In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, prove that space with fixed degree is finite dimensional. We also analogous results for ancient functions. other without any condition, sharp dimensional estimates Schrödinger degree.

Introduction and statement of results. This announcement describes an elementary method of constructing harmonic maps in some cases not covered by the general existence theory. Recall that given smooth Riemannian manifolds N and M, with N compact, then the energy functional E:H(N,M) -> R is defined on a suitable manifold of maps H(iV, M ) and is given by E{ f ) = \ \n \df | . A map ƒ is said to...

We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x|2−n and observing the patterns that emerge. This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollar...

Harmonic plus noise models have been successfully applied to a broad range of speech processing applications, including, among others, low bit-rate speech coding, and speech restoration and transformation. In conventional methods, the frequencies, the relative phases and the amplitudes of the pitch-harmonic components are assumed to be piecewise constants over an analysis frame. This assumption...

In a recent paper [1] Cariñena et al analyzed a non-polynomial onedimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schrödinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained m...

This paper suggests a new approach to the problem of harmonic signal estimation. The idea is to model the harmonic signal as a function of the state of a second order nonlinear ordinary differential equation (ODE). The function of the right hand side of the nonlinear ODE is parameterized with a polynomial model. A Kalman filter and an extended Kalman filter are then developed. The proposed meth...

regular polytopes, mean value property, finite reflection groups We compute certain polynomial invariants for the finite reflection groups of the types H3, H4 and F4. Using this result, we explicitly determine the solution space of functions satisfying a mean value property related to the exceptional regular polytopes, namely, the icosahedron and dodecahedron in three dimensions and the 24-cell...

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate line...