We enumerate cyclically colored m-gonal plane cacti (called m-ary cacti) according to the degree distribution of vertices of each color. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. The corresponding problem for rooted m-ary cacti has been solved by Goulden and Jackson in connect...

We first give a new proof and also a new formulation for the Abhyankar-Gujar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gujar formula to give a more straightforward proof for the equivalence of the Jacobian conjecture with a special case of the vanishing conjecture of (homogeneous) quadratic differential operators with constant coefficients.

The Krĕın-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in te...

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

We give a survey of the status of some of the fundamental problems in harmonic analysis on semisimple symmetric spaces, including the description of the discrete series, the deenition of the Fourier transform, the inversion formula, the Plancherel formula and the Paley{ Wiener theorem.

A formula is given for the propagation of errors during matrix inversion. An explicit calculation for a 2 × 2 matrix using both the formula and a Monte Carlo calculation are compared. A prescription is given to determine when a matrix with uncertain elements is sufficiently nonsingular for the calculation of the covariances of the inverted matrix elements to be reliable.

The integral image approach allows the optimal computation of Haar-based features for realtime recognition of objects in image sequences. This short note gives a direction to generalize the approach to high-dimensional images. We o er a formula for optimal computation of sums on high-dimensional rectangles. Integral Image Haar-Based Features High-Dimensional Image Möbius Inversion Formula

The phase-noise spectrum determines the Allan variance by a well-known integral formula. It is shown here that unique inversion of this formula is not possible in general because the mapping from spectrum to Allan variance is not one-to-one. A necessary and sufficient condition for two distinct phase spectra to have the same Allan variance is given.

The Krĕın-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in te...