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Reduction of Kummer surfaces modulo 2 in the non-supersingular case
No full textWe obtain necessary and sufficient conditions for the good reduction ofKummer surfaces attached to abelian surfaces with non-supersingular reductionwhen the residue field is perfect of characteristic 2. In this case, goodreduction with an algebraic space model is equivalent to good reduction with ascheme model, which we explicitly construct.Comment: 31 pages, comments welcome
A categorical characterization of relative entropy on standard Borel spaces
No full textWe give a categorical treatment, in the spirit of Baez and Fritz, of relativeentropy for probability distributions defined on standard Borel spaces. Wedefine a category suitable for reasoning about statistical inference onstandard Borel spaces. We define relative entropy as a functor into Lawvere'scategory and we show convexity, lower semicontinuity and uniqueness
A Plea for Pluralism
No full textMainstream economics is one way of understanding how the economy works, but mainstream economists argue much more: that mainstream economics is the onlyway of understanding the economy. Mainstream economists should embrace pluralism for reasons suggested by John Stuart Mill: as a guard against the tyranny of the majority, a tyranny that fortifies itself against doubt not by reason but by power; even if the majority is right and the doubters wrong, engaging with doubt is a way to strengthen correct arguments; and, most likely, according to Mill, there is partial truth on the side of heterodoxy as well as on the orthodox side. The two elements of the power of mainstream economists are related: the police power over what is and what is not published in the major journals, and the role of publication in these journals in the tenure process. Pluralism is not an issue of concern to academics only. Economists of all stripes may try to construct the economy in the image of their theories, but for some time the mainstream has had the upper hand, just as it does in the academy. The push to deregulate the economy, which began in the United States during the Carter presidency, had its full flowering in the financial crisis of 2008.What will it take to allow heterodoxy into the academy? If history is any guide, innovations in economics take root when they are allied to successful political movements. One case in point is the symbiosis between Keynes’s General Theory and the New Deal and social democracy. Another is the resurgence of pre-Keynesian theory dressed up in the high-tech mathematics of New Classical theory and the coming to power of the apostles of neoliberalism in the 1980s. It’s a good bet that for a new economics to take hold in this century, it will do so in partnership, however tacit, with a new politics
A Language for Evaluating Derivatives of Functionals Using Automatic Differentiation
No full textWe present a simple functional programming language, called Dual PCF, thatimplements forward mode automatic differentiation using dual numbers in theframework of exact real number computation. The main new feature of thislanguage is the ability to evaluate correctly up to the precision specified bythe user -- in a simple and direct way -- the directional derivative offunctionals as well as first order functions. In contrast to other comparablelanguages, Dual PCF also includes the recursive operator for defining functionsand functionals. We provide a wide range of examples of Lipschitz functions andfunctionals that can be defined in Dual PCF. We use domain theory both to givea denotational semantics to the language and to prove the correctness of thenew derivative operator using logical relations. To be able to differentiatefunctionals -- including on function spaces equipped with their compact-opentopology that do not admit a norm -- we develop a domain-theoretic directionalderivative that is Scott continuous and extends Clarke's subgradient ofreal-valued locally Lipschitz maps on Banach spaces to real-valued continuousmaps on Hausdorff topological vector spaces. Finally, we show that we canexpress arbitrary computable linear functionals in Dual PCF.Comment: 19 pages, no figures, MFPS'2
Saturating automata for game semantics
No full textSaturation is a fundamental game-semantic property satisfied by strategiesthat interpret higher-order concurrent programs. It states that the strategymust be closed under certain rearrangements of moves, and corresponds to theintuition that program moves (P-moves) may depend only on moves made by theenvironment (O-moves). We propose an automata model over an infinite alphabet, called saturatingautomata, for which all accepted languages are guaranteed to satisfy a closureproperty mimicking saturation. We show how to translate the finitary fragment of Idealized Concurrent Algol(FICA) into saturating automata, confirming their suitability for modellinghigher-order concurrency. Moreover, we find that, for terms in normal form, theresultant automaton has linearly many transitions and states with respect toterm size, and can be constructed in polynomial time. This is in contrast toearlier attempts at finding automata-theoretic models of FICA, which did notguarantee saturation and involved an exponential blow-up during translation,even for normal forms.Comment: Presented at MFPS 202
Universal Address Sequence Generator for Memory Built-in Self-test
No full textThis paper presents the universal address sequence generator (UASG) formemory built-in-self-test. The studies are based on the proposed universalmethod for generating address sequences with the desired properties formultirun march memory tests. As a mathematical model, a modification of therecursive relation for quasi-random sequence generation is used. For thismodel, a structural diagram of the hardware implementation is given, of whichthe basis is a storage device for storing so-called direction numbers of thegeneration matrix. The form of the generation matrix determines the basicproperties of the generated address sequences. The proposed UASG generates awide spectrum of different address sequences, including the standard ones, suchas linear, address complement, gray code, worst-case gate delay, , nextaddress, and pseudorandom. Examples of the use of the proposed methods areconsidered. The result of the practical implementation of the UASG ispresented, and the main characteristics are evaluated
Weakening and Iterating Laws using String Diagrams
No full textDistributive laws are a standard way of combining two monads, providing acompositional approach for reasoning about computational effects in semantics.Situations where no such law exists can sometimes be handled by weakening thenotion of distributive law, still recovering a composite monad. A celebratedresult from Eugenia Cheng shows that combining monads is possible byiterating more distributive laws, provided they satisfy a coherence conditioncalled the Yang-Baxter equation. Moreover, the order of composition does notmatter, leading to a form of associativity. The main contribution of this paperis to generalise the associativity of iterated composition to weak distributivelaws in the case of monads. To this end, we use string-diagrammaticnotation, which significantly helps make increasingly complex proofs morereadable. We also provide examples of new weak distributive laws arising fromiteration.Comment: Conference version (proceedings of MFPS 2022
Representations of Domains via CF-approximation Spaces
No full textRepresentations of domains mean in a general way representing a domain as asuitable family endowed with set-inclusion order of some mathematicalstructures. In this paper, representations of domains via CF-approximationspaces are considered. Concepts of CF-approximation spaces and CF-closed setsare introduced. It is proved that the family of CF-closed sets in aCF-approximation space endowed with set-inclusion order is a continuous domainand that every continuous domain is isomorphic to the family of CF-closed setsof some CF-approximation space endowed with set-inclusion order. The concept ofCF-approximable relations is introduced using a categorical approach, whichlater facilitates the proof that the category of CF-approximation spaces andCF-approximable relations is equivalent to that of continuous domains and Scottcontinuous maps.Comment: 13pages, an interaction of Mathematics and information scienc
A note on the category of c-spaces
No full textWe prove that the category of c-spaces with continuous maps is not cartesianclosed. As a corollary the category of locally finitary compact spaces withcontinuous maps is also not cartesian closed.Comment: 5 page
Weakly meet -continuity and -continuity
No full textBased on the concept of weakly meet -continuouity put forward by Xuand Luo in \cite{qzm}, we further prove that if the subset system satisfiescertain conditions, a poset is -continuous if and only if it is weaklymeet -continuous and -quasicontinuous, which improves a relatedresult given by Ruan and Xu in \cite{sz}. Meanwhile, we provide acharacterization for the poset to be weakly meet -continuous, that is, aposet with a lower hereditary -Scott topology is weakly meet-continuous if and only if it is locally weakly meet -continuous.In addition, we introduce a monad on the new category and characterize its - algebras concretely.Comment: 11 page