The reduction of star sets

Mahler’s theory of irreducible star bodies is redeveloped and extended in a modified form. It is shown that any closed bounded star set S contains a closed irreducible star set T having the same critical determinant. Further, it is shown that, if the first set S is bounded by a finite number of algebraic surfaces, then there will be an irreducible set T which is also bounded by a finite number of algebraic surfaces.

2017 ◽  
Vol 25 (1) ◽  
pp. 5-11 ◽  
Author(s):  
Werner Georg Nowak

Abstract In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body|x1|(|x1|3 + |x2|3 + |x3|3 ≤ 1.In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body|x1|(|x1|3 + |x22 + x32)3/2≤ 1.


2017 ◽  
Vol 25 (2) ◽  
pp. 149-157
Author(s):  
Werner Georg Nowak

Abstract In the problem of (simultaneous) Diophantine approximation in ℝ3 (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body K2 : (y2 + z2)(x2 + y2 + z2) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant ∆ (Kc) of more general star bodies Kc : (y2 + z2)c/2(x2 + y2 + z2) ≤ 1 ; where c is any positive constant. These are obtained by inscribing into Kc either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of c.


Author(s):  
R. A. Crowther

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.


2019 ◽  
Vol 26 (5) ◽  
pp. 765-779 ◽  
Author(s):  
Alexios S. Antonopoulos ◽  
Athina Goliopoulou ◽  
Evangelos Oikonomou ◽  
Sotiris Tsalamandris ◽  
Georgios-Angelos Papamikroulis ◽  
...  

Background: Myocardial redox state is a critical determinant of atrial biology, regulating cardiomyocyte apoptosis, ion channel function, and cardiac hypertrophy/fibrosis and function. Nevertheless, it remains unclear whether the targeting of atrial redox state is a rational therapeutic strategy for atrial fibrillation prevention. Objective: To review the role of atrial redox state and anti-oxidant therapies in atrial fibrillation. Method: Published literature in Medline was searched for experimental and clinical evidence linking myocardial redox state with atrial fibrillation pathogenesis as well as studies looking into the role of redoxtargeting therapies in the prevention of atrial fibrillation. Results: Data from animal models have shown that altered myocardial nitroso-redox balance and NADPH oxidases activity are causally involved in the pathogenesis of atrial fibrillation. Similarly experimental animal data supports that increased reactive oxygen / nitrogen species formation in the atrial tissue is associated with altered electrophysiological properties of atrial myocytes and electrical remodeling, favoring atrial fibrillation development. In humans, randomized clinical studies using redox-related therapeutic approaches (e.g. statins or antioxidant agents) have not documented any benefits in the prevention of atrial fibrillation development (mainly post-operative atrial fibrillation risk). Conclusion: Despite strong experimental and translational data supporting the role of atrial redox state in atrial fibrillation pathogenesis, such mechanistic evidence has not been translated to clinical benefits in atrial fibrillation risk in randomized clinical studies using redox-related therapies.


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