Harmonic Space Research Articles

Banana diagrams as functions of geodesic distance

We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable – the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward – while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere.

Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.

CMB-PAInT: An inpainting tool for the cosmic microwave background

The presence of astrophysical emissions in microwave observations forces us to perform component separation to extract the Cosmic Microwave Background (CMB) signal. However, even in the most optimistic cases, there are still strongly contaminated regions, such as the Galactic plane or those with emission from extragalactic point sources, which require the use of a mask. Since many CMB analyses, especially the ones working in harmonic space, need the whole sky map, it is crucial to develop a reliable inpainting algorithm that replaces the values of the excluded pixels by others statistically compatible with the rest of the sky. This is especially important when working with Q and U sky maps in order to obtain E- and B-mode maps which are free from E-to-B leakage. In this work we study a method based on Gaussian Constrained Realizations (GCR), that can deal with both intensity and polarization. Several tests have been performed to asses the validation of the method, including the study of the one-dimensional probability distribution function (1-PDF), E- and B-mode map reconstruction, and power spectra estimation. We have considered two scenarios for the input simulation: one case with only CMB signal and a second one including also Planck PR4 semi-realistic noise. Even if we are limited to low resolution maps, Nside = 64 if T, Q and U are considered, we believe that this is a useful approach to be applied to future missions such as LiteBIRD, where the target are the largest scales.

Pulsar timing array harmonic analysis and source angular correlations

Gravitational waves (GWs) influence the arrival times of radio signals coming from pulsars. Here, we investigate the harmonic space approach to describing a pulsar’s response to GWs. We derive and discuss the “diagonalized form” of the response, which is a sum of spin-2-weighted spherical harmonics of the GW direction multiplied by normal (spin-weight 0) spherical harmonics of the pulsar direction. We show how this allows many useful objects, for example, the Hellings and Downs two-point function, to be easily calculated. The approach also provides a clear description of the gauge dependence. We then employ this harmonic approach to model the effects of angular correlations in the sky locations of GW sources (sometimes called “statistical isotropy”). To do this, we construct ensembles made up of many Gaussian subensembles. While each of the individual subsensembles breaks rotational invariance, the full ensemble is rotationally invariant. Using harmonic techniques, we compute the cosmic covariance and the total covariance of the Hellings and Downs correlation in these models. The results may be used to assess the impact of angular source correlations on the Hellings and Downs correlation, and for optimal reconstruction of the Hellings and Downs curve in models where GW sources have correlated sky locations. Published by the American Physical Society 2024

The nonlinear superposition operator on harmonic fock spaces

We describe the conditions under which the superposition operator maps one harmonic Fock space into another. We further proved that all superposition operators on the spaces are both bounded and globally Lipschitz continuous. The condition for invertibility is also characterized. In the end, we identify all the fixed points of the operator on the spaces.

Tunable UV ∼ IR frequency comb generation via high-order sideband generation

We propose the generation of a widely tunable UV-to-IR frequency comb by high-order sideband generation (HSB) spectrum emitted from semiconductors. In our theoretical simulations, we demonstrate the high-order sideband signals of two series (2m Ωseed + (2n + 1) ωdriver, and (2m + 1) Ωseed + 2 nωdriver ), where m and n are integers of a seed pulse and a driver laser frequency, respectively. The simulations also reveal the intensity of HSB scale with the driver laser power, both perturbatively and non-perturbatively. We find that the harmonic position and spacing of the high-order sideband emission can be controlled by varying the seed pulse and driver photon energies. In the experiment, we applied a visible ( ℏΩseed = 3.1 eV, ∼400 nm) seed pulse and mid-infrared (MIR, ℏωdriver = 0.4 eV, 3.1 μm) driver pulses to ZnSe target. Our experimental observations confirmed the UV (4.7 eV, 263 nm and 3.9 eV, 317 nm) HSB generation.

Winding Number Features for Vector Sketch Colorization

AbstractVector sketch software (e.g. Adobe Illustrator, Inkscape) and touch‐interactive technologies have long aided artists in the creation of resolution‐independent digital drawings that mimic the unconstrained nature of freehand sketches. However, artist intent behind stroke topology is often ambiguous, complicating traditional segmentation tasks such as coloring. For inspiration, we turn to the winding number, a classic geometric property of interest for binary segmentation in the presence of boundary data. Its direct application for multi‐region segmentation poses two main challenges: (1) strokes may not be consistently oriented to best identify perceptually salient regions; (2) for interior strokes there is no “correct” orientation, as either choice better distinguishes one of two neighboring regions. Thus, we form a harmonic feature space from multiple winding number fields and perform segmentation via Voronoi/power diagrams in this domain. Our perspective allows both for automatic fill region detection and for a semi‐automatic framework that naturally incorporates user hints and interactive sculpting of results, unlike competing automatic methods. Our method is agnostic to curve orientation and gracefully handles varying gap sizes in the sketch boundary, outperforming state‐of‐the‐art colorization methods on these “gappy” inputs. Moreover, it inherits the ability of winding numbers to specify “fuzzy” boundaries, leading to simple strategies for color diffusion and single‐parameter‐driven growing and shrinking of regions.

An efficient algorithm for time-domain acoustic scattering in three dimensions by layer potentials

In this paper, we develop a simple and fast algorithm for the time-domain acoustic scattering by a sound-soft or an impedance obstacle in three dimensions. We express the solution to the scattering problem by layer potentials, and then a time-domain boundary integral equation is derived. To numerically solve the resulting boundary integral equation, we propose a full discretization scheme by combining the convolution splines with a Galerkin method. In time, we approximate the density in a backward manner in terms of the convolution splines. In space, we project the density at each time onto the space of spherical harmonics, and then use the spatial discretization of a Nyström type on the surface of an obstacle which is homeomorphic to a sphere. A gallery of numerical examples are presented to show the efficiency of our algorithm. The stability, convergence and accuracy of the algorithm are discussed.

Horocyclic harmonic Bergman spaces on homogeneous trees

Horocyclic harmonic Bergman spaces on homogeneous trees

Ultra-Generalized Continuous Class F Power Amplifier with Finite Third-Harmonic Load Impedance

This paper proposes the ultra−generalized continuous class F (UCCF) mode, which combines the influences of the drain current conduction angle and overdriven transistor on the drain current waveform to achieve a broader finite third-harmonic load impedance space. The UCCF mode uses α to describe the magnitude of the drain current conduction angle and pcr to describe the drain current peak-clipped ratio. An analysis of the effects of pcr and α on the linearity and efficiency of the UCCF model is presented, establishing a robust theoretical foundation for achieving a balance between these two characteristics. Additionally, an examination of how pcr and α influence the load impedance design space is conducted, demonstrating that the UCCF mode not only offers a broader finite third-harmonic load impedance space but also expands the fundamental and second-harmonic load impedance design space. For practical validation, a PA with frequency of 2.05–2.65 GHz is designed based on CGH40010F. The test results show that S11 is less than −15 dB, the drain efficiency is 67.0–73.2%, and the output power is 40.1–41.0 dBm. The linearity is tested using a 5G NR (New Radio) signal with a bandwidth of 100 MHz and a peak-to-average power ratio of 8 dB at 2.35 GHz. The worst adjacent channel power ratio (ACPR) is −34.8 dBc without digital predistortion (DPD), and −57.8 dBc with DPD. An average output power (Pave) of around 32.4 dBm and an average DE (DEave) of 34.39% were obtained.

Input‐harmonic manipulated wideband high‐efficiency series of continuous mode power amplifier

SummaryIn this paper, a systematic research of the effects of second source harmonics caused by gate‐source capacitance in the design of a series of continuous mode (SCM) power amplifiers (PAs) is first presented. The research results show that, compared to the conventional SCMs that require the input second harmonic to be short circuited, SCMs under the impact of input nonlinearity have a more flexible input second harmonic manipulation space, and to further improve and maintain the performance of PAs in wideband operations, a new impedance design space is explored through analysis. For practical validation, SCMs PA considering the effects of the input nonlinearity is designed and fabricated with a 10‐W gallium nitride (GaN) device, acquiring 39.1–42 dBm output power and 60.6–71.5% drain efficiency (DE) performance at a 3 dB gain compression level operating over the 0.5–3.5 GHz frequency range with a relative bandwidth of 150%.

Harmonic analysis of discrete tracers of large-scale structure

It is commonplace in cosmology to analyze fields projected onto the celestial sphere, and in particular density fields that are defined by a set of points e.g. galaxies. When performing an harmonic-space analysis of such data (e.g. an angular power spectrum) using a pixelized map one has to deal with aliasing of small-scale power and pixel window functions. We compare and contrast the approaches to this problem taken in the cosmic microwave background and large-scale structure communities, and advocate for a direct approach that avoids pixelization. We describe a method for performing a pseudo-spectrum analysis of a galaxy data set and show that it can be implemented efficiently using well-known algorithms for special functions that are suited to acceleration by graphics processing units (GPUs). The method returns the same spectra as the more traditional map-based approach if in the latter the number of pixels is taken to be sufficiently large and the mask is well sampled. The method is readily generalizable to cross-spectra and higher-order functions. It also provides a convenient route for distributing the information in a galaxy catalog directly in harmonic space, as a complement to releasing the configuration-space positions and weights, and a route to spectral apodization. We make public a code enabling the application of our method to existing and upcoming datasets.

A wide-angle formulation of foreground filters for HI intensity mapping

Neutral hydrogen intensity mapping can in principle deliver rapid and large-volume cosmological surveys with exquisitely accurate redshifts that are determined directly from imaging. However, intensity maps suffer from very strong foreground contamination. Future surveys will require efficient data pipelines to remove the foregrounds and reveal the cosmological signal. It is expected that this cleaning will not remove the signal in substantial parts of the available Fourier space and that significant loss of signal due to imperfect cleaning will be confined to specific regions of Fourier space. This suggests a strategy which is useful for simplified estimates and rapid computations — i.e., to apply foreground filters that avoid the regions where loss of signal is significant. The standard Fourier-space power spectrum and foreground filters use a flat-sky approximation and thus exclude wide-angle correlations. We provide a new geometrical formulation of foreground filters in harmonic space, which naturally includes all wide-angle effects in the power spectrum. Foreground filtering leads to a loss of isotropy in Fourier space. In harmonic space this produces off-diagonal correlations. We derive analytical expressions for the generalised HI power spectrum and its cross-power with CMB lensing, for both single-dish and interferometer mode surveys. We show numerically that the off-diagonal contributions are negligible for the auto power. In the cross power, there is a non-negligible off-diagonal contribution, but only for a small interval of the largest available scales. For auto and cross power, the signal loss due to foreground avoidance decreases with increasing multipole (i.e. smaller scales), and the loss in interferometer mode is equal to, or slightly greater than, in single-dish mode. We find that the cross power in single-dish mode vanishes below a critical multipole, ℓ < ℓ 0. For an SKA-like survey, ℓ 0 ∼ 20 – 40 over redshifts z = 1 – 3. This feature is not seen in interferometer mode as the pertinent angular scales are larger than those allowed by the minimum baseline.

Foreground removal with ILC methods for AliCPT-1

One of the main goals of most future CMB experiments is the precise measurement of CMB B-mode polarization, whose major obstacle is the Galactic foregrounds. In this paper, we evaluate the foreground cleaning performance of the variants of the ILC method on partial sky B-modes and analyze the main sources of biases on the BB power spectrum. Specially, we compare the NILC, the cILC (in three domains) and the cMILC methods for AliCPT-1 simulations. We find that the cILC methods implemented in harmonic space and needlet space are both competent to clean different models of foregrounds, which bias the tensor-to-scalar ratio about 0.008 at maximum, and constrain the tensor-to-scalar ratio to r < 0.043(95%CL) for the AliCPT-1 configuration. We also note that the deviation of the estimated noise bias from the actual one for ILC, dubbed the noise bias error (NBE) in this paper, might make significant effects on the power spectrum for a small footprint and low signal-to-noise ratio CMB experiment. We finally obtain its relation with respect to the noise residual which fits well with the simulated results.

Harmonic State-Space Modeling and Closed-Loop Control of Single-Stage High-Frequency Isolated DC–AC Converter

The port harmonic features of the single-phase single-stage high-frequency isolated dc-ac converter is not analyzed so far and the corresponding theoretical basis for improving the current quality is still a research gap. In this paper, a linear model in frequency domain of the single-stage high-frequency isolated dc-ac converter is originally derived based on the combined harmonic state space and extended description function modeling technique. The systematic modeling process is analyzed and the accuracy of the modeling is verified by the simulation results. Combined with the model analysis, the harmonic coupling characteristics between the ac side and the dc side are clearly presented. On this basis, this paper proposes a closed-loop control strategy to reduce the decrease of the dc-side and ac-side current quality caused by the harmonic coupling at the control level. Furthermore, this paper constructs the compound controllers based on the P-type iterative learning to suppress the harmonic currents of the dc side and the ac side, respectively. The composite controller does not distinguish the harmonic frequency. Hence, the proposed controller achieves the effective suppression of the harmonic currents in real time. The effectiveness and feasibility of the proposed control strategy are verified by the experiments.

Harmonic Interaction Analysis of Inverters Based on Harmonic State-Space Model Considering Dead-Zone Effect

In the context of increasing device switching frequencies year by year, non-ideal factors such as dead-zone effects in inverters not only cause output waveform distortion, but also generate harmonics that make the interaction between inverters and power grids more complex. In order to accurately describe and analyze this phenomenon and establish a more practical grid-connected inverter model, this paper proposes and establishes a three-phase LCL grid-connected inverter harmonic state space model that considers dead-zone effects. By incorporating dead-zone effects into the modeling process, the system transfer function matrix is derived to quantify the harmonic interaction coefficient between the inverter and the power grid, and the influence mechanism of dead-zone effects on the harmonic interaction process at high switching frequencies is analyzed in depth. The necessity of considering dead-zone effects is illustrated through simulation, and the effectiveness and accuracy of the inverter HSS model considering the effects of dead-zone effects are verified through experiments.

Correlations for an anisotropic polarized stochastic gravitational wave background in pulsar timing arrays

The recent compelling observation of the nanohertz stochastic gravitational wave background has brought to light a new galactic arena to test gravity. In this paper, we derive a formula for the most general expression of the stochastic gravitational wave background correlation that could be tested with pulsar timing and future square kilometer arrays. Our expressions extend the harmonic space analysis, also often referred to as the power spectrum approach, to predict the correlation signatures of an anisotropic polarized stochastic gravitational wave background with subluminal tensor, vector, and scalar gravitational degrees of freedom. We present the first few nontrivial anisotropy and polarization signatures in the correlation and discuss their dependence on the gravitational wave speed and pulsar distances. Our results set up tests that could potentially be used to rigorously examine the isotropy of the stochastic gravitational wave background and strengthen the existing constraints on possible non-Einsteinian polarizations in the nanohertz gravitational wave regime.

Lensing convergence and anisotropic dark energy in galaxy redshift surveys

Analyses of upcoming galaxy surveys will require careful modelling of relevant observables such as the power spectrum of galaxy counts in harmonic space Cℓ(z,z′). We investigate the impact of disregarding relevant relativistic effects by considering a model of dark energy including constant sound speed ceff2, constant equation of state w, and anisotropic stress sourced by matter perturbations π. Cosmological constraints were computed using cosmic microwave background anisotropies, baryon acoustic oscillations, supernovae type Ia, and redshift space distortions. Our results are consistent with w=−1, ceff2=1, and π=0. Then, a forecast for the performance of an Euclid-like galaxy survey was carried out also adding information from other probes. Here we show that, regardless of the galaxy survey configuration, neglecting the effect of lensing convergence will lead to substantial shifts in the galaxy bias b0 and the neutrino mass ∑mν. Shifts in the dark energy sound speed and anisotropic stress also appear, but they depend on the survey configuration and hence lack robustness. While neglecting lensing convergence also leads to a Hubble constant H0 moving downwards, the significance of the shift is not big enough to play a relevant part in the current H0 tension.

Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space BHλ,0&lt;λ&lt;∞ to the weighted harmonic Zygmund space ZHβ,0&lt;β&lt;∞. The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between BHλ and ZHβ is bounded. Secondly, we will study the compactness case of the composition operator between BHλ and ZHβ. Finally, we will estimate the essential norms of the composition operator between BHλ and ZHβ.

Using spherical harmonics to solve the Boltzmann equation: an operator-based approach

ABSTRACT The transport of charged particles or photons in a scattering medium can be modelled with a Boltzmann equation. The mathematical treatment for scattering in such scenarios is often simplified if evaluated in a frame where the scattering centres are, on average, at rest. It is common therefore to use a mixed coordinate system, wherein space and time are measured in a fixed inertial frame, while momenta are measured in a ‘co-moving’ frame. To facilitate analytical and numerical solutions, the momentum dependence of the phase-space density may be expanded as a series of spherical harmonics, typically truncated at low order. A method for deriving the system of equations for the expansion coefficients of the spherical harmonics to arbitrary order is presented in the limit of isotropic, small-angle scattering. The method of derivation takes advantage of operators acting on the space of spherical harmonics. The matrix representations of these operators are employed to compute the system of equations. The computation of matrix representations is detailed and subsequently simplified with the aid of rotations of the coordinate system. The eigenvalues and eigenvectors of the matrix representations are investigated to prepare the application of standard numerical techniques, e.g. the finite volume method or the discontinuous Galerkin method, to solve the system.