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The cello suites of Johann Sebastian Bach exhibit several types of power-law scaling, the best examples of which can be considered fractal in nature. This article examines scaling with respect to the characteristics of melodic interval and its derivative, melodic moment. A new and effective method for pitch-related analysis is described and then applied to a selection of the 36 pieces that comprise the six cello suites.

A new type of iterated function systems is constructed based on different weights, and it is proved that this type of iterated function systems generates a class of bivariate continuous functions whose graphs are the fractal interpolation surfaces passing through the given interpolation points. Considering the influences of the fractal interpolation functions on different weights and basic functions, we give the corresponding error estimation formula. Finally, the calculation formula for the integral moments of this class of bivariate fractal interpolation functions is given.

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups $\mathrm{\text{O(N)}}$, $\mathrm{\text{U(N)}}$ and $\mathrm{\text{Sp(2N)}}$. For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.

This paper proposes a new approximation formula for pricing average options under Heston's stochastic volatility model. When using the formula based on the Gram-Charlier expansion, it is necessary to know any moments of an averaged underlying asset price. We try to derive an analytical solution of the moments under the Heston model. There are two key points of the derivation: One of them is to repeatedly apply change of a certain measure. Another is to sequentially solve a system of ordinary differential equations. Moreover, numerical examples support the accuracy of the proposed average option pricing formula.