Half Plane Research Articles

On the interfacial behavior of two-dimensional decagonal quasicrystal films with an adhesive layer due to thermal misfit

The interfacial behavior of a thin two-dimensional decagonal quasicrystal (QC) film bonded to a half plane with an adhesive layer is analyzed under thermal misfit. Under a perfect non-slipping contact condition at interface, a theoretical model is established based on the membrane assumption, with the governing integral–differential equations being constructed in terms of phonon interfacial shear stresses in one single and multiple films. Then, these governing equations are numerically solved by Chebyshev polynomials, obtaining the phonon interfacial shear and internal normal stresses as well as the longitudinal displacement in the QC film. Finally, the effects of neighboring films, material mismatch, adhesive layer, the geometry of QC film, and temperature variation are briefly discussed on the interfacial response. It is found that the adhesive layer plays as a softening medium which can greatly reduce the value of normal and shear stresses and weaken the singularity of shear stresses near the ends of films. These findings will be instructive to the accurate measurement and safety monitoring of QC films.

Design principles of voltage controlled DC-DC boost converters through the integration of PID and model predictive control

This paper presents the design of the transfer function between the control and output variable of a DC-DC boost converter using a combination of Proportional Integral Derivative (PID) and output-based Model Predictive Control (MPC) mechanisms for eradicating the effect of the non-minimum phase (NMP) behavior on the system. This is made possible due to the existence of a right half plane (RHP) zero. It is observed that the MPC outperformed the conventionally designed PID approach by a significant margin. Based on the computed results, it can be emphasized that the combination of output-based MPC and PID controller demonstrated a strong alignment with optimal designs, exhibiting a fast convergence rate than PID and PID-IMC. Simulation results are presented to showcase the fast transient response and a high level of robustness of the proposed control methodology.

Sensitivity of octant of θ23, CP violation and mass hierarchy in NOνA with multinucleon and detector effects

In this work, we investigate how multinucleon enhancement and RPA (Random Phase Approximation) suppression can affect the measurement of three unknown neutrino oscillation parameters — the CP-violating phase δCP, the octant of the atmospheric mixing angle θ23, and the determination of the mass hierarchy, in the appearance channel of the NOνA experiment. We include the presence of the detector effect as well in the analysis, which is crucial for capturing realistic experimental scenarios. We also conducted a comparison between the nuclear model Effective Spectral Function (calculated within the RFG model) with and without Transverse Enhancement in terms of sensitivity analysis. It is found that the analysis using our comprehensive model QE(+RPA)+2p2h along with Effective Spectral Function+Transverse Enhancement exhibits significantly enhanced sensitivity compared to the pure QE interaction process, in all the cases. Also, the higher octant of θ23, the lower half plane of δCP, and the normal mass hierarchy (HO-LHP-NH) exhibit improved sensitivity, enabling a more precise determination of the corresponding parameters. Furthermore, it is also noted that improving the performance of the detector also improves the results. Thus, including multinucleon effects and improving detector efficiency have the potential to enhance the capabilities of the NOνA (and other long baseline) experiment in conducting precise parameter studies.

Moving contact problem of a functionally graded orthotropic coated half plane

This paper develops a frictional moving contact model for a functionally graded (FG) orthotropic layer pressed by a rigid cylindrical punch. The FG orthotropic layer is fully bonded to the isotropic half-plane. The punch moves to the left on the layer at a constant subsonic velocity and a shear stress arises in the contact zone according to the Coulomb friction law. General expressions of displacements and stresses are derived with the help of the Fourier transform and Galilean transformation. Using boundary conditions, the moving contact problem is reduced to a Cauchy-type singular integral equation, the unknowns of which are contact stress and contact width. Gauss–Jacobi integration formula is used to solve the obtained singular integral equation. The effect of some parameters and material properties on the contact width, contact stress and in-plane stress are given in graphical forms and detailed numerical interpretations are presented.

Navier–Stokes Equations in the Half Space with Non Compatible Data

This paper considers the Navier–Stokes equations in the half plane with Euler-type initial conditions, i.e., initial conditions with a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we prove that the solution exists for a short time independent of the viscosity. We construct the Navier–Stokes solution through a composite asymptotic expansion involving solutions of the Euler and Prandtl equations plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, which influences the regularity of the error. The error term is the sum of a first-order Euler correction, a first-order Prandtl correction, and a further error term. The use of an analytic setting is mainly due to the Prandtl equation.

Sectorial Equidistribution of the Roots of x2 + 1 Modulo Primes

Abstract The equation $x^2 + 1 = 0\mod p$ has solutions whenever p = 2 or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of $SL_2(\mathbb{R})$. Since our Poincare series have a nontrivial dependence on their Iwasawa θ-coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work.

Analysis of stress and deformation of an exponentially graded viscoelastic coated half plane under indentation by a rigid flat punch indenter tip

This paper solves the dynamic contact problem when a rigid flat punch indents into an exponentially graded (FG) viscoelastic coated homogeneous half-plane. A harmonic vertical force is applied to the FG coating, and the solution is obtained for the stress and displacement for both the FG viscoelastic coating and the half-plane using the Helmholtz functions and the Fourier integral transform technique. By applying specific boundary conditions, the contact mechanics problem is converted into a singular integral equation of the first kind. This equation is then solved numerically using the Gauss-Chebyshev integration formulas. The analysis provides detailed insights into how various parameters—such as external excitation frequency, loss factor ratio, Young’s modulus ratio, density ratio, Poisson’s ratio, indentation load, and punch length—affect the dynamic contact stress and dynamic in-plane stress.

Polyanalytic Nonlinear Boundary Value Problems for Upper Half Plane

Polyanalytic Nonlinear Boundary Value Problems for Upper Half Plane

On the role of plastic relaxation in stress assisted grain boundary oxidation

The influence of plasticity on the high-temperature stress-assisted grain boundary oxidation of nickel-based superalloys used in applications such as turbine rotor discs is investigated using the method of discrete dislocation plasticity (DDP). The misfit stress fields of nib-shaped intrusions are captured by a continuous distribution of edge dislocations whose extra half planes represent the volumetric misfit of the oxide, which is implemented in a planar formulation of DDP by invoking the linear superposition principle. DDP simulations show that stresses generated by an intrusion several microns or more in size are large enough to generate dislocation pileups with associated stresses at the intrusion interface on the order of 1 GPa, which in turn lead to localized growth and morphology change of the intrusion by stress-assisted diffusion. This morphology change relaxes the compression stress inside the intrusion near the base, and therefore increases the fracture resistances of the intrusion. The effects of applied loading and background plasticity on the growth rate of the intrusion in defect-free and prestrained samples are predicted. It is found that applied tensile stress generally increases grain boundary oxidation, while in prestrained samples the enhancement of the intrusion growth rate by the applied load is insignificant due to dislocation pile-ups ahead of the oxidation process.

Analysis and implementation of feedback linearisation controller for fuel cell-fed boost converter

ABSTRACT Fuel cell needs to be integrated with dc-dc boost converters to supply load at elevated voltages. Due to the presence of right half plane zero (RHPZ) in the output voltage to duty cycle transfer function of the boost converter and the nonlinear V-I characteristics of the fuel cell, it necessitates the usage of nonlinear controllers. This paper proposes an effective stabilising feedback linearisation (FDBKLN) controller for fuel cell-fed boost converter to provide improved dynamic performance and tight voltage regulation for wide load variation. Performance of the controller has been tested in simulation using the developed tunable minimal order fuel cell model and validated through experiments for fuel cell-fed 600W boost converter. Stability of the proposed controller is also verified using Lyapunov theorem.

Dynamic Thermal Response of Multiple Parallel Cracks in a Half Plane under General Transient Thermal Loading

Understanding the dynamic thermal response of materials is crucial for designing effective thermal protection systems, particularly in extreme thermal environments such as transient thermal loading, extremely low/high temperature, etc. This study investigates the dynamic thermal response of a half plane containing multiple parallel cracks under transient thermal loading using a non‐Fourier, hyperbolic heat conduction model. Our findings highlight significant deviations from traditional Fourier models, enhancing the predictive capabilities for designing thermal protection systems in extreme thermal environments. The cracks are modeled as distributions of thermal dislocations, with densities determined through Fourier and Laplace transforms. By solving the resulting singular integral equations, we calculate the temperature gradient intensity factors across multiple scenarios. Additionally, we examine the effects of thermal relaxation time, loading parameters, and crack spacing on the thermal intensity factors, both in transient and steady‐state conditions. This research confirms the robustness of the hyperbolic model and its practical implications for thermal analysis in engineering applications.

Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space

<abstract><p>In this work, we investigated the local existence of the solutions to the 2D magnetohy-drodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound $ \delta > 0 $ instead of the monotonicity condition of the tangential velocity.</p></abstract>

Drivers, hitting times, and weldings in Loewner's equation

AbstractIn addition to conformal weldings , simple curves growing in the upper half plane generate driving functions and hitting times through Loewner's differential equation. While the Loewner transform and its inverse have been carefully examined, less attention has been paid to the maps . We study the continuity properties of these latter transformations and show that uniform driver convergence implies uniform hitting time convergence and uniform welding convergence, even when the corresponding curves do not converge. Welding convergence implies neither hitting time nor driver convergence, while hitting time convergence implies driver convergence in (at least) the case of constant drivers. As an application, we show that a curve of finite Loewner energy can be well approximated by an energy minimizer that matches ’s welding on a sufficiently‐fine mesh.

Conjugations on 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} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{L}}^\ extbf{2}$$\\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{H}}^\ extbf{2}$$\\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.

Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space

In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or C^{1+\alpha} ) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension BA(h) of a C^{1+\mathrm{Zygmund}} orientation-preserving diffeomorphism h of the real line, and show that the Beltrami coefficient \mu (BA(h))(x+iy) vanishes as O(y) uniformly on x near the boundary of the upper half plane if and only if h is C^{1+\mathrm{Lipschitz}} . Finally, we show this criterion is indeed true when h is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.

Simultaneous tuning of multiple PID controllers for multivariable systems using deep reinforcement learning

Traditionally, tuning of PID controllers is based on linear approximation of the dynamics between the manipulated input and the controlled output. The tuning is performed one loop at a time and interaction effects between the multiple single-input-single-output (SISO) feedback control loops is ignored. It is also well-known that if the plant operates over a wide operating range, the dynamic behaviour changes thereby rendering the performance of an initially tuned PID controller unacceptable. The design of PID controllers, in general, is based on linear models that are obtained by linearizing a nonlinear system around a steady state operating point. For example, in peak seeking control, the sign of the process gain changes around the peak value, thereby invalidating the linear model obtained at the other side of the peak. Similarly, at other operating points, the multivariable plant may exhibit new dynamic features such as inverse response. This work proposes to use deep reinforcement learning (DRL) strategies to simultaneously tune multiple SISO PID controllers using a single DRL agent while enforcing interval constraints on the tuning parameter values. This ensures that interaction effects between the loops are directly factored in the tuning. Interval constraints also ensure safety of the plant during training by ensuring that the tuning parameter values are bounded in a stable region. Moreover, a trained agent when deployed, provides operating condition based PID parameters on the fly ensuring nonlinear compensation in the PID design. The methodology is demonstrated on a quadruple tank benchmark system via simulations by simultaneously tuning two PI level controllers. The same methodology is then adopted to tune PI controllers for the operating condition under which the plant exhibits a right half plane multivariable direction zero. Comparisons with PI controllers tuned with standard methods suggest that the proposed method is a viable approach, particularly when simulators are available for the plant dynamics.

A note on Cauchy's formula

We use the correlation functions of vertex operators to give a proof of Cauchy's formula∏i=1K∏j=1N(1−xiyj)=∑μ⊆[K×N](−1)|μ|sμ{x}sμ′{y}. As an application of the interpretation, we obtain an expansion of ∏i=1∞(1−qi)i−1 in terms of half plane partitions.

Reducing the Input Capacitance to Minimize the Right-Half-Plane Zero Effect for Buck Converters in Wireless Power Receiver Systems

Reducing the Input Capacitance to Minimize the Right-Half-Plane Zero Effect for Buck Converters in Wireless Power Receiver Systems

Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane

In this paper, Cattaneo-Hristov heat diffusion is discussed in the half plane for the first time, and solved under two different boundary conditions. For the solution purpose, the Laplace, and the sine- and exponential- Fourier transforms with respect to time and space variables are applied, respectively. Since the fractional term in the problem is the Caputo-Fabrizio derivative with the exponential kernel, the solutions are in terms of time-dependent exponential and spatial-dependent Bessel functions. Behaviors of the temperature functions due to the change of different parameters of the problem are interpreted by giving 2D and 3D graphics.

Algebraic independence of values of quasi-modular forms

In this paper, we discuss the algebraic independence of special values of quasi-modular forms. We show that for three algebraically independent quasi-modular forms with algebraic Fourier coefficients, if one of e2πiz and their values at z in the upper half plane is a non-zero algebraic number, then the other three numbers are algebraically independent over Q. We also provide a condition for determining when quasi-modular forms are algebraically independent. Moreover, based on the theorem proved in the paper, we prove the algebraic independence of values of derivatives of modular forms with algebraic Fourier coefficients.