Spherical Sector Research Articles

Quaternionic spherical sectorial and dissipative operators via S-resolvent kernels

In this paper, we treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute and develop a theory of quaternionic non-negative operator, spherical sectorial operator and dissipative operator via S-resolvent kernels in quaternionic locally convex spaces (short for q.l.c.s.). The notions of quaternionic non-negative operators and quaternionic (m-)dissipative operators are introduced via S-resolvent operators and H-valued inner product on Hilbert H-bimodule. By choosing the suitable spherical sector, the spherical sectorial operator is introduced to establish the relationship with the quaternionic non-negative operator. It is crucial to note that the quaternionic operators we consider do not necessarily commute.

Near-Field Source Localization and Beamforming in the Spherical Sector Harmonics Domain

Near-Field Source Localization and Beamforming in the Spherical Sector Harmonics Domain

Computation of spherical sector harmonics norm with application to blind source separation.

Spherical microphone arrays (SMAs) are widely being used for source localization and separation. However, it is uneconomical to build a full SMA when sources are present in restricted regions of environment. Hence, a spherical sector microphone array is utilized for blind source separation for the first time. In particular, the norm of the spherical sector harmonics basis function is computed for mixing matrix estimation. The estimated steering vectors are clustered using mean-shift algorithm. The number of sources is estimated automatically from the number of clusters. The developed mathematical framework is verified using various simulations and experiments on real data.

Directional active noise control for drones

Drones have become ubiquitous in various everyday applications, but their noise pollution remains an annoying issue. In this paper, we study the possibility of applying active noise control on drones with attached speakers. We propose the active noise control algorithms designed to iteratively calculate driving signals of speakers for canceling the outgoing noise from drones, specifically focusing on the blade passage frequencies within a directional spherical sector region. Two active noise control strategies are proposed: (i) minimizing the residual signal coefficients and (ii) minimizing the acoustic potential energy within the sector region. We derive update equations for two variables: (a) the loudspeaker weights and (b) secondary source spherical sector harmonics coefficients. Considering the power and loading limitations of drones, simulations are conducted under constraints of secondary source numbers to investigate the noise reduction performance with various secondary source setups. Simulation results demonstrate that the proposed method can effectively reduce the noise level over the directional sector region with a fast convergence speed.

A Theoretical Study of Interband Absorption Spectra of Spherical Sector Quantum Dots under the Effect of a Powerful Resonant Laser.

We explore the variation of interband absorption spectra of GaAs spherical sector quantum dots (QDs) in response to a strong resonant laser, using the renormalized wave function method. Even though a spherical sector QD appears identical to a section cut from a spherical QD, it contains a controllable additional spatial parameter, the apical angle, which results in radically different wave functions and energy levels of particles, and is anticipated to exhibit novel optical properties. The obtained findings reveal that the apical angle of the dot has a considerable effect on the interband absorption spectrum. With the increase in the dot apical angle, a significant redshift of the interband absorption peaks has been identified. Increasing the pump laser detuning and dot radius yields similar results. Especially when a powerful resonant laser with tiny detuning is utilized, a dynamical coupling between electron levels arises, resulting in the formation of new interband absorption peaks. These new peaks and the former ones were similarly influenced by the aforementioned parameters. Furthermore, it is thought that the new peaks, when stimulated by a suitable laser, will produce the entangled states necessary for quantum information.

Spherical-sector harmonics domain processing for wideband source localization using spherical-sector array of directional microphones

The spherical microphone array can be uneconomical for applications where the sources arrive only from a known section of the sphere. For this reason, the spherical-sector harmonics was developed for processing spherical sector array. The orthonormal spherical sector harmonics (SSH) basis functions which accounts for the discontinuity arising from sectioning the sphere have been developed and shown to work for the array of omnidirectional microphones. In this work, the SSH basis functions are applied to far-field wideband sound source localization using spherical-sector array of first-order directional microphones (cardioid microphones). The array manifold interpolation method is used to produce the steered covariance matrix and the MUSIC algorithm applied for the direction of arrival estimation. The root-mean-square error performance of this spherical-sector array of the first-order cardioid microphones is compared against that of the omnidirectional microphones for different directions and signal-to-noise ratio.

Відновлення та зміцнення деталей і робочих органів сільськогосподарських машин, працюючих в абразивному середовищі

The working bodies of tillage agricultural machines (ploughshare, plow shelf, cultivator paw, harrow disc) work in conditions of abrasive operation, shock loads and the impact of external aggressive environment. Their repair is to restore the original size of parts and wear resistance of functional work surfaces. During the operation of tillage machines and units, the working surfaces are subjected to uneven wear, which reduces the life of parts and increases the cost of their replacement and restoration. The working bodies of tillage agricultural machines (ploughshare, plow shelf, cultivator paw, harrow disc) work in conditions of abrasive operation, shock loads and the impact of external aggressive environment. Their repair is to restore the original size of parts and wear resistance of functional work surfaces. During the operation of tillage machines and units, the working surfaces are subjected to uneven wear, which reduces the life of parts and increases the cost of their replacement and restoration. The current level of operation of agricultural machinery is characterized by increased speeds and high productivity, which, in turn, leads to increased wear of parts. accordingly, the reliability of the recovered ensure the restorative parts. Surfacing makes it possible to obtain layers of any thickness and chemical composition on the working surfaces of parts; apply a welded layer with different properties, ie high hardness and wear resistance. In this work we use point (point) hardening, which consists in micro-dripping of the hard alloy into the surface structure of parts that undergo intense abrasive wear. The geometry of the hardening point is a spherical sector, which facilitates the rapid removal of heat from the surfacing bath in the base metal and the environment, so the hardening points harden at high speed and in a short time. At point strengthening of details one of the main conditions providing stable quality of process - coordination of welding current and arc voltage. The sawtooth profile of the ploughshare blade (cultivator paws) is maintained throughout its service life. providing effective cutting even with the blades worked. We will fully consider the welding stress and deformation of the steel and tungsten carbide, to ensure the using performance of products. About tungsten carbide, we use the most advanced production equipment and the most reasonable production process during the material selection, burdening, pressing and sintering. We always fully consider the special requirements of different products on the strength, hardness, grain size of tungsten carbide.

A principal stress cap model for stresses in square silos, with some examples of exponential stress

The principal stress cap approach has been used to model stresses in square hoppers, based upon an orthogonal grid upon a spherical surface sector, bounded by “great circles”. The model agrees with Rotter's correlation for lateral variation of wall normal stress for concave, active systems at uniform state of stress. However, systems of convex, passive stress do not conform to Rotter. The data of Jarrett{Geotechnique, 1995, 45, 95–104} and Brown{Chemical Engineering Science, 2000, 55, 4399–4413} have been modelled successfully, both qualitatively and quantitatively. Regions of the exponential stress regime have been found in the corner regions of the silo, alongside conventional Janssenian stress at the wall mid-points. Equivalent data of Jarrett and Brown are in reasonable agreement when plotted in dimensionless form. • Stresses in square hoppers have been modelled. • Principal stress cap approach is easy-to-use and visual model. • The model has been fitted to 2 sources of experimental data. • The data and the model show regions of exponential stress. • Horizontal variations in wall normal stress are seen.

Optimal beamformer design in spherical sector harmonics domain

Beamforming using spherical microphone arrays (SMAs) with processing in spherical harmonics (SH) domain is widely being studied. This is due to the ease of array processing in SH domain with no spatial ambiguity. Despite the widespread applications of SMA in beamforming, its size makes it inconvenient for practical or commercial use. Building of SMA over a rigid sphere is also a challenging task. Additionally, use of the entire sphere comes at the cost of more microphones and signals to process. Hence, it is uneconomic to use full SMA when sources are only present in restricted regions of the environment. Attempts have been made in literature to use hemispherical microphone array for beamforming. Acoustic image principle was utilized to enable application of SH but with greater computational complexity. In this paper, the use of a spherical sector microphone array is proposed for beamforming. An orthonormal spherical sector harmonics (S2H) basis function is used for data model development. An ideal direction-invariant beampattern using S2H basis function is utilized. Subsequently, to make the beamformer robust against sensor noise with a sharp directivity pattern at the direction of arrival and zero elsewhere, a new maximum directivity and white noise gain (MD-WNG) beamformer is designed for a spherical sector microphone array. Performance analysis of the proposed beamformer is presented using various experiments on directivity pattern and array gain. The proposed method is additionally compared with several existing beamforming techniques. The comparison reveals that the S2H based beamforming provides reasonably better results compared to the existing methods.

Acoustic Levitation for Large Particle Based on Concave Spherical Transducer Arrays

With the advantages of non-contact, compatibility, no chemical effect and safety, acoustic levitation of object has great potential to apply in chemistry, biology, medicine field and micro robots. Suspension to large particle higher than the half wavelength( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.5\lambda $ </tex-math></inline-formula> ) of sound wave, is a challenge technology in micro-nanoscale manipulation. An increased large quantity probes, double sided and four sided phased array with virtual vortex can provide higher capture power, control ability and stability of particle fixation and the manipulation. However, for a limited confined space, the additional probes and sided phased array emission and receiving device is hard to be established and applied. Additional probes and sided ultrasonic phased array requires higher processing ability of MCU (Micro Control Unit) and the quantities of transducers. Further, multiparticles suspension is rare to be reported. In this study, a relatively simple suspension device of one sided concave spherical transducer array is designed, aiming to levitate a large particle and multi-particles. Based on the one sided acoustic field simulation of ultrasonic array, the optimized structure of open spherical sector (OSS) is constructed to establish levitation device. Experimental platform integrating FPGA controller, amplifier and impedance matching is developed to drive the transducer arrays, where a polystyrene particle with 6 mm ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.71\lambda $ </tex-math></inline-formula> ) diameter is successfully levitated. The virtual twin trap sound pressure of transducer array is measured to evaluate the particle stability. Experiments demonstrate the correctness of virtual twin trap acoustic field analysis, the feasibility in the manipulation to the large particle and multi-particles by acoustic trap.

Existence of Nonradial Domains for Overdetermined and Isoperimetric Problems in Nonconvex Cones

In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain D on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.

Quantum Revivals in Curved Graphene Nanoflakes.

Graphene nanostructures have attracted a lot of attention in recent years due to their unconventional properties. We have employed Density Functional Theory to study the mechanical and electronic properties of curved graphene nanoflakes. We explore hexagonal flakes relaxed with different boundary conditions: (i) all atoms on a perfect spherical sector, (ii) only border atoms forced to be on the spherical sector, and (iii) only vertex atoms forced to be on the spherical sector. For each case, we have analysed the behaviour of curvature energy and of quantum regeneration times (classical and revival) as the spherical sector radius changes. Revival time presents in one case a divergence usually associated with a phase transition, probably caused by the pseudomagnetic field created by the curvature. This could be the first case of a phase transition in graphene nanostructures without the presence of external electric or magnetic fields.

Підвищення технічного ресурсу і довговічності робочих органів ґрунтообробних сільськогосподарських машин

The working bodies of tillage agricultural machines (ploughshare, plow shelf, cultivator paw, harrow disc) work in conditions of abrasive operation, shock loads and the impact of external aggressive environment. Their repair is to restore the original size of parts and wear resistance of functional work surfaces. During the operation of tillage machines and units, the working surfaces are subjected to uneven wear, which reduces the life of parts and increases the cost of their replacement and restoration. The working bodies of tillage agricultural machines (ploughshare, plow shelf, cultivator paw, harrow disc) work in conditions of abrasive operation, shock loads and the impact of external aggressive environment. Their repair is to restore the original size of parts and wear resistance of functional work surfaces. During the operation of tillage machines and units, the working surfaces are subjected to uneven wear, which reduces the life of parts and increases the cost of their replacement and restoration. The current level of operation of agricultural machinery is characterized by increased speeds and high productivity, which, in turn, leads to increased wear of parts. accordingly, the reliability of the recovered machines will depend on the wear resistance and fatigue strength of the recovered parts. The leading place in the repair of machines is occupied by welding and surfacing processes, which account for about 80% of all restorative parts. Surfacing makes it possible to obtain layers of any thickness and chemical composition on the working surfaces of parts; apply a welded layer with different properties, ie high hardness and wear resistance. In this work we use point (point) hardening, which consists in micro-dripping of the hard alloy into the surface structure of parts that undergo intense abrasive wear. The geometry of the hardening point is a spherical sector, which facilitates the rapid removal of heat from the surfacing bath in the base metal and the environment, so the hardening points harden at high speed and in a short time. At point strengthening of details one of the main conditions providing stable quality of process - coordination of welding current and arc voltage. The sawtooth profile of the ploughshare blade (cultivator paws) is maintained throughout its service life. providing effective cutting even with the blades worked.

Dynamic analysis of a spherical sandwich sector with piezoelectric face sheets and FG-CNT core subjected to low-velocity impact

Dynamic analysis of a spherical sandwich sector with piezoelectric face sheets and FG-CNT core subjected to low-velocity impact

A hybrid Eulerian-Eulerian/Eulerian-Lagrangian method for dense-to-dilute dispersed phase flows

Problems involving a transition between dense and dilute dispersed phase regimes often require hybridization of Eulerian-Eulerian (EE) and Eulerian-Lagrangian (EL) methods for accurate and efficient computational modeling. A hybrid EE-EL formulation is developed in this work from first principles, which asymptotes to well-established EE and EL methods in limiting conditions. A smooth and dynamic transition criterion and a corresponding algorithm for conversion between the two representations of the dispersed phase are developed. To use EE and EL in their respective regions of effectiveness, the transition criterion is designed as a function of local volume fraction and local kinetic energy of random uncorrelated motion of particles. Several one-dimensional numerical tests are conducted to analyze limits of the hybrid EE-EL method in dense (up to 65% volume loading) and dilute conditions, at first, without any conversion between EE and EL, and then the ability of the conversion algorithm to smoothly transition from EE to EL and vice versa is evaluated using prespecified and dynamically computed transition criteria for one- and two-dimensional tests. Particle evolution in two-dimensional frozen turbulence is simulated to evaluate the method's ability to dynamically transition from EE to EL and vice versa in regions of particle trajectory crossing. Finally, the hybrid method is used for simulating dispersion of an initially dense particle cloud in a three-dimensional spherical sector blast.

Spherical sector harmonics representation of sound fields using a microphone array over spherical sector.

Spherical microphone arrays (SMAs) are widely used for sound recording and analysis, with processing being done in the spherical harmonics (SH) domain. This is due to the ease of array processing in the SH domain without spatial ambiguity. However, it is uneconomical to construct a full SMA when sources are present in restricted regions of the environment. Additionally, the use of a full SMA comes at the cost of more microphone signals to process. Attempts have been made to use hemispherical microphone arrays on the basis of the acoustic image principle, enabling application of SH but with greater computational complexity. In this paper, the use of a spherical sector microphone array instead of a full SMA is proposed. An orthonormal spherical sector harmonics (S2H) basis function is developed for accurate representation of pressure over the sector. The orthonormality of the S2H function is established using orthogonality of shifted associated Legendre polynomials and a scaled exponential function. An addition theorem for S2H basis functions is derived. The S2H basis function is applied to the decomposition of a sound field over a sector array. The S2H basis function has potential applications to brain source localization and physiological shape description.

S$^2$H Domain Processing for Acoustic Source Localization and Beamforming Using Microphone Array on Spherical Sector

Spherical microphone arrays (SMAs) are widely used for the localization of acoustic sources and beamforming. However, building an SMA over a rigid sphere is a challenging task, and it is uneconomic to build full SMAs when the sources are in some restricted regions of the environment. This paper addresses the issue of multiple source localization and beamforming using a microphone array on a spherical sector. The discontinuity over the boundary is handled by solving the Helmholtz equation over the sector. As the sound pressure is present only on the microphones over that sector, novel orthonormal spherical sector harmonics (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> H) basis functions are developed. The basis functions defined over the sector are in generalized form, for which the usual spherical harmonics are a special case. To ensure orthonormality of S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> H basis functions, the orthogonality of the shifted associated Legendre polynomials and scaled exponential functions are established for the same sector. The new S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> H basis functions are utilized for far-field data model development. Subsequently, multiple signal classification and minimum variance distortionless response spectra are formulated in the S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> H domain for wideband and narrowband source localization. An ideal direction-invariant beampattern using S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> H basis functions is also derived. The performance of the proposed methods is analyzed using various source localization experiments. The acoustic image principle for beamforming is extended herein for source localization. Additionally, the proposed methods are compared with acoustic image principle-based source localization using spherical harmonics.

Curvature induced stabilization of vortices on magnetic spherical sector shells

Finite element micromagnetics are used to simulate the vortex and onion states on permalloy spherical shells with different curvatures. To distinguish between the effects of size and curvature (covaried in the typical semi-spherical or crescent shaped caps) a family of spherical sectors where the volume is varied at constant curvature is considered. This family is parametrized by intersecting cone angle. Permalloy is chosen as a typical case of an easy-surface magnet in which the demagnetization field bounds the magnetization to follow the curvature of the surface. The thickness of the shell is set to 3 nm, i.e. comparable to the exchange length, so that the local magnetization does not vary along the thickness of the shell. It is found that as the curvature increases the vortex states become favorable down to smaller sizes. This is mainly due to the increased exchange energy cost of the onion states at larger curvatures. Furthermore, it is found that at larger curvatures, stronger in-plane fields are required to destabilize the vortex states and turn them into onion states.

Einstein-Maxwell-scalar black holes: The hot, the cold and the bald

The phenomenon of spontaneous scalarisation of charged black holes (BHs) has recently motivated studies of various Einstein-Maxwell-scalar models. Within these models, different classes of BH solutions are possible, depending on the non-minimal coupling function f(ϕ), between the scalar field and the Maxwell invariant. Here we consider the class wherein both the (bald) electrovacuum Reissner-Nordström (RN) BH and new scalarised BHs co-exist, and the former are never unstable against scalar perturbations. In particular we examine the model, within this subclass, with a quartic coupling function: f(Φ)=1+αΦ4. The domain of existence of the scalarised BHs, for fixed α, is composed of two branches. The first branch (cold scalarised BHs) is continuously connected to the extremal RN BH. The second branch (hot scalarised BHs) connects to the first one at the minimum value of the charge to mass ratio and it includes overcharged BHs. We then assess the perturbative stability of the scalarised solutions, focusing on spherical perturbations. On the one hand, cold scalarised BHs are shown to be unstable by explicitly computing growing modes. The instability is quenched at both endpoints of the first branch. On the other hand, hot scalarised BHs are shown to be stable by using the S-deformation method. Thus, in the spherical sector this model possesses two stable BH local ground states (RN and hot scalarised). We point out that the branch structure of BHs in this model parallels the one of BHs in five dimensional vacuum gravity, with [Myers-Perry BHs, fat rings, thin rings] playing the role of [RN, cold scalarised, hot scalarised] BHs.

A new notion of error bounds: necessary and sufficient conditions

In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space $${\mathbb {R}}^n$$ satisfying $$0\in F$$ . It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Thera (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in $${\mathbb {R}}^n$$ then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.