Harmonic Forms Research Articles

Four-manifolds with harmonic 2-forms of constant length

It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to $$\mathbb {CP}_{2}$$ with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on $$\mathbb {CP}_{2}$$ .

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of f -minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable f -harmonic 1-forms; in particular, in dimension 2, the procedure gives an index estimate in terms of the genus of the surface.

Hodge theory for intersection space cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincar\'e duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.

Analysis of weighted p-harmonic forms and applications

In this paper, we study weighted [Formula: see text]-harmonic forms on smooth metric measure space [Formula: see text] with a weighted Sobolev or a weighted Poincaré inequality. When [Formula: see text] is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the [Formula: see text]-Laplacian. For general [Formula: see text] we also obtain various splitting and vanishing theorems when the weighted curvature operator of [Formula: see text] is bounded below. As applications, we conclude Liouville property for weighted [Formula: see text]-harmonic functions and [Formula: see text]-harmonic maps.

Realisation and Approbation of Conditionally Constant Coefficients Method for Loss Counter Measuring Tools

To date, electrical power in a form of higher-order current harmonics and voltage are reported to be worsening substantially in power lines of industrial and non-commercial consumers. This fact relates to the change of power consumer types to use different household and industrial power receivers. Higher-order harmonic current components cause extra losses in transformers. As a consequence, electrical power delivery is less efficient and losses increase. Therefore, it is necessary to measure harmonic components of current and voltage in distribution networks. The paper presents a measuring tool of loss counter developed on the base of constant coefficients method. Performance and relative accuracy were studied experimentally using a variable-speed drive with the results reported in the paper. The developed measuring tool meets the standards of admitted relative accuracy and is applicable to implement conditionally constant coefficients method in order to calculate excessive losses to be caused by non-sinusoidal loads in transformers of distribution networks.

Deformation of cavitation bubbles at their strong expansion and collapse in a streamer

Deformation of initially spherical cavitation bubbles is considered in the case of their strong single expansion and subsequent collapse in a streamer consisting of three bubbles located in one straight line. The central bubble is equidistant from the others. The influence of the distance between the bubbles is studied in the range, in which the non-sphericity of the central bubble during the collapse does not exceed 20%. The expansion of the bubbles and the most part of their collapse is governed by the ordinary differential equations of the second order in the radii of the bubbles, the positions of their centers on the line of their interaction, and the amplitudes of their axisymmetric non-sphericity in the form of spherical harmonics. In doing so, the interaction between bubbles is taken into account, the assumptions of weak liquid compressibility, homobaricity of bubbles, and their small non-sphericity are used. The final of the collapse of each bubble is governed by the partial differential equations of gas dynamics. In doing so, the interaction of bubbles is ignored, but the axial symmetry of bubbles, liquid compressibility, non-uniformity of vapor, heat conductivity of vapor and liquid are taken into account, wide-range equation of states for vapor and liquid are applied. It is shown that with decreasing the distance between the bubbles, the non-sphericity of the central bubble at the end of its collapse first slowly increases from zero and then quite rapidly grows. Along with that, the central bubble non-sphericity level of 1% is attained at the distance between the bubbles equal to about 20 times their radius at the moment of their maximal expansion. Simulation of the final high-speed stage of the bubble collapse by the model destined to the low-speed stage of expansion and collapse leads to significant overestimation of the non-sphericity of the central bubble at the end of its collapse.

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.

Asymptotic behavior for $${\mathcal {H}}$$-holomorphic cylinders of small area

$${\mathcal {H}}$$-holomorphic curves are solutions of a modified pseudoholomorphic curve equation involving a harmonic 1-form as perturbation term. Following Hofer et al. (Dyn Syst Ergod Theory 22(5):1451–1486, 2002), we establish an asymptotic behavior of a sequence of finite energy $${\mathcal {H}}$$-holomorphic cylinders with small d$$\alpha $$-energies. Our results can be seen as a first step toward establishing the compactness of the moduli space of $${\mathcal {H}}$$-holomorphic curves, which in turn, due to the program initiated in Abbas et al. (Comment Math Helv 80:771–793, 2005), can be used for proving the generalized Weinstein conjecture.

The problem of Lagrange in discrete field theory

A discrete Lagrange problem in a fibered manifold on a cellular complex is defined as a Lagrangian density and a constraint submanifold in the 1-jets space of the manifold. After defining the concepts of admissible section and infinitesimal admissible variation, the aim of these problems is to find admissible sections that are critical for the action functional of the Lagrangian density with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that such critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established for the discrete Lagrange problems considered. Finally, the whole theory is illustrated with an example of geometric interest: the discrete harmonic 1-forms on the elliptic or hyperbolic discrete plane, for which we obtain two explicit variational integrators: one with initial conditions (Cauchy problem) and other with boundary conditions (Dirichlet problem).

Borcherds lifts of harmonic Maass forms and modular integrals

We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonic Maass forms of weight $1/2$ whose non-holomorphic part is allowed to be of exponential growth at $i\infty$. We determine the singularities of the lift and compute its Fourier expansion. It turns out that the lift is continuous but not differentiable along certain geodesics in the upper half-plane corresponding to the non-holomorphic principal part of the input. As an application, we obtain a generalization to higher level of the weight $2$ modular integral of Duke, Imamoglu and T\'oth. Further, we construct automorphic products associated to harmonic Maass forms.

Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Abstract Let j ⁢ ( z ) {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane ℍ {\mathbb{H}} . It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [ ℚ ( τ ) : ℚ ] ≠ 2 {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2} , then j ⁢ ( τ ) {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on Γ 0 ⁢ ( N ) {\Gamma_{0}(N)} . In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let T m {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i ⁢ ∞ {i\infty} are algebraic, and that f has its poles only at cusps equivalent to i ⁢ ∞ {i\infty} . We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that N ≥ 23 {N\geq 23} and N ∉ { 23 , 29 , 31 , 41 , 47 , 59 , 71 } {N\notin\{23,29,31,41,47,59,71\}} , then f ( T m . τ ) {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.

Closed Range Estimates for $$\bar{\partial }_b$$ on CR Manifolds of Hypersurface Type

The purpose of this paper is to establish sufficient conditions for closed range estimates on (0, q)-forms, for some fixed q, $$1 \le q \le n-1$$ , for $$\bar{\partial }_b$$ in both $$L^2$$ and $$L^2$$ -Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak Y(q), is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szegö projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak Y(q) is an easier condition to verify than earlier, less general conditions.

A Mass-accreting Gamma Doradus Pulsator with a Synchronized Core in Kepler Eclipsing Binary KIC 7385478

Abstract The short-period (P ≈ 1.7 days), Algol-type eclipsing binary KIC 7385478 consists of an F-type primary star (M 1 ≈ 1.71M ⊙) and an evolved K-type secondary (M 2 ≈ 0.37M ⊙). We study the variability of the Kepler light curve and attribute many frequency peaks in the Fourier spectrum to the spot modulation. These frequencies are in the form of orbital harmonics and are highly variable in amplitude. They are most likely from the mass-accreting primary star. In addition, we identify a series of prograde dipole g modes from the primary star that show a quasi-linear period spacing pattern and are very stable in amplitude. The period spacing pattern reveals an asymptotic period spacing value in agreement with fundamental parameters of the primary star and also implies that the near-convective-core rotation rate is almost the same as the orbital period. Thus, both the surface and the core of this Gamma Dor pulsator have synchronized with the binary orbit. We find that a lower stellar mass ≈1.50M ⊙ and higher effective temperature are needed in order to be compatible with the asteroseismic constraints from single-star evolutionary models.

Forelli type theorem in harmonic map forms

In the current paper, we give and prove a formulation of the Forelli Theorem on a starlike domain in C n \mathbb {C}^n . We also generalize the Forelli Theorem from a function to a map with a real Riemannian manifold as its target manifold.

Pairs of eta-quotients with dual weights and their applications

Let D be the differential operator defined by D:=12πiddz. This induces a map Dk+1:M−k!(Γ0(N))→Mk+2!(Γ0(N)), where Mk!(Γ0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k!(Γ0(N)) and Mk+2!(Γ0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1z)b1⋯η(ditz)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.

Research on Energy Response Characteristics of Rock under Harmonic Vibro-Impacting Drilling

PurposeThere are still a number of challenges to be addressed for the harmonic vibro-impacting drilling technology to become the norm in the industry. The change mechanism of energy response of rock under harmonic vibro-impacting is one of the problems that needs to be solved, which is also an indispensable part of the rock-breaking mechanism of the technology. Thus, this study focuses on the modelling of energy response of rock under harmonic vibro-impacting.MethodsModelling of energy response of rock with and without damping under harmonic vibro-impacting is undertaken and the corresponding equations of energy dissipation are presented in this study. Subsequently, the results of numerical calculation are given and the effects of natural frequency of rock, impacting frequency, impacting force and other parameters of energy response of rock are analyzed. Finally, a practical case is given to demonstrate the rock-breaking effect under harmonic vibro-impacting drilling.ResultsThe results show that energy response of rock with damping decays regularly over time in the harmonic form under harmonic vibro-impacting. Also, it increases with the increase of natural frequency and mass of rock, impacting frequency and force, and is not affected by the stiffness of the rock. In addition, the penetration rate in practice has been greatly improved under harmonic vibro-impacting.ConclusionsBased on the analysis undertaken, the variation mechanism of energy response of rock under harmonic vibro-impacting was found and the rock-breaking effect of the technology was verified. The study on the modelling of energy response of rock under harmonic vibro-impacting can provide a theoretical basis for applying harmonic vibro-impacting drilling technology.

Adiabatic limit and the Frölicher spectral sequence

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Fr\"olicher spectral sequence of any compact complex manifold supporting an SKT metric $\omega$ (i.e. such that $\partial\bar\partial\omega=0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose torsion operator $\tau$ and its adjoint vanish on $\Delta''$-harmonic forms of positive degrees up to $\mbox{dim}_\C X$. Besides the pseudo-differential Laplacian inducing a Hodge theory for $E_2$ that we constructed in earlier work and Demailly's Bochner-Kodaira-Nakano formula for Hermitian metrics, a key ingredient is a general formula for the dimensions of the vector spaces featuring in the Fr\"olicher spectral sequence in terms of the asymptotics, as a positive constant $h$ decreases to zero, of the small eigenvalues of a rescaled Laplacian $\Delta_h$, introduced here in the present form, that we adapt to the context of a complex structure from the well-known construction of the adiabatic limit and from the analogous result for Riemannian foliations of \'Alvarez L\'opez and Kordyukov.

Differential operators on polar harmonic Maass forms and elliptic duality

Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

Insights into the electromagnetic properties of the resonant slow-wave circuits from the resonant cavity perspective

A multigap resonant cavity can simply be regarded as either a slow-wave structure (SWS) with shorted ends or as a resonator. However, two distinct physical images will be obtained when characterizing the electromagnetic properties of such circuits. In particular, dispersion is no longer a necessary concept from the resonant cavity perspective. The periodicity of the circuit and the synchronization characteristics with the beam are incorporated in the longitudinal electric field shape. Thus, we present insights from the resonant cavity perspective by taking the resonant rectangular grating SWS as an example. It is found that even if the fields are originally expressed as normal modes, they can be transformed into the space harmonic form. This is thought to be the result of the periodicity of the circuit, which provides a connection between the multigap resonant cavity and the SWS. We present the findings and discuss their physical significance.

On the Measurement of Lightning Horizontal Electric Fields

This paper analyses the inherent difficulties in measuring the horizontal electric field from lightning and proposes some solutions to mitigate the contamination effect of the overwhelming vertical electric field. The analysis is carried out with a set of time-domain equations for the calculation of the electromagnetic fields from lightning. As these equations are scattered in several publications, in this paper they are compiled and presented in a harmonized form. Moreover, one new expression for the vertical electric field slightly below the ground surface is presented. It is shown that the horizontal electric field at the surface of lossy ground can be measured with minimum disturbance caused by the vertical electric field if the field sensor is placed slightly below the ground surface. This paper also shows that the horizontal electric field above the ground surface could be measured with the help of a pair of field sensors conveniently arranged so that the influence of the vertical electric field on the measurement is cancelled.