Framboid Shapes

The original idea that framboids were generally spherical was due to the limitations of the contemporary optical microscopic methods. Later scanning microscopic investigations showed that many framboids were at least partly faceted and some display polygonal icosahedral forms. This is significant since the assumption of framboid sphericity informed earlier explanations of how they could form. It cannot be assumed, for example, that framboids necessarily require a precursor template, such as a spherical space or spherical organic globule, to develop. There is a continuum in original framboid shapes between ellipsoid, oblate spheroids, prolate spheroids, and spheroids. Irregularly curved shapes are common, especially in clusters of framboids, and result from deformation under the influence of gravity, analogous to soft sediment deformation. Framboidal icosahedra have varying triangular faces and are similar to the pseudo-icosahedral habit of pyrite macrocrystals. Framboids with mixtures of curved and faceted faces are common and these may result in part by local organized internal microcrystal domains. Various framboid clusters have been described as polyframboids, but the term is strictly reserved to spherical clusters of framboids. The constituent framboids may number 100–200 in these polyframboids, and they commonly show evidence of soft-sediment deformation.

Spherical space in the Newtonian limit: The cosmological constant

We compute the cosmological constant of a spherical space in the limit of weak gravity. To this end, we use a duality developed by the present authors in a previous work. This duality allows one to treat the Newtonian cosmological fluid as the probability fluid of a single particle in nonrelativistic quantum mechanics. We apply this duality to the case when the spacetime manifold on which this quantum mechanics is defined is given by [Formula: see text]. Here, [Formula: see text] stands for the time axis and [Formula: see text] is a 3-dimensional sphere endowed with the standard round metric. A quantum operator [Formula: see text] satisfying all the requirements of a cosmological constant is identified, and the matrix representing [Formula: see text] within the Hilbert space [Formula: see text] of quantum states is obtained. Numerical values for the expectation value of the operator [Formula: see text] in certain quantum states are obtained, which are in good agreement with the experimentally measured cosmological constant.

Necessary condition for the existence of a simple closed geodesic on a regular tetrahedron in the spherical space

In the spherical space the curvature of the tetrahedron’s faces equals 1, and the curvature of the whole tetrahedron is concentrated into its vertices and faces. The intrinsic geometry of this tetrahedron depends on the value α of faces angle, where π/3 < α ⩽ 2π/3. The simple (without points of self-intersection) closed geodesic has the type (p,q) on a tetrahedron, if this geodesic has p points on each of two opposite edges of the tetrahedron, q points on each of another two opposite edges, and (p+q) points on each edges of the third pair of opposite one. For any coprime integers (p,q), we present the number αp, q (π/3 < αp, q < 2π/3) such that, on a regular tetrahedron in the spherical space with the faces angle of value α > αp, q, there is no simple closed geodesic of type (p,q)

Voltage and Time Required for Irreversible Thermal Damage of Tumor Tissues during Electrochemotherapy under Thomson Effect

The essential target of the tumor’s treatment is how to destroy its tissues. This work is dealing with the thermal damage of the tumor tissue due to the thermoelectrical effect based on the Thomson effect. The governing equation of tumor tissue in concentric spherical space based on the thermal lagging effect is constructed and solved when the surface of the tumor tissue is subjected to a specific electric voltage. Different voltage and resistance effects have been studied and discussed for three different types of tumor tissues. The thermal damage quantity has been calculated with varying values of voltages and times. The voltage has significant effects on the temperature and the amount of the irreversible thermal damage of the tumor. Electrotherapy is a successful treatment. This work introduces a different model to doctors who work in clinical cancer to do experiments using electricity to damage the cancer cells.