Interactions, space presentations, blocks and cross products

Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex 4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.

Conformal growth of Arabidopsis leaves

I show that Arabidopsis leaf growth can be described with good precision by a conformal map, where expansion is locally isotropic (the same in all directions) but the amount of expansion can vary with position. Data obtained by tracking leaf growth over time can be reproduced with almost 90% accuracy by such a map. The growth follows a Moebius transformation, which is a type of conformal map that would arise if there were an underlying linear gradient of growth rate. From the data one can derive the parameters that describe this linear gradient and show how it changes over time. Growth according to a conformal map has the property of maintaining the flatness of a leaf.

Assessing the Effects of Impedance Modifications Using the Moebius Transformation

The Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another. It has been observed that the equations relating acoustic or mechanical impedance modifications to responses under harmonic excitation are in the form of the Moebius transformation. Using the properties of the Moebius transformation, the impedance modification that will minimize the response at a particular frequency can be predicted provided that the modification is between two positions. To prove the utility of this method for acoustic and mechanical systems, it is demonstrated that the equations for calculation of transmission and insertion loss of mufflers, insertion loss of enclosures, and insertion loss of mounts are in the form of the Moebius transformation for impedance modifications. The method is demonstrated for enclosure insertion loss by adding a short duct in a partition introduced to an enclosure. In a similar manner, it is shown that the length or area of a bypass duct in a muffler can be tuned to maximize the transmission loss. In the final example, the insertion loss of an isolator system is improved at a particular frequency by adding mass to one side of the isolator.

Exploiting the Complex Plane: The Moebius Transformation and Vibro-Acoustic Optimization

If a mechanical or acoustical impedance modification is introduced between two positions, the effect of that modification can be plotted in the complex plane at a given frequency. It has been shown that the mechanical or acoustical response will trace a circle in the complex plane for straight-line modifications to impedance in the complex plane. In that case, the equations relating the response to an impedance modification are in a form consistent with the Moebius transformation, which maps straight lines or circles in one complex domain into straight lines or circles in another complex plane. It is demonstrated that the equations for muffler transmission loss and mount insertion loss are in a form consistent with the Moebius transformation for certain design changes. Accordingly, the usefulness of this linear system property will be illustrated for both muffler and mount design.

On a Theorem of Schwarz Type for Quasiconformal Mappings in Space

A space ring R is defined as a domain whose complement in the Moebius space consists of two components. The modulus of R can be defined in variously different but essentially equivalent ways (see e.g. Gehring [3] and Krivov [5]), which is denoted by mod R. Following Gehring [2], we refer to a homeomorphism y(x) of a space domain D as a k-quasiconformal mapping, if the modulus conditionis satisfied for all bounded rings R with their closure , where y(R) denotes the image of R by y = y(x). Then, it is evident that the inverse of a k-quasi-conformal mapping is itself k-quasiconformal and that a k1-quasiconformal mapping followed by a k2-quasiconformal one is k1k2-quasiconformal. It is also well known that the restriction of a Moebius transformation to a space domain is equivalent to a 1-quasiconformal mapping of its domain.