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 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
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Author Archives: Charles Li
The 100th post
This is the 100th post of this blog. As a very senior (translated as old) blogger here, I take the liberty of posting the 100th post. The honor should have gone to other people, most notably KKK, who wrote almost … Continue reading
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Uniform distribution of polynomials mod 1
Last week Lam Wai Kit posted an excellent introductory article of ergodic theory and its applications to number theory. As stated in the note, one consequence of Birhoff recurrence theorem is that for any nonconstant polynomial , the fractional part … Continue reading
Posted in Number Theory
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Fourier Analysis and Number Theory III: The Poisson summation formula as a trace formula
Yin and Yang are important concept in the Chinese philosophy. They usually describe two opposite things which are not only complement of each other but also have deep interrelationship. In mathematics, we called them duality. For example (1) primes vs … Continue reading
Posted in Fourier analysis, Number Theory
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Fourier Analysis and Number Theory II: The Poisson Summation Formula and The Functional Equation of The RiemannZeta Function
Riemann’s tenpagelong paper “Über die Anzahl der Primzahen unter einer gegebener Gröβe” has great influence on modern number theory. In the paper, he established two important properties of the Riemannzeta function (the summation is absolutely convergent for ).
Posted in Fourier analysis, Number Theory
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Fourier Analysis and Number theory I: The selfdual function e^{\pi x^2}
During a lecture at UCLA, Serge Lang asked what is the most important function in mathematics. This question is quite personal and every person certainly has his own opinion. In fact, a professor spoke out loud that the constant function … Continue reading
Posted in Fourier analysis, Number Theory
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