Harmonic Space Research Articles

English

We study spaces of Holder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Holder type space of harmonic functions with prescribed modulus of continuity ω = ω(h) and the second is the variable exponent harmonic Holder space with the continuity modulus |h|λ(·). We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

The harmonic oscillator energy level spacing for neutrons and protons in nuclei

Approximate expressions of hw for neutrons and protons separately, as functions of the neutron number Ν and the proton number Ζ respectively, are derived. The dependence of hωn{hωp) on N(Z) is established using a rather recently proposed semi-phenomenological density distribution based on the separation energies of the last neutron or proton. The corresponding curves of hω show "discontinuities in the slope" at the closed shells throughout the periodic table. The difference hωn — hωΛ is also discussed

Determination of the Harmonic Oscillator Energy Level Spacing for Atomic Clusters

The harmonic oscillator energy level spacing Κω for atomic clusters as a function of the particle number Ν is expressed analytically in terms of the parameters of a Woods-Saxon (or Symmetrized Woods-Saxon) potential which approximates the effective spherical self-consistent jellium model potential. The expressions derived depend an the particular scheme adopted to approximate the potential by the harmonic oscillator one and on the assumed dependence of the potential radius R on N. It is also observed, considering the case of sodium clusters,that for large Ν the expressions of Ηω are in good agreement with the well known expression of Ηω in terms of the Wigner-Seitz radius.

Laser generation and amplification of TE and TM modes in a semiconductor optical GaAs waveguide with distributed feedback generated by a space charge wave

We propose a scheme for generating and amplifying coupled optical TE and TM waves in the mid-IR range provided by a grating formed by a space charge wave in a planar waveguide based on GaAs and GaAlAs semiconductors. An undamped harmonic space charge wave with the frequency Ω=ωm−ωn (where ωm and ωn are the frequencies of the forward and backward optical eigenwaves) can be formed in a longitudinal constant electric field in the predomain mode. The coupling coefficients for the optical TE and TM modes of the perturbed n-GaAs waveguide have been obtained. For pairs of the TE and TM modes of the same indices (m = n), the reflectance and transmittance dependences on the carrier concentration and pump level have been found. Generation of the forward and backward waves at a wavelength of 10.6 μm is shown for TE0 and TM2 modes at the gain factor of γ ≈ 152 cm−1 and 155 cm−1, respectively. The obtained results can be used to create semiconductor two-frequency coherently synchronized oscillators based on space charge wave–optical mode interaction.

On the integral representation of finely superharmonic functions

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $\mathcal{P}$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom $D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $\mathcal{S(U)}$ of nonnegative finely superharmonic functions in $U$ a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.

Classification of Electronic Devices With Power-Swept Signals Using Harmonic Radar

A novel approach for classification of electronic devices using a harmonic radar is proposed in this article. We transmit single-tone time-varying signals with varying power to the electronic circuits under test (ECUTs). The novelty of the proposed approach stems from representing the harmonic response of the ECUT in the harmonic and statistical feature spaces. Besides, as a new approach, we sweep the transmitting power to capture the nonlinear behavior relevant to nonlinear I-V characteristics of the ECUT. Furthermore, we utilize both the time and frequency content of the received harmonic response. Statistical and Fourier domain features of the received power levels at harmonic frequencies are determined and used for classification. k-nearest neighbors is chosen as the classification method. We demonstrate the effectiveness of our method in the presence of noise through Monte Carlo simulations with different signal-to-noise ratio values. Extensive simulation studies show that our new approach of power-swept signals and new features we have developed are very effective at classifying nonlinear devices.

A Novel Fractional Harmonic $d\text{--}q$ Domain Based Power Line Signaling Technique for Power Converters in a Microgrid

Traditionally, the communication between power converters connected in a microgrid is established using a wired method or a wireless droop method. The shortcomings of these methods have opened up the possibility of power-line-based communication between converters. However, the conventional power line communication systems have the disadvantages of using high-frequency data carrying signals that are prone to attenuation. These systems also make use of explicit couplers, transmitters, and amplifiers that adds to the overall cost and infrastructure. In this paper, a novel fractional harmonic $d\text{--}q$ domain based power line communication technique is proposed. A switched fractional harmonic current space vector is used to carry data. The primary control loop and the modulation algorithm of the converters are used for encoding and decoding of data. The proposed technique thus overcomes the disadvantages of the conventional power line communication systems and at the same time carries forward the advantages of it. The proposed technique is validated with sufficient experimental test results.

An algebraic approach to calculate Franck–Condon factors

An algebraic approach based on unitary algebras to calculate Franck–Condon factors (FCF’s) is presented. The method is based on the unitary group approach, which consists in adding a scalar boson to the $$\nu $$-D harmonic oscillator space, taking advantage of the transformation brackets connecting the energy, coordinate and momentum representations. In this scheme the solutions are given in terms of an expansion of harmonic oscillator functions. In this way the overlaps are expressed in terms of simple scalar product of the eigenvectors. As a benchmark to illustrate our approach the case of two 1D-Morse potentials is presented. Discussions concerned with both non-Condon contributions and the harmonic limit are included. FCF’s involving the 1D-Morse and asymmetric double Morse potentials are also considered. As an application, the FCF’s involving the S–S stretching in the $$\hbox {S}_2\hbox {O}$$ molecule are described in terms of two Morse potentials.

Atomic decomposition for the harmonic Fock spaces in the plane

The main purpose of this paper is to extend the certain functional-analytic properties from the analytic Fock spaces to the harmonic Fock spaces in the plane. Firstly, we found the pointwise estimate for the arbitrary function in the harmonic Fock space Hαp, where 1≤p≤∞. Then, we describe the atomic decomposition for Hαp.

Berton’s Ludic Pedagogy and the Subdominant Otherwise: Tension and Compromise in the Early Paris Conservatoire Curriculum

The founding of the Paris Conservatoire in 1795 was intended to bring standardization and meritocracy to French musical pedagogy. However, major compromises were reached in the setting of the Conservatoire’s curriculum, aimed at synthesizing the diverse pedagogical approaches of its professors. Charles-Simon Catel’s Traite d’Harmonie (1803), the Conservatoire’s official harmony textbook, was one such compromise, written to bring an end to ongoing disagreements between the pro-Rameau and anti-Rameau factions on the institution’s faculty. This paper examines the music-theoretical debates that underpinned the Catel compromise by placing Catel’s Traite in dialogue with the music and pedagogy of composer Henri-Montan Berton. Although largely unknown today, Berton was one of the most prolific opera composers of the Revolution; in 1795, he was appointed harmony professor to the Conservatoire, where he served on the committee which unanimously approved Catel’s treatise. Comparing Berton’s Rameau-inspired conception of harmonic space with Catel’s more pragmatic harmonic doctrine, this paper will present an archaeology of a single chord, the subdominant, tracing its use from Berton’s pedagogical games to his Revolutionary-era operas. This paper argues that the subdominant chord occupied a much more privileged position in Berton’s work than it did in Catel’s Traite d’Harmonie, Berton frequently employing harmonic forms that seem to emphasize the subdominant at the expense of the more powerful dominant. Thus, Berton’s treatment of the subdominant, both in the opera house and in the classroom, reveals just how far Catel’s treatise was removed from the musical practices of the Conservatoire’s faculty.

Harmonic Besov spaces with small exponents

ABSTRACTWe study harmonic Besov spaces on the unit ball of , where 0<p<1 and . We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We show that the dual of harmonic Besov space is weighted Bloch space under certain volume integral pairing for 0<p<1 and . Our other results are about growth at the boundary and atomic decomposition.

Characterizations of Bloch-Type Spaces of Harmonic Mappings

We study the Banach space BHα (α&gt;0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)&lt;∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α&gt;0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α&gt;1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0&lt;α&lt;1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

Generalized Single-Phase Harmonic State Space Modeling of the Modular Multilevel Converter With Zero-Sequence Voltage Compensation

The modular multilevel converter (MMC) has attracted extensive research in recent years. An appropriate model is necessary to analyze stability or to design MMC controllers. Several published MMC models have been derived in single-phase form to simplify the modeling mathematics. However, little attention is given to the zero-sequence voltage, which introduces coupling in the single-phase model and leads to significant error. In this paper, after revealing the mechanism behind the zero-sequence voltage, a more accurate generalized single-phase MMC model is proposed, which eliminates the zero-sequence voltage coupling effect and is linearized based on harmonic state space (HSS) theory to precisely characterize the internal harmonic features of MMC. A systematic HSS modeling process is presented for both open-loop and closed-loop conditions. And the proposed model is generalized as it can incorporate different control strategy by controller transfer function substitution. Hence it is valuable to analyze MMC stability and dynamics. Model effectiveness is verified by simulation results.

Dark Energy Survey Year 1 results: measurement of the galaxy angular power spectrum

ABSTRACT We use data from the first-year observations of the DES collaboration to measure the galaxy angular power spectrum (APS), and search for its BAO feature. We test our methodology in a sample of 1800 DES Y1-like mock catalogues. We use the pseudo-Cℓ method to estimate the APS and the mock catalogues to estimate its covariance matrix. We use templates to model the measured spectra and estimate template parameters firstly from the Cℓ’s of the mocks using two different methods, a maximum likelihood estimator and a Markov Chain Monte Carlo, finding consistent results with a good reduced χ2. Robustness tests are performed to estimate the impact of different choices of settings used in our analysis. Finally, we apply our method to a galaxy sample constructed from DES Y1 data specifically for LSS studies. This catalogue comprises galaxies within an effective area of 1318 deg2 and 0.6 &amp;lt; z &amp;lt; 1.0. We find that the DES Y1 data favour a model with BAO at the $2.6 \sigma$ C.L. However, the goodness of fit is somewhat poor, with χ2/(d.o.f.) = 1.49. We identify a possible cause showing that using a theoretical covariance matrix obtained from Cℓ’s that are better adjusted to data results in an improved value of χ2/(dof) = 1.36 which is similar to the value obtained with the real-space analysis. Our results correspond to a distance measurement of DA(zeff = 0.81)/rd = 10.65 ± 0.49, consistent with the main DES BAO findings. This is a companion paper to the main DES BAO article showing the details of the harmonic space analysis.

Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)

Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.

The Aemulus Project. I. Numerical Simulations for Precision Cosmology

Abstract The rapidly growing statistical precision of galaxy surveys has led to a need for ever more precise predictions of the observables used to constrain cosmological and galaxy formation models. The primary avenue through which such predictions will be obtained is suites of numerical simulations. These simulations must span the relevant model parameter spaces, be large enough to obtain the precision demanded by upcoming data, and be thoroughly validated in order to ensure accuracy. In this paper, we present one such suite of simulations, forming the basis for the Aemulus Project, a collaboration devoted to precision emulation of galaxy survey observables. We have run a set of 75 (1.05 h −1 Gpc)3 simulations with mass resolution and force softening of and 20 h −1 kpc, respectively, in 47 different wCDM cosmologies spanning the range of parameter space allowed by the combination of recent cosmic microwave background, baryon acoustic oscillation, and Type Ia supernova results. We present convergence tests of several observables including spherical overdensity halo mass functions, galaxy projected correlation functions, galaxy clustering in redshift space, and matter and halo correlation functions and power spectra. We show that these statistics are converged to 1% (2%) or to the sample variance of the statistic, whichever is larger, for halos with more than 500 (200) particles, respectively, and scales of r &gt; 200 h −1 kpc in real space or k ∼ 3 h Mpc−1 in harmonic space for z ≤ 1. We find that the dominant source of uncertainty comes from varying the particle loading of the simulations. This leads to large systematic errors for statistics using halos with fewer than 200 particles and scales smaller than k ∼ 4 h Mpc−1. We provide the halo catalogs and snapshots detailed in this work to the community at https://AemulusProject.github.io.

(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n

We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].

New Quantum Structure of Space-Time

We start from quantum theory (instead of general relativity) to approach quantum gravity within a minimal setting and promote the space-time coordinates to quantum non-commuting operators. Comparison to the harmonic oscillator global phase space is enlighting. The phase space instanton (X, P = iT) describes the hyperbolic quantum space-time structure and generates the quantum light cone. The classical Minkowski space-time null generators X = T dissapear at the quantum level replaced by four Planck scale hyperbolae and a new quantum Planck scale vacuum region emerges. We describe the quantum Rindler and quantum Schwarzshild-Kruskal space-time structures. The horizons and the r=0 singularity are quantum mechanically erased. The four Kruskal regions merge inside a single quantum Planck scale "world". The quantum space-time structure consists of hyperbolic discrete levels of odd numbers (2n + 1) (in Planck units ), n = 0, 1, 2.... . The mass levels are the square root of (2n + 1). Large n are semiclassical tending towards a classical continuum space-time. Low n are quantum, the lowest mode (n = 0) being the Planck scale. Two dual branches are present covering the whole mass spectrum: from the largest astrophysical objects to the quantum elementary particles passing by the Planck mass. Black holes belong to both branches. (Abridged)

Nearly hyperharmonic functions and Jensen measures

Let $(X,\mathcal H)$ be a $\mathcal P$-harmonic space and assume for simplicity that constants are harmonic. Given a numerical function $\varphi$ on $X$ which is locally lower bounded, let \begin{equation*} J_\varphi(x):=\sup\{\int^\ast \varphi\,d\mu(x)\colon \mu\in \mathcal J_x(X)\}, \qquad x\in X, \end{equation*} where $\mathcal J_x(X)$ denotes the set of all Jensen measures $\mu$ for $x$, that is, $\mu$ is a compactly supported measure on $X$ satisfying $\int u\,d\mu\le u(x)$ for every hyperharmonic function on $X$. The main purpose of the paper is to show that, assuming quasi-universal measurability of $\varphi$, the function $J_\varphi$ is the smallest nearly hyperharmonic function majorizing $\varphi$ and that $J_\varphi=\varphi \vee \hat J_\varphi$, where $\hat J_\varphi$ is the lower semicontinuous regularization of $J_\varphi$. So, in particular, $J_\varphi $ turns out to be at least measurable as $\varphi$. This improves recent results, where the axiom of polarity was assumed. The preparations about nearly hyperharmonic functions on balayage spaces are closely related to the study of strongly supermedian functions triggered by J.-F. Mertens more than forty years ago.

A simple approach to solve the time independent Schröedinger equation for 1D systems

A simple algebraic approach based on the well known angular momentum SU(2) algebra is presented to describe 1D systems for arbitrary potentials. The approach is based on the dimension increase of the 1D harmonic oscillator space through the addition of a scalar boson, keeping constant the total number of bosons. In this new space the realization of the coordinate and momentum correspond to components of the angular momentum algebra, which in turn define the coordinate and momentum representation bases. This remarkable result allows the Hamiltonian matrix representationto be expressed in terms of a diagonal matrix characterizing the potential we are considering. The solutions are obtained as an expansion of harmonic oscillator functions by purely algebraic means. As an example of our approach the Morse and asymmetric double Morse potentials are considered.