Permutations destroying arithmetic progressions in finite cyclic groups

A permutation \pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).Comment: 11 pages, no figure

The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments

In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.Comment: 26 pages, 2 figure

A Unified Analysis of Stochastic Optimization Methods Using Jump System Theory and Quadratic Constraints

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).Comment: To Appear in Proceedings of the Annual Conference on Learning Theory (COLT) 201

A variant of the multi-agent rendezvous problem

The classical multi-agent rendezvous problem asks for a deterministic algorithm by which $n$ points scattered in a plane can move about at constant speed and merge at a single point, assuming each point can use only the locations of the others it sees when making decisions and that the visibility graph as a whole is connected. In time complexity analyses of such algorithms, only the number of rounds of computation required are usually considered, not the amount of computation done per round. In this paper, we consider $\Omega(n^2 \log n)$ points distributed independently and uniformly at random in a disc of radius $n$ and, assuming each point can not only see but also, in principle, communicate with others within unit distance, seek a randomised merging algorithm which asymptotically almost surely (a.a.s.) runs in time O(n), in other words in time linear in the radius of the disc rather than in the number of points. Under a precise set of assumptions concerning the communication capabilities of neighboring points, we describe an algorithm which a.a.s. runs in time O(n) provided the number of points is $o(n^3)$. Several questions are posed for future work.Comment: 18 pages, 3 figures. None of the authors has any previous experience in this area of research (multi-agent systems), hence we welcome any feedback from specialist

The Hegselmann-Krause dynamics on the circle converge

We consider the Hegselmann-Krause dynamics on a one-dimensional torus and provide the first proof of convergence of this system. The proof requires only fairly minor modifications of existing methods for proving convergence in Euclidean space.Comment: 9 pages, 2 figures. Version 2: A small error in the proof of Theorem 1.1 is corrected and an acknowledgement added. Bibliography update

What active labor market policy works in a recession?

This paper discusses the case for expanding active labor market policy in recession. We find that there is reasonable case for relying more heavily on certain kinds of programs. The argument is tied to the varying size of the lock-in effect in boom and recession. If programs with relatively large lock-in effects should ever be used, they should be used in a downturn. The reason is simply that the cost of forgoing search time is lower in recession. We also provide new evidence on the relative effectiveness of different kinds of programs over the business cycle. In particular we compare an on-the-job training scheme with (traditional) labor market training. We find that labor market training is relatively more effective in recession. This result is consistent with our priors since labor market training features relative large lock-in effects.Active labor market policy; business cycle; unemployment

The Be X-ray Binary Outburst Zoo

Be X-ray binaries are among the best known transient high-energy sources. Their outbursts are commonly classified into a simple scheme of 'normal' and 'giant' outbursts, but a closer look shows that actual outbursts do not always follow this simple scheme. Recent data show a variety of properties, like pre-flares, shifts of the outburst peaks with respect to the periastron, multi-peaked outbursts etc. We present results from a systematic study of a large number of outbursts monitored by various space missions, comparing outburst properties and their relation to system parameters and current theoretical understanding.Comment: Proceedings of "An INTEGRAL view of the high-energy sky (the first 10 years)" the 9th INTEGRAL Workshop, October 15-19, 2012, Paris, France, in Proceedings of Science (INTEGRAL 2012), Eds. A. Goldwurm, F. Lebrun and C. Winkler, (http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=176), id PoS(INTEGRAL 2012)01

Building Global Knowledge Pipelines The Role of Temporary Clusters

Business people and professionals come together regularly at trade fairs, exhibitions, conventions, congresses, and conferences. Here, their latest and most advanced findings, inventions and products are on display to be evaluated by customers and suppliers, as well as by peers and competitors. Participation in events like these helps firms to identify the current market frontier, take stock of relative competitive positions and form future plans. Such events exhibit many of the characteristics ascribed to permanent spatial clusters, albeit in a temporary and intensified form. These short-lived hotspots of intense knowledge exchange, network building and idea generation can thus be seen as temporary clusters. The present paper compares temporary clusters with permanent clusters and other types of inter-firm interactions. If regular participation in temporary clusters can satisfy a firm’s need to learn through interaction with suppliers, customers, peers and rivals, why is the phenomenon of permanent spatial clustering of similar and related economic activity so pervasive? The answer, it is claimed, lies in the restrictions imposed upon economic activity when knowledge and ideas are transformed into valuable products and services. The paper sheds new light on how interaction among firms in current clusters coincides with knowledge-intensive pipelines between firms in different regions or clusters. In doing so, it offers a novel way of understanding how interfirm knowledge relationships are organized spatially and temporally.Economic geography, knowledge, clusters, temporary clusters, trade fairs, conventions, pipelines

What active labor market policy works in a recession?

This paper discusses the case for expanding active labor market policy in recession. We find that there is reasonable case for relying more heavily on certain kinds of programs. The argument is tied to the varying size of the lock-in effect in boom and recession. If programs with relatively large lock-in effects should ever be used, they should be used in a downturn. The reason is simply that the cost of forgoing search time is lower in recession. We also provide new evidence on the relative effectiveness of different kinds of programs over the business cycle. In particular we compare an on-the-job training scheme with (traditional) labor market training. We find that labor market training is relatively more effective in recession. This result is consistent with our priors since labor market training features relative large lock-in effects.Active labor market policy; business cycle; unemployment

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node $\mathbf {v}^0$ to the all-ones node $\mathbf {v}^1$ tends respectively to 0, 1 and 1, as $n$ tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of $\mathbf {v}^0$ is set to some $\alpha=\alpha_n\in[0,1]$. We prove that there is a very sharp threshold at $\alpha_n=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$. As a corollary we prove significant concentration, for $\alpha$ below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$ existing tends to $1$ provided the drift parameter $\theta=\theta_n$ satisfies $n\theta_n\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.Comment: Published in at http://dx.doi.org/10.1214/13-AAP949 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org