A distributional limit law for the continued fraction digit sum

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalized linearly we determine a large deviation asymptotic.Comment: 14 pages, 1 figur

Separated at Birth: Jet Maximization, Axis Minimization, and Stable Cone Finding

Jet finding is a type of optimization problem, where hadrons from a high-energy collision event are grouped into jets based on a clustering criterion. As three interesting examples, one can form a jet cluster that (1) optimizes the overall jet four-vector, (2) optimizes the jet axis, or (3) aligns the jet axis with the jet four-vector. In this paper, we show that these three approaches to jet finding, despite being philosophically quite different, can be regarded as descendants of a mother optimization problem. For the special case of finding a single cone jet of fixed opening angle, the three approaches are genuinely identical when defined appropriately, and the result is a stable cone jet with the largest value of a quantity J. This relationship is only approximate for cone jets in the rapidity-azimuth plane, as used at the Large Hadron Collider, though the differences are mild for small radius jets.Comment: 7 pages, 2 tables; v2: references added; v3: small clarifications and table 2 added to match journal versio

Is Determinism with regard to the Spheres of Law or Nature Consistent?

Probably the most important problem for lawyers is the relationship between cases that confront them and the rules of the legal order that they have to apply. Lawyers want to know whether a certain case falls under a certain rule or, more generally, to which set of cases a certain rule has to be applied. If they are able to answer this question, then those lawyers can tell you fairly precisely what the content of a certain rule is

Energy cooperation between the EU and Switzerland Partners by destiny in search of a new model. IES Policy Paper Issue 2020/01 • January 2020

The gradual integration of EU energy policy has implications for the national energy policies of its neighbors. While members of the European Economic Area (EEA) and the Energy Community implement large parts of the EU’s energy acquis, other third countries are also affected. Switzerland—physically integrated in the European energy grid but lacking a formalized mechanism of regulatory adaptation with the EU—is an interesting case in this respect, not least because of its implications for a UK-EU relationship post-Brexit. Currently, an EU-Switzerland electricity agreement is being negotiated but its conclusion remains highly uncertain. This briefing paper highlights that either outcome—with or without an electricity agreement—has important implications for the Swiss energy transition, Swiss access to European bodies of energy policy-making, and Swiss renewable investors. Yet, even without an electricity agreement, interdependence between the Swiss and EU electricity systems will increase, creating pressure to find alternative forms of cooperation

Semi-Streaming Algorithms for Annotated Graph Streams

Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require $\Omega(n^2)$ space to solve. One of the few bright spots has been the development of semi-streaming algorithms for a handful of graph problems -- these algorithms use space $O(n\cdot\text{polylog}(n))$. In the annotated data streaming model of Chakrabarti et al., a computationally limited client wants to compute some property of a massive input, but lacks the resources to store even a small fraction of the input, and hence cannot perform the desired computation locally. The client therefore accesses a powerful but untrusted service provider, who not only performs the requested computation, but also proves that the answer is correct. We put forth the notion of semi-streaming algorithms for annotated graph streams (semi-streaming annotation schemes for short). These are protocols in which both the client's space usage and the length of the proof are $O(n \cdot \text{polylog}(n))$. We give evidence that semi-streaming annotation schemes represent a substantially more robust solution concept than does the standard semi-streaming model. On the positive side, we give semi-streaming annotation schemes for two dynamic graph problems that are intractable in the standard model: (exactly) counting triangles, and (exactly) computing maximum matchings. The former scheme answers a question of Cormode. On the negative side, we identify for the first time two natural graph problems (connectivity and bipartiteness in a certain edge update model) that can be solved in the standard semi-streaming model, but cannot be solved by annotation schemes of "sub-semi-streaming" cost. That is, these problems are just as hard in the annotations model as they are in the standard model.Comment: This update includes some additional discussion of the results proven. The result on counting triangles was previously included in an ECCC technical report by Chakrabarti et al. available at http://eccc.hpi-web.de/report/2013/180/. That report has been superseded by this manuscript, and the CCC 2015 paper "Verifiable Stream Computation and Arthur-Merlin Communication" by Chakrabarti et a