Set Of Irrationals Research Articles

Race, Gender, and Religion

AbstractThis article focuses on anti-Muslim racism as a discourse that collapses race and religion and cannot be reduced to phobia. It is instead about a racial project of accumulation based on European superiority and how cultural racism upholds the European civilizational project. The author argues that Islamophobia should be traced back to colonial modernity, its regimes of othering, and its perception of Islam as Mohammedanism that conceals its nature as a fetishistic, primitive, barbaric, patriarchal, and irrational set of beliefs. To illustrate anti-Muslim racism, the author elaborates briefly on three interconnected ideas: the construction of Islam as a unified religious and cultural mindset, its fetishistic character, and its enigmatic image of the woman to reflect on how Islam is presented as the antonym of Western civilization.

What if ideal advice conflicts? A dilemma for idealizing accounts of normative practical reasons

One of the deepest and longest-lasting debates in ethics concerns a version of the Euthyphro question: are choiceworthy things choiceworthy because agents have certain attitudes toward them or are they choiceworthy independent of any agents’ attitudes? Reasons internalists, such as Bernard Williams, Michael Smith, Mark Schroeder, Sharon Street, Kate Manne, Julia Markovits, and David Sobel answer in the first way. They think that all of an agent’s normative reasons for action are grounded in facts about that agent’s pro-attitudes (e.g., her desires, valuing states, normative judgments). According to the most popular brand of internalism, idealizing internalism, an agent’s reasons are grounded, not in her actual pro-attitudes, but rather in what her pro-attitudes would be in suitably idealized conditions. Idealizing internalists presuppose that, for any agent with an irrational set of attitudes, there is one uniquely rational set that that agent would have if she were to undergo the relevant idealizing process. I argue that this assumption is false and that it raises two puzzles for idealizing internalism: one about the existence of practical reasons and another about their normative weight. I argue that idealizing internalists have an adequate solution to the first puzzle but not the second. Indeed, when they try to solve the second puzzle, they confront a dilemma. This second puzzle and the associated dilemma thus constitutes a powerful, but so far unnoticed, difficulty for idealizing internalism.

Invariant subspaces for Bishop operators and beyond

Bishop operators Tα acting on L2[0,1) were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits of nilpotent operators. Moreover, by means of arithmetical techniques along with a theorem of Atzmon, the set of irrationals α∈(0,1) for which Tα is known to possess non-trivial closed invariant subspaces is considerably enlarged, extending previous results by Davie [11], MacDonald [21] and Flattot [14]. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon's Theorem cannot be applied. Finally, upon applying arithmetical bounds obtained, we deduce local spectral properties of Bishop operators proving, in particular, that neither of them satisfy Dunford's property (C).

Matematik Öğretmenlerinin İrrasyonel Sayılara Yönelik Kavram Bilgilerinin İncelenmesi

The objective of this study was to examine mathematics teachers’ conceptual understanding of irrational numbers. Data were gathered from semi-structured interviews with eight primary school mathematics teachers in Central Anatolia, Turkey. The interviews carried on with the teachers lasted about 45 minutes. To be able to record the interviews, permissions were taken from the teachers. After the interviews had been completed, they were decoded and begun to be analyzed. The data were analyzed via content analysis method. At the stage during which the interview form was prepared, at first the studies in the literature and the books prepared by the Ministry of National Education (MNE) for the secondary and high schools were examined.The results show that teachers have difficulty defining and recognizing irrational numbers, placing them on the number line, and doing operations with them. The teachers gave intuitive answers instead of using formal mathematics knowledge. Teachers’ knowledge about irrational numbers is insufficient and they have misconceptions. The teachers’ definitions, knowing of irrational numbers, showing the exact place of irrational numbers on number line and the difficulties they faced while doing operations on irrational set are taken into consideration so as to recommend providing studies involving a large number of participants.

Notes from the Editors, January 2014

buy this issueOn November 16, 2013, Paul Krugman published a piece on his New York Times blog entitled "Secular Stagnation, Coalmines, Bubbles, and Larry Summers," consisting of an extended commentary on former Clinton Treasury Secretary and Obama economic advisor Lawrence Summers's November 8 presentation to the IMF's Economic Forum.… Krugman, in following up on Summers's IMF speech, highlighted Alvin Hansen's theory of secular stagnation in the 1930s to '50s.… [acknowledging that] long-term economic stagnation…was now "the norm" for the economy, not the exception.… Writing in a fashion that could have come straight out of Monthly Review at any point in the last forty years, he declared: "We now know that the economic expansion of 2003–2007 was driven by a bubble. You can say the same about the later part of the 90s expansion; and you can in fact say the same about the later years of the Reagan expansion, which was driven at that point by runaway thrift institutions and a large bubble in commercial real estate." But in trying to understand how stagnation itself came about and created this whole irrational set of economic conditions, Krugman…failed to draw attention to the much more important problem of investment under conditions of overcapacity and mature industry, as well as the whole question of monopolistic/oligopolistic capitalism—all of which were taken seriously at some level by Hansen, and were developed in a far more radical way by socialist thinkers such as Michał Kalecki, Joseph Steindl, Paul Baran, and Paul Sweezy.This article can also be found at the Monthly Review website, where most recent articles are published in full.Click here to purchase a PDF version of this article at the Monthly Review website.

On codings and dynamics of planar piecewise rotations

In this paper, we investigate codings and dynamics of the piecewise rotations belonging to a sub-class of piecewise isometries. We show that the irrational set is empty for some piecewise rational rotations under some assumptions, while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space. We further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation.

Dynamical behaviors and codings of invertible planar piecewise isometric systems

In this paper, the authors discuss some important sets, including singular sets, exceptional sets, irrational sets, etc., generated by invertible planar piecewise isometric maps. They first extend the coding map onto the entire phase space and characterize singular sets and exceptional sets by the number of codings. Then they reveal the relation among these sets, and show that an irrational set is contained in an exceptional set. And they confirm the existence of admissible rational codings under some conditions (including a necessary and sufficient condition and a sufficient condition). They also discuss dynamical properties of the discontinuity line segments which determine the dynamics of a piecewise isometric system. Finally, they list some open problems for further research.

The arithmetic-geometric scaling spectrum for continued fractions

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to infinity.

Are creationists rational?

Creationism is usually regarded as an irrational set of beliefs. In this paper I propose that the best way to understand why individual learners settle on any mature set of beliefs is to see that as the developmental outcome of a series of “fast and frugal” boundedly rational inferences rather than as a rejection of reason. This applies to those whose views are opposed to science in general. A bounded rationality model of belief choices both serves to explain the fact that folk traditions tend to converge on “anti-modernity”, and to act as a default hypothesis, deviations from which we can use to identify other, arational, influences such as social psychological, economic and individual dispositions. I propose some educational and public policy strategies that might decrease the proportion of learners who find creationism and anti-science in general a rational choice.

The Abductive Loop: Tracking Irrational Sets

We argue from the Church-Turing thesis (Kleene Mathematical logic. New York: Wiley 1967) that a program can be considered as equivalent to a formal language similar to predicate calculus where predicates can be taken as functions. We can relate such a calculus to Wittgenstein’s first major work, the Tractatus, and use the Tractatus and its theses as a model of the formal classical definition of a computer program. However, Wittgenstein found flaws in his initial great work and he explored these flaws in a new thesis described in his second great work; the Philosophical Investigations. The question we address is “can computer science make the same leap?” We are proposing, because of the flaws identified by Wittgenstein, that computers will never have the possibility of natural communication with people unless they become active participants of human society. The essential difference between formal models used in computing and human communication is that formal models are based upon rational sets whereas people are not so restricted. We introduce irrational sets as a concept that requires the use of an abductive inference system. However, formal models are still considered central to our means of using hypotheses through deduction to make predictions about the world. These formal models are required to continually be updated in response to peoples’ changes in their way of seeing the world. We propose that one mechanism used to keep track of these changes is the Peircian abductive loop.

Seeking Allies: Modelling How Listeners Choose Their Musical Friends

In this paper we describe in some detail a formal computer model of inferential discourse based on a belief system. The key issue is that a logical model in a computer, based on rational sets, can usefully model a human situation based on irrational sets. The background of this work is explained elsewhere, as is the issue of rational and irrational sets (Billinge and Addis, in: Magnani and Dossena (eds.), Computing, philosophy and cognition, 2004; Stepney et al., Journey: Non-classical philosophy—socially sensitive computing in journeys non-classical computation: A grand challenge for computing research, 2004). The model is based on the Belief System (Addis and Gooding, Proceedings of the AISB’99 Symposium on Scientific Creativity, 1999) and it provides a mechanism for choosing queries based on a range of belief. We explain how it provides a way to update the belief based on query results, thus modelling others’ experience by inference. We also demonstrate that for the same internal experience, different models can be built for different actors.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

An Asymptotic Formula for a Trigonometric Sum of Vinogradov

We obtain a representation formula for the trigonometric sum f(m, n)≔ ∑m−1a=1|sin(πan/m)|sin(πa/m) and deduce from it the upper bound f(m, n)<(4/π2)mlogm+ (4/π2)(γ−log(π/2)+2CG)m+O(m/logm), where CG is the supremum of the function G(t)≔∑∞k=1log|2sinπkt|/(4k2−1), over the set of irrationals. The coefficients on both the main term and the second term are shown to be best possible. This improves earlier bounds for f(m, n). It is conjectured that CG=G(2)≈ 0.236. We also obtain the following asymptotic formula: If α is a real algebraic integer of degree 2 with 0<α<1, then for any rational approximation n/m of α with 0<n<m we have f(m, n)=(4/π2)mlogm+(4/π2)(γ−log(π/2)+2G(α))m+ Oα (|α−nm|1/2mlogm).

„Volksmedizin" als historisches Konstrukt

The article investigates some of the ideas held by the non-specialist general public as to the causes of smallpox. By using medical reports of early 19th century physicians on smallpox epidemics, it is possible to analyse some of these popular theories, such as the possibility of material transmission of the disease from person to person, the role played by feelings of „disgust when confronted with a person suffering from the disease, the potential implications of innate impurities in the blood or of rotten blood left in the umbilical cord after birth, and so on. These ideas are then compared to the explanations given by contemporary doctors. The comparison shows that lay and academic explanations were actually very similar and were closely related to each other in a number of ways. This is demonstrated by looking at several examples of the mutual exchange of ideas between both types of explanation for the etiology of smallpox. Texts which try to emphasize the differences between lay and academic interpretations of the disease should therefore be understood as part of a historical construction of „folk medicine as a symbolic and supposedly irrational set of ideas and practices, which have been handed down from generation to generation in an unchanging fashion.

On functions that are trivial cocycles for a set of irrationals. II

Two results are obtained about the topological size of the set of irrationals for which a given function is a trivial cocycle. An example of a continuous function which is a coboundary with non- L 1 L^1 cobounding function is constructed.

Can chemicals be loved?—A problem for 2000

Public perception of the value of chemicals in daily life depends on an often irrational set of conclusions reached on partial or defective information often derived from single issue pressure groups. Benefits are consistently undervalued, clean water and uncontaminated food are taken for granted and changes in the pattern of disease are assumed to have occurred as a result of other kinds of intervention. If we are to ensure a proper evaluation of the place of chemicals in our lives then we will have to persuade opinion formers of the need for rational debate, inform the scientifically literate part of the population about hazard, risk, the difficult concept of safety and the dangers of mistaking association for causation. Changes in science in the field of toxicology give us a more reasonable ground for this debate than has existed in the past and the way in which these can be exploited will be discussed.

Lead Retention and Complexation in a Magnesium Smectite (Hectorite)

The retention and complexation of Pb 2+ in the interlayer of a smectite clay (hectorite) was investigated. Based on XRD results, the Ca-Pb exchanged clays most likely exhibit random interstratification. A random alternation of one-water-layer (∼ 12.5 Å) and two-water-layer (∼ 15.5 Å) hydrates as well as an irrational set of basal reflections are observed. At a specified humidity, the basal spacings and IR intensity of the water-bending mode decrease as the lead content in the clays increases. The Ca-Pb exchange isotherm clearly shows increasing selectivity of the clay for Pb 2+ over Ca 2+ with increasing Ca 2+ concentrations in the clay. The thermodynamic equilibrium constant is 5.18 and the standard free energy change is -4.08 kJ/mol at 298 K. The higher adsorption specificity for Pb 2+ over Ca 2+ could be partly due to the difference in ionic potentials and hydration energies between Pb 2+ and Ca 2+. However, in the solid retrieved from the exchange solution, an infrared spectroscopy study has shown the formation of a lead hydroxycarbonate complex. The role eventually played by this complex in enhancing the selectivity is unknown.

Phase transitions on strange irrational sets.

Nonanalyticities in the generalized dimensions of fractal sets of physical interest are interpreted as phase transitions. We apply the thermodynamical formalism to the fractal set formed by the irrational winding parameter values of critical circle maps and introduce and investigate in detail several distinct fractal measures on this set. The thermodynamic functions associated with different measures are distinct: We discover that, in all cases that we study, they exhibit phase transitions. The numerical estimates of the Hausdorff dimension from various versions of the thermodynamical formalism and a variety of circle maps yield ${D}_{H}$=0.8701\ifmmode\pm\else\textpm\fi{}0.0003 and are consistent with the conjectured universality of ${D}_{H}$.

On Functions that are Trivial Cocycles for a Set of Irrationals

On Functions that are Trivial Cocycles for a Set of Irrationals

On functions that are trivial cocycles for a set of irrationals

The main result of this paper is that the set of irrationals, for which a given function is a trivial cocycle, must be of the first category, unless the function is the exponential of a trigonometric polynomial.