We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T , and diverges at a critical value of N/T , that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T , expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T , and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or nonexistent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.

Abstract. For a finite group G let Γ(G) denote the graph defined on the nonidentity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph Γ(G) contains a Hamiltonian cycle for many finite groups G.

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In a projective plane PG(2, K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614-1624, 2004, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672-688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa's 3-nets (Adv. Geom. 10:287-310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples.If p is larger than the order of the group, the above classification holds in characteristic p > 0 apart from three possible exceptions Alt 4 , Sym 4 , and Alt 5 .Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in

Prediction on a numeric scale, i.e., regression, is one of the most prominent machine learning tasks with various applications in finance, medicine, social and natural sciences. Due to its simplicity, theoretical performance guarantees and successful real-world applications, one of the most popular regression techniques is the k nearest neighbor regression. However, k nearest neighbor approaches are affected by the presence of bad hubs, a recently observed phenomenon according to which some of the instances are similar to surprisingly many other instances and have a detrimental effect on the overall prediction performance. This paper is the first to study bad hubs in context of regression. We propose hubness-aware nearest neighbor regression schemes. We evaluate our approaches on publicly available real-world datasets from various domains. Our results show that the proposed approaches outperform various other regressions schemes such as kNN regression, regression trees and neural networks. We also evaluate the proposed approaches in the presence of label noise because tolerance to noise is one of the most relevant aspects from the point of view of real-world applications. In particular, we perform experiments under the assumption of conventional Gaussian label noise and an adapted version of the recently proposed hubness-proportional random label noise.

Matthews and Michel \cite{Michel} investigated the minimum distances of certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field $\mathbb F_{q^2}(\HC)$. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree $3$ place of $\mathbb F_{q^2}(\HC)$, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian $1$-point codes, as well as with estimates due to Xing and Chen \cite{XC}

Abstract. In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian 2-group by S 5 . The construction uses the description of the structure of such loops given by M. Aschbacher (2005). These results answer some questions of M. Aschbacher.

We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the 15 groups of order 81, there are 5 nonassociative Moufang loops of order 81, 2 of which are commutative.