The Ring of Quasimodular Forms for a Cocompact Group

We describe the additive structure of the graded ring $\widetilde{M}_*$ of quasimodular forms over any discrete and cocompact group \Gamma \subset \rm{PSL}(2, \RM). We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight $k$. This number is constant for $k$ sufficiently large and equals \dim_{\CM}(I / I \cap \widetilde{I}^2), where $I$ and $\widetilde{I}$ are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that $\widetilde{M}_*$ is contained in some finitely generated ring $\widetilde{R}_*$ of meromorphic quasimodular forms with $\dim \widetilde{R}_k = O(k^2),$ i.e. the same order of growth as $\widetilde{M}_*.$Comment: 22 pages, 1 figur

A M\"untz-Collocation spectral method for weakly singular volterra integral equations

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method

A note on "optimal resource allocation for security in reliability systems"

In a recent paper by Azaiez and Bier [Azaiez, M.N., Bier, V.M., 2007. Optimal resource allocation for security in reliability systems. European Journal of Operational Research 181, 773–786], the problem of determining resource allocation in series-parallel systems (SPSs) is considered. The results for this problem are based on the results for the least-expected cost failure-state diagnosis problem. In this note, it is demonstrated that the results for the least-expected cost failure-state diagnosis problem for SPSs in Azaiez and Bier (2007) are incorrect. In addition relevant results that were not cited in the paper are summarized

Numerical simulation of the non-Newtonian mixing layer

This work is a continuing effort to advance our understanding of the effects of polymer additives on the structures of the mixing layer. In anticipation of full nonlinear simulations of the non-Newtonian mixing layer, we examined in a first stage the linear stability of the non-Newtonian mixing layer. The results of this study show that, for a fluid described by the Oldroyd-B model, viscoelasticity reduces the instability of the inviscid mixing layer in a special limit where the ratio (We/Re) is of order 1 where We is the Weissenberg number, a measure of the elasticity of the flow, and Re is the Reynolds number. In the present study, we pursue this project with numerical simulations of the non-Newtonian mixing layer. Our primary objective is to determine the effects of viscoelasticity on the roll-up structure. We also examine the origin of the numerical instabilities usually encountered in the simulations of non-Newtonian fluids

Object-based assessment of tree attributes of Acacia tortilis in Bou-Hedma, Tunisia

Acacia tortilis subsp. raddiana represents the most important woody species in the pre-Saharan zone. It is the only forest tree persisting on the edge of the desert. Due to tree/environment interactions, canopy sub-habitats arise, enabling an increased storage of soil water, soil nutrients and soil oxygen. Depending on their density, they can also reduce erosion and reverse desertification. Soil erosion and desertification are the main problems faced by the UNESCO Biosphere Reserve in South-Tunisia (Bou-Hedma National Park). The restoration of the original woodland cover to combat desertification (particularly) by afforestation and reforestation of Acacia tortilis goes hand in hand with a climate change in the Biosphere Reserve, also influencing rural population outside the Biosphere Reserve. In order to study the different effects of woodland restoration in Bou-Hedma, the number of Acacia trees and their attributes have to be known. High resolution satellite imagery (GeoEye-1), was used with a GEOBIA approach. Field measurement of bole diameter, crown diameter and tree height were collected at > 400 locations. After segmentation, correlations with > 200 object features and tree attributes were calculated. For crown diameter and tree height, high correlations were observed with the features area and GLCM Entropy Layer 4 (90 degrees). Relations between these features and measured tree attributes were modeled, resulting in RMSE values of resp. 1.47 m and 1.62 m for crown diameter estimation and 0.92 m for tree height. The results show that a GEOBIA working strategy is suitable for estimating tree attributes in open forests in semi-arid regions