Classifying elements of the Brauer group of a variety X over a p-adic field by the p-adic accuracy needed to evaluate them gives a filtration on Br X. We relate this filtration to that defined by... Show moreClassifying elements of the Brauer group of a variety X over a p-adic field by the p-adic accuracy needed to evaluate them gives a filtration on Br X. We relate this filtration to that defined by Kato’s Swan conductor. The refined Swan conductor controls how the evaluation maps vary on p-adic discs: this provides a geometric characterisation of the refined Swan conductor. We give applications to rational points on varieties over number fields, including failure of weak approximation for varieties admitting a non-zero global 2-form. Show less
In analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to... Show moreIn analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to make mistakes, i.e., to accept false claims or reject true claims. Further, probabilistic proofs may have multiple rounds of interaction between the prover and the verifier, in which case they are also referred to as interactive proofs. These two relaxations revolutionized the theory of proofs. For instance, by trading absolute certainty for high probability and allowing interaction, it is possible to prove claims without revealing anything beyond their correctness, i.e., in zero-knowledge. Nowadays, zero-knowledge proofs are widely deployed; they are for instance essential in the public-key infrastructures (PKIs) that manage digital identities and secure communication channels on the internet. Especially the theory of Σ-protocols provides a well-understood basis for the modular design of zero-knowledge proof systems in a wide variety of application domains. However, recently a new folding mechanism was introduced as a drop-in replacement for Σ-protocols, significantly reducing the communication costs in many practical scenarios. In this dissertation, we show that the folding mechanism can be cast as a significant strengthening, rather than a replacement, of Σ-protocol theory, thereby reconciling it with the established theory. In addition, we close several gaps in the theory of probabilistic proofs that were exposed due to the introduction of these efficiency improvements. Show less
Combinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well... Show moreCombinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well-known examples of this class of games, as well as game of go. Though these games are by no means simple, there does exist a beautiful mathematical framework for their analysis. Using this theory, it is possible to efficiently determine the outcome of a given position of a game without having to explicitly compute the results for every possible move. Moreover, the theory provides a measure of how profitable a given position is to either player, often denoted by the `value’ of a position. An example application of the theory is research on endgames in go.However, not all games are combinatorial games. The game of poker, for example, introduces hidden information. In practice, impressive results have been obtained for these non-combinatorial games using artificial intelligence, but theory and understanding are perhaps lacking. In this thesis, the main question we address is whether the existing theory for combinatorial games can be adapted or extended to non-combinatorial ones. Show less
Grafieken zijn heel geschikt om veel en complexe informatie weer te geven: ze bieden in één oogopslag overzicht en duidelijkheid. Bovendien hebben ze een objectieve uitstraling waardoor ze het vaak... Show moreGrafieken zijn heel geschikt om veel en complexe informatie weer te geven: ze bieden in één oogopslag overzicht en duidelijkheid. Bovendien hebben ze een objectieve uitstraling waardoor ze het vaak goed doen als argument bij een bewering: "De cijfers liegen niet." Behalve als ze dat wel doen, en dat komt helaas maar al te vaak voor. Daarom hier een lesje in het doorprikken van nepnieuws met misleidende grafieken. Show less
After collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector... Show moreAfter collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector lattices and Banach lattices have direct limits, and thatthese coincide with direct limits of the systems in naturally associated other cate-gories. For those categories where the general constructions do not work to establishthe existence of general direct limits, we describe the basic structure of those directlimits that do exist. A direct system in the category of Banach lattices and contractivealmost interval preserving vector lattice homomorphisms has a direct limit. Whenthe Banach lattices in the system all have order continuous norms, then so does theBanach lattice in a direct limit. This is used to show that a Banach function space overa locally compact Hausdorff space has an order continuous norm when the topologieson all compact subsets are metrisable and (the images of) the continuous compactlysupported functions are dense. Show less
Meer, N. van der; Werf, V. van der; Brinkman, W.P.; Specht, M.M. 2023
Datavisualisatie kan cijfers in één oogopslag inzichtelijk maken, maar deze top-16 van wapenexporterende landen in NRC Handelsblad is eerder verwarrend dan verhelderend, en ook niet helemaal correct.
