Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields

Relax and Localize: From Value to Algorithms

We show a principled way of deriving online learning algorithms from a minimax analysis. Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones. Our framework also captures such "unorthodox" methods as Follow the Perturbed Leader and the R^2 forecaster. We emphasize that understanding the inherent complexity of the learning problem leads to the development of algorithms. We define local sequential Rademacher complexities and associated algorithms that allow us to obtain faster rates in online learning, similarly to statistical learning theory. Based on these localized complexities we build a general adaptive method that can take advantage of the suboptimality of the observed sequence. We present a number of new algorithms, including a family of randomized methods that use the idea of a "random playout". Several new versions of the Follow-the-Perturbed-Leader algorithms are presented, as well as methods based on the Littlestone's dimension, efficient methods for matrix completion with trace norm, and algorithms for the problems of transductive learning and prediction with static experts

On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot\sqrt[2k]{2}^n$ and $c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c'\cdot1.246^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.Comment: 13 page

On the equivalence of linear sets

Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $\Sigma^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $\Sigma_1,\, \Sigma_2 \subset \Sigma^*$ such that there is a $(t-3)$-space $\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $\Sigma_i$ from center $\Gamma$ and there exists no collineation $\phi$ of $\Sigma^*$ such that $\Gamma^{\phi}=\Gamma$ and $\Sigma_1^{\phi}=\Sigma_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented. The final version is to appear in Designs, Codes and Cryptograph

Synthesis equivalence of triples

This working paper describes a framework for compositional supervisor synthesis, which is applicable to all discrete event systems modelled as a set of deterministic automata. Compositional synthesis exploits the modular structure of the input model, and therefore works best for models consisting of a large number of small automata. State-space explosion is mitigated by the use of abstraction to simplify individual components, and the property of synthesis equivalence guarantees that the final synthesis result is the same as it would have been for the non-abstracted model. The working paper describes synthesis equivalent abstractions and shows their use in an algorithm to compute supervisors efficiently. The algorithm has been implemented in the DES software tool Supremica and successfully computes modular supervisors, even for systems with more than 1014 reachable states, in less than 30 seconds