Irrationality Measures Research Articles

Rational approximations to {\@root 3 \of {2}} and other algebraic numbers revisited

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $\theta(k,l;x)$ and $\psi(k,l;x)$ for $k=1,3,4,6$ are also presented.

Propriétés diophantiennes du développement en cotangente continue de Lehmer

Diophantine Properties of Lehmer’s Continued Cotangent Developments. This article deals with an algorithm devised by Lehmer which enables us to write any real positive number as the sum of an alternating series of cotangents of integers nν, ν ≥ 0, in a unique way. We continue the work begun by Lehmer and continued by Shallit: amongst other things, we give explicitly the link between the rational approximations of a given real number coming from this algorithm and the usual convergents of the same real number and we produce a quasi-optimal bound for the growth of the sequence (nν)ν ≥ 0 associated to an algebraic number. We also determine the regular continued fractions of an exceptional class of continued cotangent developments, which enables us to produce optimal irrationality measures of these expansions.

The Varied Dimensions of Law, Policy, and Treatment of Convicted Sex Offenders

A seemingly endless struggle continues to exist over policy prescriptions for convicted sex offenders. The oversimplified answer is to enact more punitive measures, however futile they may be, to appease a misinformed populace. This review essay looks at three titles that examine the policy justifications for sex offenders and explain the expensive and irrational measures taken in both American and British society to try to gain some control over this class of criminal and mental health cases.

Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y2 = x3 + Ax + B, where A and B are integers, has no rational points of finite order if A is sufficiently large relative to B (at least if one assumes the abc Conjecture of Masser and Oesterlé). In the present article we show, perhaps surprisingly, that the rational torsion on the above curve is also quite restricted if B is sufficiently large relative to A. In particular, we demonstrate that for any ε > 0 there is a constant cε such that if A and B are integers satisfying |B| > cε |A|6+ε, then the elliptic curve defined above has no rational torsion points, other than a possible point of order 2 (again making use of the abc Conjecture in some cases). We then extend this by proving similar results for elliptic curves admitting non-trivial ℚ-isogenies, elliptic curves written in other forms, and elliptic curves over certain number fields. Curiously, the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers.

Diophantine properties of real numbers generated by finite automata

We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.

On conditional irrationality measures for values of the digamma function

The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function.

A Geometric Proof That e Is Irrational and a New Measure of Its Irrationality

The proof leads in section 3 to a new measure of irrationality for e, that is, a lower bound on the distance from e to a given rational number, as a function of its denominator. A connection with the greatest prime factor of a number is discussed in section 4. In section 5 we compare the new irrationality measure for e with a known one, and state a numbertheoretic conjecture that implies the known measure is almost always stronger. The new measure is applied in section 6 to prove a special case of a result from [24], leading to another conjecture. Finally, in section 7 we recall a theorem of G. Cantor that can be proved by a similar construction.

A Geometric Proof that 𝑒 Is Irrational and a New Measure of Its Irrationality

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Irrationality measures are given for the values of the series $$\sum_{n=0}^{\infty} t^{n}/W_{an+b}$$ , where $$a,b\in\mathbb{Z}^+, 1\le b\le a, (a,b)=1$$ and W n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers $$ \sum_{n=0}^\infty \frac{1}{f_{an+b}},\quad \sum_{n=0}^\infty \frac{1}{l_{an+b}}, $$ where f n and l n are the Fibonacci and Lucas numbers, respectively.

New irrationality measures for 𝑞-logarithms

The three main methods used in diophantine analysis of q q -series are combined to obtain new upper bounds for irrationality measures of the values of the q q -logarithm function \[ ln q ⁡ ( 1 − z ) = ∑ ν = 1 ∞ z ν q ν 1 − q ν , | z | ⩽ 1 , \ln _{q}(1-z)=\sum _{\nu =1}^{\infty }\frac {z^{\nu }q^{\nu }}{1-q^{\nu }}, \qquad |z|\leqslant 1, \] when p = 1 / q ∈ Z ∖ { 0 , ± 1 } p=1/q\in \mathbb {Z}\setminus \{0,\pm 1\} and z ∈ Q z\in \mathbb {Q} .

Ramanujan-type formulae and irrationality measures of some multiples of

An explicit construction of simultaneous Padé approximations for generalized hypergeometric series and formulae for the quantities , , in terms of these series are used for estimates of irrationality measures of these multiples of . Other possible applications are also discussed.

Sinn und Unsinn der Antibiotikatherapie respiratorischer Infekte

Whether an antibiotic successfully eradicates pathogens depends on the pathogens involved, on pharmacokinetics and bioavailability in the target tissue, and on the antimicrobial resistance of the pathogen. Other determinants are drug interactions, individual risk factors, age and compliance with respect to correct dosage and duration of therapy. In many cases, antimicrobial therapy is begun on an empirical basis, because the responsible pathogen can be identified in only half of all respiratory infections. The eradication of the pathogen has to be the first aim if treatment is to be curative and the development of resistance prevented. Long-term prevention of antimicrobial resistance will require a more critical prospective evaluation of the prescription of antibiotics. This paper considers rational and irrational measures in the antimicrobial therapy of respiratory infections.

Irrationality measure of sequences

The new concept of an irrationality measure of sequences is introduced in this paper by means of the related irrational sequences. The main results are two criteria characterising lower bounds for the irrationality measures of certain sequences. Applications and several examples are included.

IRRATIONAL BELIEFS AND SOCIAL CONSTRUCTIONISM: CORRELATIONS WITH ATTITUDES ABOUT REALITY, BELIEFS ABOUT PEOPLE, AND COLLECTIVE SELF-ESTEEM

Rational Emotive Behavior Therapy (REBT) is theoretically consistent with the postmodern social constructionist assumption of a reciprocal causality between the individual and society. In a test of this claim, REBT irrationalities as measured by the Irrational Belief Scale and by the Survey of Personal Beliefs were correlated with Attitudes toward Reality, Cynicism, Beliefs in the Conventional Goodness of People, and Collective Self-Esteem. All measures of irrationality identified the postmodern social constructionism recorded by the Attitudes toward Reality Scale as rational. Especially the Irrational Belief Scale identified Cynicism and Beliefs in the Conventional Goodness of People as irrational and Collective Self-Esteem as rational. The Survey of Personal Beliefs described Collective Self-Esteem as irrational and therefore suggested that the conceptual framework of REBT can promote negative assessments of collectivistic forms of adjustment. Most importantly, however, these data confirmed that REBT includes potentials that are compatible with postmodern social constructionism.

Reanalysis of Thickness Loss Data for Weathering Steel

The current ASTM Standard Guide G101 has a method for estimating the atmospheric corrosion resistance of low-alloy steels from the content of five elements: Cu, Ni, Cr, Si, and P. The ASTM method was developed by fitting a quadratic model to thickness loss data from specimens that were made from 270 different steel compositions and were exposed to the atmosphere for 15.5 years. The quadratic structure resulted in a model that provides an irrational measure of the effect of the most important alloying element. Additionally, the use of the correlation coefficient as the single criterion for selecting predictor variables led to inaccurate coefficients. The reanalysis of the writers, using multiple criteria for selecting the best predictor variables, yields more accurate coefficients and a level of explained variance of nearly 80%. While the multiple criteria analysis of the writers does not sacrifice prediction accuracy to achieve rationality, the problem inherent to the quadratic model cannot be overcome altogether in low-alloy steels containing more than about 0.40% copper. The corrosion index of Guide 101 should not be used to decide whether a low-alloy steel qualifies as a weathering steel.

Heine's basic transform and a permutation group for q-harmonic series

Let p = 1/q ∈ Z {0,±1}. The new bound μ(hp(1)) 6 2.46497868 . . . for the irrationality measure of the q-harmonic series hp(1) = ∑∞ n=1 q n/(1 − qn) is proved. The essential ingredient of the proof is a permutation group originated by classical transform of the Heine series. 2000 Mathematics Subject Classification. 11J82, 33D15.

Alexithymia, Irrational Beliefs, and the Rational-Emotive Explanation of Emotional Disturbance

In recent discussions of Rational-Emotive Behavior Therapy (REBT), Ellis has argued that unconscious processes supplement irrational beliefs in explaining psychopathology. The hypothesis of this investigation was that these unconscious processes reflect alexithymia. University students responded to the three factors from the Toronto Alexithymia Scale, the Malouff and Schutte Belief Scale (MSBS), 5 subscales from the Survey of Personal Beliefs, and instruments recording depression, anxiety, and neuroticism. As hypothesized, alexithymia and irrational beliefs predicted greater emotional disturbance while also displaying some small positive correlations with each other. In multiple regressions, both types of variables combined to predict emotional disturbance, with alexithymia explaining the greater amount of variance. Alexithymic difficulties in identifying emotions had uniquely negative mental health implications. The MSBS and the Low Frustration Tolerance subscale were the most noteworthy measures of a pathogenic irrationality. These data supported the claim that the unconscious processes of REBT could be conceptualized in terms of alexithymia.

Birational transformations and values of the Riemann zeta-function

In his proof of Apery’s theorem on the irrationality of ζ(3), Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to ζ(2) and ζ(3). Beukers’ method was subsequently improved by Dvornicich and Viola, by Hata, and by Rhin and Viola. We give here a survey of our recent results ([RV2] and [RV3]) on the irrationality measures of ζ(2) and ζ(3) based upon a new algebraic method involving birational transformations and permutation groups acting on double and triple integrals of Beukers’ type. In the last two sections we give a constructive method to obtain the relevant birational transformations for triple integrals from the analogous transformations for double integrals, and we also apply such a method to get birational transformations acting on quadruple integrals of Vasilyev’s type.

Irrationality Measures for the Series of Reciprocals from Recurrence Sequences

Using Padé approximations of Heine's q-hypergeometric series we obtain new irrationality measures for the values of the series ∑ ∞ n=1 t n / W n , where W n is a Fibonacci or Lucas type arithmetical form satisfying the recurrence W n+2=rW n+1+sW n r, s∈ Q * with initial values W 1, W 2 and t∈ Q *.

Irrationality Measures for the Series of Reciprocals from Recurrence Sequences

Using Padé approximations of Heine's q-hypergeometric series we obtain new irrationality measures for the values of the series ∑∞n=1tn/Wn, where Wn is a Fibonacci or Lucas type arithmetical form satisfying the recurrenceWn+2=rWn+1+sWnr, s∈Q* with initial values W1, W2 and t∈Q*.