resulting in a lower induced velocity allowing for flow to travel upward, and reaching the rotor disk plane. The acoustic spectrum appears at all harmonics of the blade passing frequency for the 0% and 5% overlap configurations. In contrast, the acoustic spectrum is distinctly present only at even harmonics for the 15% and 25% overlap cases. A noise reduction benefit of 3-5 dBA is noticed in the full configuration cases having rotor overlap compared to the 0% overlap configuration. Acoustic destructive interference is observed in the directivity plot only for the 15% and 25% overlap configurations. Overall, this paper reveals that the fuselage has a significant impact on the performance, aerodynamics, and aeroacoustics of the side-by-side UAM vehicle in hover.

SPIRou spectropolarimetry of the T Tauri star TW Hydrae: magnetic fields, accretion, and planets

ABSTRACT In this paper, we report near-infrared observations of the classical T Tauri star TW Hya with the SPIRou high-resolution spectropolarimeter and velocimeter at the 3.6-m Canada–France–Hawaii Telescope in 2019, 2020, 2021, and 2022. By applying Least-Squares Deconvolution (LSD) to our circularly polarized spectra, we derived longitudinal fields that vary from year to year from –200 to +100 G, and exhibit low-level modulation on the 3.6 d rotation period of TW Hya, despite the star being viewed almost pole-on. We then used Zeeman–Doppler Imaging to invert our sets of unpolarized and circularly polarized LSD profiles into brightness and magnetic maps of TW Hya in all four seasons, and obtain that the large-scale field of this T Tauri star mainly consists of a 1.0–1.2 kG dipole tilted at about 20° to the rotation axis, whereas the small-scale field reaches strengths of up to 3–4 kG. We find that the large-scale field is strong enough to allow TW Hya to accrete material from the disc on the polar regions at the stellar surface in a more or less geometrically stable accretion pattern, but not to succeed in spinning down the star. We also report the discovery of a radial velocity signal of semi-amplitude $11.1^{+3.3}_{-2.6}$ m s−1 (detected at 4.3σ) at a period of 8.3 d in the spectrum of TW Hya, whose origin may be attributed to either a non-axisymmetric density structure in the inner accretion disc, or to a $0.55^{+0.17}_{-0.13}$MꝜ candidate close-in planet (if orbiting in the disc plane), at an orbital distance of 0.075 ± 0.001 au.