Harmonic Functions Research Articles

Exploring the depths of degenerate hyper-harmonic numbers in view of harmonic functions

Exploring the depths of degenerate hyper-harmonic numbers in view of harmonic functions

Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Radius Problems for the New Product of Planar Harmonic Mappings

Radius Problems for the New Product of Planar Harmonic Mappings

Research of stochastic stability of constructions parametric vibrations by the Monte-Carlo method

The stochastic parametric vibrations of the elastic systems behave to the statistical dynamics section of the nonlinear systems. Under stochastic parametric influence the vibrations of the elastic systems, which is a random process more frequent are not stabilized, but their amplitudes are fading or unlimited are growing. Therefore important is stability of stochastic vibrations, which can be examined as stability by probability, on average or by the moment functions of different orders. The mathematical models which describe the stochastic parametric vibrations of the elastic systems are building by analytical or numeral approaches. Researches of stochastic stability of parametric vibrations are executing by probabilistic methods. Stability of parametric vibrations of modern constructions under the action of operating random loads are not enough investigated. In the article the numeral method of research of dynamic stability of the elastic systems under stochastic parametric influence was presented. The mathematical model of stochastic parametric vibrations of construction in the form of a reduсed finite element model was formed. Matrixes of the reduced model were obtained using calculated procedures of finite element analysis software NASTRAN. The dynamic loading as stationary ergodic random process with the spectral density was presented in the form of a finite amount of harmonic functions. Stochastic stability of the constructions was investigated using the Monte Carlo method and the Runge Kutta method of direct numeral integration of the equations of reduсed model. Stochastic stability as stability by probability of appearance of tendency to fading or unlimited growth of amplitude of parametric vibrations was examined during the time interval in space of the random parameters of loading. Dynamic stability of a I-beam with corrugated wall under narrow-band parametric influence was investigated using the numeral method.

Removable Singularities of Harmonic Functions on Stratified Sets

Removable Singularities of Harmonic Functions on Stratified Sets

Robust methods for multiscale coarse approximations of diffusion models in perforated domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

F-Biharmonic Submanifolds in Space Forms and f-Biharmonic Riemannian Submersions from 3-Manifolds

f-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of f-biharmonic curves in a space form. We also obtain a complete classification of proper f-biharmonic isometric immersions of a developable surface in R3 by proving that a proper f-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into R3 exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as a dual notion of isometric immersions (i.e., submanifolds). We also study f-biharmonicity of Riemannian submersions from 3-manifolds by using the integrability data. Examples are given of proper f-biharmonic Riemannian submersions and f-biharmonic surfaces and curves.

Harmonic maps on the tangent bundle according to the ciconia metric

The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection 
 π : TM → M, where M denotes a Riemannian manifold and TM its associated tangent bundle. Additionally, we delve into the harmonicity of vector fields ξ ∈ χ (M) , treated as mappings from M to TM. Moreover, our exploration extends to situations such as the case involving the map π : (TM, ˜g) → (M2n, J, g), where (M2n, J, g) represents an anti-paraKähler manifold and (TM, ˜g) its tangent bundle with the ciconia metric. In this context, we delve into the harmonicity relations between the ciconia metric ˜g and the Sasaki metric Sg, examining their mutual interactions. Furthermore, we delve into the Schoutan-Van Kampen connection and the Vranceanu connection, both associated with the Levi-Civita connection of the ciconia metric. We also undertake the computation of the mean connections for the Schoutan-Van Kampen and Vranceanu connections, thereby shedding light on their properties. Finally, our investigation extends to the second fundamental form of the identity mapping from 
 (TM, ˜g) to (TM,∇m) and (TM, ∇∗m). Here ∇m and ∇∗m represent the mean connections associated with the Schoutan-Van Kampen and Vranceanu connections, respectively.

Uncertainty quantification and stochastic response prediction for excavator systems under extreme operating conditions

The quantification of extreme operating conditions and their corresponding response predictions is a considerable challenge regarding the operation and maintenance of electromechanical equipment, such as dynamic excavator systems. This paper proposes a novel nonstationary modulation function model based on kinematic and dynamic analyses driven by on-site measurement data acquired from excavators. Extreme conditions are simulated through Gaussian white noise and impulse functions, and harmonic functions are overlaid on them to fit uncertain excitations, thus quantifying real random excitations. A key component of this method is its detailed statistical analysis of the probability density function (PDF) of the system response via Monte Carlo simulation. The results show that, compared with the measured results, the mean error of the system response PDFs predicted by this methodology is extremely low, and the developed approach can effectively capture the heavy-tailed characteristics of random responses under extreme loads. The proposed method utilizes a limited number of physical experiments to generate statistical results for large samples, providing the dual benefit of cost-effectiveness and high accuracy.

Quantitative stability of harmonic maps from $${\mathbb {R}}^2$$ to $${\mathbb {S}}^2$$ with a higher degree

Quantitative stability of harmonic maps from $${\mathbb {R}}^2$$ to $${\mathbb {S}}^2$$ with a higher degree

Approaching certain central-field Schrödinger problems through the harmonic oscillator green's function

We propose a method to analytically solve the Schrödinger equation with central potentials. By means of Sturm–Liouville expansion, the Green's function (GF) of the Schrödinger differential operator can be represented in the GF of the isotropic harmonic oscillator. Applying this straightforward approach, we solve the Schrödinger problem for the Coulomb potential, for the Morse potential, and for the respective inverted potentials. Using the obtained results, we formulate a method to analytically solve Schrodinger equations for multi-well potentials, demonstrating that classically forbidden regions can be treated as inverted harmonic oscillators, whereas harmonic oscillators or Morse potentials can approximate the wells.

Multi-Robot Exploration Employing Harmonic Map Transformations

Robot Exploration can be used to autonomously map an area or conduct search missions in remote or hazardous environments. Using multiple robots to perform this task can improve efficiency for time-critical applications. In this work, a distributed method for multi-robot exploration using a Harmonic Map Transformation (HMT) is presented. We employ SLAM to construct a map of the unknown area and utilize map merging to share terrain information amongst robots. Then, a frontier allocation strategy is proposed to increase efficiency. The HMT is used to safely navigate the robots to the frontiers until the exploration task is complete. We validate the efficacy of the proposed strategy via tests in simulated and real-world environments. Our method is compared to other recent schemes for multi-robot exploration and is shown to outperform them in terms of total path distance.

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}. For each d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 4$$\\end{document}, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

Interannual variations in the seasonal cycle of extreme precipitation in Germany and the response to climate change

Abstract. Annual maxima of daily precipitation sums can be typically described well with a stationary generalized extreme value (GEV) distribution. In many regions of the world, such a description does also work well for monthly maxima for a given month of the year. However, the description of seasonal and interannual variations requires the use of non-stationary models. Therefore, in this paper we propose a non-stationary modeling strategy applied to long time series from rain gauges in Germany. Seasonal variations in the GEV parameters are modeled with a series of harmonic functions and interannual variations with higher-order orthogonal polynomials. By including interactions between the terms, we allow for the seasonal cycle to change with time. Frequently, the shape parameter ξ of the GEV is estimated as a constant value also in otherwise instationary models. Here, we allow for seasonal–interannual variations and find that this is beneficial. A suitable model for each time series is selected with a stepwise forward regression method using the Bayesian information criterion (BIC). A cross-validated verification with the quantile skill score (QSS) and its decomposition reveals a performance gain of seasonally–interannually varying return levels with respect to a model allowing for seasonal variations only. Some evidence can be found that the impact of climate change on extreme precipitation in Germany can be detected, whereas changes are regionally very different. In general, an increase in return levels is more prevalent than a decrease. The median of the extreme precipitation distribution (2-year return level) generally increases during spring and autumn and is shifted to later times in the year; heavy precipitation (100-year return level) rises mainly in summer and occurs earlier in the year.

Neuromagnetic representation of musical roundness in chord progressions.

Musical roundness perception relies on consonance/dissonance within a rule-based harmonic context, but also on individual characteristics of the listener. The present work tackles these aspects in a combined psychoacoustic and neurophysiological study, taking into account participant's musical aptitude. Our paradigm employed cadence-like four-chord progressions, based on Western music theory. Chord progressions comprised naturalistic and artificial sounds; moreover, their single chords varied regarding consonance/dissonance and harmonic function. Thirty participants listened to the chord progressions while their cortical activity was measured with magnetoencephalography; afterwards, they rated the individual chord progressions with respect to their perceived roundness. Roundness ratings differed according to the degree of dissonance in the dominant chord at the progression's third position; this effect was pronounced in listeners with high musical aptitude. Interestingly, a corresponding pattern occurred in the neuromagnetic N1m response to the fourth chord (i.e., at the progression's resolution), again with somewhat stronger differentiation among musical listeners. The N1m magnitude seemed to increase during chord progressions that were considered particularly round, with the maximum difference after the final chord; here, however, the musical aptitude effect just missed significance. The roundness of chord progressions is reflected in participant's psychoacoustic ratings and in their transient cortical activity, with stronger differentiation among listeners with high musical aptitude. The concept of roundness might help to reframe consonance/dissonance to a more holistic, gestalt-like understanding that covers chord relations in Western music.

New Auxiliary Principle Technique for General Harmonic Directional Variational Inequalities

This paper explores the utilisation of harmonic variational inequalities to establish the minimum value among two locally Lipschitz continuous harmonic convex functions. This investigation introduces novel classes of harmonic directed variational inequalities, particularly focusing on scenarios like harmonic complementarity and related optimization challenges. The study proposes and analyses various inertial iterative strategies for addressing harmonic directed variational inequalities through the auxiliary principle technique. It examines convergence criteria under specific weak conditions, emphasising the simplicity of the approach compared to other methodologies. The findings presented herein have broad applicability in the context of harmonic variational inequalities and optimization problems, though they are limited to theoretical exploration. Further research is required to implement these strategies numerically.

Some questions about complex harmonic functions

Some questions about complex harmonic functions

Infrared Maritime Small-Target Detection Based on Fusion Gray Gradient Clutter Suppression

The long-distance ship target turns into a small spot in an infrared image, which has the characteristics of small size, weak intensity, limited texture information, and is easily affected by noise. Moreover, the presence of heavy sea clutter, including sun glints that exhibit local contrast similar to small targets, negatively impacts the performance of small-target detection methods. To address these challenges, we propose an effective detection scheme called fusion gray gradient clutter suppression (FGGCS), which leverages the disparities in grayscale and gradient between the target and its surrounding background. Firstly, we designed a harmonic contrast map (HCM) by using the two-dimensional difference of Gaussian (2D-DoG) filter and eigenvalue harmonic mean of the structure tensor to highlight high-contrast regions of interest. Secondly, a local gradient difference measure (LGDM) is designed to distinguish isotropic small targets from background edges with local gradients in a specific direction. Subsequently, by integrating the HCM and LGDM, we designed a fusion gray gradient clutter suppression map (FGGCSM) to effectively enhance the target and suppress clutter from the sea background. Finally, an adaptive constant false alarm threshold is adopted to extract the targets. Extensive experiments on five real infrared maritime image sequences full of sea glints, including a small target and sea–sky background, show that FGGCS effectively increases the signal-to-clutter ratio gain (SCRG) and the background suppression factor (BSF) by more than 22% and 82%, respectively. Furthermore, its receiver operating characteristic (ROC) curve has an obviously more rapid convergence rate than those of other typical detection algorithms and improves the accuracy of small-target detection in complex maritime backgrounds.

Stability conditions and Teichmüller space

AbstractWe consider a 3-Calabi–Yau triangulated category associated to an ideal triangulation of a marked bordered surface. Using the theory of harmonic maps between Riemann surfaces, we construct a natural map from a component of the space of Bridgeland stability conditions on this category to the enhanced Teichmüller space of the surface. We describe a relationship between the central charges of objects in the category and shear coordinates on the Teichmüller space.

Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form

We prove the parabolic strong maximum principle and the Harnack inequality for classical solutions to degenerate second order partial differential equations of the formLu:=∑i,j=1m0∂xi(aij∂xju)+∑j=1m0bj∂xju+cu+∑i,j=1Nbijxj∂xiu−∂tu=f, where 1≤m0≤N, A0=(aij)i,j=1,…,m0 is a bounded, symmetric and uniformly positive matrix and the matrix B:=(bij)i,j=1,…,N has real constant entries. Moreover, we assume low regularity (i.e. Hölder continuity) on the coefficients aij, bj and c, for i,j=1,…,m0. We point out the proofs of our main results only rely on the classical theory developed for harmonic functions.