Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

It is well known that the complexity of the Delaunay triangulation of $n$ points in $\RR ^d$, i.e., the number of its simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In particular, in $\RR ^3$, the number of tetrahedra can be quadratic. Put another way, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the three-dimensional Delaunay triangulation of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^{1.8})$ for general polyhedral surfaces and $O(n\sqrt{n})$ for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points.     

The correlation dimension of the Sierpinski carpet

We compute the correlation dimension of a measure defined on a general Sierpinski carpet. We relate this function to a free energy function associated to a partition composed of ‘nearly squares’ and well fitted to the planar Cantor set. Actually, we prove that these functions are real analytic on , are strictly increasing and are strictly concave (respectively linear in the degenerate case). This is an example of a two-dimensional dynamical system contracting in the two directions with different ratios. We first study measures of Gibbsian type before generalizing to Markovian measures.     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.     

Itaconic acid hybrids as potential anticancer agents

Molecular Diversity - In this paper, we report the synthesis of novel hybrids 2–14 based on itaconic acid and fluoroaniline, pyridine, indole and quinoline scaffolds. Itaconic acid is a...     

Discovery of ( ±)-3-(1 H -pyrazol-1-yl)-6,7-dihydro-5 H -[1,2,4]triazolo[3,4- b ][1,3,4] thiadiazine derivatives with promising in vitro anticoronavirus and antitumoral activity

A new series of (?±)-(3-(3,5-dimethyl-1H-pyrazol-1-yl)-6-phenyl-6,7-dihydro-5H-[1,2,4]triazolo[3,4-b][1,3,4]thiadiazin-7-yl)(phenyl)methanones were efficiently synthesized starting from 4-amino-5-hydrazinyl-4H-1,2,4-triazole-3-thiol 1, acetyl acetone 2, various aromatic and heterocyclic aldehydes 3 and phenacyl bromides 4. All the newly synthesized compounds were tested for their antiviral and antitumoral activity. It was shown that subtle structural variations on the phenyl moiety allowed to tune biological properties toward antiviral or antitumoral activity. Mode-of-action studies revealed that the antitumoral activity was due to inhibition of tubulin polymerization.

Graphic abstract

     

Maxima and minima of the displacement components for the Lamb modes

This paper revisits the vanishing of the transverse component of the particle displacement vector in free surfaces of an isotropic homogeneous plate, for both symmetric and antisymmetric Lamb waves. Drawing on well-known analytical expressions from Viktorov's book [(1967) Rayleigh and Lamb Waves: Physical Theory Applications, Chap. II, pp. 67-121], two distinct frequency-thickness product expressions, in cases where this vanishing occurs, are derived: one for the symmetric modes and another for the antisymmetric modes. At these frequency-thickness products, phase and group velocities have appreciable values which are discussed herein. It appears that these velocities depend on the transverse bulk wave velocity only. This is the specific condition of the Lame? modes. Moreover, theoretical and experimental investigations of displacements in the surface of a plate in air have been carried out. The theoretical part shows that the normal and transverse displacements have, respectively, a local maximum and a local minimum in the vicinity of these frequency-thickness products. The experimental part corroborates the presence of the local maximum of the S(0) Lamb mode for various materials.     

Electronic core levels of hydroxyls at the surface of chromia related to their XPS O 1s signature: A DFT + U study

The Cr 2p and O 1s binding energy (BE) levels have been calculated by first principles methods for different models of hydroxylated (0001)-Cr₂O₃ surfaces. Several surface terminations have been considered. The calculations allow us to reproduce the O 1s shifts between O in oxide and OH groups. It is found that two main effects account for the OH binding energy shifts. On the one hand, the increased covalency of the O–H bond with respect to the Cr–O bond, lowers the electronic O (1s and 2p) energy, and in consequence the BE of the core levels (O 1s) are higher. On the other hand, the lower the OH coordination number, the higher the valence and core levels energy, and the lower the BE. Consequently, mono-coordinated hydroxyls have a binding energy near that of O^2? in the oxide (ΔBE_OH–O = ? 0.2–0.0 eV). Two-fold coordinated hydroxyls have a slightly higher BE (ΔBE_OH–O = + 0.3 eV). Three-fold coordinated OH groups have a higher binding energy (ΔBE_OH–O = + 0.6?0.7 eV), corresponding to that experimentally measured for OH groups. Finally, water adsorbed above OH groups exhibits a still higher BE (ΔBE_HOH–O = + 0.9–1.0 eV). The ΔBE are slightly under-estimated under the initial state approximation, and overestimated under the final state (Z + 1) approximation.     

Feasibility of using ultrasound contrast agents to increase the size of thermal lesions induced by non-focused transducers: in vitro demonstration in tissue mimicking phantom

Miniature flat ultrasound transducers have shown to be effective for a large variety of thermal therapies, but the associated superficial heating implicates developing original strategies in order to extend therapeutic depth. The goal of the present paper is to use ultrasound contrast agents (UCA) to increase remote attenuation and heating. Theoretical simulations demonstrated that increasing attenuation from 0.27 to 0.8 Np/cm at 10 MHz beyond a distance of 18 mm from the transducer should result in longer thermal damages due to protein coagulation in a tissue mimicking phantom. Contrast agents (BR14, Bracco, Plan-les-Ouates, Switzerland) were embedded in thermo-sensitive gel and attenuations ranging from 0.27 to 1.33 Np/cm were measured at 10 MHz for concentrations of BR14 between 0 and 4.8%. Thermal damages were then induced in several gels, which had different layering configurations. Thermal damages, 12.8 mm in length, were obtained in homogeneous gels. When mixing contrast agents at a concentration of 3.2% beyond a first 18 mm-thick layer of homogeneous gel, the thermal damages reached 21.5 mm in length. This work demonstrated that contrast agents can be used for increasing attenuation remotely and extending therapeutic depth induced by a non-focused transducer. Additional work must be done in vivo in order to verify the remote-only distribution of bubbles and associated increase in attenuation.     

Evolutionary dynamics of imatinib-treated leukemic cells by stochastic approach

The evolutionary dynamics of a system of cancerous cells in a model of chronic myeloid leukemia (CML) is investigated by a statistical approach. Cancer progression is explored by applying a Monte Carlo method to simulate the stochastic behavior of cell reproduction and death in a population of blood cells which can experience genetic mutations. In CML front line therapy is represented by the tyrosine kinase inhibitor imatinib which strongly affects the reproduction of leukemic cells only. In this work, we analyze the effects of a targeted therapy on the evolutionary dynamics of normal, first-mutant and cancerous cell populations. Several scenarios of the evolutionary dynamics of imatinib-treated leukemic cells are described as a consequence of the efficacy of the different modelled therapies. We show how the patient response to the therapy changes when a high value of the mutation rate from healthy to cancerous cells is present. Our results are in agreement with clinical observations. Unfortunately, development of resistance to imatinib is observed in a fraction of patients, whose blood cells are characterized by an increasing number of genetic alterations. We find that the occurrence of resistance to the therapy can be related to a progressive increase of deleterious mutations.      

Ab initio study of the chemical states of water on Cr2O3(0 0 0 1): From the isolated molecule to saturation coverage

The reactivity of the (0 0 0 1)-Cr–Cr₂O₃ surface towards water was studied by means of periodic DFT + U. Several water coverages were studied, from 1.2H₂O/nm² to 14.1H₂O/nm², corresponding to ¼, 1, 2 and 3 water/Cr at the (0 0 0 1)-Cr₂O₃ surface, respectively. With increasing coverage, water gradually completes the coordination sphere of the surface Cr atoms from 3 (dry surface) to 4 (1.2 and 4.7H₂O/nm²), 5 (9.4H₂O/nm²) and 6 (14.1H₂O/nm²). For all studied coverages, water replaces an O atom from the missing above plane. At coverages 1.2 and 4.7H₂O/nm², the Cr–O_s (surface oxygen) acid–base character and bond directionality govern the water adsorption. The adsorption is molecular at the lowest coverage. At 4.7H₂O/nm², molecular and dissociative states are isoenergetic. The activation energy barrier between the two states being as low as 12 kJ/mol, allowing protons exchanges between the OH groups, as evidenced by ab inito molecular dynamics at room temperature. At coverages of 9.4 and 14.1H₂O/nm², 1D- (respectively, 2D-) water networks are formed. The resulting surface terminations are –Cr(OH)₂ and –Cr(OH)₃– like, respectively. The increased stability of those terminations as compared to the previous ones are due to the stabilization of the adsorbed phase through a H-bond network and to the increase in the Cr coordination number, stabilizing the Cr (t_2g) orbitals in the valence band. An atomistic thermodynamic approach allows us to specify the temperature and water pressure domains of prevalence for each surface termination. It is found that the –Cr(OH)₃-like, –Cr(OH)₂ and anhydrous surfaces may be stabilized depending on (T, P) conditions. Calculated energies of adsorption and OH frequencies are in good agreement with published experimental data and support the full hydroxylation model, where the Cr achieves a 6-fold coordination, at saturation.