Fourier inversion for multidimensional characteristic functions

A density functionf(x),xR ⁿ is said to bepiecewise smooth if for eachxR ⁿ , the mean value function is piecewiseC with compact support. (d is normalized surface measure on the unit sphere). The Fourier transform is with spherical partial sum . Theorem. For suchf, lim _r f _R (x)=M ₀+f(x) if and only ifrM _r f(x) hask=[(n–3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim where 0 is uniquely determined.     

Finite element modeling of acousto-mechanical coupling in the cat middle ear

The function of the middle ear is to transfer acoustic energy from the ear canal to the cochlea. An essential component of this system is the tympanic membrane. In this paper, a new finite element model of the middle ear of the domestic cat is presented, generated in part from cadaver anatomy via microcomputed tomographic imaging. This model includes a layered composite model of the eardrum, fully coupled with the acoustics in the ear canal and middle-ear cavities. Obtaining the frequency response from 100 Hz to 20 kHz is a computationally challenging task, which has been accomplished by using a new adaptive implementation of the reduced-order matrix Padé-via-Lanczos algorithm. The results are compared to established physiological data. The fully coupled model is applied to study the role of the collagen fiber sublayers of the eardrum and to investigate the relationship between the structure of the middle-ear cavities and its function. Three applications of this model are presented, demonstrating the shift in the middle-ear resonance due to the presence of the septum that divides the middle-ear cavity space, the significance of the radial fiber layer on high frequency transmission, and the importance of the transverse shear modulus in the eardrum microstructure.     

Brownian Continuity Modulus Via Series Expansions

Beginning with the series representation in terms of Haar functions, we give a simplified proof of the Lévy modulus of continuity for standard Brownian motion.     

The first eigenvalue of a spherical cap

We find upper and lower bounds for the first eigenvalue of the Laplacian on the two-sphere from which a disk has been removed, with Dirichlet conditions imposed on the resulting boundary. When the radius of the disk tends to zero our lower bound is sharper than that obtained by Del Grosso, Gerardi, and Marchetti in the preceding paper.     

Perturbation theory with instantons

We develop “perturbation theory” rules for calculating the effect of instantons in a pure Yang-Mills theory with no fermions, in the “dilute gas” approximation in which the N-instanton solution is assumed to be the sum of N widely separated one-instanton solutions. These rules are then used to compute the gluon propagator and proper vertex function including all orders of the instanton interaction but only to lowest order in the gluon coupling. It is to be expected that such an approximation is valid only for momenta q larger than the physical mass μ. The result is that in this regime instantons cause variations in the propagator and vertex of the form where b is the coefficient in the expansion of the β function: β = bg³ + 3. .