On the spectrum of stiffness matrices arising from isogeometric analysis

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of \(\mathbb C\) . All the analysis is related to the notion of symbol in the Toeplitz setting and is carried out both for the cases of 1D and 2D problems.     

A class of piecewise interpolating functions based on barycentric coordinates

Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolant function by means of polynomial pieces and ensure that some regularity conditions are guaranteed at the break-points. In this work, we propose a novel class of piecewise interpolating functions whose expression depends on the barycentric coordinates and a suitable weight function. The underlying idea is to specialize to the 1D settings some aspects of techniques widely used in multi-dimensional interpolation, namely Shepard’s, barycentric and triangle-based blending methods. We show the properties of convergence for the interpolating functions and discuss how, in some cases, the properties of regularity that characterize the weight function are reflected on the interpolant function. Numerical experiments, applied to some case studies and real scenarios, show the benefit of our method compared to other interpolant models.     

A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots.     

Irradiation of Amelanotic Melanoma Cells with 532 nm High Peak Power Pulsed Laser Radiation in the Presence of the Photothermal Sensitizer Cu(II)-Hematoporphyrin: A New Approach to Cell Photoinactivation

Cu(II)-hematoporphyrin (CuHp) was efficiently accumulated by B78H1 amelanotic melanoma cells upon incubation with porphyrin concentrations up to 52 microM. When the cells incubated for 18 h with 13 microM CuHp were irradiated with 532 nm light from a Q-switched Nd: YAG laser operated in a pulsed mode (10 ns pulses, 10 Hz) a significant decrease in cell survival was observed. The cell photoinactivation was not the consequence of a photodynamic process, as CuHp gave no detectable triplet signal upon laser flash photolysis excitation and no decrease in cell survival was observed upon continuous wave irradiation. Thus, it is likely that CuHp sensitization takes place by photothermal pathways. The efficiency of the photoprocess was modulated by different parameters; thus, while varying the amount of added CuHp in the 3.25-26 microM range had little effect, pulse energies larger than 50 mJ and irradiation times of at least 120 s were necessary to induce a cell inactivation of about 50%. The porphyrin-cell incubation time prior to irradiation had a major influence on cell survival, suggesting that the nature of the CuHp microenvironment can control the efficiency of photothermal sensitization.     

Optimal and one-complemented subspaces

Let X be a real Banach space and let V ⊂ X be a closed linear subspace. In [4, Prop. 5] it has been proven that if X is strictly convex, reflexive and smooth and V is an optimal subset of X then V is one-complemented in X. In this note we would like to extend this result to non-smooth Banach spaces. In particular, we show that any existence subspace of c,c _o and l ₁ is one-complemented. Also some results concerning non-smooth Musielak-Orlicz sequence spaces equipped with the Luxemburg norm will be presented. The first author was supported by Polish State Committee for Scientific Research Grant KBN 1 PO3A O10 26. Authors’ addresses: Grzegorz Lewicki, Department of Mathematics, Jagiellonian University, 30-059 Kraków, Reymonta 4, Poland; Giulio Trombetta, Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende, Cosenza, Italy     

Quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented. Dedicated to Prof. Mariano Gasca on the occasion of his 60th birthday.     

On the order of global branches

We give both a topological and an algebraic definition of the order of a global branch of a variety along a subvariety. Then we show that these two definitions agree. Finally we compare this order with the orders of the related local branches.

---

Sunto Diamo due definizioni, una topologica e una algebrica, di ordine di un ramo globale di una varietà lungo una sottovarietà. Poi mostriamo che queste due definizioni coincidono. Infine confrontiamo questo ordine con gli ordini dei relativi rami locali.

---

---

Partially supported by CNR funds.