Harmony theory has been essential in composing, analysing, and performing music for centuries. Since Western tonal harmony exhibits a considerable amount of structure and regularity, it lends itself to formalisation. In this paper we present HARMTRACE, a system that, given a sequence of symbolic chord labels, automatically derives the harmonic function of a chord in its tonal context. Among oth...

In music, harmony refers to a pleasant combination of sounds. In mathematics, a harmonic function is a sine function obtained by projecting a circular motion on a diameter, and harmonic analysis is the theory of the development of periodic functions into harmonic components, or the theory of similar developments. The occurrence of the same word in musical and mathematical contexts is neither a ...

Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let φ : M → N be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and φ has finite energy, then φ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that |df | is bounded, and ...

In computer graphics, smooth data reconstruction on 2D or 3D manifolds usually refers to subdivision problems. Such a method is only valid based on dense sample points. The manifold usually needs to be triangulated into meshes (or patches) and each node on the mesh will have an initial value. While the mesh is refined the algorithm will provide a smooth function on the redefined manifolds. Howe...

Gabor jets are a set of filters that are used to extract the local frequency information from the face images. These filters are generally linear filter with impulse responses defined by a harmonic function and a Gaussian function. The Fourier transform of a Gabor filter’s impulse response is the convolution of the Fourier transform of the harmonic function and the Fourier transform of the Gaus...

A function f = u + iv defined in the domain D ⊂ C is harmonic in D if u, v are real harmonic. Such functions can be represented as f = h+ ḡ where h, g are analytic in D. In this paper the class of harmonic functions constructed by the Hadamard product in the unit disk, and properties of some of its subclasses are examined.

Recursion formulae of the N-particle partition function, the occupation numbers and its fluctuations are given using the single-particle partition function. Exact results are presented for fermions and bosons in a common one-dimensional harmonic oscillator potential, for the three-dimensional harmonic oscillator approximations are tested. Applications to excited nuclei and Bose-Einstein condens...

If u(x, y) is an infinity harmonic function, i.e., a viscosity solution to the equation −∆∞u = 0 in Ω ⊂ Rm+1 then the function v(x, z) = u(x, ‖z‖) is infinity harmonic in the set {(x, z) : (x, ‖z‖) ∈ Ω} (provided u(x,−y) = u(x, y)).

We derive via the interaction "representation" the many-body wave function for harmonically confined electrons in the presence of a magnetostatic field and perturbed by a spatially homogeneous time-dependent electric field-the Generalized Kohn Theorem (GKT) wave function. In the absence of the harmonic confinement - the uniform electron gas - the GKT wave function reduces to the Kohn Theorem wa...