19 December, 2024 in expository, math.MG | Tags: cosmic distance ladder, quaternions, spherical trigonometry | by Terence Tao | 21 comments Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and...
19 December, 2024 in expository, math.MG | Tags: cosmic distance ladder, quaternions, spherical trigonometry | by Terence Tao | 24 comments Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and...
A torsor is, in informal terms, an algebraic structure that’s like a group with the identity element “forgotten”. The precise definition (whose connection to the informal notion is not meant to be obvious!) is: a torsor is a set equipped with a group acting on such that for any two...
>>> a, b = 6, 9 # Typical unpacking >>> a, b (6, 9) >>> (a, b = 16, 19) # Oops File "<stdin>", line 1 (a, b = 16, 19) ^ SyntaxError: invalid syntax >>> (a, b := 16, 19) # This prints out a weird 3-tuple (6, 16, 19) >>> a # a is still ...
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>>> (a := 6, 9) == ((a := 6), 9) True >>> x = (a := 696, 9) >>> x (696, 9) >>> x[0] is a # Both reference same memory location True Similarly, (a, b := 16, 19) is equivalent to (a, (b := 16), 19) which is nothing but a 3-tuple.▶...
Now with a metric whose signature is not {1,1,...}, but say {-1,1,...}, we would have to write it as g_{pq} = \eta^{st} y_{s,p} y_{t,q} where...
Proof: We may assume is non-empty, as the claim is trivial otherwise. For any , define the quantity and let be a subset of that maximizes . Since and when , we see that . If the are distinct and , then we also have when , thus in this case we have . Let be the set (3...
Again, the example of (4) shows that a cluster of two solutions is certainly possible; the convexity argument only kicks in once one has a cluster of three or more solutions. To finish the proof of Theorem 1, one has to show that any two solutions to (1) in the region of interest...
Similarly, (a, b := 16, 19) is equivalent to (a, (b := 16), 19) which is nothing but a 3-tuple.▶ Strings can be tricky sometimes1.>>> a = "some_string" >>> id(a) 140420665652016 >>> id("some" + "_" + "string") # Notice that both the ids are same. 140420665652...