A summary of Euler’s work on the pentagonal number theorem

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions.     

The love of truth

It is frequently said that belief aims at truth, in an explicitly normative sense—that is, that one ought to believe the proposition that p if, and only if, p is true. This truth norm is frequently invoked to explain why we should seek evidential justification in our beliefs, or why we should try to be rational in our belief formation—it is because we ought to believe the truth that we ought to follow the evidence in belief revision. In this paper, I argue that this view is untenable. The truth norm clashes with plausible evidential norms in a wide range of cases, such as when we have excellent but misleading evidence for a falsehood or no evidence for a truth. I will consider various ways to resolve this conflict and argue that none of them work. However, I will ultimately attempt to vindicate the love of truth, by arguing that knowledge is the proper epistemic goal. The upshot is that we should not aim merely to believe the truth; we should aim to know it.     

Kuhn and coherentist epistemology

The paper challenges a recent attempt by Jouni-Matti Kuukkanen to show that since Thomas Kuhn’s philosophical standpoint can be incorporated into coherentist epistemology, it does not necessarily lead to: (Thesis 1) an abandonment of rationality and rational interparadigm theory comparison, nor to (Thesis 2) an abandonment of convergent realism. Leaving aside the interpretation of Kuhn as a coherentist, we will show that Kuukkanen’s first thesis is not sufficiently explicated, while the second one entirely fails. With regard to Thesis 1, we argue that Kuhn’s view on inter-paradigm theory comparison allows only for (what we shall dub as) ‘the weak notion of rationality’, and that Kuukkanen’s argument is thus acceptable only in view of such a notion. With regard to Thesis 2, we show that even if we interpret Kuhn as a coherentist, his philosophical standpoint cannot be seen as compatible with convergent realism since Kuhn’s argument against it is not ‘ultimately empirical’, as Kuukkanen takes it to be.     

The problem of contextuality and the impossibility of experimental metaphysics thereof

Recently a new impulse has been given to the experimental investigation of contextuality. In this paper we show that for a widely used definition of contextuality there can be no decisive experiment on the existence of contextuality. To this end, we give a clear presentation of the hidden variable models due to Meyer, Kent and Clifton (MKC), which would supposedly nullify the Kochen–Specker theorem. Although we disagree with this last statement, the models do play a significant role in the discussion on the meaning of contextuality. In fact, we introduce a specific MKC-model of which we show that it is non-contextual and completely in agreement with quantum mechanical predictions. We also investigate the possibility of other definitions of non-contextuality—with an emphasis on operational definitions—and argue that any useful definition relies on the specification of a theoretical framework. It is therefore concluded that no experimental test can yield any conclusions about contextuality on a metaphysical level.     

Leibniz on the requisites of an exact arithmetical quadrature

In this paper we will try to explain how Leibniz justified the idea of an exact arithmetical quadrature. We will do this by comparing Leibniz's exposition with that of John Wallis. In short, we will show that the idea of exactitude in matters of quadratures relies on two fundamental requisites that, according to Leibniz, the infinite series have, namely, that of regularity and that of completeness. In the first part of this paper, we will go deeper into three main features of Leibniz's method, that is: it is an infinitesimal method, it looks for an arithmetical quadrature and it proposes a result that is not approximate, but exact. After that, we will deal with the requisite of the regularity of the series, pointing out that, unlike the inductive method proposed by Wallis, Leibniz propounded some sort of intellectual recognition of what is invariant in the series. Finally, we will consider the requisite of completeness of the series. We will see that, although both Wallis and Leibniz introduced the supposition of completeness, the German thinker went beyond the English mathematician, since he recognized that it is not necessary to look for a number for the quadrature of the circle, given that we have a series that is equal to the area of that curvilinear figure.     

Optimal prediction with nonstationary ARFIMA model

We propose two methods to predict nonstationary long‐memory time series. In the first one we estimate the long‐range dependent parameter d by using tapered data; we then take the nonstationary fractional filter to obtain stationary and short‐memory time series. In the second method, we take successive differences to obtain a stationary but possibly long‐memory time series. For the two methods the forecasts are based on those obtained from the stationary components. Copyright © 2007 John Wiley & Sons, Ltd.     

Forecasting volatility with outliers in GARCH models

In this paper, we detect and correct abnormal returns in 17 French stocks returns and the French index CAC40 from additive‐outlier detection method in GARCH models developed by Franses and Ghijsels (1999) and extended to innovative outliers by Charles and Darné (2005). We study the effects of outlying observations on several popular econometric tests. Moreover, we show that the parameters of the equation governing the volatility dynamics are biased when we do not take into account additive and innovative outliers. Finally, we show that the volatility forecast is better when the data are cleaned of outliers for several step‐ahead forecasts (short, medium‐ and long‐term) even if we consider a GARCH‐t process. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of <Emphasis Type="Italic">Dictyostelium</Emphasis> TACC reveals differential interactions with CP224 and unusual dynamics of <Emphasis Type="Italic">Dictyostelium</Emphasis> microtubules

We have localized TACC to the microtubule-nucleating centrosomal corona and to microtubule plus ends. Using RNAi we proved that Dictyostelium TACC promotes microtubule growth during interphase and mitosis. For the first time we show in vivo that both TACC and XMAP215 family proteins can be differentially localized to microtubule plus ends during interphase and mitosis and that TACC is mainly required for recruitment of an XMAP215-family protein to interphase microtubule plus ends but not for recruitment to centrosomes and kinetochores. Moreover, we have now a marker to study dynamics and behavior of microtubule plus ends in living Dictyostelium cells. In a combination of live cell imaging of microtubule plus ends and fluorescence recovery after photobleaching (FRAP) experiments of GFP-α-tubulin cells we show that Dictyostelium microtubules are dynamic only in the cell periphery, while they remain stable at the centrosome, which also appears to harbor a dynamic pool of tubulin dimers.     

The influence of Spinoza’s concept of infinity on Cantor’s set theory

Georg Cantor, the founder of set theory, cared much about a philosophical foundation for his theory of infinite numbers. To that end, he studied intensively the works of Baruch de Spinoza. In the paper, we survey the influence of Spinozean thoughts onto Cantor’s; we discuss Spinoza’s philosophy of infinity, as it is contained in his Ethics; and we attempt to draw a parallel between Spinoza’s and Cantor’s ontologies. Our conclusion is that the study of Spinoza provides deepening insights into Cantor’s philosophical theory, whilst Cantor can not be called a ‘Spinozist’ in any stricter sense of that word.     

Knowing what would happen: The epistemic strategies in Galileo's thought experiments

While philosophers have subjected Galileo's classic thought experiments to critical analysis, they have tended to largely ignored the historical and intellectual context in which they were deployed, and the specific role they played in Galileo's overall vision of science. In this paper I investigate Galileo's use of thought experiments, by focusing on the epistemic and rhetorical strategies that he employed in attempting to answer the question of how one can know what would happen in an imaginary scenario. Here I argue we can find three different answers to this question in Galileo later dialogues, which reflect the changing meanings of ‘experience’ and ‘knowledge’ (scientia) in the early modern period. Once we recognise that Galileo's thought experiments sometimes drew on the power of memory and the explicit appeal to ‘common experience’, while at other times, they took the form of demonstrative arguments intended to have the status of necessary truths; and on still other occasions, they were extrapolations, or probable guesses, drawn from a carefully planned series of controlled experiments, it becomes evident that no single account of the epistemological relationship between thought experiment, experience and experiment can adequately capture the epistemic variety we find Galileo's use of imaginary scenarios. To this extent, we cannot neatly classify Galileo's use of thought experiments as either ‘medieval’ or ‘early modern’, but we should see them as indicative of the complex epistemological transformations of the early seventeenth century.     

Isaac Newton and Augustan Anglo-Latin poetry

Although many historians of science acknowledge the extent to which Greek and Roman ideals framed eighteenth-century thought, many classical references in the texts they study remain obscure. Poems played an important role not only in spreading ideas about natural philosophy, but also in changing people’s perceptions of its value; they contributed to Newton’s swelling reputation as an English hero. By writing about Latin poetry, we focus on the intersection of two literary genres that were significant for eighteenth-century natural philosophy, but seem alien to modern science. We classify Augustan Latinate scientific poetry by considering the audiences for whom the poems were intended. We distinguish three broad categories. One type of poetry was circulated amongst gentlemanly scholars (we look particularly at Tripos verses, and laments for Queen Caroline). A second group comprised poetry written specifically to promote or criticise Newton and his books, particularly the Principia (we look at versions of Pope’s epitaph, and Halley’s Lucretian poem). After Newton’s death, a third type of poetry became increasingly significant, included in collections of poems rather than in texts of natural philosophy. Overall, we show how the classical past was vital for creating the scientific future.     

Predictability of Equity Models

In this study, we verify the existence of predictability in the Brazilian equity market. Unlike other studies in the same sense, which evaluate original series for each stock, we evaluate synthetic series created on the basis of linear models of stocks. Following the approach of Burgess (Computational Finance, 1999; 99, 297–312), we use the ‘stepwise regression’ model for the formation of models of each stock. We then use the variance ratio profile together with a Monte Carlo simulation for the selection of models with potential predictability using data from 1 April 1999 to 30 December 2003. Unlike the approach of Burgess, we carry out White's Reality Check (Econometrica, 2000; 68, 1097–1126) in order to verify the existence of positive returns for the period outside the sample from 2 January 2004 to 28 August 2007. We use the strategies proposed by Sullivan, Timmermann and White (Journal of Finance, 1999; 54, 1647–1691) and Hsu and Kuan (Journal of Financial Econometrics, 2005; 3, 606–628) amounting to 26,410 simulated strategies. Finally, using the bootstrap methodology, with 1000 simulations, we find strong evidence of predictability in the models, including transaction costs. Copyright © 2015 John Wiley & Sons, Ltd.     

Debunking material induction

In this paper, we present an explanatory objection to Norton's material theory of induction, as applied to predictive inferences. According to the objection we present, there is an explanatory disconnect between our beliefs about the future and the relevant future facts. We argue that if we recognize such a disconnect, we are no longer rationally entitled to our future beliefs.     

Trading zones and interactional expertise

The phrase ‘trading zone’ is often used to denote any kind of interdisciplinary partnership in which two or more perspectives are combined and a new, shared language develops. In this paper we distinguish between different types of trading zone by asking whether the collaboration is co-operative or coerced and whether the end-state is a heterogeneous or homogeneous culture. In so doing, we find that the voluntary development of a new language community—what we call an inter-language trading zone—represents only one of four possible configurations. In developing this argument we show how different modes of collaboration result in different kinds of trading zone, how different kinds of trading zone may be ‘nested’ inside each other and discuss how a single collaboration might move between different kinds of trading zone over time. One implication of our analysis is that interactional expertise is a central component of at least one class of trading zone.     

Dogma and dreams: experimental lessons for epilepsy mechanism chasers

Epilepsy mechanism chasers face one major difficulty. Since we don’t know how the normal brain works, we can’t start to understand how the diseased brain fails. Most of today’s hypotheses are based on what we think about ‘normal’ brain function, which may lead to misconceptions, as will be developed here. Furthermore, since there are many different types of epilepsies, some mechanisms may only be relevant to some epilepsies. Here, I shall focus on temporal lobe epilepsy (TLE) the most common form of partial epilepsy in adults. TLE is often drug resistant, as are 30–40% of all forms of epilepsies. The failure of drug-treatments most likely reflects our lack of knowledge of the underlying mechanisms.Received 10 January 2005; received after revision 3 March 2005; accepted 23 March 2005     

Statistical estimation of optimal portfolios for non‐Gaussian dependent returns of assets

This paper discusses the asymptotic efficiency of estimators for optimal portfolios when returns are vector‐valued non‐Gaussian stationary processes. We give the asymptotic distribution of portfolio estimators ? for non‐Gaussian dependent return processes. Next we address the problem of asymptotic efficiency for the class of estimators ?. First, it is shown that there are some cases when the asymptotic variance of ? under non‐Gaussianity can be smaller than that under Gaussianity. The result shows that non‐Gaussianity of the returns does not always affect the efficiency badly. Second, we give a necessary and sufficient condition for ? to be asymptotically efficient when the return process is Gaussian, which shows that ? is not asymptotically efficient generally. From this point of view we propose to use maximum likelihood type estimators for g, which are asymptotically efficient. Furthermore, we investigate the problem of predicting the one‐step‐ahead optimal portfolio return by the estimated portfolio based on ? and examine the mean squares prediction error. Copyright © 2008 John Wiley & Sons, Ltd.     

The Foundations of Projective Geometry in Italy from De Paolis to Pieri

 In this paper we examine the contributions of the Italian geometrical school to the Foundations of Projective Geometry. Starting from De Paolis' work we discuss some papers by Segre, Peano, Veronese, Fano and Pieri. In particular we try to show how a totally abstract and general point of view was clearly adopted by the Italian scholars many years before the publication of Hilbert's Grundlagen. We are particularly interested in the interrelations between the Italian and the German schools (mainly the influence of Staudt's and Klein's works). We try also to understand the reason of the steady decline of the Italian school during the twentieth century. (Received Febuary 25, 2000)     

Regulation of the H<Superscript>+</Superscript>-ATP synthase by IF1: a role in mitohormesis

The mitochondrial H⁺-ATP synthase is a primary hub of cellular homeostasis by providing the energy required to sustain cellular activity and regulating the production of signaling molecules that reprogram nuclear activity needed for adaption to changing cues. Herein, we summarize findings regarding the regulation of the activity of the H⁺-ATP synthase by its physiological inhibitor, the ATPase inhibitory factor 1 (IF1) and their functional role in cellular homeostasis. First, we outline the structure and the main molecular mechanisms that regulate the activity of the enzyme. Next, we describe the molecular biology of IF1 and summarize the regulation of IF1 expression and activity as an inhibitor of the H⁺-ATP synthase emphasizing the role of IF1 as a main driver of energy rewiring and cellular signaling in cancer. Findings in transgenic mice in vivo indicate that the overexpression of IF1 is sufficient to reprogram energy metabolism to an enhanced glycolysis and activate reactive oxygen species (ROS)-dependent signaling pathways that promote cell survival. These findings are placed in the context of mitohormesis, a program in which a mild mitochondrial stress triggers adaptive cytoprotective mechanisms that improve lifespan. In this regard, we emphasize the role played by the H⁺-ATP synthase in modulating signaling pathways that activate the mitohormetic response, namely ATP, ROS and target of rapamycin (TOR). Overall, we aim to highlight the relevant role of the H⁺-ATP synthase and of IF1 in cellular physiology and the need of additional studies to decipher their contributions to aging and age-related diseases.     

Borelli’s edition of books V–VII of Apollonius’s Conics,and Lemma 12 in Newton’s Principia

To solve the direct problem of central forces when the trajectory is an ellipse and the force is directed to its centre, Newton made use of the famous Lemma 12 (Principia, I, sect. II) that was later recognized equivalent to proposition 31 of book VII of Apollonius’s Conics. In this paper, in which we look for Newton’s possible sources for Lemma 12, we compare Apollonius’s original proof, as edited by Borelli, with those of other authors, including that given by Newton himself. Moreover, after having retraced its editorial history, we evaluate the dissemination of Borelli's edition of books V-VII of Apollonius’s Conics before the printing of the Principia.

     

Fluctuation and response in biology

In 1905, Albert Einstein proposed that the forces that cause the random Brownian motion of a particle also underlie the resistance to macroscopic motion when a force is applied. This insight, of a coupling between fluctuation (stochastic behavior) and responsiveness (non-stochastic behavior), founded an important branch of physics. Here we argue that his insight may also be relevant for understanding evolved biological systems, and we present a ‘fluctuation–response relationship’ for biology. The relationship is consistent with the idea that biological systems are similarly canalized to stochastic, environmental, and genetic perturbations. It is also supported by in silico evolution experiments, and by the observation that ‘noisy’ gene expression is often both more responsive and more ‘evolvable’. More generally, we argue that in biology there is (and always has been) an important role for macroscopic theory that considers the general behavior of systems without concern for their intimate molecular details.