2,115 research outputs found
The Problem of Motion: The Statistical Mechanics of Zitterbewegung
Around 1930, both Gregory Breit and Erwin Schroedinger showed that the
eigenvalues of the velocity of a particle described by wavepacket solutions to
the Dirac equation are simply c, the speed of light. This led Schroedinger
to coin the term Zitterbewegung, which is German for "trembling motion", where
all particles of matter (fermions) zig-zag back-and-forth at only the speed of
light. The result is that any finite speed less than , including the state
of rest, only makes sense as a long-term average that can be thought of as a
drift velocity. In this paper, we seriously consider this idea that the
observed velocities of particles are time-averages of motion at the speed of
light and demonstrate how the relativistic velocity addition rule in one
spatial dimension is readily derived by considering the probabilities that a
particle is observed to move either to the left or to the right at the speed of
light.Comment: Knuth K.H. 2014. The problem of motion: the statistical mechanics of
Zitterbewegung. Bayesian Inference and Maximum Entropy Methods in Science and
Engineering, Amboise, France, Sept 2014, AIP Conference Proceedings, American
Institute of Physics, Melville N
A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings
Stable matching is a classical combinatorial problem that has been the
subject of intense theoretical and empirical study since its introduction in
1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new
upper bound on , the maximum number of stable matchings that a stable
matching instance with men and women can have. It has been a
long-standing open problem to understand the asymptotic behavior of as
, first posed by Donald Knuth in the 1970s. Until now the best
lower bound was approximately , and the best upper bound was . In this paper, we show that for all , for some
universal constant . This matches the lower bound up to the base of the
exponent. Our proof is based on a reduction to counting the number of downsets
of a family of posets that we call "mixing". The latter might be of independent
interest
Matching under Preferences
Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory.
Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs.
Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process.
Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully
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