Reconstruction approximating method by biquadratic splines of offset surfaces holes

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples.

In vitro proton magnetic resonance spectroscopy at 14T for benign and malignant ovary: Part I, signal processing by the nonparametric fast Padé transform

The present study deals with two different kinds of time signals, encoded by in vitro proton magnetic resonance spectroscopy (MRS) with a high external static magnetic field, 14.1T (Bruker 600 MHz spectrometer). These time signals originate from the specific biofluid samples taken from two patients, one with benign and the other with malignant ovarian cysts. The latter two diagnoses have been made by histopathologic analyses of the samples. Histopathology is the diagnostic gold standard in medicine. The obtained results from signal processing by the nonparametric derivative fast Padé transform (dFPT) show that a number of resonances assignable to known metabolites are considerably more intense in the malignant than in the benign specimens. Such conclusions from the dFPT include the recognized cancer biomarkers, lactic acid and choline-containing compounds. For example, the peak height ratio for the malignant-to-benign samples is about 18 for lactate, Lac. This applies equally to doublet Lac(d) and quartet Lac(q) resonating near 1.41 and 4.36 ppm (parts per million), respectively. For the choline-containing conglomerate (3.19-3.23 ppm), the dFPT with already low-derivative orders (2nd, 3rd) succeeds in clearly separating the three singlet component resonances, free choline Cho(s), phosphocholine PC(s) and glycerophosphocholine GPC(s). These constituents of total choline, tCho, are of critical diagnostic relevance because the increased levels, particularly of PC(s) and GPC(s), are an indicator of a malignant transformation. It is gratifying that signal processing by the dFPT, as a shape estimator, coheres with the mentioned histopathology findings of the two samples. A very large number of resonances is identifiable and quantifiable by the nonparametric dFPT, including those associated with the diagnostically most important low molecular weight metabolites. This is expediently feasible by the automated sequential visualization and quantification that separate and isolate sharp resonances first and subsequently tackle broad macromolecular lineshape profiles. Such a stepwise workflow is not based on subtracting nor annulling any part of the spectrum, in sharp contrast to controversial customary practice in the MRS literature. Rather, sequential estimation exploits the chief derivative feature, which is a faster peak height increase of the thin than of the wide resonances. This is how the dFPT simultaneously improves resolution (linewidth narrowing) and reduces noise (background flattening). Such a twofold achievement makes the dFPT-based proton MRS a high throughput strategy in tumor diagnostics as hundreds of metabolites can be visualized/quantified to offer the opportunity for a possible expansion of the existing list of a handful of cancer biomarkers.

The $$\gamma$$ function in quantum theory II. Mathematical challenges and paradoxa

While the square root of Dirac’s $$\delta$$ δ is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called $$\gamma (x)$$ γ ( x ) which permits a quasi-classical treatment of simple systems like the H atom or the 1D harmonic oscillator for which accurate quantum mechanical energies were previously reported. The so-defined $$\gamma (x)$$ γ ( x ) is neither a traditional function nor a distribution, and it remains to be seen that any consistent mathematical approaches can be set up to deal with it rigorously. A straightforward use of $$\gamma (x)$$ γ ( x ) generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.