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Data Scientist @ Explorer Media


1d
revised Terence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.
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1d
answered Terence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.
2d
asked Limsup question $ \sum_{n=1}^N \sum_{m=1}^N x_{m-n} \leq N^2 \bigg( \limsup_{n \to \infty} x_n + o(1) \bigg) $
2d
answered Is there a relationship between curl and area?
Jan
27
revised How to find the inverse arc in the configuration space
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Jan
27
revised How to find the inverse arc in the configuration space
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Jan
27
answered How to find the inverse arc in the configuration space
Jan
26
revised Is there a generalization of the Lagrange polynomial to 3D?
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Jan
26
answered Is there a generalization of the Lagrange polynomial to 3D?
Jan
26
comment Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$
this is a $5120$ dimensional representation!
Jan
26
revised How can one find intermediate digits of a root of an algebraic equation?
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Jan
26
revised How can one find intermediate digits of a root of an algebraic equation?
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Jan
26
comment How can one find intermediate digits of a root of an algebraic equation?
@YvesDaoust oops! Numerical algorithms for finding roots of polynomials... to high precision can be very difficult. It's unlikely we can get $10^9$ decimal places even with supercomputer, unless we have special information. $$ . $$ Historical note: These algorithms come from a time when no computers were available. Halley's algorithm above is named from Sir Edmund Halley with the comet named after him.
Jan
26
comment How can one find intermediate digits of a root of an algebraic equation?
@YvesDaoust As this equation is algebraic yes we do. However, numerically finding roots of algorithms is very difficult. I believe my solution is closest to Newton's method which solves $f(x) = 0$ with the recursion: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ these have known convergence issues, and we might try to use something more sophisticated like Halley's method or Jenkins-Traub.
Jan
23
comment How can one find intermediate digits of a root of an algebraic equation?
@YvesDaoust that's the beautify of it. using the equation itself as information we deduce more and more continued fraction digits of our number. see Enrico Bombieri's continued fractions and algebraic numbers. our number is definitely algebric, e.g. let $x = z^{12345 \times 3456}$.
Jan
23
revised How can one find intermediate digits of a root of an algebraic equation?
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Jan
23
revised How can one find intermediate digits of a root of an algebraic equation?
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Jan
23
comment How can one find intermediate digits of a root of an algebraic equation?
@YvesDaoust I think continued fractions might be more appropriate, given his equation and their good approximation properties.
Jan
23
revised How can one find intermediate digits of a root of an algebraic equation?
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Jan
23
answered How can one find intermediate digits of a root of an algebraic equation?