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Sharpening Littlewood subordination principle with univalent symbol

AbstractLet be a subharmonic function on the open unit disc , centered at the origin of the complex plane, and let be a holomorphic function such that . A classical result, known as Littlewood subordination principle, states , where and are integral means over the circle of radius centered at the origin, of the functions and , respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function is univalent, by proving that where . We also list some applications of this result including an improved variant of Rogosinski theorem with univalent symbol.

Spiral complete coverage path planning based on conformal slit mapping in multi-connected domains

The generation of smoother and shorter spiral complete coverage paths in multi-connected domains is a crucial research topic in path planning for robotic cavity machining and other related fields. Traditional methods for spiral path planning in multi-connected domains typically incorporate a subregion division procedure that leads to excessive subregion bridging, requiring longer, more sharply turning, and unevenly spaced spirals to achieve complete coverage. To address this issue, this paper proposes a novel spiral complete coverage path planning method using conformal slit mapping. It takes advantage of the fact that conformal slit mapping can transform multi-connected domains into regular disks or annuluses without the need for subregion division. Firstly, a slit mapping calculation technique is proposed for segmented cubic spline boundaries with corners. Secondly, a spiral path spacing control method is developed based on the maximum inscribed circle radius between adjacent conformal slit mapping iso-parameters. Thirdly, the spiral coverage path is derived by offsetting iso-parameters. Numerical experiments indicate that our method shares a comparable order-of-magnitude in computation time with the traditional PDE-based spiral complete coverage path method, but it excels in optimizing total path length, smoothness, and spacing consistency. Finally, we performed experiments on cavity milling and dry runs to compare the new method with the traditional PDE-based method in terms of machining duration and steering impact, respectively. The comparison reveals that, with both algorithms achieving complete coverage, the new algorithm reduces machining time and steering impact by 12.34% and 22.78%, respectively, compared with the traditional PDE-based method.

Prediction of the non-SUSY AdS conjecture on the lightest neutrino mass revisited

We study the constraint of the nonsupersymmetric (non-SUSY) anti–de Sitter (AdS) conjecture on the three-dimensional vacua obtained from the compactification of the Standard Model coupled to Einstein gravity on a circle where the three-dimensional components of the four-dimensional metric are general functions of both noncompact and compact coordinates. We find from studying the wave function profile of the three-dimensional metric in the compactified dimension that the radius of the compactified dimension must be quantized. Consequently, the three-dimensional vacua are constrained by not only the non-SUSY AdS conjecture but also the quantization rule of the circle radius, leading to both upper and lower bounds for the mass of the lightest neutrino as 2≤mν/Λ4<3, where Λ4≃5.06×10−84 GeV2 is the observed cosmological constant. This means that the lightest neutrino should have a mass around 10−32 eV or it would be approximately massless. With this prediction, we reconstruct the light neutrino mass matrix that is fixed by the neutrino oscillation data and in terms of three new mixing angles and six new phases for both the normal ordering and inverted ordering. In the situation where the light neutrino mass matrix is Hermitian, we calculate its numerical value in the 3σ range. Published by the American Physical Society 2024

Modern Approach in Pattern Recognition Using Circular Fermatean Fuzzy Similarity Measure for Decision Making with Practical Applications

The circular Fermatean fuzzy (CFF) set is an advancement of the Fermatean fuzzy (FF) set and the interval-valued Fermatean fuzzy (IVFF) set which deals with uncertainty. The CFF set is represented as a circle of radius ranging from 0 to 2 with the center at the degree of association (DA) and degree of nonassociation (DNA). If multiple people are involved in making decisions, the CFF set, as an alternative to the FF and IVFF sets, can deal with ambiguity more effectively by encircling the decision values within a circle rather than taking an average. Using algorithms, a pattern can be observed computationally or visually. Machine learning algorithm utilizes pattern recognition as an instrument for identifying patterns and also similarity measure (SM) is a beneficial pattern recognition tool used to classify items, discover variations, and make future predictions for decision making. In this work, we introduce the CFF cosine and Dice similarity measures (CFFDMs and CFFSMs), and their properties are studied. Unlike traditional approaches of decision making, which emphasize a single number, the proposed CFFSMs observe the pattern over the circular region to help in dealing with uncertainty more effectively. We introduce an innovative decision-making method in the FF setting. Available bank loans and applicants’ eligibility levels are represented as CFF set using their FF criteria and are taken as loan patterns and customer eligibility patterns. The loan is allocated to the applicant by measuring the CFFCSM and CFFDSM between the two patterns. Also, laptops are suggested to the customers by measuring the similarity between specification pattern and requirement pattern. The correctness and consistency of the proposed models are ensured by comparison analysis and graphical simulations of the input and similarity CFFNs.

3D deformation measurement of rotating blades based on concentric ring calibration and GPU-SIFT feature point searching

Image grayscale matching and camera calibration are two key technologies for measuring the three-dimensional deformation of rotating blades. To address the issues of low accuracy in extracting corresponding points caused by unclear grayscale boundaries of traditional chessboard calibration patterns and the high computational complexity of image matching algorithms in high-speed and high-resolution scenarios, this paper proposes a method for measuring the three-dimensional deformation of rotating blades based on concentric circle calibration and GPU-SIFT feature point search. Firstly, a concentric circle calibration method based on arc segment combination for extracting circle center is proposed. By fitting the circular calibration pattern with extracted marker points, errors introduced by unclear grayscale boundaries of chessboard patterns are avoided, thereby improving the accuracy of calibration parameters. Secondly, an image matching algorithm based on GPU-SIFT feature point search is presented. By optimizing the storage location of scale space process data, the performance of the SIFT algorithm is maximized, leading to improved image matching speed. Lastly, the calculation of displacement and strain fields is achieved using the P3P theory and local least squares fitting method, respectively. The experimental results show that the proposed system and method for 3D deformation measurement achieve a strain measurement accuracy of 80 με. Moreover, they successfully measure the surface deformation of turbofan blades under the conditions of 3000 rpm and 6000 rpm. Through this method, it can improve the deformation measurement capability of large engine manufacturers for high-speed rotating blades and provide reliable technical support.

Dynamics of energy transport on hydromagnetic Casson slippery nanoflow over curved surface

The study of fluid flow over curved surfaces is crucial in various engineering applications, such as designing aircraft wings, turbines, and submarines. Curved surfaces are being explored in various biomedical applications, such as designing stents for blood vessels and implants for bones and joints. Concerning the present applications of curved stretching sheets on fluid dynamics along with trihybrid nanofluids, this study is unique and fills the research gaps and offers solutions to several issues. This work looks at the flow through the boundary layer of an electrically conductive trihybrid nanofluid and the convection heat exchange of a Casson fluid across a curved stretched surface encircled within a circle of radius R. The study considers the effects of thermal radiation using the non-linearized Rosseland approximation, as well as a magnetic field, and hydromagnetic slip. The flow as well as the transfer of heat problem is mathematically described by curvilinear coordinates. Using the combination of the shooting technique and the Runge-Kutta method, similar solutions to the modeled partial differential equations are produced, and the set of non-linear ordinary differential equations with a boundary value solver is implemented through the MATLAB program. The study finds the influence of several limits on critical characteristics such as fluid velocity, coefficient of skin friction, pressure, temperature, and rate of heat exchange over a surface. The findings are shown in tables and graphs. Additionally, a comparative analysis between the current findings and those found in the literature is provided. Blood-containing nanoparticles (GO-SWCNTs-MWCNTs) on curved surfaces could improve drug delivery effectiveness, growth of synthetic tissues or organs with complex structures and effective cancer therapy treatment.

Simultaneous Chandra and HST observations of the quiescent neutron star low-mass X-ray binaries in 47 Tucanae

ABSTRACT We present simultaneous Chandra X-ray Observatory and Hubble Space Telescope observations of three certain (X5, X7, W37) and two likely (X4, W17) quiescent neutron star low-mass X-ray binaries (qLMXBs) in the globular cluster 47 Tuc. We study these systems in the X-ray, optical, and near-ultraviolet (NUV) using the simultaneous data and additional non-contemporaneous HST data. We have discovered a blue and variable NUV counterpart to W17. We have not securely identified the eclipsing qLMXB W37 in the optical or NUV. Deeper high-resolution imaging is needed to further investigate the faint NUV excess near the centre of the W37 error circle. We suggest that a previously identified optical astrometric match to X7 is likely the true counterpart. The H α emission and the location of the counterpart in the colour–magnitude diagram, indicate that the secondary is probably a non-degenerate, H-rich star. This is consistent with previous results from fitting X7’s X-ray spectrum. In X4, the simultaneous X-ray and optical behaviour supports the earlier suggestion that the X-ray variability is driven by changes in accretion rate. The X-ray eclipses in X5 coincide with minima in the optical/NUV light curves. Comparison of the 47 Tuc qLMXBs with the cataclysmic variables (CVs) in the cluster confirms that overall the qLMXBs have larger X-ray to optical flux ratios. Based on their optical/NUV colours, we conclude that the accretion discs in the qLMXBs are less prominent than in CVs. This makes the ratio of X-ray flux to excess blue-optical flux a powerful discriminator between CVs and qLMXBs.

Applicability analysis of burners arrangement for an improved entrained-flow fine slag gasifier

Fine slag with high carbon content is unsuitable for brick-making or road paving and is usually deposited in piles or landfills. The atmospheric entrained-flow gasification technology can make use of the unburned carbon in fine slag, thereby resolving the problem of wasted resources and environmental pollution caused by waste fine slag. The arrangement of burners has an impact on the flow field within the gasifier, which subsequently affects the gasification performance. It is important to find a suitable burner arrangement for different gasification raw materials. This study investigates the gas–solid flow characteristics in 10,000 Nm3/h fine slag entrained-flow gasifiers using a combination of two-phase experiments and numerical simulations. The flow fields were compared in detail for three different burner arrangements. The results indicate a clear central tangent circle under the co-biased burner arrangement, which leads to the highest particle concentration and swirl intensity near the membrane wall. This condition is most conducive to slag formation. No rotating flow field is generated in the gasifier under the opposed and counter-biased burner arrangements. The high-temperature region is located near the top of the chamber. Additionally, the residence time of fine slag is short. These factors make it unsuitable for powder gasification. The results will be helpful to the design and operation of entrained-flow powder-feedstock gasifiers with multiple burners.

DATA ON THE RELATIONSHIP OF POETS OF THE 15TH–16TH CENTURIES FROM “TARIKH-I RASHIDI”

The author of the article sets herself a goal – to highlight the relationship between Turkic-speaking and Iranian-speaking poets of the 15th–16th centuries, recorded in the “Tarikh-i Rashidi” by Mirza Muhammad Haidar. This question is poorly understood. This is proved the novelty of the idea of research. The article, written on the basis of data from a medieval source, helps to expand the relationship in Central Asian literary circles, a genre of poetry that was popular during this period. The picture of the relationship of poets is one of the aspects of the relationship in the culture of the peoples of Central Asia. The researcher used the publication “Tarikh-i Rashidi”, carried out by source scholars of Uzbekistan. As a result of the analysis of data from this work, it was established that the muʻamma genre was quite popular during the study period. The names of the masters of compiling charades, works on this genre are given. The data of “Tarikh-i Rashidi” were compared with data from “Majalis an-Nafais” by ʻAli Shir Navaʻi. It is established that some young poets of the era of ʻAli Shir Nava’i by the time of writing “Tarikh-i Rashidi” became famous poets in Herat. In the cultural center of Khorasan – Heart, poetry developed along with treatises. The relationship between the poets of Herat circle proves the path of development of the literature of Iranian-speaking and Turkic-speaking peoples of Central Asia, being an indicator of the relationship in the culture of the peoples of studying region.

Spin Angular Momentum at the Focus of a Superposition of an Optical Vortex and a Plane Wave with Linear Polarizations

In this paper, tight focusing of a superposition of a vortex laser beam with topological charge n with linear polarization and a plane wave with the same linear polarization directed along the horizontal axis is considered. Using the Richards–Wolf formalism, analytical expressions are obtained for the intensity distribution and longitudinal projection of the spin angular momentum in the focal plane. It is shown that for even and odd numbers n, the intensity and the spin angular momentum have different symmetries: for even n they are symmetric about both Cartesian axes, and for odd n they are symmetric only about the vertical axis. The intensity distribution has n local maxima at the focus, and it is nonzero on the optical axis for any n. The distribution of the longitudinal spin angular momentum (spin density) in the focal plane has (n + 2) subwavelength regions with a positive spin angular momentum and (n + 2) regions with a negative spin angular momentum, the centers of which alternately lie on a circle of a certain radius with a center on the optical axis. This spin distribution with different signs demonstrates the spin Hall effect at the focus. Negative and positive spins are mutually compensated, and the total spin is equal to zero at the focus. We have shown that by changing the topological charge of the optical vortex, it is possible to control the spin Hall effect at the focus, that is, to change the number of regions with spins of different signs.

The General Parametric Equation of Pythagoras Theorem and The General Connectedness Theorem

In this article, the Quarter Squares Rule is used to prove that it also satisfies the Pythagoras Theorem. Using this proof, it will be shown that there are parametric equations among sides and the radius of inner circle of a right triangle. Thus, it is proved that a quadratic equation which has some definite properties is connected with a right triangle. After that, firstly, utilizing this connection, it will be reached the general parametric equation of The Pythagoras Theorem. Secondly, it is going to be identified the general connectedness theorem. Further, it is going to be given relation between golden ratio and Earth’s axial tilt angle as an interesting example.

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.

Multi-objective optimization of twisted tri-lobed tube geometry for enhanced heat transfer and reduced resistance in turbulent flow

This study presents a detailed multi-objective optimization analysis of turbulent flow and heat transfer within a twisted tri-lobed tube, employing an integrated approach that combines Central Composite Design (CCD), Response Surface Methodology (RSM), Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto frontier analysis, Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP), and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). The investigation focuses on optimizing key design parameters: Reynolds number (Re), twist pitch (P), and minor circle radius (r), targeting Nusselt number (Nu), Poiseuille number (fRe), and Performance Evaluation Criterion (PEC) as optimization objectives. Our major findings reveal that the developed second-order regression model precisely predicts the influences of these parameters on Nu, fRe, and PEC, with sensitivity analysis showing Re's strong positive impact on Nu and fRe, P's negative effect on fRe, and r's minimal impact. The study notably highlights the significant Nu enhancement with Re increase. Utilizing multi-objective optimization, we identified a Pareto optimal set, and through LINMAP and TOPSIS methods, derived compromise solutions that optimally balance heat transfer efficiency, flow resistance, and overall performance. These outcomes underscore the critical importance of parameter selection to meet specific engineering needs, offering a strategic framework for enhancing twisted tri-lobed tube designs.

How anatomic should anatomic total shoulder arthroplasty be? Evaluation of humeral head reconstruction with the best-fit circle

IntroductionHumeral implant designs for anatomic total shoulder arthroplasty (aTSA) focus on anatomic reconstruction of the articular segment. Likewise, the pathoanatomy of advanced glenohumeral osteoarthritis (GHOA) often results in humeral head deformity. We hypothesized the anatomic reconstruction of the humeral head in aTSA risks overstuffing the glenohumeral joint. Methods97 cases (52 females) of primary GHOA in patients treated with aTSA were evaluated. Preoperative computed tomography (CT) scans were used to classify glenoid morphology according to the Walch classification. Coronal plane images in the plane of the humerus were used to determine the anatomic perfect circle as described by Youderian et al. Humeral head thinning was determined as the distance from the center of rotation (COR) of the perfect circle to the nearest point along the humeral articular surface. aTSA was modeled with a predicted anatomic humeral head and a simulated 4 mm polyethylene glenoid component. The change in the position of the native humerus was determined. Wilcoxon Rank Sum tests were used to evaluate differences in humeral head thinning and humeral lateralization between monoconcave and biconcave glenoid morphologies. Spearman’s rank correlation coefficients were used to assess the relationship between humeral head thinning with preoperative active forward elevation and external rotation. ResultsThe mean radius of the perfect circle was 25.0±2.1 mm. There was a mean thinning of 2.4±2.0 mm (Range -1.7–8.3). Mean percent thinning of the humeral head was 9.4%±7.7%. The mean humeral lateralization was 6.4±2.0 mm. Humeral head thinning was not significantly associated with active forward elevation (r=-0.15, p=0.14) or active external rotation (r= -0.12, p=0.25). There were no significant differences in the percentage of humeral head thinning (p=0.324) or humeral lateralization (p=0.350) between concentric and eccentric glenoid wear patterns ConclusionsUtilization of the perfect circle as a guide in aTSA may risk excessive lateralization of the humerus and overstuffing the glenohumeral joint. This may have implications for subscapularis repair and healing, as well as glenoid implant and rotator cuff longevity. These findings call into question whether recreation of normal glenohumeral anatomy in aTSA is appropriate for all patients. Humeral head reconstruction in aTSA should account for glenohumeral joint volume and soft tissue contracture.

Choriocapillaris flow features in children with myopic anisometropia

AimsTo examine differences between the eyes in choriocapillaris perfusion and choroidal thickness in children with myopic anisometropia.MethodsIn this observational and prospective study, 46 children with myopic anisometropia were enrolled. Choriocapillaris...

Statistical Inference in Circular Structural Model and Fitting Circles to Noisy Data

It is well known that commonly used algorithms for circle fitting perform poorly when sampling distribution of the points is not symmetric with respect to the circle center, for example, when the points are sampled from a circle arc. To overcome this difficulty we introduce and study a parametric circular structural model. In this model the points on the circumference are assumed to be sampled according to the von Mises distribution with unknown concentration and mean direction parameters. We develop maximum likelihood and method of moments estimators of the circle center and radius, and study their statistical properties. In particular, we show that the proposed maximum likelihood estimator is asymptotically normal and efficient. We also develop a test of uniformity for the sampling distribution along the circle. Based on the derived theoretical results we propose a numerically stable circle fitting algorithm, investigate its accuracy in a simulation study, and illustrate its behavior in a real data example. Supplementary materials for this article are available online.

Studying GPR's direct and reflected waves

AbstractAs the transmitter and receiver (Tx and Rx, respectively) are located in close proximity during a typical ground‐penetrating radar (GPR) survey, the powerful signal generated by the Tx and which is then recorded by the Rx at various time delays, can be saturated at early times (i.e., this is the direct wave (DW) signal reaching the Rx). This often causes the masking of shallow targets, complicating data interpretation. In this study, our aim is to examine the spatial distribution of the electromagnetic signals around the Tx, attempting to locate areas where the DW becomes minimum, whereas the signal strength from subsurface targets (i.e., reflected wave – RW) remains ideally unchanged. The position of these local minima in the DW signal could give rise to advantageous Tx–Rx configurations, where clear reflections from subsurface targets lying at shallow depths can be obtained with the least possible involvement of the DW. To perform such a study, we carried out static field measurements over a flat lying reflector as well as numerical simulations in a reflection, common‐offset mode around a transmitting antenna. In the field, we also collected wide‐angle reflection–refraction data to determine the GPR wave velocity in the uppermost layer. GPR signals were recorded by the Rx around the Tx in three concentric circles of various radii (i.e., varying the Tx/Rx separation), using a specific angular step and varying the Tx/Rx polarization each time. The synthetic data were produced using a three‐dimensional finite‐difference time‐domain modelling tool. Field and numerically simulated data were analysed and compared to study the behaviour of both the DW and RW events around the Tx when changing the Tx/Rx distance, their respective angular position, as well as their relative polarization/orientation.

Development of a digital star-shot analysis system for comparing radiation and imaging isocenters of proton treatment machine.

To directly compare the radiation and imaging isocenters of a proton treatment machine, we developed and evaluated a real-time radiation isocenter verification system. The system consists of a plastic scintillator (PI-200, Mitsubishi Chemical Corporation, Tokyo, Japan), an acrylic phantom, a steel ball on the detachable plate, Raspberry Pi 4 (Raspberry Pi Foundation, London, UK) with camera module, and analysis software implemented through a Python-based graphical user interface (GUI). After kV imaging alignment of the steel ball, the imaging isocenter defined as the position of the steel ball was extracted from the optical image. The proton star-shot was obtained by optical camera because the scintillator converted proton beam into visible light. Then the software computed both the minimum circle radius and the radiation isocenter position from the star-shot. And the deviation between the imaging isocenter and radiation isocenter was calculated. We compared our results with measurements obtained by Gafchromic EBT3 film (Ashland, NJ, USA). The minimum circle radii were averaged 0.29 and 0.41mm while the position deviations from the radiation isocenter to the laser marker were averaged 0.99 and 1.07mm, for our system and EBT3 film, respectively. Furthermore, the average position difference between the radiation isocenter and imaging isocenter was 0.27mm for our system. Our system reduced analysis time by 10min. Our system provided automated star-shot analysis with sufficient accuracy, and it is cost-effective alternative to conventional film-based method for radiation isocenter verification.

Urban Complexity and the Dynamic Evolution of Urban Land Functions in Yiwu City: A Micro-Analysis with Multi-Source Big Data

The diversification of business forms leads to functional and spatial complexity in cities. The efficient determination of the complexity of an urban system is the basis for the scientific monitoring of the multi-functional aggregation within cities. Previous studies on the urban spatial structure were limited by the difficulty of collecting micro-data and the high time cost, and they focused on the macro-spatial structure, lacking fine-grained investigations of the micro-spatial structure. Additionally, high-resolution remote sensing images, which mainly rely on the textural characteristics of the spectrum of ground objects, cannot detect the social and economic functions of ground objects. Thus, it is difficult to meet the actual needs of urban planning and management. The purpose of this paper is to automatically identify the spatial heterogeneity and temporal variation of urban land use functions in the context of complex urban systems. The TF-IDF (term frequency–inverse document frequency) algorithm, a machine learning classification algorithm, and other methods are applied to identify the urban functions and distribution characteristics of the main urban area based on the POI (point of interest) data and urban form data. The results show the following: (1) From 2012 to 2022, all types of land use in Yiwu city grew at different rates, with logistics and warehousing space growing the fastest, which is in line with Yiwu’s goal of building a national logistics center for trade and services. (2) The residential area has a spatial structure with a dense central circle and a scattered periphery extending from northeast to southwest and from east to west. (3) The commercial service sector shows clear spatial differentiation between the core and the periphery. The commercial functional areas of Niansanli, Houzhai, and Chengxi, where the number of commercial POIs is relatively small, are located at the intersection of the administrative subdistricts near the city center, indicating that the commercial economic activities of the downtown subdistrict have a certain spillover effect on adjacent subdistricts. (4) The public facilities of each subdistrict are generally located in the core of each subdistrict, which ensures better convenience and accessibility. (5) Industrial land with a large total area that is scattered and mixed with urban residential land gradually tends to be centralized, forming an industrial belt around the city. This study comprehensively considers the aggregation relationship between urban buildings and land use and improves the accuracy of land identification and functional zoning.

Some preconditioning techniques for a class of double saddle point problems

SummaryIn this paper, we describe and analyze the spectral properties of several exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the (F)GMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.