Nikiya Anton Bettey 2021: 09-21-2021-1440 - Schumann resonances (SR) Upper-atmospheric lightning

Tuesday, September 21, 2021

09-21-2021-1440 - Schumann resonances (SR) Upper-atmospheric lightning

The Schumann resonances (SR) are a set of spectrum peaks in the extremely low frequency (ELF) portion of the Earth's electromagnetic field spectrum. Schumann resonances are global electromagnetic resonances, generated and excited by lightning discharges in the cavity formed by the Earth's surface and the ionosphere.^[1]

he limited dimensions of the Earth cause this waveguide to act as a resonant cavity for electromagnetic waves in the ELF band. The cavity is naturally excited by electric currents in lightning. Schumann resonances are the principal background in the part of the electromagnetic spectrum^[2] from 3 Hz through 60 Hz,^[3] and appear as distinct peaks at extremely low frequencies (ELF) around 7.83 Hz (fundamental), 14.3, 20.8, 27.3 and 33.8 Hz.^[4]

In the normal mode descriptions of Schumann resonances, the fundamental mode is a standing wave in the Earth–ionosphere cavity with a wavelength equal to the circumference of the Earth. The lowest-frequency mode has highest intensity, and the frequency of all modes can vary slightly owing to solar-induced perturbations to the ionosphere (which compress the upper wall of the closed cavity)^{[citation needed]} amongst other factors. The higher resonance modes are spaced at approximately 6.5 Hz intervals (as may be seen by feeding numbers into the formula), a characteristic attributed to the atmosphere's spherical geometry. The peaks exhibit a spectral width of approximately 20% on account of the damping of the respective modes in the dissipative cavity.^{[citation needed]}

https://en.wikipedia.org/wiki/Schumann_resonances

Upper-atmospheric lightning or ionospheric lightning are terms sometimes used by researchers to refer to a family of short-lived electrical-breakdown phenomena that occur well above the altitudes of normal lightning and storm clouds. Upper-atmospheric lightning is believed to be electrically induced forms of luminous plasma. The preferred usage is transient luminous event (TLE), because the various types of electrical-discharge phenomena in the upper atmosphere lack several characteristics of the more familiar tropospheric lightning.

Transient luminous events have also been observed in far-ultraviolet images of Jupiter's upper atmosphere, high above the altitude of lightning-producing water clouds.^[1]^[2]

ELVES[edit]

ELVES (Emission of Light and Very Low Frequency perturbations due to Electromagnetic Pulse Sources) often appear as a dim, flattened, expanding glow around 400 km (250 mi) in diameter that lasts for, typically, just one millisecond.^[26] They occur in the ionosphere 100 km (62 mi) above the ground over thunderstorms. Their color was unknown for some time, but is now believed to be red. ELVES were first recorded on another shuttle mission, this time recorded off French Guiana on October 7, 1990.^[15] That ELVES was discovered in the Shuttle Video by the Mesoscale Lightning Experiment (MLE) team at Marshall Space Flight Center, AL led by the Principal Investigator, Otha H."Skeet" Vaughan, Jr.^{[citation needed]}

ELVES is a whimsical acronym for Emissions of Light and Very Low Frequency Perturbations due to Electromagnetic Pulse Sources.^[27] This refers to the process by which the light is generated; the excitation of nitrogen molecules due to electron collisions (the electrons possibly having been energized by the electromagnetic pulse caused by a discharge from an underlying thunderstorm).^{[citation needed]}

Trolls, Pixies, Ghosts, and Gnomes[edit]

Trolls[edit]

TROLLs (Transient Red Optical Luminous Lineaments) occur after strong sprites, and appear as red spots with faint tails, and on higher-speed cameras, appear as a rapid series of events, starting as a red glow that forms after a sprite tendril, that later produces a red streak downward from itself. They are similar to jets.^[28]^[29]

Pixies[edit]

Pixies were first observed during the STEPS program during the summer of 2000, a multi-organizational field program investigating the electrical characteristics over thunderstorms on the High Plains. A series of unusual, white luminous events atop the thunderstorm were observed over a 20-minute period, lasting for an average of 16 milliseconds each. They were later dubbed 'pixies'. They are less than 100 meters across. They are not related to lightning.^[28]

Ghosts[edit]

Ghosts (Green emissions from excited Oxygen in Sprite Tops) are faint, green glows that appear after red sprites, fading away in milliseconds.^[30] While they were initially documented by photographer Randy Halverson in 2014, ^[31] Ghosts were officially discovered by professional storm chaser Hank Schyma and Paul M Smith in 2019.^[32] No scientific journals exist on these phenomena yet due to their recent classification as a unique Upper-atmospheric phenomenon, but they have been observed on footage. Their green glow has been hypothesized by its observers to come from excited oxygen atoms, much as how auroras appear green, however, this is unconfirmed.^{[citation needed]}

Gnomes[edit]

Gnomes are small, brief spikes of light that point upward from a thunderstorm cloud's anvil top, caused as strong updrafts push moist air above the anvil. They last for only a few microseconds.^[28] They are about 200 meters wide, and are a maximum of 1 kilometer in height. Their color is unknown as they have only been observed in black-and-white footage.^[33]

US National Space Weather Program[edit]

The purpose of the US National Space Weather Program is to focus research on the needs of the affected commercial and military communities, to connect the research and user communities, to create coordination between operational data centers and to better define user community needs. NOAA operates the National Weather Service's Space Weather Prediction Center.^[11]

The concept was turned into an action plan in 2000,^[12] an implementation plan in 2002, an assessment in 2006^[13] and a revised strategic plan in 2010.^[14] A revised action plan was scheduled to be released in 2011 followed by a revised implementation plan in 2012.

https://en.wikipedia.org/wiki/Space_weather

Solar energetic particles (SEP) are high-energy particles coming from the Sun. They were first observed in the early 1940s. They consist of protons, electrons and HZE ions with energy ranging from a few tens of keV to many GeV (the fastest particles can reach a large fraction of the speed of light, as in a "ground-level enhancement", a sudden increase in cosmic ray intensity observed by ground‐based detectors first observed by Scott Forbush). They are of particular interest and importance because they can endanger life in outer space (especially particles above 40 MeV).

Solar energetic particles can originate either from a solar-flare site or by shock waves associated with coronal mass ejections (CMEs). However, only about 1% of CMEs produce strong SEP events^{[citation needed]}.

SEPs are also of interest because they provide a good sample of solar material. Despite the nuclear fusion occurring in the core, the majority of solar material is representative of the material that formed the solar system. By studying SEP's isotopic composition, scientists can indirectly measure the material that formed the solar system.

Two main mechanisms of acceleration are possible: diffusive shock acceleration (DSA, an example of second-order Fermi acceleration) or the shock-drift mechanism. SEPs can be accelerated to energies of several tens of MeV within 5–10 solar radii (5% of the Sun–Earth distance) and can reach Earth in a few minutes in extreme cases. This makes prediction and warning of SEP events quite challenging.

In March 2021, NASA reported that scientists had located the source of several SEP events, potentially leading to improved predictions in the future.^[1]^[2]

https://en.wikipedia.org/wiki/Solar_energetic_particles

https://en.wikipedia.org/wiki/Space_weather

https://en.wikipedia.org/wiki/Dynatron_oscillator

https://en.wikipedia.org/wiki/Quantum_tunnelling

https://en.wikipedia.org/wiki/Schumann_resonances

https://en.wikipedia.org/wiki/Quantum_tunnelling

https://en.wikipedia.org/wiki/Polar_route

A polar route is an aircraft route across the uninhabited polar ice cap regions. The term "polar route" was originally applied to great circle navigation routes between Europe and the west coast of North America in the 1950s.^[1]

https://en.wikipedia.org/wiki/Polar_route

https://en.wikipedia.org/wiki/Schumann_resonances

Quantum tunnelling or tunneling (US) is the quantum mechanical phenomenon where a wavefunction can propagate through a potential barrier.

The transmission through the barrier can be finite and depends exponentially on the barrier height and barrier width. The wavefunction may disappear on one side and reappear on the other side. The wavefunction and its first derivative are continuous. In steady-state, the probability flux in the forward direction is spatially uniform. No particle or wave is lost. Tunneling occurs with barriers of thickness around 1–3 nm and smaller.^[1]

Some authors also identify the mere penetration of the wavefunction into the barrier, without transmission on the other side as a tunneling effect. Quantum tunneling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires potential energy.

Quantum tunneling plays an essential role in physical phenomena, such as nuclear fusion.^[2] It has applications in the tunnel diode,^[3] quantum computing, and in the scanning tunneling microscope.

The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century.^[4]

Quantum tunneling is projected to create physical limits to the size of the transistors used in microelectronics, due to electrons being able to tunnel past transistors that are too small.^[5]^[6]

Tunneling may be explained in terms of the Heisenberg uncertainty principle in that a quantum object can be known as a wave or as a particle in general. In other words, the uncertainty in the exact location of light particles allows these particles to break rules of classical mechanics and move in space without passing over the potential energy barrier.

Quantum tunnelling may be one of the mechanisms of proton decay.^[7]^[8]^[9]

https://en.wikipedia.org/wiki/Quantum_tunnelling

Sigizmund Aleksandrovich Levanevsky (Russian: Сигизмунд Александрович Леваневский, Polish: Zygmunt Lewoniewski; 15 May [O.S. 2 May] 1902 – 13 August 1937) was a Soviet pioneer of long-range flight who was awarded the title Hero of the Soviet Union in 1934 for his role in the SS Chelyuskin rescue.

Sigizmund Levanevsky

Native name	Russian: Сигизмунд Александрович Леваневский Polish: Zygmunt Lewoniewski
Born	15 May [O.S. 2 May] 1902 St. Petersburg, Russian empire
Died	13 August 1937 (aged 35) Arctic Ocean
Allegiance	Soviet Union
Service/branch	Soviet Army before 1925 Soviet Air Force since 1925
Years of service	1918 - 1930
Battles/wars	October Revolution Civil war in Russia
Awards	Hero of the Soviet Union

https://en.wikipedia.org/wiki/Sigizmund_Levanevsky

A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

For most pairs of distinct points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles. These great circles are the geodesics of the sphere.

The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space R^{n + 1}.

A great circle divides the sphere in two equal hemispheres

https://en.wikipedia.org/wiki/Great_circle

In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true diameter).^[1]

This term applies to opposite points on a circle or any n-sphere.

An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word antipodes, meaning "opposite (the) feet", as the true word singular is antipus.

https://en.wikipedia.org/wiki/Antipodal_point

In mathematics, an $n$ -sphere is a topological space that is homeomorphic to a standard $n$ -sphere, which is the set of points in $(n + 1)$ -dimensional Euclidean space that are situated at a constant distance $r$ from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit $n$ -sphereor simply the $n$ -sphere for brevity. In terms of the standard norm, the $n$ -sphere is defined as

S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\},

and an $n$ -sphere of radius $r$ can be defined as

S^{n}(r)=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=r\right\}.

The dimension of $n$ -sphere is $n$ , and must not be confused with the dimension $(n + 1)$ of the Euclidean space in which it is naturally embedded. An $n$ -sphere is the surface or boundary of an $(n + 1)$ -dimensional ball.

In particular:

the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,
a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere,
the two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere,
the three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere,
the $n - 1$ dimensional boundary of a ( $n$ -dimensional) $n$ -ball is an $(n - 1)$ -sphere.

For $n \geq 2$ , the $n$ -spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected $n$ -dimensional manifolds of constant, positive curvature. The $n$ -spheres admit several other topological descriptions: for example, they can be constructed by gluing two $n$ -dimensional Euclidean spaces together, by identifying the boundary of an $n$ -cube with a point, or (inductively) by forming the suspension of an $(n - 1)$ -sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold consisting of two points, which is not even connected.

https://en.wikipedia.org/wiki/N-sphere

In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points.

The space SX is sometimes called the unreduced, unbased, or free suspension of X, to distinguish it from the reduced suspension ΣX of a pointed spacedescribed below.

The reduced suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies. In homotopy theory, the phenomena which are preserved under suspension, in a suitable sense, make up stable homotopy theory.

Suspension of a circle. The original space is in blue, and the collapsed end points are in green.

https://en.wikipedia.org/wiki/Suspension_(topology)

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪_f Y (sometimes also written as X +_f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally,

X\cup _{f}Y=(X\amalg Y)/\sim

where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, X ∪_f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction.

Intuitively, one may think of Y as being glued onto X via the map f.

https://en.wikipedia.org/wiki/Adjunction_space

A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all $b,c\in X,$ the inequality

d_{Y}(f(b),f(c))\leq K\cdot (d_{X}(b,c))^{\alpha }

holds. Any Hölder continuous function is uniformly continuous. The particular case $\alpha =1$ is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality

d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)

holds for any $b,c\in X.$ ^[14] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.

Continuous functions between topological spaces[edit]

Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function

f:X\to Y

between two topological spaces X and Y is continuous if for every open set $V\subseteq Y,$ the inverse image

f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology $T_{X}$ ), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions

f:X\to T

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Continuity at a point[edit]

Continuity at a point: For every neighborhood V of

f(x),

, there is a neighborhood U of x such that

f(U)\subseteq V

The translation in the language of neighborhoods of the $(\varepsilon, \delta)$ -definition of continuity leads to the following definition of the continuity at a point:

A function $f:X\to Y$ is continuous at a point $x\in X$ if and only if for any neighborhood $V$ of $f(x)$ in $Y$ , there is a neighborhood $U$ of $x$ such that $f(U)\subseteq V.$

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and $f^{{-1}}(V)$ is the largest subset $U$ of $X$ such that $f(U)\subseteq V,$ this definition may be simplified into:

A function $f:X\to Y$ is continuous at a point $x\in X$ if and only if $f^{{-1}}(V)$ is a neighborhood of $x$ for every neighborhood $V$ of $f(x)$ in $Y$ .

As an open set is a set that is a neighborhood of all its points, a function $f:X\to Y$ is continuous at every point of $X$ if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above $\varepsilon -\delta$ definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If however the target space is a Hausdorff space, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Given $x\in X,$ a map $f:X\to Y$ is continuous at $x$ if and only if whenever ${\mathcal {B}}$ is a filter on $X$ that converges to $x$ in $X,$ which is expressed by writing ${\mathcal {B}}\to x,$ then necessarily $f({\mathcal {B}})\to f(x)$ in $Y.$ If ${\mathcal {N}}(x)$ denotes the neighborhood filter at $x$ then $f:X\to Y$ is continuous at $x$ if and only if $f({\mathcal {N}}(x))\to f(x)$ in $Y.$ ^[15] Moreover, this happens if and only if the prefilter $f({\mathcal {N}}(x))$ is a filter base for the neighborhood filter of $f(x)$ in $Y.$ ^[15]

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

Inverse image[edit]

Let f be a function from X to Y. The preimage or inverse image of a set B ⊆ Y under f, denoted by $f^{-1}[B]$ , is the subset of X defined by

f^{-1}[B]=\{x\in X\,|\,f(x)\in B\}.

Other notations include $f -1 (B)$ ^[5] and $f - (B)$ .^[6] The inverse image of a singleton, denoted by f⁻¹[{y}] or by f⁻¹[y], is also called the fiber or fibre over y or the level set of y. The set of all the fibers over the elements of Y is a family of sets indexed by Y.

For example, for the function f(x) = x², the inverse image of {4} would be {−2, 2}. Again, if there is no risk of confusion, f⁻¹[B] can be denoted by f⁻¹(B), and f⁻¹can also be thought of as a function from the power set of Y to the power set of X. The notation f⁻¹ should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of B under f is the image of B under f⁻¹.

Notation for image and inverse image[edit]

The traditional notations used in the previous section can be confusing. An alternative^[7] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation[edit]

$f^{\rightarrow }:{\mathcal {P}}(X)\rightarrow {\mathcal {P}}(Y)$ with $f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}$
$f^{\leftarrow }:{\mathcal {P}}(Y)\rightarrow {\mathcal {P}}(X)$ with $f^{\leftarrow }(B)=\{a\in X\;|\;f(a)\in B\}$

Star notation[edit]

$f_{\star }:{\mathcal {P}}(X)\rightarrow {\mathcal {P}}(Y)$ instead of $f^{\rightarrow }$
$f^{\star }:{\mathcal {P}}(Y)\rightarrow {\mathcal {P}}(X)$ instead of $f^{\leftarrow }$

Other terminology[edit]

An alternative notation for f[A] used in mathematical logic and set theory is f "A.^[8]^[9]
Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomainof f.

Examples[edit]

f: {1, 2, 3} → {a, b, c, d} defined by $f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\a,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.$
The image of the set {2, 3} under f is f({2, 3}) = {a, c}. The image of the function f is {a, c}. The preimage of a is f⁻¹({a}) = {1, 2}. The preimage of {a, b} is also {1, 2}. The preimage of {b, d} is the empty set {}.
f: R → R defined by f(x) = x².
The image of {−2, 3} under f is f({−2, 3}) = {4, 9}, and the image of f is R⁺ (the set of all positive real numbers and zero). The preimage of {4, 9} under f is f⁻¹({4, 9}) = {−3, −2, 2, 3}. The preimage of set N = {n ∈ R | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
f: R² → R defined by f(x, y) = x² + y².
The fibre f⁻¹({a}) are concentric circles about the origin, the origin itself, and the empty set, depending on whether a > 0, a = 0, or a < 0, respectively. (if a> 0, then the fiber f⁻¹({a}) is the set of all (x, y) ∈ R² satisfying the equation of the origin-concentric ring x² + y² = a.)
If M is a manifold and π: TM → M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces T_x(M) for x∈M. This is also an example of a fiber bundle.
A quotient group is a homomorphic image.

https://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image

In differential geometry, the tangent bundle of a differentiable manifold $M$ is a manifold $TM$ which assembles all the tangent vectors in $M$ . As a set, it is given by the disjoint union^{[note 1]} of the tangent spaces of $M$ . That is,

{\begin{aligned}TM&=\bigsqcup _{x\in M}T_{x}M\\&=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M\\&=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\&=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}\end{aligned}}

where $T_{x}M$ denotes the tangent space to $M$ at the point $x$ . So, an element of $TM$ can be thought of as a pair $(x,v)$ , where $x$ is a point in $M$ and $v$ is a tangent vector to $M$ at $x$ .

There is a natural projection

\pi : TM \twoheadrightarrow M

defined by $\pi (x,v)=x$ . This projection maps each element of the tangent space $T_{x}M$ to the single point $x$ .

The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of $TM$ is a vector field on $M$ , and the dual bundle to $TM$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of $M$ . By definition, a manifold $M$ is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold $M$ is framed if and only if the tangent bundle $TM$ is stably trivial, meaning that for some trivial bundle $E$ the Whitney sum $TM\oplus E$ is trivial. For example, the n-dimensional sphere Sⁿ is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).

https://en.wikipedia.org/wiki/Tangent_bundle

In mathematics, a differentiable manifold $M$ of dimension n is called parallelizable^[1] if there exist smooth vector fields

\{V_1, \dots,V_n\}

on the manifold, such that at every point $p$ of $M$ the tangent vectors

\{V_{1}(p),\dots ,V_{n}(p)\}

provide a basis of the tangent space at $p$ . Equivalently, the tangent bundle is a trivial bundle,^[2] so that the associated principal bundle of linear frames has a global section on $M.$

A particular choice of such a basis of vector fields on $M$ is called a parallelization (or an absolute parallelism) of $M$ .

https://en.wikipedia.org/wiki/Parallelizable_manifold

In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for E_x. The general linear group acts naturally on F(E ) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bundle (where k is the rank of E ).

The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle.

https://en.wikipedia.org/wiki/Frame_bundle

In mathematics, a principal homogeneous space,^[1] or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that x·g = y, where · denotes the (right) action of G on X). An analogous definition holds in other categories, where, for example,

G is a topological group, X is a topological space and the action is continuous,
G is a Lie group, X is a smooth manifold and the action is smooth,
G is an algebraic group, X is an algebraic variety and the action is regular.

https://en.wikipedia.org/wiki/Principal_homogeneous_space

In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is a related shape, a toroid.

Real-world objects that approximate a torus of revolution include swim rings and inner tubes. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses.

A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S¹ × S¹, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S¹ in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.

In the field of topology, a torus is any topological space that is homeomorphic to a torus.^[1] A coffee cup and a doughnut are both topological tori with genus one.

An example of a torus can be constructed by taking a rectangular strip of flexible material, for example, a rubber sheet, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Möbius strip).

https://en.wikipedia.org/wiki/Torus

Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with inertial confinement fusion. The magnetic approach began in the 1940s and absorbed the majority of subsequent development.

Fusion reactions combine light atomic nuclei such as hydrogen to form heavier ones such as helium, producing energy. In order to overcome the electrostatic repulsion between the nuclei, they must have a temperature of tens of millions of degrees, creating a plasma. In addition, the plasma must be contained at a sufficient density for a sufficient time, as specified by the Lawson criterion (triple product).

Magnetic confinement fusion attempts to use the electrical conductivity of the plasma to contain it through interaction with magnetic fields. The magnetic pressureoffsets the plasma pressure. Developing a suitable arrangement of fields that contain the fuel without excessive turbulence or leaking is the primary challenge of this technology.

https://en.wikipedia.org/wiki/Magnetic_confinement_fusion

The Coulomb barrier, named after Coulomb's law, which is in turn named after physicist Charles-Augustin de Coulomb, is the energy barrier due to electrostaticinteraction that two nuclei need to overcome so they can get close enough to undergo a nuclear reaction.

Potential energy barrier[edit]

This energy barrier is given by the electrostatic potential energy:

U_{\text{coul}}=k{{q_{1}\,q_{2}} \over r}={1 \over {4\pi \epsilon _{0}}}{{q_{1}\,q_{2}} \over r}

where

k is Coulomb's constant = 8.9876×10⁹ N·m²·C⁻²;

ε₀ is the permittivity of free space;

q₁, q₂ are the charges of the interacting particles;

r is the interaction radius.

A positive value of U is due to a repulsive force, so interacting particles are at higher energy levels as they get closer. A negative potential energy indicates a bound state (due to an attractive force).

The Coulomb barrier increases with the atomic numbers (i.e. the number of protons) of the colliding nuclei:

U_{\text{coul}}={{k\,Z_{1}\,Z_{2}\,e^{2}} \over r}

where e is the elementary charge, 1.60217653×10⁻¹⁹ C, and Z_i the corresponding atomic numbers.

To overcome this barrier, nuclei have to collide at high velocities, so their kinetic energies drive them close enough for the strong interaction to take place and bind them together.

According to the kinetic theory of gases, the temperature of a gas is just a measure of the average kinetic energy of the particles in that gas. For classical ideal gases the velocity distribution of the gas particles is given by Maxwell–Boltzmann. From this distribution, the fraction of particles with a velocity high enough to overcome the Coulomb barrier can be determined.

In practice, temperatures needed to overcome the Coulomb barrier turn out to be smaller than expected due to quantum mechanical tunnelling, as established by Gamow. The consideration of barrier-penetration through tunnelling and the speed distribution gives rise to a limited range of conditions where fusion can take place, known as the Gamow window.

The absence of the Coulomb barrier enabled the neutron's discovery by James Chadwick in 1932.^[1]^[2]

References[edit]

^ Chadwick, James (1932). "Possible existence of a neutron". Nature. 129 (3252): 312. Bibcode:1932Natur.129Q.312C. doi:10.1038/129312a0.
^ Chadwick, James (1932). "The existence of a neutron". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 136(830): 692–708. Bibcode:1932RSPSA.136..692C. doi:10.1098/rspa.1932.0112.

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Genesis[edit]

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