The Laplace transform on time scales revisited

In this work, we reexamine the time scale Laplace transform as defined by Bohner and Peterson [M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transform: Unification and extension, Methods Appl. Anal. 9 (1) (2002) 155-162]. In particular, we give conditions on the class of functions which have a transform, develop an inversion formula for the transform, and further, we provide a convolution for the transform. The notion of convolution leads to considering its algebraic structure—in particular the existence of an identity element—motivating the development of the Dirac delta functional on time scales. Applications and examples of these concepts are given.     

Comments on the primary and scatter dose-spread kernels used for convolution methods

The basic primary and scatter dose-spread kernels used for convolution methods are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom. However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. Using the differential primary and scatter concept, this paper proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under conditions that the beam is parallel, the incident beam intensity is uniform within and zero outside the field, and the primary beam attenuation coefficient along raylines is not a function of depth and off-axis distance.     

A fitting algorithm for real coefficient polynomial rooting

A new algorithm for computing all roots of polynomials with real coefficients is introduced. The principle behind the new algorithm is a fitting of the convolution of two subsequences onto a given polynomial coefficient sequence. This concept is used in the initial stage of the algorithm for a recursive slicing of a given polynomial into degree-2 subpolynomials from which initial root estimates are computed in closed form. This concept is further used in a post-fitting stage where the initial root estimates are refined to high numerical accuracy. A reduction of absolute root errors by a factor of 100 compared to the famous Companion matrix eigenvalue method based on the unsymmetric QR algorithm is not uncommon. Detailed computer experiments validate our claims.     

On log-normal convolutions: An analytical–numerical method with applications to economic capital determination

We put forward an efficient algorithm for approximating the sums of independent and log-normally distributed random variables. Namely, by combining tools from probability theory and numerical analysis, we are able to compute the cumulative distribution functions of the just-mentioned sums to a high precision and in a relatively short computing time. We illustrate the effectiveness of the new method in the contexts of the individual and collective risk models, aggregate economic capital determination, and economic capital allocation.     

Estimating Precision in Functional Images

Abstract

Functional imaging of biologic parameters like in vivo tissue metabolism is made possible by Positron Emission Tomography (PET). Many techniques have been suggested for extracting such images from dynamic time-course sequences of reconstructed PET scans. Quantitating the precision of these estimates is important for drawing inferences on the biologic parameters. Analytic variance formulas are not immediate owing to the nonlinear methods used in extraction. The usual resampling approach is infeasible because each image reconstruction in PET is a computationally demanding solution to a high-dimensional linear inverse problem. We suggest an alternative simulation approach that approximates the distribution of reconstructed PET scans and performs a parametric bootstrap in the imaging domain. Results on a simplified model chosen to match the characteristics of PET reconstruction are very encouraging. Mixture analysis is used to estimate functional images; however, the suggested approach is general enough to extend to other techniques or imaging methods.     

Computing the distribution of the Robinson-Foulds distance

With the exponential growth of genome databases, the importance of phylogenetics has increased dramatically over the past years. Studying phylogenetic trees enables us not only to understand how genes, genomes, and species evolve, but also helps us predict how they might change in future. One of the crucial aspects of phylogenetics is the comparison of two or more phylogenetic trees. There are different metrics for computing the dissimilarity between a pair of trees. The Robinson-Foulds (RF) distance is one of the widely used metrics on the space of labeled trees. The distribution of the RF distance from a given tree has been studied before, but the fastest known algorithm for computing this distribution is a slow, albeit polynomial-time, O(l⁵) algorithm. In this paper, we modify the dynamic programming algorithm for computing the distribution of this distance for a given tree by leveraging the number-theoretic transform (NTT), and improve the running time from O(l⁵) to O(l³ log l), where l is the number of tips of the tree. In addition to its practical usefulness, our method represents a theoretical novelty, as it is, to our knowledge, one of the rare applications of the number-theoretic transform for solving a computational biology problem.     

On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

On the least values of -norms for the Kontorovich–Lebedev transform and its convolution

We establish analogs of the Hausdorff–Young and Riesz–Kolmogorov inequalities and the norm estimates for the Kontorovich–Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in L_p spaces, 1p∞. Boundedness properties of the Kontorovich–Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich–Lebedev operator is equal to . It confirms, for instance, by the known Plancherel-type theorem for this transform when p=2.     

Weighted Young Inequalities for Convolutions

Sufficient conditions for some weighted Young inequalities type are obtained.AMS Subject Classification (1991): 42A85, 42B20, 47G10