mindlessselfindulge
Greycel
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Integrate ln(x + 1)/(x^2 + 1) from 0 to 1
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Integrate ln(x + 1)/(x^2 + 1) from 0 to 1
(pi/8)*ln(2)
one can use integration under the integral sign -- e.g., consider ln(ax + 1) / (1 + x^2)
Do you mean that, e.g., a_i = a_k for some i != k or that, e.g., a_i = b_k for some i & k or both?but all a_i, b_i, and c_i are not necessarily distinct
Do you mean that, e.g., a_i = a_k for some i != k or that, e.g., a_i = b_k for some i & k or both?
technically they are then multisetsThe sets A, B, C
Also, as writ, it suggests that s_1 = {a_1, b_1, c_1} and s_2 = {a_2, b_2, c_2} and {a_1, b_2, b_3} ∉ S. I take it that's not intended?Let S be a size n set of tuples s_i = {a_i, b_i, c_i}.
Substitute u = x − a, use cos(u + a) = cos(u)cos(a) − sin(u)sin(a) and the fact that the antiderivative of tan(u) is ln|sec(u)| + C. Technically we can choose a different constant of integration for every connected component of the domain of tan(u). Lastly we should rewrite the result in terms of x of course.Integrate ∫(cos x)/cos(x-a) dx
technically they are then multisets
Also, as writ, it suggests that s_1 = {a_1, b_1, c_1} and s_2 = {a_2, b_2, c_2} and {a_1, b_2, b_3} ∉ S. I take it that's not intended?
x-a =tIntegrate ∫(cos x)/cos(x-a) dx
better would be to say that n_a is the cardinality of image of S under the projection sending (a, b, c) to (b, c) etc., but I get what you mean nown_a is the number of distinct pairs {b_i, c_i} found in S
Not immediately seeing how to prove this. Got a hint?Prove that n^2 <= (n_a * n_b * n_c).
using the subadditivity of the nth root this is trivialGiven nonnegative reals a_1, ... , a_n and b_1, ... , b_n prove:
GM(a_1, ... , a_n) + GM(b_1, ... , b_n) <= GM(a_1 + b_1, ... , a_n + b_n)
GM = geometric mean
using the subadditivity of the nth root this is trivial
I confused subadditivity with superadditivity so this is bogus. My badusing the subadditivity of the nth root this is trivial
Use gram-schimdt processfind a set of orthonormal vectors from the independent vectors a_{1}=(1,-1,0) , a_{2}=(0,1,-1) and a_{3}=(1,0,-1) How many non-zero orthonormal vectors are obtained?
Use approximations to reduce to the (trivial) case where one of the functions is an inducator function with finite support.Prove that the convolution of any two square integrable functions on R is a continuous function vanishing at infinity.
I dunno nigga. Doubt it. Do you think you can pass through Fourier transforms to establish properties of a convolution of two square integrable functions that don't hold true for an arbitrary function in C0(R)?Bonus: Can every continuous function vanishing at infinity on R be expressed as a convolution of two square integrable functions?
4What is 2+2
I didn't know you were a mathcel.Use approximations to reduce to the (trivial) case where one of the functions is an inducator function with finite support.
bruh I was very good at this in highschool, I forgot it all nowIntegrate ln(x + 1)/(x^2 + 1) from 0 to 1
I take it you mean compact support in lieu of finite support. I'm sure something along these lines works, but I don't think this is nearly as trivial as you portray it. E.g., what breaks when I try to do this when one of the functions is integrable and the other one is essentially bounded?Use approximations to reduce to the (trivial) case where one of the functions is an inducator function with finite support.
I know you canDo you think you can pass through Fourier transforms to establish properties of a convolution of two square integrable functions that don't hold true for an arbitrary function in C0(R)?
Did they teach you double digits in school yet? Do you know what 14 + 23 is?71x187x7916931 to the power of 6 divded by 16
Integrate ln(x + 1)/(x^2 + 1) from 0 to 1
Right, the nontrivial part lies in the technicalities of how you 'approximate' your function (say f) with indicator functions (say f_n), and establishing that such an approximation exists. A pointwise approximation isn't sufficient for instance. You want the existence of a uniform approximation in the sense that the L2-norm of (f - f_n) approaches 0.I take it you mean compact support in lieu of finite support. I'm sure something along these lines works, but I don't think this is nearly as trivial as you portray it. E.g., what breaks when I try to do this when one of the functions is integrable and the other one is essentially bounded?
Indeed. You also need that f_n * g → f * g in a pretty strong sense (which necessitates that f_n → f in a pretty strong sense) to be able to deduce the vanishing at infinity of f * g from f_n * g so you could say that establishing the continuity of f ↦ f * g w.r.t. the right topologies is nontrivial as well.Right, the nontrivial part lies in the technicalities of how you 'approximate' your function (say f) with indicator functions (say f_n), and establishing that such an approximation exists. A pointwise approximation isn't sufficient for instance. You want the existence of a uniform approximation in the sense that the L2-norm of (f - f_n) approaches 0.
I meant "what part of your argument breaks when I try to apply it to the case where one is integrable and the other is essentially bounded". I think we've established the answer thereto en passant, so no need to answer it anymore.If one function is integrable and the other is bounded, well, everything breaks no? Even the 'trivial' case of where the integrable function is a compactly supported indicator function doesn't work. Just take the bounded function to be 1 everywhere.
3?find a set of orthonormal vectors from the independent vectors a_{1}=(1,-1,0) , a_{2}=(0,1,-1) and a_{3}=(1,0,-1) How many non-zero orthonormal vectors are obtained?
2. Notice that a_1 + a_2 = a_3.
Thanks for the correction, can't believe I overlooked that2. Notice that a_1 + a_2 = a_3.
PS @im done there are some problems with how you phrased the question
this is intended to be constructive feedback
- "non-zero orthonormal vectors" is tautological as orthoNORMAL vectors have a norm of 1 and therefore cannot be zero
- since a_1 + a_2 = a_3 they are only PAIRWISE independent
- the phrase "a set of orthonormal vectors from ..." is somewhat ambiguous; better would be "an orthonormal basis of the span of ..."
- your question can be far more concisely phrased as "find the dimension of the span of a_1, a_2 and a_3"; concise phraseology is generally preferred