4,584 research outputs found

    Multidisciplinary perspectives on Artificial Intelligence and the law

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    This open access book presents an interdisciplinary, multi-authored, edited collection of chapters on Artificial Intelligence (‘AI’) and the Law. AI technology has come to play a central role in the modern data economy. Through a combination of increased computing power, the growing availability of data and the advancement of algorithms, AI has now become an umbrella term for some of the most transformational technological breakthroughs of this age. The importance of AI stems from both the opportunities that it offers and the challenges that it entails. While AI applications hold the promise of economic growth and efficiency gains, they also create significant risks and uncertainty. The potential and perils of AI have thus come to dominate modern discussions of technology and ethics – and although AI was initially allowed to largely develop without guidelines or rules, few would deny that the law is set to play a fundamental role in shaping the future of AI. As the debate over AI is far from over, the need for rigorous analysis has never been greater. This book thus brings together contributors from different fields and backgrounds to explore how the law might provide answers to some of the most pressing questions raised by AI. An outcome of the Católica Research Centre for the Future of Law and its interdisciplinary working group on Law and Artificial Intelligence, it includes contributions by leading scholars in the fields of technology, ethics and the law.info:eu-repo/semantics/publishedVersio

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Slopes of modular forms and geometry of eigencurves

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    Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's ⌊k−1p+1⌋\lfloor\frac{k-1}{p+1}\rfloor-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.Comment: 97 pages; comments are welcom

    On the essential torsion finiteness of abelian varieties over torsion fields

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    The classical Mordell-Weil theorem implies that an abelian variety AA over a number field KK has only finitely many KK-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension Kcyc=KQabK^{\rm cyc}=K\mathbb{Q}^{\mathrm{ab}} by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety AA over the infinite algebraic extension KBK_B obtained by adjoining the coordinates of all torsion points of an abelian variety BB. Assuming the Mumford-Tate conjecture, and up to a finite extension of the base field KK, we give a necessary and sufficient condition for the finiteness of A(KB)torsA(K_B)_{\rm tors} in terms of Mumford--Tate groups. We give a complete answer when both abelian varieties have dimension both three, or when both have complex multiplication.Comment: 35 page

    Mirror symmetry for Dubrovin-Zhang Frobenius manifolds

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    Frobenius manifolds were formally defined by Boris Dubrovin in the early 1990s, and serve as a bridge between a priori very different fields of mathematics such as integrable systems theory, enumerative geometry, singularity theory, and mathematical physics. This thesis concerns, in particular, a specific class of Frobenius manifolds constructed on the orbit space of an extension of the affine Weyl group defined by Dubrovin together with Youjin Zhang. Here, we find Landau-Ginzburg superpotentials, or B-model mirrors, for these Frobenius structures by considering the characteristic equation for Lax operators of relativistic Toda chains as proposed by Andrea Brini. As a bonus, the results open up various applications in topology, integrable hierarchies, and Gromov-Witten theory, making interesting research questions in these areas more accessible. Some such applications are considered in this thesis. The form of the determinant of the Saito metric on discriminant strata is investigated, applications to the combinatorics of Lyashko-Looijenga maps are given, and investigations into the integrable systems theoretic and enumerative geometric applications are commenced

    Enumerating Regular Languages with Bounded Delay

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    Deformation theory of G-valued pseudocharacters and symplectic determinant laws

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    We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group, where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters. We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from small characteristics. We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously, we obtain that tensor products of determinant laws exist and are compatible with tensor products of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor product of Taylor’s pseudocharacters. We generalize some of the results of [Che14] to general reductive groups, in particular we show that the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show, that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p. One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension n(2n + 1)[F : Qp]. We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected reductive groups G come from a semisimple representation. We introduce a formal scheme and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to conjugacy and Frobenius automorphisms. At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under an involution. We prove the expected basic properties in that we show that symplectic determinant laws over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s pseudocharacters and show that it is an isomorphism over reduced rings

    Investigating the learning potential of the Second Quantum Revolution: development of an approach for secondary school students

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    In recent years we have witnessed important changes: the Second Quantum Revolution is in the spotlight of many countries, and it is creating a new generation of technologies. To unlock the potential of the Second Quantum Revolution, several countries have launched strategic plans and research programs that finance and set the pace of research and development of these new technologies (like the Quantum Flagship, the National Quantum Initiative Act and so on). The increasing pace of technological changes is also challenging science education and institutional systems, requiring them to help to prepare new generations of experts. This work is placed within physics education research and contributes to the challenge by developing an approach and a course about the Second Quantum Revolution. The aims are to promote quantum literacy and, in particular, to value from a cultural and educational perspective the Second Revolution. The dissertation is articulated in two parts. In the first, we unpack the Second Quantum Revolution from a cultural perspective and shed light on the main revolutionary aspects that are elevated to the rank of principles implemented in the design of a course for secondary school students, prospective and in-service teachers. The design process and the educational reconstruction of the activities are presented as well as the results of a pilot study conducted to investigate the impact of the approach on students' understanding and to gather feedback to refine and improve the instructional materials. The second part consists of the exploration of the Second Quantum Revolution as a context to introduce some basic concepts of quantum physics. We present the results of an implementation with secondary school students to investigate if and to what extent external representations could play any role to promote students’ understanding and acceptance of quantum physics as a personal reliable description of the world
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