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xorshammer.com | ||
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thehousecarpenter.wordpress.com
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| | | | | Given two integers $latex n$ and $latex k$, where $latex k$ is non-negative, an ordered $latex k$-partition of $latex n$ is a $latex k$-tuple $latex (m_1, \dotsc, m_k)$ of positive integers such that $latex \displaystyle m_1 + \dotsb + m_k = n. $ For example, the ordered 3-partitions of 5 are $latex (1, 1, 3)$,... | |
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thatsmaths.com
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| | | | | The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that... | |
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algorithmsoup.wordpress.com
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| | | | | The ``probabilistic method'' is the art of applying probabilistic thinking to non-probabilistic problems. Applications of the probabilistic method often feel like magic. Here is my favorite example: Theorem (Erdös, 1965). Call a set $latex {X}&fg=000000$ sum-free if for all $latex {a, b \in X}&fg=000000$, we have $latex {a + b \not\in X}&fg=000000$. For any finite... | |
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profmattstrassler.com
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| | | One of the most prominent theoretical physicists of our time, Professor Joe Polchinski of the University of Santa Barbara, who has made lasting contributions to our understanding of quantum field theory, of gravity, and of string theory, gave a couple of talks at the Institute for Advanced Study in Princeton this week. The two presentations... | ||
