Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions

The area of a polygon is the region occupied by a polygon. Polygons can be regular and irregular. The basic polygons used in geometry are triangle, square, rectangle, pentagon, hexagon, etc. All of these polygons have their own area. For regular polygons, it is easy to find their area, as the dimensions are defined and known to us. For example, the area of a square can be easily determined if we know the length of one of its sides, since all its sides are equal. For the computation of area, there are pre-defined formulas for squares, rectangles, circle, triangles, trapeziums etc. However, for an irregular polygon, the area is calculated by viewing it as a combination of 2 or more regular polygons. The area of any given polygon whether it a triangle, square, quadrilateral, rectangle, parallelogram or rhombus, hexagon or pentagon, is defined as the region occupied by it in a two-dimensional plane. The areas or formulas for areas of different types of polygon depends on their shapes. Suppose, to find the area of the triangle, we have to know the length of its base and height. For finding the area of a polygon which is not regular or its formula is not defined, we split the figure into triangles, squares, trapezium etc. The purpose is to visualize the given geometry as a combination of geometries for which we know how to calculate the area. We then calculate the area for each of the part and then add them up to obtain the area of the polygon. Therefore, and within this context, the objective of the present work was to develop a new methodology to calculate the area of any polygon using trapezoids. The results obtained showed that the developed methodology can be efficiently used to calculate the area of regular and irregular polygons.

23 - The Simplest Methods of Constructing Pleasing Patterns

The accuracy of the Java simulation by the Runge-Kutta method for the charge motion in electric and magnetic fields has been investigated in comparison with the analytical solution. The error of the simulation depends on the time increment, h , used for the numerical calculation. If we use an increment that is larger than the boundary value, the simulation results in a non-accurate image of the charge motion. In this case, the simulation almost results in an underestimation, that is, a motion that is smaller than the real motion. The boundary increment is proportional to the mass of the charge, m , and is inversely proportional to the charge, q , and the magnetic field, B 0 . The empirical results conclude that the image of the charge motion can be obtained accurately by Java simulation using h < 0.2 m / qB 0 .

We consider a disk-shaped cold atom Bose-Einstein condensate with repulsive atom-atom interactions within a circular trap, described by a two-dimensional time-dependent Gross-Pitaevskii equationwith cubic nonlinearity and a circular box potential. In this setup, we discuss the existence of a type of stationary nonlinear waves with propagation-invariant density profiles, consisting of vortices located at the vertices of a regular polygon with or without an antivortex at its center. These polygons rotate around the center of the system and we provide approximate expressions for their angular velocity. For any size of the trap, we find a unique regular polygon solution that is static and is seemingly stable for long evolutions. It consists of a triangle of vortices with unit charge placed around a singly charged antivortex, with the size of the triangle fixed by the cancellation of competing effects on its rotation. There exist other geometries with discrete rotational symmetry that yield static solutions, even if they turn out to be unstable. By numerically integrating in real time the Gross-Pitaevskii equation, we compute the evolution of the vortex structures and discuss their stability and the fate of the instabilities that can unravel the regular polygon configurations. Such instabilities can be driven by the instability of the vortices themselves, by vortex-antivortex annihilation or by the eventual breaking of the symmetry due to the motion of the vortices.

Improper control of cathode spot (CS) motion can lead to uneven cathode erosion and shorter life, so magnetic fields are often used to direct CS motion. Here, we incorporate an axial magnetic field component into a model for CS retrograde motion based on the difference between the plasma kinetic and self-magnetic pressures. We consider the motion of the positive space charge associated with retrograde motion to generate a current perpendicular to the axial magnetic field, introducing an additional component to CS motion. The predicted angle of CS motion agrees well with the experimental data and a prior model based on electron backflow to the cathode surface.

Metode numerik dikembangkan untuk menghitung beban perlawanan kolom meruncing keujung dari penampang melintang poligon teratur dengan volume konstan dan kedua ujungnya dijepit. Keruncingan linier, parabolik dan sinusiodal dipertimbangkan dalam contoh numerik. Dari hasil numerik, kolom yang paling kuat adalah dengan tipe keruncingan dan jumlah sisi penampang melintang poligon teratur dapat diketahui. Kata kunci : Beban perlawanan, volume konstan, kolom yang paling kuat, keruncingan kolom

The magnetic fields influence the biological systems by several mechanisms. The magnetic field induces the translation and rotation movements of the electric charges. The effects of a stationary magnetic field on the germination and initial stages of growth of Lettuce seeds (Lactuca sativa var.longifolia) have been studied. The seeds were exposed to a magnetic field strength (125 and 250mT) for different time including 0, 1, 6, 12 and 24 hour. The results revealed a significant difference (P<0.01) of Lettuce seeds traits exposed to stationary magnetic field. By conducting the results, germination parameters in 0 and 24 h were underneath the 1, 6 and 12 h exposed to both 125 and 250mT of magnetic field. Germination rate at both 125 and 250mT were increased. Root length at 250mT was rapidly raised and then rapidly reduced than 125mT. Peroxidase activity rose with increasing to exposed magnetic field. The decline in peroxidase activity above 12 h is seen for 250mT. Maximum peroxidase activity at 125mT was recorded incubated 24 h exposed to 125mT magnetic field. Finally, magnetic treatment produces an increase in the germination and first stages of growth of lettuce. So, the magnetic field as nondestructive growth factors to accelerate germination and early growth of lettuce can be used.

The motion of non-relativistic test charges is studied for a class of fields (arising, for example, in cavity resonators) which oscillate with a high frequency ω comparable with or greater than the local peak gyromagnetic frequency ωg of the charges. Thus the recently developed methods based upon ω « ωg are supplemented whenever the (variable) magnetic field so confines the charges that the total field may be regarded as locally homogeneous. Rapidly converging expansions involving integral order Bessel functions Jn(ωg/ω) as coefficients are used to obtain asymptotic expressions for the kinetic energy limited by a simplified collision term in Lorentz's equations of motion. Kinetic energy resonances, reminiscent of the action of the betatron, arise for certain values of the critical ratio ωg/ω, notably near the peaks of [(ωg/ω)J1(ωg/ω)]2.

Gastric H(+),K(+)-ATPase is shown to transport 2 mol of H(+)/mol of ATP hydrolysis in isolated hog gastric vesicles. We studied whether the H(+) transport mechanism is due to charge transfer and/or transfer of hydronium ion (H(3)O(+)). From transport of [(18)O]H(2)O, 1.8 mol of water molecule/mol of ATP hydrolysis was found to be transported. We performed a molecular dynamics simulation of the three-dimensional structure model of the H(+),K(+)-ATPase alpha-subunit at E(1) conformation. It predicts the presence of a charge transfer pathway from hydronium ion in cytosolic medium to Glu-345 in cation binding site 2 (H(3)O(+)-Lys-164 -Gln-161-Glu-345). No charge transport pathway was formed in mutant Q161L, E345L, and E345D. Alternative pathways (H(3)O(+)-Gln-161-Glu-345) in mutant K164L and (H(3)O(+)-Arg-105-Gln-161-Gln-345) in mutant E345Q were formed. The H(+),K(+)-ATPase activity in these mutants reflected the presence and absence of charge transfer pathways. We also found charge transfer from sites 2 to 1 via a water wire and a charge transfer pathway (H(3)O(+)-Asn-794 -Glu-797). These results suggest that protons are charge-transferred from the cytosolic side to H(2)O in sites 2 and 1, the H(2)O comes from cytosolic medium, and H(3)O(+) in the sites are transported into lumen during the conformational transition from E(1)PtoE(2)P.

Optically Mapping the Movement of Discrete Gating Charges in Shaker

The object of this note is to study the motion of a charged particle entering the magnetic field due to a steady current inside a plasma column. Even if the particles of the plasmas stream mutually interact, their distribution across the stream can be such that the interaction forces are balanced by the transverse pressure gradient in the plasma 1. It is shown here that the test charge entering the plasma stream remains bounded to the stream between two coaxial cylindrical surfaces and further that under suitable conditions it may remain trapped inside a cylindrical box. FISSER and KIPPENHAHN 2 have discussed the general problem of the motion of a charged particle in an axially symmetric magnetic field in configuration space. The following is a particular case of that problem. HERTWECK 3 has investigated the motion of a test charge in the magnetic field due to a line current. His results are applicable here if the test charge remains outside the plasma stream

Superconductivity occurs in systems that have a lot of negative charge: the highly negatively charged (CuO2)= planes in the cuprates, negatively charged (FeAs)− planes in the iron arsenides, and negatively charged B − planes in magnesium diboride. And, in the nearly filled (with negative electrons) bands of almost all superconductors, as evidenced by their positive Hall coefficient in the normal state. No explanation for this charge asymmetry is provided by the conventional theory of superconductivity, within which the sign of electric charge plays no role. Instead, the sign of the charge carriers plays a key role in the theory of hole superconductivity, according to which metals become superconducting because they are driven to expel negative charge (electrons) from their interior. This is why NIS tunneling spectra are asymmetric, with larger current for negatively biased samples. The theory also offers a compelling explanation of the Meissner effect: as electrons are expelled towards the surface in the presence of a magnetic field, the Lorentz force imparts them with azimuthal velocity, thus generating the surface Meissner current that screens the interior magnetic field. In type II superconductors, the Lorentz force acting on expelled electrons that do not reach the surface gives rise to the azimuthal velocity of the vortex currents. In the absence of applied magnetic field, expelled electrons still acquire azimuthal velocity, due to the spin–orbit interaction, in opposite direction for spin-up and spin-down electrons: the “Spin Meissner effect.” This results in a macroscopic spin current flowing near the surface of superconductors in the absence of applied fields, of magnitude equal to the critical charge current (in appropriate units). Charge expulsion also gives rise to an interior outward-pointing electric field and to excess negative charge near the surface. In strongly type II superconductors this physics should give rise to charge inhomogeneity and spin currents throughout the interior of the superconductor, to large sensitivity to (non-magnetic) disorder and to a strong tendency to phase separation.

In Chapter I, we studied primarily individual points in the plane, or at most finite sets of points: for example, the vertices of triangles, parallelograms, and regular polygons. The sole exception was a straight line, which we encountered in defining the product of a vector by a number. In this second chapter, we take up the description of lines in coordinate form. We will first show that the simplest possible relation between coordinates x and y yields a line. We then define the graph of a function and functions themselves. We will also obtain the equation of a line from the geometric properties of lines. We will do the same for the circle, the ellipse, the hyperbola, and the parabola, defining these curves first by their geometric properties. After this, we take up the question of defining a line parametrically. We define the winding number of a closed curve with respect to the origin. In the last section of this chapter, we present a geometric interpretation of the behavior of polynomials with complex coefficients. Using the winding number, we prove the fundamental theorem of algebra: a polynomial of degree n admits n roots.KeywordsCoordinate AxisClosed CurveFundamental TheoremParametric EquationParametric RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let E denote a Euclidean space with dimension p ⩾ 2; a periodic orbit of length n ⩾ 2 for a C1-diffeomorphism f:E → E; and ρ(f) the number defined by means of the classical triple norm associated with a Euclidean norm ‖ · ‖ in the set of linear endomorphisms of E. I have proved in (Dehove 2007 Nonlinearity 20 2191–203) that one has optimally . The object of this paper is to show that if , then the orbit is located on the vertices of a regular polygon, on the convex hull of which the diffeomorphism f coincides with a rotation of an angle . This involves several new results in geometry, formulated and proved in the paper.

2-dimensional simulation of the migration behavior of grain boundary for regular polygons is carried out by using the finite element method. In the finite element calculation, a straight line element is applied to represent the grain boundaries where surface tension exerts. The driving force for boundary migration is the free energy reduction per unit volume atoms crossing the boundary. The polygons with edges less than 6 shrinks and finally disappears, while those with edges more than 6 grows. The grain boundary shape between triple points at steady state can be approximated by circle arcs. The migration velocity of the grain boundary is inversely proportional to the grain radius, indicating that a square of the radius is proportional to the time. The proportional constant increases with the number of edges of the regular polygons. In the case of a regular trigonal grain, the predicted annihilation time is consistent well with that from the theoretical model.