Half Plane Research Articles

Ces\`{a}ro-Type Operator on the Dirichlet Space of the Upper Half Plane

We construct a Ces\`{a}ro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Ces\`{a}ro-type operator.

Detecting safe operational regimes of synchronous motor‐generator pair for wind integration: A non‐linear perspective

AbstractIn this article, the stability of a synchronous motor generator pair (SMGP) used for improving the inertia of grid‐connected renewable energy systems is investigated. The useful operational regime for different sets of system parameters is identified, such as electromagnetic torque, damping co‐efficient, and inertia by employing bifurcation analysis to detect stability boundaries. For the first time, the existence of bi‐stable regimes for the SMGP with non‐linear stability analysis is revealed. The authors' analysis unravels the possibility of the system getting transited to unsafe operation even when the system is in the linearly stable region. The existence of the bistable regime indicates the possibility of the system becoming unstable even when the eigenvalues are in the left half plane. The authors also identify globally stable and globally unstable regimes in the parameter space. The safe operating range of inertia and damping co‐efficient values helps in the design of a suitable MGP set that is robust to frequency deviations, even with a low inertia source. With the recommended values of electromagnetic torque, the authors' analysis provides a safe operational regime for power generation from renewable energy sources.

Strong Continuity of Composition Semigroups on the Generalized Bloch Spaces of the Upper Half Plane

We investigate strong continuity of composition semigroups on the generalized Bloch spaces of the upper half plane. These composition semigroups are induced by automorphisms of the upper half plane as classified into three distinct groups in [3].

Spectral-Domain Method of Moments for the Modal Analysis of Line Waveguides

A rigorous full-wave modal analysis based on the method of moments in the spectral domain is presented for line waveguides constituted by two-part impedance planes with arbitrary anisotropic surface impedances. An integral equation is formulated by introducing an auxiliary current sheet on one of the two half planes and extending the impedance boundary condition of the complementary half plane to hold on the entire plane. The equation is then discretized with the method of moments in the spectral domain, by employing exponentially weighted Laguerre polynomials as entire-domain basis functions and performing a Galerkin testing. Numerical results for both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bound</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">leaky</i> line waves are presented and validated against independent results, obtained for isotropic surface impedances with the analytical Sommerfeld–Maliuzhinets method and for the general anisotropic case with a commercial electromagnetic simulator. The proposed approach is computationally efficient, can accommodate the presence of spatial dispersion, and offers physical insight into the modal propagation regimes.

On the Location of Chromatic Zeros of Series-Parallel Graphs

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, giving density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)&gt;3/2$ and we show there exists an open region $U$ containing the interval $(0,32/27)$ such that $U\setminus\{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs.&#x0D; We also disprove a conjecture of Sokal by showing that for each large enough integer $\Delta$ there exists a series-parallel graph for which all vertices but one have degree at most $\Delta$ and whose chromatic polynomial has a zero with real part exceeding $\Delta$.

Propagators and widths of physical and purely virtual particles in a finite interval of time

We study the free and dressed propagators of physical and purely virtual particles in a finite interval of time τ and on a compact space manifold Ω, using coherent states. In the free-field limit, the propagators are described by the entire function (ez– 1 – z)/z2, whose shape on the real axis is similar to the one of a Breit-Wigner function, with an effective width around 1/τ. The real part is positive, in agreement with unitarity, and remains so after including the radiative corrections, which shift the function into the physical half plane. We investigate the effects of the restriction to finite τ on the problem of unstable particles vs resonances, and show that the muon observation emerges from the right physical process, differently from what happens at τ = ∞. We also study the case of purely virtual particles, and show that, if τ is small enough, there exists a situation where the geometric series of the self-energies is always convergent. The plots of the dressed propagators show testable differences: while physical particles are characterized by the usual, single peak, purely virtual particles are characterized by twin peaks.

Formation and propagation mechanism of complex stacking fault in 180 μm thick 4H-SiC epitaxial layers

The formation mechanism and propagation behavior of a complex stacking fault (SF) in thick silicon carbide epitaxial layers was investigated using depth-resolved section X-ray topography and ultraviolet photoluminescence (UVPL) imaging. The SF is initiated in the substrate by deflection of a threading screw dislocation onto two basal half planes creating a Frank fault. This fault propagates into the epitaxial layer and interacts with several threading mixed dislocations, which creates both Frank and Shockley SFs on the same basal plane. UVPL images using different spectral filters show various SF type conversions occurring during propagation of the complex SF.

The effective crack extension and its implications on the single cantilever beam test for sandwich composites

The response of Single Cantilever Beam sandwich specimens may be analyzed effectively through one-dimensional theories accounting for the actual deformation of the core and its effects on fracture parameters and compliance. The model of a beam on an elastic Winkler foundation describes the detaching facesheet as an Euler–Bernoulli beam resting, in the bonded region, on an elastic foundation which generates oscillating deflections with characteristic length [Formula: see text]. Modified beam theory (MBT) describes the debonded part of the face sheet as a cantilever beam, built-in at the debond tip, with an effective length which is longer than the actual detached length, by a quantity [Formula: see text]. If [Formula: see text] the two models predict the same energy release rate and similar load-point displacements. Effective crack extension and characteristic length depend on the thicknesses and elastic constants of facesheets and core and may be derived through experimental measurements of the specimen compliance at various increments of crack growth, as prescribed by experimental protocols. In the theoretical analyses, [Formula: see text] is typically assumed, either by directly using formulas proposed for other specimens or by calibrating their parameters to case specific finite element results. In this paper, we investigate the applicability of such formulas to a wide selection of sandwich composites for naval and aeronautical applications having thicknesses or materials other than those used for the original fitting; we then highlight their limitations and effects on the design of experimental protocols for material characterization. The applicability of a recent isotropic-elasticity based formula for [Formula: see text] is then verified using extensive data from the literature and novel finite element (FE) results for typical sandwich composites. Finally, a novel elasticity solution for the load-point displacement of a layer on a half plane is derived which supports the test data reduction procedure for [Formula: see text].

Tri-State Coupled Inductor Based High Step-Up Gain Converter Without Right Hand Plane Zero

This brief proposes a novel tri-state Coupled Inductor (CI) based high step-up gain converter without Right Half Plane (RHP) zero. The basic Coupled Inductor based Boost Converter (CIBC) can achieve high step-up gain by changing the turn ratio of CI and duty cycle. However, the control to output voltage transfer function of basic CIBC converter in CCM has RHP zero hence non-minimum phase behavior with limited bandwidth and unstable operation of the converter when step changes in duty cycle. RHP zero can be eliminated from the basic CIBC by providing freewheeling period in converter operation hence additional degree of freedom to control energy transfer to output capacitor. Thus, the converter achieves stable operation and improves the bandwidth. The small-signal model of the proposed converter is derived to verify the claim. All theoretical findings are verified by experimental results. Experimental results are provided from implemented laboratory prototype of input voltage of 24 V to output voltage of 100 V and 200 W power.

Frictional contact problem of a coated half plane pressed by a rigid punch with coupled stress elasticity

Frictional contact problem of a coated half plane pressed by a rigid punch with coupled stress elasticity

The a posteriori finite element method (APFEM), a framework for efficient parametric study and Bayesian inferences

Stochastic methods have recently been the subject of increased attention in Computational Mechanics for their ability to account for the stochasticity of both material parameters and geometrical features in their predictions. Among them, the Galerkin Stochastic Finite Element Method (GSFEM) was shown to be particularly efficient and able to provide accurate output statistics, although at the cost of intrusive coding and additional theoretical algebraic efforts. In this method, distributions of the stochastic parameters are used as inputs for the solver, which in turn outputs nodal displacement distributions in one simulation. Here, we propose an extension of the GSFEM—termed the A posteriori Finite Element Method or APFEM—where uniform distributions are taken by default to allow for parametric studies of the inputs of interest as a postprocessing step after the simulation. Doing so, APFEM only requires the knowledge of the vertices of the parameter space. In particular, one key advantage of APFEM is its use in the context of Bayesian inferences, where the random evaluations required by the Bayesian setting (usually done through Monte Carlo) can be done exactly without the need for further simulations. Finally, we demonstrate the potential of APFEM by solving forward models with parametric boundary conditions in the context of (i) metamaterial design and (ii) pitchfork bifurcation of the buckling of a slender structure; and demonstrate the flexibility of its use for Bayesian inference by (iii) inferring friction coefficient of a half plane in a contact mechanics problem and (iv) inferring the stiffness of a brain region in the context of cancer surgical planning.

Frictional half plane contacts subject to varying normal load and bulk tension under various load paths with application to fretting fatigue experiments

In many fretting fatigue experiments the loads applied to the contact are sinusoidal and in phase, creating a straight line path through load space. Although this form of loading is a good representation of many of the load histories present in engineering assemblies, the usefulness of this form of load cycle when investigating the causes of fretting fatigue damage is limited. In this paper we first review solutions for uncoupled half plane contacts under bulk tension and normal loading, then develop some novel load paths with properties that are desirable in fretting fatigue experiments. Throughout this paper the results are illustrated via application to a Hertzian contact, although the principles apply to any contact capable of representation using half-plane theory.

Magnetohydrodynamic squeezing micropolar nanofluid flow confined in parallel disks with implication of Maxwell-Cattaneo law

Time-dependent electrically conducting squeezing flow of micropolar nanofluid is analyzed between two parallel disks. The bottom porous disk is fixed while the upper disk moved orthogonally along axial direction. Nanofluid features such as thermophoresis and the Brownian motion are considered using the Buongiorno theory of nanoparticles. Cattaneo-Christov model (CCM) is executed in the expression of energy equation. Uniform suction/injection is considered at bottom disk. Inclined magnetic field is accounted at an acute angle along axial direction. The constitutive model is normalized by utilizing the similarity functions. The numerical solution of flow model is developed by MATLAB built-in bvp4c function. The physical parameters are explored through graphical depiction, two-dimensional contours, three-dimensional flow phenomenon and streamlines. Since, the micropolar parameters tend to move the fluid particles in opposite way. Therefore, micro-rotational field has an increasing tendency in the left half region and decreasing behavior in the right half plane of the central region. It is obtained that the temperature field is boost up against increased squeezing Reynolds number, parameter of thermal relaxation stress and Prandtl number. Brownian movement and thermophoretic factors have a reverse phenomenon on concentration curves.

High Step-Up Gain Converter Based on Two-Phase Interleaved Coupled Inductor Without Right-Hand Plane Zero

This paper proposed a novel two-phase interleaved Coupled Inductor (CI) based high step-up gain converter without Right Half Plane (RHP) zeros for a wide operating range of duty cycle. A coupled inductor between input and output is used in the proposed converter to eliminate RHP zero. The energy of CI is transferred to output capacitor when switch is ON to eliminate RHP zero. Furthermore, the converter has a wide operating range to eliminate RHP zero with a coefficient coupling of CI near one. Thus, the converter achieves higher bandwidth and a wide range of control over output voltage. The small-signal model of proposed converter is derived to verify the proposed claim. The single loop voltage control is sufficient to regulate output voltage of the converter with a good stability margin and faster dynamic response for a wide range of load changes. In addition, proposed converter has a high voltage step-up gain and low voltage stress on switching devices compared to the boost converter. Moreover, interleaved operation at input side reduces ripple and increases power density. All theoretical findings are verified by simulation and experimental results. Experimental results are provided from implemented laboratory prototype of input voltage of 24 V to output voltage of 100 V and 200 W power.

Difference of composition operators on the half plane

Difference of composition operators on the half plane

Plane Potential Field Outside a Symmetric T-Shaped Profile

We consider a new plane problem of the potential theory outside a symmetric T-shaped profile. In view of the symmetry of the profile, we reduce the analyzed problem to a mixed boundary-value problem in a half plane with perpendicular cut whose solution is constructed by the method of partial domains with the use of polar coordinates and the Mellin integral transformation. The original problem is reduced to a system of two Wiener–Hopf equations whose solution is reduced to a completely regular infinite system of linear algebraic equations. The solution of the infinite system is approximately found by the method of reduction. The obtained explicit expressions for the unknown harmonic function make it possible to efficiently determine its values for all possible values of the arguments.

Uniform Asymptotic Expansions for Gegenbauer Polynomials and Related Functions via Differential Equations Having a Simple Pole

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order n that are uniformly valid for unbounded complex values of the argument z, including the real interval $$0 \le z \le 1$$ in which the zeros in the right half plane are located: symmetry extends the results to the left half plane. The approximations are derived from the differential equation satisfied by these polynomials, and other independent solutions are also considered. For large n this equation is characterized by having a simple pole, and expansions valid at this singularity involve Bessel functions and slowly varying coefficient functions. The expansions for these functions are simpler than previous approximations, in particular being computable to a high degree of accuracy. Simple explicit error bounds are derived which only involve elementary functions, and thereby provide a simplification of previous expansions and error bounds associated with differential equations having a large parameter and simple pole.

Following the Higgs mode across the BCS-BEC crossover in two dimensions

Although substantial effort has been dedicated to analyzing the Higgs (amplitude) mode in superconducting systems, there are relatively few studies of the Higgs peak's dispersion and width, quantities which are observable in spectroscopic measurements. These properties can be obtained from the location of the pole of the amplitude (Higgs) susceptibility in the lower half plane of complex frequency. We analyze the behavior of the Higgs mode in a two-dimensional neutral fermionic superfluid at $T=0$ throughout the crossover from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) behavior. This occurs when the dressed chemical potential $\ensuremath{\mu}$ changes sign from positive to negative. For $\ensuremath{\mu}&gt;0$, we find a pole in the Higgs susceptibility in the lower half plane of frequency and demonstrate that it leads to a well-defined peak in spectroscopic probes. For $\ensuremath{\mu}&lt;0$, the pole still exists but is ``hidden,'' not giving rise to a peak in spectroscopic probes. Extending this analysis to a charged superconductor, we find that the Higgs mode is unaffected by the long-range Coulomb interaction.

The Picard group of vertex affinoids in the first Drinfeld covering

AbstractLetFbe a finite extension of${\mathbb Q}_p$. Let$\Omega$be the Drinfeld upper half plane, and$\Sigma^1$the first Drinfeld covering of$\Omega$. We study the affinoid open subset$\Sigma^1_v$of$\Sigma^1$above a vertex of the Bruhat–Tits tree for$\text{GL}_2(F)$. Our main result is that$\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that$\text{Pic}({\mathbf Y})[p] = 0$for${\mathbf Y}$the Deligne–Lusztig variety of$\text{SL}_2\!\left({\mathbb F}_q\right)$. One formal consequence is a description of the representation$H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$of$\text{GL}_2(\mathcal{O}_F)$as thep-adic completion of$\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.

Performance Improvement by Sampling Position Approach in Dual Loop Digital ACM Controlled DC-DC Boost Converter

In DC-DC boost converter mostly discontinuous voltage ripple is associated with Equivalent Series Resistance (ESR) of the output side capacitor. This voltage ripple makes the design of digital controllers challenging in deciding the appropriate sampling position in one switching period. The necessity of Right Half Plane Zero (RHPZ) elimination, which impacts the stability is managed by shifting the sampling position to OFF-time duration. This is mostly analyzed in the frequency domain. The proposed work provides the time-domain analysis in consideration of ESR impacting the RHPZ. A dual loop Digital Average Current Mode (DACM) controller which has two sampling position freedom is considered in this brief. The ESR value of the output capacitor for extracting the benefits of RHPZ elimination is proposed with help of the discrete-time modeling technique. An analytical discussion about the influence of RHPZ elimination with time-domain characteristics such as, voltage deviation, peak-to-peak inductor current oscillations, and efficiency is presented. Finally, two 60W boost converters prototype with different output capacitor (C) and its ESR <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r_{C})$ </tex-math></inline-formula> values are developed to validate the proposed analysis.