Some sets are more infinite than other sets. Cantor's Theorem kindly informs us that there are two kinds of infinity, the countable infinity of natural numbers and its much bigger brother, the infinity of the real numbers.
Some still hope that Cantor's Continuum Hypothesis will be proved on a bright sunny day, albeit nobody can say what a mathematical proof is or even what a number is. It`s about time to make a valiant attempt to resolve this embarrassing problem that has been a running sore for far too long. See the opportunity as a chance for everlasting fame.
Some still hope that Cantor's Continuum Hypothesis will be proved on a bright sunny day, albeit nobody can say what a mathematical proof is or even what a number is. It`s about time to make a valiant attempt to resolve this embarrassing problem that has been a running sore for far too long. See the opportunity as a chance for everlasting fame.
How to see Comet Neowise from the UK: Comet NEOWISE delights observers across the world - skyatnightmagazine
Earthquakes, illnesses, red rain, and even the births of two-headed animals have been blamed on comets.
'Harbingers of Doom' they've been called. We can be thankful it's just a little terrible weather! 😀
'Harbingers of Doom' they've been called. We can be thankful it's just a little terrible weather! 😀
Much gloomy nonsense about Doomsday with this Comet "Fireball"! Annular eclipse (Ring of Fire...) over the "Holy Land" this year. And asteroid might hit us later in the year... 😱
Comet Neowise and '''asteroid approaching Earth''' are signs of APOCALYPSE, crackpot YouTube preacher warns
The Sun and the Beano were required reading when I was at college. Best to avoid serious reading and television... only makes you worry. 😀
I found our musings about infinity and geometric progressions back on Page 299.
Conway was the man: Quanta Magazine
Life, Death and the Monster (John Conway) - Numberphile - YouTube
Natural numbers and the real numbers having different order of infinity. And then, where I get muddled:
It seems the algebraics can be mapped to the naturals. So that's OK. 😎
Apparently Continuum hypothesis - Wikipedia is neither provable nor disproveable within the axioms of Set Theory. A Kurt Godel thing.
Comet Neowise and '''asteroid approaching Earth''' are signs of APOCALYPSE, crackpot YouTube preacher warns
The Sun and the Beano were required reading when I was at college. Best to avoid serious reading and television... only makes you worry. 😀
I found our musings about infinity and geometric progressions back on Page 299.
Conway was the man: Quanta Magazine
Life, Death and the Monster (John Conway) - Numberphile - YouTube
Natural numbers and the real numbers having different order of infinity. And then, where I get muddled:
It seems the algebraics can be mapped to the naturals. So that's OK. 😎
Apparently Continuum hypothesis - Wikipedia is neither provable nor disproveable within the axioms of Set Theory. A Kurt Godel thing.
Lot to get through here. 😱
My favourite TinTin adventure was Destination Moon. We used to get that one in kid's hour before the 18.00 Hrs News that the boring grown-ups watched.
TinTin's loveable dog Snowy got stuck in a crater with his oxygen running out. Barking for help did no good. No sound on the Moon. A worrisome moment. I think it worked out allright the next night, but dramatic. Kept me awake worrying. 😱
Also enjoyable is Asterix in Britain. The likeable Gaul discovers the British were obsessed with a strange game called Rugby, and Beer. Still got this one on the go, as it happens.
BTW, September the 1st is doomsday. Allegedly.
Derivation of the Basic Wave Equation:
Perspective and Explanation:
A wave can be imagined as a sine curve travelling alone a straight line.
Mathematically this can be very basically expressed as:
y = A.Sin(2πft)............[f: frequency, t: time, A: amplitude]
But this formula form does not take account of the displacement along the wave's path of travel. *We need another dimension to fully describe the wave, and this is the distance along the path of travel, x. As it is, the formula only applies to arbitrary points along the line of travel of the wave, assuming the wave is continuous.
For any wave, v = fλ.....[v: wave velocity, λ: wavelength]
==> λ = v/f
To include wavelength along the path of travel, we need to convert x into an angle because we are dealing with a sine function.
In x there are x/λ complete wavelengths which translate to an angle of 2πx/λ.
Substituting for λ, we get:
2πx/(v/f) = 2πfx/v
Therefore, the angle to be passed to the sine function will be:
2πft + 2πfx/v = 2πf(t + x/v)
If we also want to account for a phase shift at t = 0, we can add a constant, Φ, to this expression.
Φ + 2πf(t + x/v)
The wave equation becomes:
y = A.Sine[Φ + 2πf(t + x/v)]
The Differential Wave Equation:
The Quantum Mechanical Schroendinger wave equation is a differential equation that has its remote roots based on the differential equation we are going to derive.
Let us start our journey with the wave equation derived just above.
Our little equation is:
y = A.Sine[Φ + 2πf(t + x/v)]
Let us now use Differentiation. We will continue to derive up to the second order differentials both with respect to time and x.
We will do it here:
Differentiating with respect to time:
dy/dt = 2πfA.Cosine[Φ + 2πf(t + x/v)].....[1st order]
d^2y/dt^2 = -(2πf)^2.A.Sine[Φ + 2πf(t + x/v)] ...[2nd order]
Differentiating with respect to x:
dy/dx = (2πf/v).A.cosine[Φ + 2πf(t + x/v)].....[1st order]
d^2y/dx^2 = -(2πf/v)^2.A.Sine[Φ + 2πf(t + x/v)]....[2nd order]
The wave equation is expressed in terms of the second order differentials.
Inspecting both second order differential equations, we immediately notive there is a Sine common term with exactly the same expression in brackets. We will make this subject first, and the aim, is to have two differential equations expressing the same thing. The latter will provide us the ingredients for both sides of the differential wave equation.
So, making the Sine(...) subject we get:
For the first one:
Sine[Φ + 2πf(t + x/v)] = -[d^2y/dt^2]/[(2πf)^2.A]
And, for the second one:
Sine[Φ + 2πf(t + x/v)] = -[d^2y/dx^2]/[(2πf/v)^2.A]
The differential wave equation is formed by the right hand side expression of the latest two equations.
Therefore:
-[d^2y/dt^2]/[(2πf)^2.A] = -[d^2y/dx^2]/[(2πf/v)^2.A]
Simplying by removing redundant terms:
d^2y/dt^2 = [d^2y/dx^2]/[(1/v)^2]
Further simplifying:
d^2y/dt^2 = v^2.[d^2y/dx^2]
The latter is the wave equation for a single dimension.
*refers to the author and the reader
Perspective and Explanation:
A wave can be imagined as a sine curve travelling alone a straight line.
Mathematically this can be very basically expressed as:
y = A.Sin(2πft)............[f: frequency, t: time, A: amplitude]
But this formula form does not take account of the displacement along the wave's path of travel. *We need another dimension to fully describe the wave, and this is the distance along the path of travel, x. As it is, the formula only applies to arbitrary points along the line of travel of the wave, assuming the wave is continuous.
For any wave, v = fλ.....[v: wave velocity, λ: wavelength]
==> λ = v/f
To include wavelength along the path of travel, we need to convert x into an angle because we are dealing with a sine function.
In x there are x/λ complete wavelengths which translate to an angle of 2πx/λ.
Substituting for λ, we get:
2πx/(v/f) = 2πfx/v
Therefore, the angle to be passed to the sine function will be:
2πft + 2πfx/v = 2πf(t + x/v)
If we also want to account for a phase shift at t = 0, we can add a constant, Φ, to this expression.
Φ + 2πf(t + x/v)
The wave equation becomes:
y = A.Sine[Φ + 2πf(t + x/v)]
The Differential Wave Equation:
The Quantum Mechanical Schroendinger wave equation is a differential equation that has its remote roots based on the differential equation we are going to derive.
Let us start our journey with the wave equation derived just above.
Our little equation is:
y = A.Sine[Φ + 2πf(t + x/v)]
Let us now use Differentiation. We will continue to derive up to the second order differentials both with respect to time and x.
We will do it here:
Differentiating with respect to time:
dy/dt = 2πfA.Cosine[Φ + 2πf(t + x/v)].....[1st order]
d^2y/dt^2 = -(2πf)^2.A.Sine[Φ + 2πf(t + x/v)] ...[2nd order]
Differentiating with respect to x:
dy/dx = (2πf/v).A.cosine[Φ + 2πf(t + x/v)].....[1st order]
d^2y/dx^2 = -(2πf/v)^2.A.Sine[Φ + 2πf(t + x/v)]....[2nd order]
The wave equation is expressed in terms of the second order differentials.
Inspecting both second order differential equations, we immediately notive there is a Sine common term with exactly the same expression in brackets. We will make this subject first, and the aim, is to have two differential equations expressing the same thing. The latter will provide us the ingredients for both sides of the differential wave equation.
So, making the Sine(...) subject we get:
For the first one:
Sine[Φ + 2πf(t + x/v)] = -[d^2y/dt^2]/[(2πf)^2.A]
And, for the second one:
Sine[Φ + 2πf(t + x/v)] = -[d^2y/dx^2]/[(2πf/v)^2.A]
The differential wave equation is formed by the right hand side expression of the latest two equations.
Therefore:
-[d^2y/dt^2]/[(2πf)^2.A] = -[d^2y/dx^2]/[(2πf/v)^2.A]
Simplying by removing redundant terms:
d^2y/dt^2 = [d^2y/dx^2]/[(1/v)^2]
Further simplifying:
d^2y/dt^2 = v^2.[d^2y/dx^2]
The latter is the wave equation for a single dimension.
*refers to the author and the reader
Mileage can vary.
I think it is highly speculative; Based on some theories for which there exists no experimental verification.
In simple words: Wild guesses depending of favorite unprovable theories.
I think, we cannot know any better.
I think it is highly speculative; Based on some theories for which there exists no experimental verification.
In simple words: Wild guesses depending of favorite unprovable theories.
I think, we cannot know any better.
Actually, I am quite comfortable with that which is unknown.
So was John Horton Conway, who was the Dick Feynman of mathematics in our current age:
Life, Death and the Monster (John Conway) - Numberphile - YouTube
I have a wild theory that "God" or whoever is currently politically correct has set us a Universe that is endlessly interesting and unsolvable.
Me, I am a geometer. We can solve the problem of the Dirac anti-electron in 4 lines with Matrices on the back of a beermat. You may be an algebra person. In which case it takes you a LOT more lines. 😱
Call me lazy. But lazy and simple has something going for it.
So was John Horton Conway, who was the Dick Feynman of mathematics in our current age:
Life, Death and the Monster (John Conway) - Numberphile - YouTube
I have a wild theory that "God" or whoever is currently politically correct has set us a Universe that is endlessly interesting and unsolvable.
Me, I am a geometer. We can solve the problem of the Dirac anti-electron in 4 lines with Matrices on the back of a beermat. You may be an algebra person. In which case it takes you a LOT more lines. 😱
Call me lazy. But lazy and simple has something going for it.
Gödel was one of the few who could comprehensively penetrate the abstruse formal axiomatic system. Mathematical theorems cannot be proved.
I could spend a few hours in a bar with Leonard Euler, Isaac Newton, Karl Friederik Gauss, (who is way beyond my pay grade as it goes) even Sophie Germain or David Hilbert.
Mathematical abstraction fascinates me.
But did you enjoy John Horton?
Life, Death and the Monster (John Conway) - Numberphile - YouTube
He wasn't rubbish, was he? 😎
Mathematical abstraction fascinates me.
But did you enjoy John Horton?
Life, Death and the Monster (John Conway) - Numberphile - YouTube
He wasn't rubbish, was he? 😎
Sorry. Missed that one. Godel didn't prove that nothing is provable. Just that certain things are neither provable nor disprovable. Fermat's Last Theorem was considered to fall into that category. But Thanks to Andrew Wiles, and possibly theories of Elliptic Curves, we live in the Happy Uplands where these ideas now extend into Trancendental numbers.
I rest my case. 🙂
It`s rather people like Aristotle, Hegel, Kant, Frege, Russell, Gödel, Tarski. Well, maybe nothing for you.
Mathematicians do not want to hear about Gödel’s Incompleteness Theorems or Tarski’s Undefinability Theorem, saying that mathematical theorems cannot be proved. Proof is everything. Proof is the truth. An unproven mathematical theorem is virtually worthless.
Mathematicians do not want to hear about Gödel’s Incompleteness Theorems or Tarski’s Undefinability Theorem, saying that mathematical theorems cannot be proved. Proof is everything. Proof is the truth. An unproven mathematical theorem is virtually worthless.
Mate, I think you are taking this far more seriously than it warrants. 😕
A Mathematical Englishman, a Welshman and an Irish man travel to Scotland, 🙂
They look out of the train window and see a field of Brown cows.
The Irishman concludes that all Cows in Scotland are Brown.
The Welshman concludes that all the cows he has seen so far are Brown.
The English Mathematician concludes that there exist Cows in Scotland, of which one side is Brown.
It's a hard discipline. 😀
A Mathematical Englishman, a Welshman and an Irish man travel to Scotland, 🙂
They look out of the train window and see a field of Brown cows.
The Irishman concludes that all Cows in Scotland are Brown.
The Welshman concludes that all the cows he has seen so far are Brown.
The English Mathematician concludes that there exist Cows in Scotland, of which one side is Brown.
It's a hard discipline. 😀
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