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 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
 A curious identity on the median triangle
 27 lines on a smooth cubic surface
 Weighted HsiungMinkowski formulas and rigidity of umbilic hypersurfaces
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Category Archives: Probability
Why is a² + b² ≥ 2ab ?
This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading
Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability
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Two little remarks on sphere
1. “Take a sphere of radius in dimensions, large; then most points inside the sphere are in fact very close to the surface.” David Ruelle Let and . Let The fraction of the volume of to the volume of is … Continue reading
From combinatorics to entropy
Let and . Then by Stirling’s formula. This is probably wellknown to people who have studied statistical physics, but some of us including myself may not be very aware of this kind of relation. I wonder if this was the … Continue reading
Martingale Theory III: Optional stopping theorem
This is a sequel to Martingale Theory II: Conditional Expectation. Our aim here is to prove the main theorems about discrete time martingales. More advanced probability texts (e.g. those on stochastic calculus) assume that these theorems are well known to … Continue reading
Posted in Probability
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Potential theory on finite sets
Motivation Consider the following Dirichlet problem on the unit square : on where is a given function on the boundary. To get an approximate solution, we may replace the unit square by a grid. Let be large and set . … Continue reading
Posted in Potential theory, Probability
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Estimating the probability of grad school admission
We start with a naive model of the situation. Model 1. Suppose that I apply for schools , , …, . I estimate that I will be admitted to school with probability . Thus, if denotes the event that I am admitted … Continue reading
Posted in Miscellaneous, Probability
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An indicator approach to discrete probability
The purpose of this elementary post is to illustrate that much of discrete probability can be analyzed in terms of indicator functions, linearity and independence. These ideas are wellknown in probabilitly, but the following approach is seldom seen in elementary … Continue reading
Posted in Probability
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