In this paper we study a class of measures, called harmonic measures, that one can associate to a dynamical system consisting og a space X and a finitely generated group of transformations. If the group of transformations is infinite cyclic, then these measures are invariant. We show how the theory of classical dynamical systems with invariant measure can be extended to the case of harmonic mea...

this paper presents time domain fundamental solutions for the extended biot's dynamic formulations of two-dimensional (2d) unsaturated poroelasticity. unsaturated porous media is considered as a porous media in which the voids are saturated with two immiscible fluids, i.e. liquid and gas. at first, the corresponding explicit laplace transform domain fundamental solution is obtained in term...

Given a list of N numbers, the maximum can be computed in N iterations. During these N iterations, the maximum gets updated on average as many times as the Nth harmonic number. We first use this fact to approximate the Nth harmonic number as a side effect. Further, using the fact the Nth harmonic number is equal to the natural logarithm of N plus a constant that goes to zero with N, we approxim...

A multi-mode harmonic transform coding (MMHTC) for speech and music signals is proposed. Its structure is organized as a linear prediction model with an input of harmonic and transform-based excitation. The proposed coder also utilizes harmonic prediction and an improved quantizer of excitation signal. To efficiently quantize the excitation of music signals, the modulated lapped transform (MLT)...

A multi-mode harmonic transform coding (MMHTC) for speech and music signals is proposed. Its structure is organized as a linear prediction model with an input of harmonic and transform-based excitation. The proposed coder also utilizes harmonic prediction and an improved quantizer of excitation signal. To efficiently quantize the excitation of music signals, the modulated lapped transform (MLT)...

where τ( f ) is the tension field of f and dvg is the volume form of M. It is clear that E2( f |Ω) = 0 on any compact domain if and only if f is a harmonic map. Thus E2 provides a measure for the extent to which f fails to be harmonic. If f is a critical point of (1.1) over every compact domain, then f is called a biharmonic map or 2-harmonic maps. Jiang [10] proved that f is biharmonic if and ...

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields. As an important tool the Hardy-Littlewood inequality have been playing critical roles in many mathematics, including potential analysis, partial differential equations and the theory of elas...