Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our main results is a general theorem which under the hypothesis AD implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's (α,β,ρ) game on R is determined from AD alone, but on Rn for n≥3 we show that AD does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt's (α,β,ρ) game on R has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from AD

The thesis investigates the deep conceptual structure of the distinction between creation and application of law. Legal philosophers either take the distinction for granted (and so do the vast majority of legal scholars in general) or, when they address it directly, it is so for the only purpose of criticising it as just another upshot of legal formalism. The latter approach suggests the distinction is either unsound or, at the very least, useless. The thesis argues that supporters of this stance do not realise the implications of their positions – which transcend legal-theoretical discourse and raise serious problems in both political and constitutional theory. The thesis’s first chapter purports to show that our models of constitutional democracy – as a complex set of institutional power-constraining mechanisms - are ultimately grounded on the distinction between creation and application of law. Hence the theoretical unsoundness of the distinction would undermine the very foundations of such democracies. The thesis argues that if judges are always creating the law, it follows that nothing like the ‘application of law’ is possible and, as a result, there is little or no justification for the practice of having statutes and other institutionalised forms of law. In this regard, Chapter 2 argues that realist theories of adjudication cannot make sense of one of law’s most important features, namely its normativity. More generally, undergirding a certain conception of our current adjudicative practices is the idea of (judicial) discretion, which is an essentially legal concept. In Chapter 3 I discuss this idea, comparing how it is conceived and used in both jurisprudence and administrative law (where such notion has been originally developed, at least in civil-law jurisdictions). The chapter aims at showing that an appropriate conception of discretion plays a pivotal role in grounding an analytical distinction between creation and application of law. Chapter 4 fleshes out the deeper philosophical assumptions of such distinction and, more specifically, it puts forward a conception of ‘law as communication’ which highlights the inescapable relation between law and language, and the parallel relation between philosophy of law and philosophy of language. In this chapter, I also argue that philosophy of language can and should play a role in understanding what law is, but that, at the same time, law is a unique communicative phenomenon, whose characteristics call for an original theoretical analysis. Finally, Chapter 5 brings together the several threads of the argument and presents the analytical model of the distinction between creation and application of law. The conclusion is that the creation/application distinction, thus conceived, is necessary not just in order to defend our liberal political practices - but more fundamentally, to account for law’s own raison d'être as a power-constraining device.

The subject of this dissertation is nominal GDP (NGDP) targeting. In the wake of the Great Recession, some economists have proposed using some form of NGDP target to replace current monetary policy. We evaluate the desirability of NGDP targets based upon their ability to deliver unique and \learnable" equilibria and their welfare gains in the presence of nancial frictions. In the second chapter, we assess the determinacy and E-stability conditions for simple interest rate rules which respond to NGDP's deviation from target in a simple three-equation New Keynesian model. The rules under consideration target either NGDP level or growth, and can either be contemporaneous, one period ahead, or two periods ahead. We also allow for dierent types of information sets for the agents. In the third chapter, we compare welfare loss in consumption equivalent terms for NGDP targets with more conventional monetary policy in a New Keynesian model which features nancial frictions. Finally, in the fourth chapter we continue our analysis from chapter one but now allow for strictly positive trend inflation. We present findings for the relationship between trend inflation and the determinacy and E-stability of the equilibrium when using interest rate rules that target NGPD.

Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets.

This dissertation studies monetary-fiscal policy interactions and adaptive learning applications in regime-switching DSGE models. A common thread through my research is understanding how policymakers may be affected by the interaction of policy regime change and agents' beliefs about past, current or future policy in general equilibrium. The work I present in this dissertation shows that conventional and unconventional policy outcomes, as well as the existence, uniqueness and expectational stability of rational expectations solutions, depend heavily on the expectational effects of time-varying policy. These findings suggest that uncertainty over future fiscal policy may curb the effectiveness of monetary policy, or otherwise constrain the actions of central bankers. In carrying out this research agenda, my work also examines the relationship between determinacy and expectational stability in a general class of Markov-switching DSGE models.

No description available.

The production of long, thin polymeric fibers is a main objective of the textile industry. Melt-spinning is a particularly simple and effective technique. In this work, we shall discuss the equations of melt-spinning in viscous and viscoelastic flow. These quasilinear hyperbolic equations model the uniaxial extension of a fluid thread before its solidification. We will address the following topics: first we shall prove existence, uniqueness, and regularity of solutions. Our solution strategy will be developed in detail for the viscous case. For non-Newtonian and isothermal flows, we shall outline the general ideas. Our solution technique consists of energy estimates and fixed-point arguments in appropriate Banach spaces. The existence result for a simple transport equation is the key to understanding the quasilinear case. The second issue of this exposition will be the stability of the unforced frost line formation. We will give a rigorous justification that, in the viscous regime, the linearized equations obey the ``Principle of Linear Stability''. As a consequence, we are allowed to relate the stability of the associated strongly continuous semigroup to the numerical resolution of the spectrum of its generator. By using a spectral collocation method, we shall derive numerical results on the eigenvalue distribution, thereby confirming prior results on the stability of the steady-state solution. / Ph. D.

No description available.

Rice have highly derived florets borne on a short branch called ‘spikelet’ comprised of a pair of rudimentary glumes and sterile lemma (empty glumes) that subtends a single fertile floret. The floral organs consist of a pair of lodicules, six stamens and a central carpel that are enclosed by a pair of bract-like organs, called lemma and palea. A progressive reprogramming of meristem identity during the floral development of flowers, on branches on the inflorescence, is correlated with changes in transcriptional status of regulatory genes that execute cascades of distinct developmental events. On the other hand phytohormones such as auxin and cytokinin that are critical in predetermining the sites of new organ primordia emergence and in maintaining the size or populations of meristems. Molecular genetic analyses of mutants have expanded the repository of genes regulating floral organ specification and identity, yet the finer mechanistic details on process downstream to these regulatory genes and co-ordination with phytohormone signalling pathways needs further investigation. One aim of the study presented in this thesis is to develop a tool that would display of spatial description of dynamic auxin or cytokinin accumulation in developing rice inflorescence and floral meristems and to evaluate auxin distribution defects of OsMADS1-RNAi florets using this tool. Additionally, we aim to understand the regulatory effects on OsMADS1 on candidate floral organ and meristem fate determining genes during two temporal phases of flower development to decipher other regulatory cascades controlled by OsMADS1. Spatial distribution profile of phytohormones in young and developing meristems of rice Cytokinin promotes meristem activity (Su et al., 2011) while auxin accumulation, directed by auxin efflux transport PIN proteins predicts sites of new organ initiation (Reinhardt et al., 2003; van Mourik et al., 2012). Previous studies in the lab deciphered that OsMADS1 exerts positive regulatory effects on genes in auxin pathways and repressive effects on cytokinin signaling and biosynthetic genes (Khanday et al., 2013). Thus, the need for a reliable system to understand auxin and cytokinin activity in live inflorescence and floral meristems of rice motivated us to raise promoter: reporter tools to map the spatial and temporal phytohormone distribution. Confocal live imaging conditions in primary roots of IR4DR-GFP and DR5-CyPet lines was performed and responsiveness of the DR5 elements to auxin was authenticated. Auxin maxima were distinctly seen in the epidermal and sub-epidermal cells of inflorescence branch primordia anlagen and apices of newly emerged branch primordia. As floral organs were being initiated, on the floret meristem, we discerned the sequential appearance of auxin accumulation at sites of organ primordia while apices of early floral meristems (FM) showed low auxin content. We clearly detect canalization of auxin streams marking regions of vascular inception. Using this live imaging system we probed auxin patterns and levels in malformed and indeterminate OsMADS1-RNAi florets and we observed a significant reduction in the levels of auxin. Two oppositely positioned peaks of auxin were noted in the persistent FM of OsMADS1-RNAi florets, a pattern similar to auxin dynamics at sites of rudimentary glume primordia on the wild-type (WT) spikelet meristem. These studies were followed up with immunohistochemistry (IHC) on fixed tissues for “PIN” transport proteins that suggest PIN convergence towards organ initiation sites, regions where auxin accumulation was clearly visualized by the IR4DR5-GFP and DR5-CyPet reporters. IHC experiments that detected GFP, in fixed tissues of TCSn-mGFP ER (WT) and TCSn-mGFP ER;OsMADS1-RNAi (OsMADS1-RNAi) inflorescence and florets showed an ectopic increase in the domain of cells with cytokinin response in OsMADS1-RNAi florets, compared to that of WT. Intriguingly, cytokinin responsive cells persisted in the central FM of OsMADS1-RNAi florets that might partially account for some of the FM indeterminacy defects seen in these florets. A correlative observation of these different imaging data hint at some exclusive patterns of the IR4DR5/DR5 and TCSn reporters that in turn lead us to speculate that a cross talk between auxin and cytokinin distribution may contribute to the precise phyllotaxy of lateral organs in rice inflorescence. Studies on novel targets of OsMADS1 in floral organ identity and meristem determinacy Loss of OsMADS1 function results in rice florets with miss specified floral organs and an indeterminate carpel produces new abnormal florets. Despite having several mutants in OsMADS1, mechanisms of how OsMADS1 regulates meristem maintenance and termination is not well understood. Global expression profile in OsMADS1-RNAi vs. WT tissues encompassing a wide range of developing florets (0.2 to 2cm panicles), gave an overview of OsMADS1 functions in many aspects of floret development. Here, a gene-targeted knockout of OsMADS1 named - osmads1ko (generated in a collaborative study) was characterized and found to display extreme defects in floral organs and an indeterminate FM. Strikingly, in addition to loss of determinacy, FM reverts to a prior developmental fate of inflorescence on whose new rachis are leaf-like malformed florets. We suggest these phenotypes reflect the null phenotype of OsMADS1 and its role in meristem fate maintenance. We tested gene expression levels for some proven targets of OsMADS1 (Khanday et al., 2013) and utilized panicles in two developmental phases- young early FMs (panicles of 0.2 to 0.5 cm) and older florets with organ differentiation (panicles of 0.5 to 1cm). We observed temporally different effects on the regulation of OsMADS34 that together with histology of young osmads1ko inflorescences suggest that the mutant is impeded for spikelet to floral meristem transition. In addition, OsMADS1 had a positive regulatory effect on genes implicated for lemma and palea organ identity such as OsIDS1, OsDH1, OsYABBY1, OsMADS15, OsMADS32, OsDP1 and OsSPL16 in both young and old panicles while OsIG1 was negatively regulated in both phases of development. MADS-box genes important for carpel and ovule development - OsMADS13 and OsMADS58 were had significantly reduced expression in florets undergoing organ differentiation. OsMADS1 positively regulated several other non MADS-box developmental genes - OsSPT, OsHEC2 and OsULT1, whose Arabidopsis homologs control carpel development and FM determinacy. These genes are de-regulated in later stages of osmads1ko floret development and are unaffected in younger panicles. Finally, OsMADS1 continually activated meristem maintenance genes - OsBAM2-like and OsMADS6 while the activation of OSH1 in early floral meristems was later altered to a repressive effect in developing florets. Perhaps such dynamic temporal effects on meristem genes are instrumental in the timely termination of the floral meristem after floral organ differentiation. More importantly, we show that regulation of many of these genes is directly affected by OsMADS1, through our studies on expression levels before and after chemical induction of OsMADS1-GR protein in amiRNAOsMADS1 florets. Further, some key downstream targets were re-affirmed by studying expression status in transgenic lines, with the OsMADS1-EAR repressive protein variant. These results provide new insights into the developmentally phased roles of OsMADS1 on floral meristem regulators and determinants of organ identity to form a determinate rice floret. Gene networks regulated by OsMADS1 during early flower development To identify global targets in early floret meristems, we determined the differential RNA transcriptome in osmads1ko tissues as compared to wild-type tissues. These data revealed regulators of inflorescence architecture, floral organ identity including MADS-box floral homeotic factors, factors for meristem maintenance, auxin response, transport and biosynthesis as some of the important functional classes amongst the 2725 differentially expressed genes (DEGs). Integrating DEGs with OsMADS1 ChIP-seq data (prior studies from our lab) we deciphered direct vs. indirect and positive vs. negatively regulated targets of OsMADS1. These datasets reveal an enrichment for functional categories such as metabolic processes, signaling, RNA transcription and processing, hormone metabolism and protein modification. Using Bio-Tapestry plot as a tool we present a visualization of a floral stage-specific regulatory network for genes with likely functional roles in meristem specification and in organ development. Further, to examine if indirect targets regulated by OsMADS1 could be mediated through transcription factors (that are themselves direct targets), we constructed a small network with the transcription factors OSH1, OSH15 and OsYABBY1 as key nodal genes and we predicted their downstream effects. Taken together, these analyses provide examples of the complex networks that OsMADS1 controls during the process of rice floret development. In summary, we surmise that defect in phytohormone distribution in OsMADS1 knockdown florets results in irregular patterns of lateral organ primordia emergence. In addition, the derangements in the developmentally stage specific expression of floral meristems identity and organ identity genes culminates in miss-specified and irregularly patterned abnormal organs in Osmads1 florets. Thus, our study highlights the versatility of OsMADS1 in regulating components of hormone signaling and response, and its effects on various floral development regulators results in the formation of a single determinate floret on the spikelet. References: Khanday I, Yadav S.R, and Vijayraghavan U. (2013). Plant Physiol 161, 1970–1983. van Mourik S , Kaufmann K, van Dijk AD, Angenent G.C, Merks R.M.H, Molenaar J. (2012). PLOS One 1, e28762 Reinhardt D, Pesce E, Stieger P, Mandel T, Baltensperger K, Bennett M, Traas J, Friml J and Kuhlemeier C. (2003). Nature 426, 255-260 Su Y, Liu Y and Zhang X. (2011) Mol Plant 4, 616–625

Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.