Abstract and Explicit Constructions of Jacobian Varieties

Abelian varieties, in particular Jacobian varieties, have long attracted interest in mathematics. Their influence pervades arithmetic geometry and number theory, and understanding their construction was a primary motivator for Weil in his work on developing new foundations for algebraic geometry in the 1930s and 1940s. Today, these exotic mathematical objects find applications in cryptography and computer science, where they can be used to secure confidential communications and factor integers in subexponential time. Although in many respects well-studied, working in concrete, explicit ways with abelian varieties continues to be difficult. The issue is that, aside from the case of elliptic curves, it is often difficult to find ways of modelling and understanding these objects in ways amenable to computation. Often, the approach taken is to work ``indirectly'' with abelian varieties, in particular with Jacobians, by working instead with divisors on their associated curves to simplify computations. However, properly understanding the mathematics underlying the direct approach --- why, for instance, one can view the degree zero divisor classes on a curve as being points of a variety --- requires sophisticated mathematics beyond what is usually understood by algorithms designers and even experts in computational number theory. A direct approach, where explicit polynomial and rational functions are given that define both the abelian variety and its group law, cannot be found in the literature for dimensions greater than two. In this thesis, we make two principal contributions. In the first, we survey the mathematics necessary to understand the construction of the Jacobian of a smooth algebraic curve as a group variety. In the second, we present original work with gives the first instance of explicit rational functions defining the group law of an abelian variety of dimension greater than two. In particular, we derive explicit formulas for the group addition on the Jacobians of hyperelliptic curves of every genus g, and so give examples of explicit rational formulas for the group law in every positive dimension

The multifunctional NS1 protein of influenza A viruses

The non-structural (NS1) protein of influenza A viruses is a non-essential virulence factor that has multiple accessory functions during viral infection. In recent years, the major role ascribed to NS1 has been its inhibition of host immune responses, especially the limitation of both interferon (IFN) production and the antiviral effects of IFN-induced proteins, such as dsRNA-dependent protein kinase R (PKR) and 2'5'-oligoadenylate synthetase (OAS)/RNase L. However, it is clear that NS1 also acts directly to modulate other important aspects of the virus replication cycle, including viral RNA replication, viral protein synthesis, and general host-cell physiology. Here, we review the current literature on this remarkably multifunctional viral protein. In the first part of this article, we summarize the basic biochemistry of NS1, in particular its synthesis, structure, and intracellular localization. We then discuss the various roles NS1 has in regulating viral replication mechanisms, host innate/adaptive immune responses, and cellular signalling pathways. We focus on the NS1-RNA and NS1-protein interactions that are fundamental to these processes, and highlight apparent strain-specific ways in which different NS1 proteins may act. In this regard, the contributions of certain NS1 functions to the pathogenicity of human and animal influenza A viruses are also discussed. Finally, we outline practical applications that future studies on NS1 may lead to, including the rational design and manufacture of influenza vaccines, the development of novel antiviral drugs, and the use of oncolytic influenza A viruses as potential anti-cancer agents.Publisher PDFPeer reviewe

How Genealogies Can Affect the Space of Reasons

Can genealogical explanations affect the space of reasons? Those who think so commonly face two objections. The first objection maintains that attempts to derive reasons from claims about the genesis of something commit the genetic fallacy—they conflate genesis and justification. One way for genealogies to side-step this objection is to focus on the functional origins of practices—to show that, given certain facts about us and our environment, certain conceptual practices are rational because apt responses. But this invites a second objection, which maintains that attempts to derive current from original function suffer from continuity failure—the conditions in response to which something originated no longer obtain. This paper shows how normatively ambitious genealogies can steer clear of both problems. It first maps out various ways in which genealogies can involve non-fallacious genetic arguments before arguing that some genealogies do not invite the charge of the genetic fallacy if they are interpreted as revealing the original functions of conceptual practices. However, they then incur the burden of showing that the conditions relative to which practices function continuously obtain. Taking its cue from the genealogies of E. J. Craig, Bernard Williams, and Miranda Fricker, the paper shows how model-based genealogies can avoid continuity failures by identifying bases of continuity in the demands we face

Implicit Bias and the Idealized Rational Self

The underrepresentation of women, people of color, and especially women of color—and the corresponding overrepresentation of white men—is more pronounced in philosophy than in many of the sciences. I suggest that part of the explanation for this lies in the role played by the idealized rational self, a concept that is relatively influential in philosophy but rarely employed in the sciences. The idealized rational self models the mind as consistent, unified, rationally transcendent, and introspectively transparent. I hypothesize that acceptance of the idealized rational self leads philosophers to underestimate the influence of implicit bias on their own judgments and prevents them from enacting the reforms necessary to minimize the effects of implicit bias on institutional decision-making procedures. I consider recent experiments in social psychology that suggest that an increased sense of one’s own objectivity leads to greater reliance on bias in hiring scenarios, and I hypothesize how these results might be applied to philosophers’ evaluative judgments. I discuss ways that the idealized rational self is susceptible to broader critiques of ideal theory, and I consider some of the ways that the picture functions as a tool of active ignorance and color-evasive racism

Ur-Priors, Conditionalization, and Ur-Prior Conditionalization

Conditionalization is a widely endorsed rule for updating one’s beliefs. But a sea of complaints have been raised about it, including worries regarding how the rule handles error correction, changing desiderata of theory choice, evidence loss, self-locating beliefs, learning about new theories, and confirmation. In light of such worries, a number of authors have suggested replacing Conditionalization with a different rule — one that appeals to what I’ll call “ur-priors”. But different authors have understood the rule in different ways, and these different understandings solve different problems. In this paper, I aim to map out the terrain regarding these issues. I survey the different problems that might motivate the adoption of such a rule, flesh out the different understandings of the rule that have been proposed, and assess their pros and cons. I conclude by suggesting that one particular batch of proposals, proposals that appeal to what I’ll call “loaded evidential standards”, are especially promising

Epistemic Teleology: Synchronic and Diachronic

According to a widely held view of the matter, whenever we assess beliefs as ‘rational’ or ‘justified’, we are making normative judgements about those beliefs. In this discussion, I shall simply assume, for the sake of argument, that this view is correct. My goal here is to explore a particular approach to understanding the basic principles that explain which of these normative judgements are true. Specifically, this approach is based on the assumption that all such normative principles are grounded in facts about values, and the normative principles that apply to beliefs in particular are grounded in facts about alethic value––a kind of value that is exemplified by believing what is true and not believing what is false. In this chapter, I shall explain what I regard as the best way of interpreting this approach. In doing so, I shall also show how this interpretation can solve some problems that have recently been raised for approaches of this kind by Selim Berker, Jennifer Carr, Michael Caie, and Hilary Greaves

Real Hypercomputation and Continuity

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of "Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS vol.352

A robust implementation of the Carathéodory-Fejér method

Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carathéodory-Fejér (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the chebfun software system and illustrate its use with several examples. Our implementation handles both polynomial and rational approximation and substantially improves upon earlier published software