The topographic development and areal parametric characterization of a stratified surface polished by mass finishing

Mass finishing is amongst the most widely used finishing processes in modern manufacturing, in applications from deburring to edge radiusing and polishing. Processing objectives are varied, ranging from the cosmetic to the functionally critical. One such critical application is the hydraulically smooth polishing of aero engine component gas-washed surfaces. In this, and many other applications the drive to improve process control and finish tolerance is ever present. Considering its widespread use mass finishing has seen limited research activity, particularly with respect to surface characterization. The objectives of the current paper are to; characterise the mass finished stratified surface and its development process using areal surface parameters, provide guidance on the optimal parameters and sampling method to characterise this surface type for a given application, and detail the spatial variation in surface topography due to coupon edge shadowing. Blasted and peened square plate coupons in titanium alloy are wet (vibro) mass finished iteratively with increasing duration. Measurement fields are precisely relocated between iterations by fixturing and an image superimposition alignment technique. Surface topography development is detailed with ‘log of process duration’ plots of the ‘areal parameters for scale-limited stratified functional surfaces’, (the Sk family). Characteristic features of the Smr2 plot are seen to map out the processing of peak, core and dale regions in turn. These surface process regions also become apparent in the ‘log of process duration’ plot for Sq, where lower core and dale regions are well modelled by logarithmic functions. Surface finish (Ra or Sa) with mass finishing duration is currently predicted with an exponential model. This model is shown to be limited for the current surface type at a critical range of surface finishes. Statistical analysis provides a group of areal parameters including; Vvc, Sq, and Sdq, showing optimal discrimination for a specific range of surface finish outcomes. As a consequence of edge shadowing surface segregation is suggested for characterization purposes

On certain diophantine equations of diagonal type

In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\in\{(2,4,8,8), (2,8,4,8)\}$. In the case $(p,q,r,s)=(4,4,4,4)$, we present some new parametric solutions of the equation $x^4-y^4=4(z^4-w^4)$.Comment: 16 pages, revised version will appear in the Journal of Number Theor

Oka manifolds: From Oka to Stein and back

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert's classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations of Oka manifolds, the functorial properties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov's ellipticity. We survey the current status of the theory in terms of known examples of Oka manifolds, mention open problems and outline the proofs of the main results. In the appendix by F. Larusson it is explained how Oka manifolds and Oka maps, along with Stein manifolds, fit into an abstract homotopy-theoretic framework. The article is an expanded version of the lectures given by the author at the Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive exposition of Oka theory is available in the monograph F. Forstneric, Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56, Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci. Toulouse Math. (6), vol. 22, no. 4. This version is identical with the published tex

Human hepatic cell behavior on polysulfone membrane with double porosity level

In the membrane-based bioartificial livers developed up to now,the hepatic cells were located either in the fibers lumen or in the cartridge,with limited capacity for cell hosting.Here,we designed a polysulfone (PSU) membrane with a double porosity level:(i)surface macroporosity emerging in macrochambers accessible to hepatic cell colonization;(ii)microporosity to ensure gas and molecule transfers between macrochambers and supernatant,as well as potential immune barrier. ESEM and X-ray tomography confirmed that macrochambers accessed the membrane surface and were inter-connected. Biocompatibility and performances of this PSU membrane with double porosity level were compared to classical semi-permeable structures,following cell organization,cell proliferation and liver specific activities over a 9 days incubation. Macrochambers were colonized by hepatic cells, leading to higher albumin synthesis compared to control.Therefore,this membrane with double porosity appeared as a promising support to offer an inner 3D environment adequate to cell proliferation to form a liver-like tissue

A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe