Возбуждение и ионизация частицы в одномерной потенциальной яме нулевого радиуса предельно коротким световым импульсом

The Migdal sudden perturbation approximation is used to solve the problem of excitation and ionization particles in a one-dimensional potential of zero radius with an extremely short pulse. There is has only one energy level in such a one-dimensional the delta-shaped potential well. It is shown that for pulse durations shorter than the characteristic period of oscillations of the wave function of the particle in the bound state, the population of the level (and the probability of ionization) is determined by the ratio of the electric the area of ​​the pulse to the characteristic “scale” of the area inversely proportional to the area of ​​localization of the particle in a bound state.

Genomes of keystone Mortierella species lead to better in silico prediction of soil mycobiome functions from Taiwan's offshore islands

The ability to correlate the functional relationship between microbial communities and their environment is critical to understanding microbial ecology. There is emerging knowledge on island biogeography of microbes but how island characteristics influence functions of microbial community remain elusive. Here, we explored soil mycobiomes from nine islands adjacent to Taiwan using ITS2 amplicon sequencing. Geographical distances and island size were positively correlated to dissimilarity in mycobiomes, and we identified 56 zero-radius operational taxonomic units (zOTUs) that were ubiquitously present across all islands, and as few as five Mortierella zOTUs dominate more than half of mycobiomes. Correlation network analyses revealed that seven of the 45 hub species were part of the ubiquitous zOTUs belonging to Mortierella, Trichoderma, Aspergillus, Clonostachys and Staphylotrichum. We sequenced and annotated the genomes of seven Mortierella isolates, and comparative predictions of KEGG orthologues using PICRUSt2 database updated with new genomes increased sequence reads coverage by 62.9% at the genus level. In addition, genes associated with carbohydrate and lipid metabolisms were differentially abundant between islands which remained undetected in the original database. Predicted functional pathways were similar across islands despite their geographical separation, difference in differentially abundant genes and composition. Our approach demonstrated the incorporation of the key taxa genomic data can improve functional gene prediction results and can be readily applied to investigate other niches of interests.

Cavity Expansion in Cohesive-Frictional Soils with Limited Dilation

The problem of cavity expansion from zero radius has no characteristic length and therefore possesses a similarity solution, in which the cavity pressure remains constant and the continuing deformation is geometrically self-similar. In this case, the incremental velocity approach first used by Hill (1950) to analyze cavity expansion in Tresca materials can be extended to derive a solution for limiting pressure of cavity expansion in other types of material. In this article, a rigorous semi-analytical solution is derived, following Hill's incremental velocity method, for the expansion of cavities from zero initial radius in cohesive-frictional soils with limited dilation. In particular, the radius of the elastic-plastic interface c is used in this article as the time scale and the solution for the limit pressure has been presented. Solutions are evaluated for a number of cases representative of a range of cohesive-frictional and dilatant soils. A comparison is also made between the solutions presented here and previous solutions for cohesive-frictional soils that have unlimited (on-going) plastic dilation. In particular, the influence of limited plastic dilation on the cavity limit pressure is identified and discussed.

Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

Spherically Symmetric Exact Vacuum Solutions in Einstein-Aether Theory

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlevè-Gullstrand, and Schwarzschild coordinates, in which the aether is always comoving, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter c14 in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit (c14=0). However, as long as c14≠0, a marginally trapped throat with a finite non-zero radius always exists, and on one side of it the spacetime is asymptotically flat, while on the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

NEW SOLUTION TO THE DROPLET KERNEL MODEL

The droplet model of the nucleus is revived, for which an exact solution for an incompressible fluid is obtained using the hydrodynamic potential solution obtained from the Schrödinger equation. Moreover, for an incompressible fluid, there are formulas for the pressure or potential. There is the main part of the hydrodynamic potential, which is obtained by replacing the modulus of the inverse difference of vectors by the difference in moduli of the values of the vectors. The bulk of the potential is expressed in a finite formula with singularities. A formula is obtained for the integral containing the modulus of the difference between the exact values of the vectors minus the main part of the potential. This difference defines a continuous correction with the features taken into account. The main part of the potential at the boundary of the nucleus turned out to be infinitely large with an imaginary part, locking particles in the nucleus. In this case, the real part of the main potential decreases with decreasing radius, becomes negative, and determines the bound state. At half the radius of the nucleus, there is a linear term along the radius. At the zero radius, there is an infinite negative potential with an imaginary part. An expression for the quantum of the emitted energy is obtained. Note that the added mass was not used due to the rotational regime of the nucleus. An algorithm for calculating the spectrum of the kernel is proposed, and each state of the action of the kernel sn corresponds to n calculated frequencies, determined by n angles in the configuration space. The main space is n + 1 dimensional, and each dimension of space has its own energy. But without special means, the potential of the nucleus tends to infinity. It is necessary to introduce the imaginary degree of roughness of the corners, in expressions containing singularities, then the infinities disappear.

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

In this paper, we completely solve the $L^{2}\to L^{r}$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^{p}\to L^{4}$ extension estimate for non-zero radii spheres over finite fields, which improves the previous result of the first and second authors significantly.

Theoretical Investigation of the Bending Process of the Pre-Strained Metal Sheet

During the bending operation of the thin sheet materials by the punch with the near-to-zero radius the special technological operation should be carried out. It means that the metal sheet obtained a certain thinning value, which is usually done in the form of the channel-concentrator or groove by pre-drawing operation in a cold state. It follows to the pre-straining and strengthening of the material. The authors investigated the strain hardened sheet's area after roll forming process theoretically, and obtained the strain-stress distribution inside the sheet during the bending operation. It was found out that the increase of the prior deformation during pre-straining in the bend layer follows to the increase of the radial and tangential stresses and displacement of the neutral axis inside the blank during bending operation. As a result, the bending moment changes its values depends on the punch radius and strain hardening.