Punctured sphere

A subset A of a topological space X is compact if every open cover of A contains a finite subcover. For example, a sphere is compact but if we remove a point from the sphere, the generated punctured sphere is no longer compact. Missing a single point from a sphere may not appear all that important, but due to the resulting lack of compactness, it has important topological consequences. 

In an attempt to obtain a 1 1 Diels Alder adduct, small quantities were sealed into a steel bomb and incubated in an oven at 140°C. After a time there was a violent explosion, converting the oven to an irregular perforated sphere [1]. The editor suspects that the bursting bomb released flammable vapours that ignited in the cubic oven, distorting the shapnel punctured body. 

The post-mortem investigation discovered a metallic sphere, 1.52 mm in diameter (about the size of a pin s head), just below the surface of the skin at the site of the puncture mark. The sphere, made of an alloy of platinum (90%) and iridium (10%), had two holes in it, of diameter 0.35 mm, that extended across the entire sphere. This would allow about 0.28mm of a substance to be stored. The pellet was checked by the Government Chemical Defense Establishment at Porton Down but no traces of any substance could be found. Despite this, owing to the symptomatology and the very high toxicity required for so small a dose, the Coroner was satisfied that the cause of the poisoning was ricin. The pellet found in Mr. Markov was placed in the Black Museum at New Scotland Yard but has since been removed to serve as a piece... 

The Euler-Poincare characteristic x, which is related to the number of holes (or missing points ), with and without boundary, of the surface. For example, a sphere with perforations has x = 2 — whereas x = —for a punctured torus, with s 0. 

Theorem 5.4.4 (see [63]). On a two-dimensional sphere of class whose geodesic Sow has an additional integral quadratic in momenta and independent of the energy integral, in some isothermic coordinates z = x + ty set on a sphere without a point (with a point punctured), the metric is necessarily of the form A(x, y)[dx + dy ), where the function A has one of the forms exhibited pelow ... 

At the same time we would like to make the following instructive remark. The necessary and sufficient conditions described above are not yet enough effective in the following context. We have not yet received the answer to the question of how one can find out whether or not a given metric on a sphere or a torus admits an additional integral. According to Theorem 5.4.4, this will be the case if and only if tfi certain isothermic coordinates (on a sphere with a punctured point) a metric under investigation is reduced to one of the two forms indicated in the theorem. But to answer this question one should find those suitable isothermic coordinates or, conversely, prove that in no isothermic coordinates (on a sphere with a punctured point) does this metric admit the canonical representation of one of the two types specified in the theorem. 

These investigations have demonstrated that cut-spherical POPC GVs formed in an alternating electric field for 2-A h are stable enough for microinjection experiments. Other morphological structures cannot be used for such experiments, such as the mushroom structures. Spherical GVs-preferably the cut-spheres-can be touched and punctured by a microneedle furthermore, after withdrawing the microneedle the GVs still retain their spherical shape. 

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