Related papers: But some are more equal than others
In the late 1980s, Sam Edwards proposed a possible statistical-mechanical framework to describe the properties of disordered granular materials. A key assumption underlying the theory was that all jammed packings are equally likely. In the…
To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. When the third named author was four years old he asked his father, the fifth named…
The BK inequality (\cite{BK85}) says that,for product measures on $\{0,1\}^n$, the probability that two increasing events $A$ and $B$ `occur disjointly' is at most the product of the two individual probabilities. The conjecture in…
If the human race comes to an end relatively shortly, then we have been born at a fairly typical time in history of humanity. On the other hand, if humanity lasts for much longer and trillions of people eventually exist, then we have been…
Apportionment refers to the well-studied problem of allocating legislative seats among parties or groups with different entitlements. We present a multi-level generalization of apportionment where the groups form a hierarchical structure,…
In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct…
A cycle C={v_1,v_2,....,v_1} in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from v_{i+1} to v_i. In this short paper, we show that for every…
According to the media, in spring of this year the experiment CDF at Fermilab has made most likely ("this result has a 99.7 percent chance of being correct", Discovery News) a great discovery ("the most significant in physics in half a…
Recent research highlighted the scaling property of human and animal mobility. An interesting issue is that the exponents of scaling law for animals and humans in different situations are quite different. This paper proposes a general…
A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. Expectation (resp. fractional expectation) threshold $q$ (resp.…
We prove that the conjecture made by Peter Frankl in the late 1970s is true. In other words for every finite union-closed family which contains a non?empty set, there is an element that belongs to at least half of its m
An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same…
A 1971 conjecture of Graham (later repeated by Erd\H{o}s and Graham) asserts that every set $A \subseteq \mathbb{F}_p \setminus \{0\}$ has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size $|A|…
Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and…
Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…
A traditional "Farmer Ted" calculus problem is to minimize the perimeter of a rectangular chicken coop given the area N, so that as little as possible will be spent on the fencing. But what if N is an integer, and we are only allowed to…
The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erd\H{o}s, Alfr\'{e}d R\'{e}nyi, and Vera…
One of the greatest contributors of the 20th century among all academician in the field of statistical finance, M. F. M. Osborne published in 1956 [6] an essential paper and proposed to treat the question of stock market motion through the…
In 1996, my brilliant student John Noonan, discovered, and proved that there are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one is such…
The best-of-both-worlds paradigm advocates an approach that achieves desirable properties both ex-ante and ex-post. We launch a best-of-both-worlds fairness perspective for the important social choice setting of approval-based committee…