Analysis Fact
@AnalysisFact
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Daily tweets about real and complex analysis and related topics. From @JohnDCook .
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'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
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"All mathematics is divided into three parts: cryptography, hydrodynamics, and celestial mechanics."
1/4
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''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
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''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
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870
Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
12
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840
All continuous functions f : R → (0, ∞) satisfying
f(x + y) = f(x)f(y)
are of the form f(x) = a^x.
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
9
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800
''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
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The dot product of two vectors equals the product of the lengths of the two vectors and the cosine of the angle between them.
This is a theorem in 2D, and the definition of θ in higher dimensions.
🧵 (1/3)
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
9
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Absolutely correct, but a gross violation of convention:
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'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
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626
Apprentice: Everything is linear.
Journeyman: Nothing is linear.
Master: A lot of things are locally approximately linear.
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632
The Hamiltonian is the Legendre transform of the Lagrangian.
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584
It's easier to understand the δ-ϵ definition of continuity by inserting a phrase that isn't logically necessary.
Instead of saying "for every ϵ > 0, …" it's easier to understand "for every ϵ > 0, no matter how small, …"
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581
''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
10
88
555
“Half of the plane ℝ ² is to the left of the vertical line x = 0 and half is to the right.”
“OK, sure.”
“Half the plane is to the left of x = 42 and half is to the right of x = 57.”
“Wait, what?!”
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50
549
'Mathematics is the science that uses easy words for hard ideas.' -- Edward Kasner
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'Mathematics is the science that uses easy words for hard ideas.' -- Edward Kasner
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447
'Mathematics is an experimental science, and definitions do not come first, but later on.' -- Oliver Heaviside
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442
Apply the Fourier transform 4 times and you end up back where you started.
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
5
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434
In one sense coordinate systems are simple. But as with almost anything, you can go deeper.
This is a 1365 page book, and specific coordinate systems begin on page 985.
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Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
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430
Gauss proved the Fundamental Theorem of Algebra in his doctoral dissertation in 1799.
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"The most painful thing about mathematics is how far away you are from being able to use it after you have learned it." -- J. R. Newman
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419
Gauss proved the Fundamental Theorem of Algebra in his doctoral dissertation in 1799.
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Let f be a convex function on [a, b]. Then Hermite's inequality says f at the average of a and b is bounded by the average of f over the interval [a, b], which is bounded by the average of f over the pair of points {a, b}.
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422
The derivative of an odd function is an even function.
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'Mathematics is an experimental science, and definitions do not come first, but later on.' -- Oliver Heaviside
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59
391
Gabriel's horn: a surface with finite volume but infinite surface area.
If it were a can of paint, it couldn't hold enough paint to paint itself!
This post generalizes the paradox, then resolves it.
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'Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.' -- Joseph Fourier
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379
Every continuous function of many variables is a finite composition of continuous functions of two variables. -- Kolmogorov
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36
379
'Mathematics is an experimental science, and definitions do not come first, but later on.' -- Oliver Heaviside
4
56
383
Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
5
173
375
Bolzano-Weierstrass: Every bounded sequence of real numbers has a convergent subsequence.
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The cross product 𝗮 × 𝗯 of two vectors, 𝗮 and 𝗯, is perpendicular to both.
The length of the cross product equals the area spanned a parallelogram whose sides are 𝗮 and 𝗯.
This area is minimized when θ = 0 and maximized when θ = π/2.
5
51
364
Gauss proved the Fundamental Theorem of Algebra in his doctoral dissertation in 1799.
5
25
368
Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.
6
269
360
'Young men should prove theorems, old men should write books.' -- G. H. Hardy
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359
Gödel (1940): The continuum hypothesis is not false. Cohen (1964): The continuum hypothesis is not true. (It's undecidable.)
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'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
4
52
354
'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
6
60
357
''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
7
130
342
'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
12
102
340
Gauss proved the Fundamental Theorem of Algebra in his doctoral dissertation in 1799.
7
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336
The Riemann zeta function has an infinite sum and an infinite product.
Which is a better approximation: N terms of the sum or N terms of the product?
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335
Why not just define anything divided by zero to be ∞?
You could, but then the derivative of every function would be ∞.
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''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
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101
336
The function exp(1/(x^2 - 1)) goes to 0 quickly and smoothly as x goes to 1, if you're looking at just the real axis.
In the complex plane, the function is going nuts at 1. (Picard's theorem)
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The sum of 1/n over positive integers diverges.
But if you remove from the sum those values of n containing a given digit, say 7, then the sum converges.
The analogous theorem holds in any base.
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Stoke's theorem: The integral of curl F over a surface equals the integral of F around the boundary of that surface.
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Young's inequality for strictly increasing, continuous functions f.
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