May 22 2024

Isometry and Homotopy

Published by under Mathematics,Nuclear Physics,Quantum Field Theory at 10:51 am

In Chemistry, “This phenomenon of two or more compounds having the same molecular formula but different structures is called isomerism.” ([1], pg. 405) In a positive charge groupoid, traveling through an open gamma ray field, coming from a given nucleus, there is an isometry. Though the homotopy comes across “an m-dimensionable manifold M of class Cr” ([2], pg. 646), the gravitons have different phases as the groupoid travels on, in an outer tangent space, and “the homotopy problem is equivalent to an extension problem.” ([3], pg. 175)

“16.2. LEMMA.  If (E,S) is a cell and its boundary, then (E X 0)(S X I) is a retract of E X I.” ([3], pg. 84)

Thus, “π is an infinite cyclic group,” ([3], pg. 199). Steenrod calls S “A system S of coordinates” ([3], pg. 22), or “a bundle of coefficients.” ([3], pg. 190) In this case, with “E X I“, I is an isotopy, and the zero space is the tangent space. There is “no left distributive law” ([3], pg. 122), because orbital electrons absorb gravitons at various rates.

[1] Hein, Morris, “Foundations of College Chemistry, Fourth Edition”, DICKENSON PUBLISHING COMPANY, INC., c. 1977

[2] Whitney, Hassler, “Differential Manifolds”, The Annals of Mathematics, Second Series, Vol. 37. No. 3 (Jul., 1936) pp. 645-680

[3] Steenrod, Norman, “The Topolgy of Fibre Bundles”, Princeton University Press, c. 1951

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May 18 2024

Emitting Cell Phone, Radio, and Television EM Waves

Let us say there is an imaginary horizontal disk centered on a vertical, unshielded emitting antenna. Cosine waves of various frequencies and amplitudes go out in all directions centered on the disk. As a cosine wave travels away from the disk, it imbues EM waves of the same frequency perpendicular outward in a push and peel process, in pairs, backward, laterally at an acute angle. As the cosine wave comes back toward the disk, there is no push, preventing double signals. Each torus grows continually until it runs out of momentum, and in a complex set of signals there are many interspersed tori.

The amplitude of each cosine signal, as it multiplies, may not be constant throughout the torus, though frequency is. For a given location of a receiving antenna, the amplitude ratios of all the signals are the same.

The H field, just like in electrodynamics, is a magnetic field.

“ h + n(I) = dim H ≤ a(I) = i(I) + n(I), so h ≤ i(I) “      ([1], pg. 233)

“ h “ is centripetal force from a magnetic field, producing a corkscrew, and n(I) is the antenna.

24.10.  THEOREM.  “If n ≡ 0 mod 4, then πn (Rn+1) contains a cyclic group of order 2 whose non-zero element is represented by Tn+2.”   ([2], pg. 130)

The electrons in the antenna produce 4 corkscrews for each signal. Electrons tend to congregate at the outside of the antenna, so it may be 4 electrons to a signal. Outside the antenna they quickly spread. What is meant by augmented index, a(I), is that the cosine waves, as they are emitted, go out in all directions from the antenna.

[1] Bishop, Richard L. and Crittenden, Richard J., “Geometry of Manifolds”, AMS CHELSEA PUBLISHING, Copyright 1964 held by the American Mathematical Society. Reprinted with corrections by the American Mathematical Society, 2001

[2] Steenrod, Norman, “The Topolgy of Fibre Bundles”, Princeton University Press, c. 1951

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May 17 2024

Unethical

Published by under General at 03:47 pm

People use my work, especially for defense, and it is unethical not to pay me.

We would like to purchase a Condo north of Tampa, FL a few miles.

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Mar 19 2024

Spherical Groups

With spherical groups of opposite charge signs, like emitted by the proton and electron in hydrogen, why would they not annihilate each other at different spots? For one thing they are segmented and pulsed, and spread out, and come out of a charged mass in pairs. These are convex regions, and “A region X open or closed, will be called convex if any two points in X are joined by at least one path which does not leave X.” ([1], pg. 7)

Additionally, “Let X be any open region and its closure.” ([1], pg. 7) It is like the groups have AI, and know how to avoid each other and know how to come back together. In footnote “*” on page 7, attributed to K. Menger, “We may think of X as filled with substance which conducts light along paths, all the space except X being opaque.”

At least in atoms or molecules, at close range, the groups are strong enough to avoid each other, before the polarization factor takes over. As far as hitting a target, the Coulomb field travels at the speed of light squared, and more groups come through fast.

[1] Whitehead, J. H. C., “CONVEX REGIONS IN THE GEOMETRY OF PATHS”, Princeton Press, (Received 15 August 1931)

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Jan 20 2024

Conical Angle

On page 9 of the April 2007 booklet, it is proposed that the magnetic field of the electron guides the rectification of incoming gravitons “within some inclusive conical angle that is greater than zero”. As the electron grows in an atomic or molecular orbital, the magnetic field of the electron becomes stronger.

The Cauchy integral formula is used in many applications. Here we are applying it to gravity.

Since z is the direction straight out of the earth, let us call f’’(z) the gravitational force. “The Cauchy integral formula in the theorem in Sec. 50 can be extended so as to provide an integral representation of derivatives of f at z0” ([1], pg. 165). Then the formula becomes:

     f(z) = (1/2πi) ∫c [(f(s) / (s – z)] ds       ([1], pg. 166, formula (1))

Let us say z is a point at the center of the 3-dimensional electron, and (s – z) is the spin radius that starts off at 6.6676 x 10-16 m immediately after a spin flip at the end of τi. Let f(s) be the function that grows (s – z) as the orbital electron absorbs gravitons. Keep in mind that the conical angle grows also.

f’(z) is the rate at which the orbital electron absorbs gravitons, since an arc τ is often not directed at the center of the earth:

     f’(z) = (1/2πi) ∫c [(f(s) / (s – z)2] ds       ([1], pg. 166, formula (2))

f’’(z) is Newton’s second law of motion, F = ma:

     f’’(z) = (1/πi) ∫c [(f(s) / (s – z)3] ds       ([1], pg. 167, formula (4))

Since the electron is perfectly round, (s – z) still starts off at the radius of 6.6676 x 10-16 m at the beginning of an arc. As the electron grows in size in an orbital, at some point it is able to produce Pontrjagin classes, or higher k-planes, as long as f(s) is strong enough.

[1] Brown, James Ward and Churchill, Ruel V., “Complex Variables and Applications”, McGraw-Hill Higher Education, c. 2009

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Dec 22 2023

k-planes

Published by under Quantum Field Theory,String Theory at 08:50 am

As presented before, a magnetic field can bend a charge path, but not speed it up or slow it down. k-planes are produced by magnetic fields, whereby they manipulate the direction of electric fields. Due to a charged mass, k-planes are produced, and this means that electric fields of gravitons are bent into the same plane as the vector potential comes through. A vector potential can be a charged mass or a group traveling through the gamma ray field before acting on another charge.

When two graviton electric fields are combined it is called a “k-th Chern class ck(E)” ([1], pg. 309) and a “2k form γk“. When a “4k-form βk” ([1], pg. 309), it is called a “Pontrjagin class”. We may think of a Pontrjagin class as a flat picture of a mountain range with 4 mountains in the picture. It is not a sine or cosine curve, but has 4 lobes. The curve that defines the tops of the mountains can be thought of as a string. Statistically, lobes may be combined at times. There are “two elements (A, p) and (B, q)” ([2], pg. 216), and q is the distance the p moves laterally to help form a hypersurface k-plane. For example, there is a “2k-form on P” ([1], pg. 293) for a Chern class, and in any k class there is “oriented p-planes in Rp+q” ([1], pg. 271).

k-planes are created to do heavy work, and are parts of larger Lie groups which determine the chirality. As the group approaches another charge, it determines whether the charge is positive or negative. If it is of the same charge sign, the Lie groups instruct the k-planes to slap the target charge on the face. If the target is the opposite charge sign, the k-planes split, spin around, and slap the target charge on the back. After 18 years, we really need the String Theorists working on the mathematics of this.

The k-planes take up new gravitons quickly and leave others behind. It is its own entity within a Lie group, and “(g, h, σ) is effective” ([1], pg. 249), σ being the effect of the magnetic field.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

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Oct 27 2023

Pauli Exclusion Principle

With the Pauli exclusion principle, “Only two electrons (with opposite spins) can occupy a given quantum state.” ([1], pg. 798) The magnetic moments being in opposite directions help keep the separation, though it is electric current loops in each electron doing the work.

Ref: https://www.fruechtetheory.com/blog/2022/02/14/spin-drive/

The reason orbitals in an atom or molecule are limited to two electrons has to do with the fact that “σ, τ are any (local) bisections of G”. ([2], pg. 28) In mechanical engineering, σ is the symbol for stress. Likewise, σ is the symbol here for local negative charge stress in the gamma ray field, and we have “The map φ: σ → γ” ([3], pg. 226). Alternatively, there is “the sheaf of germs of local bisections of G.” ([2], pg. 133) Here we have another name for the graviton in “germ”, and a “sheaf” is a member of the h field, when two or more gravitons get combined to do the work.

With the spin of the electron, “the mapping Sσ → aσn Sσ” ([4], pg. 162) provides a matrix theory to the Coulomb field, with “cyclic groups (σi) of orders ni” ([4], pg. 130). This provides additional proof that Coulomb groups are pulsed.

Two other authors call the Pauli exclusion principle a “set of pairs (τ, J)” ([5], pg. 68), with “permutations σ” ([6], pg. 293). J is a charged mass:

https://www.fruechtetheory.com/blog/2023/06/29/vector-bosons-and-other-fleeting-field-particles/,

and in this case we are referring to an electron.

Physicists already knew most of this blog entry before it was entered. What many people do not know is the presence of a gamma ray field, though it is reasonable to know because of the gamma ray telescopes.

[1] Tipler, Paul A., “Physics”, Worth Publishers, Inc., 1976

[2] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[3] Bishop, Richard L. and Crittenden, Richard J., “Geometry of Manifolds”, AMS CHELSEA PUBLISHING, Copyright 1964 held by the American Mathematical Society. Reprinted with corrections by the American Mathematical Society, 2001

[4] Zassenhaus, Hans J., “THE THEORY OF GROUPS”, Dover Publications Inc., 1999 (Originally published by Chelsea Publishing Co., 1958)

[5] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[6] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

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Oct 16 2023

Atomic and Molecular Electron Arcs

Further to uniting Riemannian geometry, Lie groups, and symmetric spaces with gravity, τ is an atomic or molecular arc, and “τ is a segment” ([1], pg. 168). Also, “τ is minimizing” ([1], pg. 166).

Sometimes τ is called a complete orbital, and we “divide τ into a finite number of arcs, say, τ1, τ2, … , τk” ([1], pg. 191).

Ref: https://www.fruechtetheory.com/blog/2008/06/28/gravity-and-the-uncertainty-principle-2-2/

In an orbital arc the “endomorphisms A1, … , Ak are linearly independent” ([2], pg. 353), and k – 1 in this instance is the number of gravitons absorbed in an arc. “A” is the vector potential, and each time an electron absorbs a graviton in an orbital, its vector potential increases. We know that A1, … , Ak is not pulsed Lie groups in the gamma ray field, because there is no “…” after the Ak. In the same paragraph it talks about a “mapping ξ → Aξ“, therefore in a particle mass, and in groups or manifolds in the open gamma ray field, the gamma rays are blended and surjective.

If a function can be called “the growth of an orbital electron in size and charge”:

https://www.fruechtetheory.com/blog/2022/08/27/concentrated-group-action/ ,

then “γ and f point in opposite directions” ([3], pg.165).

The Φ field is within atomic and molecular orbitals, including the boundary, and Ψ is outside of the orbitals. In an emitting antenna, it is the Ψ field as well, since the electrons are free. “Φ0 is isomorphic to Ψ0 in a natural manner” ([1], pg. 193), because the gamma ray field is normally constant in the area within and around the molecule.

Often in a molecule, or any type of p orbital, the Gaussian curvature, when ¾ through the arc compared to ¼ through the arc, is negative.

In the open gamma ray field “m = dim M and n = dim Δ” ([4], pg. 155), and m – n is the number of singularities in a locality. Stoker terms it “singularity in the coordinate system” ([5], pg. 84). A singularity is when the electric and magnetic fields of a gamma ray cross over the t axis, though when near the axis it could be called a singularity also.

If the polarization factor is greater than 2, as at the surface of the sun or Jupiter, then specific nuclei likely have more mass than on the face of the earth, and electrons in atomic or molecular arcs grow larger. It could be because of these factors the value of Newton’s gravitational constant G = 6.672 x 10-11 (N-m2)/ kg2 stays the same.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[4] Boothby, William M., “An Introduction to Differentiable Manifolds and Riemannian Geometry”, Academic Press, 2003

[5] Stoker, James J., “Differential Geometry”, John Wiley & Sons, Inc., c. 1969

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Sep 30 2023

Michelson-Morley experiment

It has been said that “The electric field travels faster the denser a gravitational field is, though the speed difference may not be discernable.” Also: “What we have is an infinitesimal zigzag pattern, though when we back out to the classical level, it does not matter for any application.”

Ref: https://www.fruechtetheory.com/blog/2023/01/17/action-of-the-electric-field/

As it turns out, the speed difference may have been indirectly discerned by the Michelson-Morley experiment in 1887, and the “infinitesimal zigzag pattern” is less for a denser gravitational field.

A picture and description of the Michelson-Morley experiment tells it was mounted on a large block of sandstone, for stability, and floated on an annular trough of mercury for rotation.

The block that the experiment was on, and the sensors and brackets in the forward direction of travel of the whole apparatus, would have helped decompress the gamma rays toward the center of the apparatus, near the forward brackets. The brackets behind would have compressed the gamma rays. What was probably happening was a slow-fast travel of the electromagnetic waves in one direction, and a fast-slow travel in the opposite direction.

ηi is when the center t of a gamma ray moves one way in an alternative direction. ξi is a smooth electric field, and ηi and ξi work together to smear electric fields of the gamma rays into electromagnetic waves of larger dimensions, as a laser, emitted cell phone wave, etc. As a tornado takes up air molecules and expels others, these waves of lower frequency than a graviton take up gravitons they reach. “Assume that ξ is affine” ([1], pg. 377), and “ξ is a (column) vector in Rn” ([2], pg. 269). The gravitons that are expelled can take off in almost any direction. “ξi and ηi are orthogonal” ([3], pg. 315], because they are independent. “t” is not an electric field, it is a singularity.

ξi is caused by the H field, though it is specific to an emitted wave, subject to “the compatibility conditions which ξ and H are obliged to satisfy” ([1], pg. 362), and “ξ is a pure translation” ([4], pg. 193).

In the Michelson-Morley experiment, in the denser gamma ray field the ξ field is more efficient and moves faster, and the wavelength is slightly shorter than average. In the less dense gamma ray field, the wavelength is slightly longer than average.

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[3] Stoker, James J., “Differential Geometry”, John Wiley & Sons, Inc., c. 1969

[4] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

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Aug 03 2023

Ricci tensor field S

Published by under Mathematics,Quantum Field Theory at 07:02 pm

When a free electron accelerates, it may be able to increase in mass, charge, and diameter for the increase in work it must do. Again, we think of an emitting antenna.

We know that “s” can stand for spin, and that electrons have spin. The electrons in the antenna may impart spin into a “tensor space Tsr “ ([1], pg. 209], where “r” is the vector away from the antenna, and “s” is the spin. What we can liken this to is a corkscrew in a gravitational field. Each corkscrew “s is a direct sum of simple ideals: s1 + … +sk” ([2], Appendix 5, pg. 279)

To send these corkscrews out in all directions from an antenna is a phenomenal amount of work. It is not absolutely necessary that accelerated free electrons expand for this to occur, though they would at least absorb gravitons at a greater rate than a free electron at rest or traveling at constant velocity in a straight line.

It is not known what percentage of these corkscrews would be left-handed. When two electrons are near each other, “(βi) is invariant by the left translation” ([2], pg. 207), and they repel each other.

Furthermore to the Ricci tensor field containing spin, there are the following two corollaries:

“Corollary 5.5   If M is a compact Riemannian manifold with vanishing Ricci tensor field, then every infinitesimal isometry of M is a parallel vector field.” ([2], pg. 251)

“Corollary 5.6   If a connected compact homogeneous Riemannian manifold M has zero Ricci tensor, then M is a Euclidean torus.” ([2], pg. 251)

As the output from an emitting antenna turns into tori, there is a “concatenation of paths” ([3], pg. 229].

Of course, as these tori wear out, they disintegrate, because of “a theorem of Weyl that any representation of a semisimple Lie algebra is completely reducible” ([2], Appendix 5, pg. 279]. We now know that these Lie algebras are reducible to gravitons.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

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