https://www.houstoniamag.com/arts-and-culture/2018/11/day-for-night-free-press-houston-sold
The problem of reconstruction from zero crossings can be stated as: given the zero crossings of a continuous signal, is it possible to reconstruct the signal (to within a constant factor)? Worded differently, what are the conditions under which a signal can be reconstructed from its zero crossings?
This problem has two parts. Firstly, proving that there is a unique reconstruction of the signal from the zero crossings, and secondly, how to actually go about reconstructing the signal. Though there have been quite a few attempts, no conclusive solution has yet been found. Ben Logan from Bell Labs wrote an article in 1977 in the Bell System Technical Journal giving some criteria under which unique reconstruction is possible. Though this has been a major step towards the solution, many people[who?] are dissatisfied with the type of condition that results from his article.
According to Logan, a signal is uniquely reconstructible from its zero crossings if:
- The signal x(t) and its Hilbert transform xt have no zeros in common with each other.
- The frequency-domain representation of the signal is at most 1 octave long, in other words, it is bandpass-limited between some frequencies B and 2B.
Further reading[edit]
- B. F. Logan, Jr. "Information in the Zero Crossings of Bandpass Signals", Bell System Technical Journal, vol. 56, pp. 487–510, April 1977.
The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions(particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.
In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number; ℓ, the azimuthal quantum number; mℓ, the magnetic quantum number; and ms, the spin quantum number. For example, if two electrons reside in the same orbital, then their n, ℓ, and mℓ values are the same; therefore their ms must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and −1/2.
Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser or atoms in a Bose–Einstein condensate.
A more rigorous statement is that, concerning the exchange of two identical particles, the total (many-particle) wave function is antisymmetric for fermions, and symmetric for bosons. This means that if the space and spin coordinates of two identical particles are interchanged, then the total wave function changes its sign for fermions and does not change for bosons.
If two fermions were in the same state (for example the same orbital with the same spin in the same atom), interchanging them would change nothing and the total wave function would be unchanged. The only way the total wave function can both change sign as required for fermions and also remain unchanged is that this function must be zero everywhere, which means that the state cannot exist. This reasoning does not apply to bosons because the sign does not change.
https://en.wikipedia.org/wiki/Pauli_exclusion_principle
Wednesday, September 15, 2021
09-15-2021-0511 - Infinite divisibility Continuum
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects such as the continuum.
https://en.wikipedia.org/wiki/Infinite_divisibility
Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variation using qualitatively different states.[1]
In physics, for example, the space-time continuum model describes space and time as part of the same continuum rather than as separate entities. A spectrum in physics, such as the electromagnetic spectrum, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths).
In contrast, quantum mechanics uses quanta, certain defined amounts (i.e. categorical amounts) which are distinguished from continuous amounts.
External links[edit]
Continuity and infinitesimals, John Bell, Stanford Encyclopedia of Philosophy
https://en.wikipedia.org/wiki/Continuum_(measurement)
Categories: ConceptsConcepts in metaphysicsConcepts in physicsConcepts in the philosophy of scienceHistory of mathematicsHistory of philosophyHistory of sciencePhilosophical conceptsPhilosophical theoriesPhilosophy of mathematicsPhilosophy of scienceTheories
A spectrum (plural spectra or spectrums)[1] is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism. As scientific understanding of light advanced, it came to apply to the entire electromagnetic spectrum. It thereby became a mapping of a range of magnitudes (wavelengths) to a range of qualities, which are the perceived "colors of the rainbow" and other properties which correspond to wavelengths that lie outside of the visible light spectrum.
https://en.wikipedia.org/wiki/Spectrum
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to interaction between the nucleus and electron clouds.
In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule.
Hyperfine structure contrasts with fine structure, which results from the interaction between the magnetic moments associated with electron spin and the electrons' orbital angular momentum. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the nucleus (or nuclei, in molecules) with internally generated electric and magnetic fields.
https://en.wikipedia.org/wiki/Hyperfine_structure
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories exist.[1]
https://en.wikipedia.org/wiki/Supersymmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement")[1] in everyday language refers to a sense of harmonious and beautiful proportion and balance.[2][3][a]
https://en.wikipedia.org/wiki/Symmetry
A mirror is an object that reflects an image.
https://en.wikipedia.org/wiki/Mirror
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