The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Bryan Caplan - One of the best experts on this subject based on the ideXlab platform.
-
rational Irrationality and the microfoundations of political failure
Public Choice, 2001Co-Authors: Bryan CaplanAbstract:Models of inefficient political failure have been criticized forimplicitly assuming the Irrationality of voters (Wittman, 1989,1995, 1999; Coate and Morris, 1995). Building on Caplan's (1999a,1999b) model of ``rational Irrationality'', the current papermaintains that the assumption of voter Irrationality is boththeoretically and empirically plausible. It then examinesmicrofoundational criticisms of four classes of political failuremodels: rent-seeking, pork-barrel politics, bureaucracy, andeconomic reform. In each of the four cases, incorporating simpleforms of privately costless Irrationality makes it possibleto clearly derive the models' standard conclusions. Moreover, itfollows that efforts to mitigate political failures will besocially suboptimal, as most of the literature implicitlyassumes. It is a mistake to discount the empirical evidence forthese models on theoretical grounds.
-
rational ignorance versus rational Irrationality
Kyklos, 2001Co-Authors: Bryan CaplanAbstract:The paper presents a model of "rational Irrationality," to explain why political and religious beliefs are marked not only by low information (as the notion of rational ignorance highlights), but also by systematic bias and high certainty. Being irrational--i.e. deviating from rational expectations--is modeled as a normal good. The reason that Irrationality in politics and religion is so pronounced is that the private repercussions of error are virtually nonexistent. The consumption of Irrationality can be efficient, but it will usually not be when the private and the social cost of Irrationality differ--for example, in elections. Copyright 2001 by WWZ and Helbing & Lichtenhahn Verlag AG
-
rational Irrationality a framework for the neoclassical behavioral debate
Eastern Economic Journal, 2000Co-Authors: Bryan CaplanAbstract:Critics of behavioral economics often argue that apparent Irrationality arises mainly because test subjects lack adequate incentives; the defenders of behavioral economics typically reply that their findings are robust to this criticism. The current paper presents a simple theoretical model of "rational Irrationality" to clarify this debate, reducing the neoclassical-behavioral dispute to a controversy over the shape of agents' wealth/Irrationality indifference curves. Many experimental anomalies are consistent with small deviations from polar "neoclassical" preferences, but even mildly relaxing standard assumptions about preferences has strong implications. Rational Irrationality can explain both standard, costly biases, as well as wealth-enhancing Irrationality, but it remains inconsistent with evidence that intensifying financial incentives for rationality makes biases more pronounced.
Walter Van Assche - One of the best experts on this subject based on the ideXlab platform.
-
little q legendre polynomials and Irrationality of certain lambert series
arXiv: Classical Analysis and ODEs, 2001Co-Authors: Walter Van AsscheAbstract:We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Pad\'e approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the Irrationality, with an upper bound of the measure of Irrationality which is as sharp as the upper bound given by Bundschuh and V\"a\"an\"anen for the harmonic series and a better upper bound than the one given by Matala-aho and V\"a\"an\"anen for the logarithm.
-
little q legendre polynomials and Irrationality of certain lambert series
Ramanujan Journal, 2001Co-Authors: Walter Van AsscheAbstract:Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős (J. Indiana Math. Soc.12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory37, 1991, 253–259; Proc. Cambridge Philos. Soc.112, 1992, 141–146) used Pade approximation and complex analysis to prove the Irrationality of these q-harmonic series and of q-analogs ln p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math.20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of Irrationality similar to Apery's proof of Irrationality of ζ(2) and ζ(3). They also obtain an upper bound for the measure of Irrationality, but better upper bounds were earlier given by Bundschuh and Vaananen (Compositio Math.91, 1994, 175–199) and recently also by Matala-aho and Vaananen (Bull. Australian Math. Soc.58, 1998, 15–31) (for ln p (2)). In this paper we show how one can obtain rational approximants for hp(1) and ln p (2) (and many other similar quantities) by Pade approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the Irrationality, with an upper bound of the measure of Irrationality which is as sharp as the upper bound given by Bundschuh and Vaananen for hp(1) and a better upper bound as the one given by Matala-aho and Vaananen for ln p (2).
Mark Van Roojen - One of the best experts on this subject based on the ideXlab platform.
-
expressivism and Irrationality
The Philosophical Review, 1996Co-Authors: Mark Van RoojenAbstract:Noncognitive analyses of evaluative discourse characterize moral discourse as primarily functioning to express attitudes that are not, strictly speaking, representational in the way that ordinary beliefs are representational. But, since expressivists must explain our practices of making evaluative judgments as we do, they owe us an explanation of the logical relations between these evaluative judgments and other judgments. For it is part of our ordinary evaluative practices to make inferences based upon and leading to evaluative judgments. The most thorny problem for this project has been to explain the logical relations between evaluative judgments and other judgments best expressed using evaluative terms in unasserted contexts, such as clauses embedded in conditionals. Because one may use evaluative terms in such unasserted contexts without expressing the attitudes usually associated with them in asserted contexts, it becomes hard to explain why there should be logical relations between the judgments expressed. The noncognitivist who has given this problem the most sustained attention over the years is Simon Blackburn,' and recently he has been joined by Allan Gibbard.2
Lihong Wang - One of the best experts on this subject based on the ideXlab platform.
-
on the Irrationality measure of log3
Journal of Number Theory, 2014Co-Authors: Qiang Wu, Lihong WangAbstract:Abstract In this paper, we obtain a new estimate of an Irrationality measure of the number log 3 . We have μ ( log 3 ) ≤ 5.1163051 with an “arithmetical method”. The previous results were μ ( log 3 ) ≤ 8.616 … by G. Rhin in 1987 and μ ( log 3 ) ≤ 5.125 … by V.H. Salikhov in 2007.
Theodore A Slaman - One of the best experts on this subject based on the ideXlab platform.
-
Irrationality exponent hausdorff dimension and effectivization
Monatshefte für Mathematik, 2018Co-Authors: Veronica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the Irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have Irrationality exponent equal to a. We give an analogous result relating the Irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and Irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
-
Irrationality exponent hausdorff dimension and effectivization
arXiv: Number Theory, 2016Co-Authors: Veronica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the Irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have Irrationality exponent equal to a. We give an analogous result relating the Irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and Irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
-
the Irrationality exponents of computable numbers
Proceedings of the American Mathematical Society, 2015Co-Authors: Veronica Becher, Yann Bugeaud, Theodore A SlamanAbstract:is satisfied by an infinite number of integer pairs (p, q) with q > 0. Rational numbers have Irrationality exponent equal to 1, irrational numbers have it greater than or equal to 2. The Thue–Siegel–Roth theorem states that the Irrationality exponent of every irrational algebraic number is equal to 2. Almost all real numbers (with respect to the Lebesgue measure) have Irrationality exponent equal to 2. The Liouville numbers are precisely those numbers having infinite Irrationality exponent. For any real number a greater than or equal to 2, Jarnik (1931) used the theory of continued fractions to give an explicit construction, relative to a, of a real number xa such that the Irrationality exponent of xa is equal to a. For a = 2, we can take x2 = √ 2. For a > 2, we construct inductively the sequence of partial quotients of xa = [0; a1, a2, . . .]. For n ≥ 1, set pn/qn = [0; a1, a2, . . . , an]. Take a1 = 2 and an+1 = bqa−2 n c, for n ≥ 1, where b·c denotes the integer part function. Then, the theory of continued fractions (see Schmidt, 1980) directly gives that the Irrationality exponent of xa is equal to a. The theory of computability defines a computable function from non-negative integers to non-negative integers as one which can be effectively calculated by some algorithm. The definition extends to functions from one countable set to another, by fixing enumerations of those sets. A real number x is computable if there is a base and a computable function that gives the digit at each position of the expansion of x in that base. Equivalently, a real number is computable if there is a computable sequence of rational numbers (rj)j≥0 such that |x− rj | < 2−j for j ≥ 0. The construction cited above shows that for any computable real number a there is a computable real number xa whose Irrationality exponent is equal to a. What of the inverse question? Are there computable numbers with non-computable Irrationality exponents? Theorem 1 gives a characterization of the Irrationality exponents of computable real numbers.
-
the Irrationality exponents of computable numbers
arXiv: Number Theory, 2014Co-Authors: Veronica Becher, Yann Bugeaud, Theodore A SlamanAbstract:We prove that a real number a greater than or equal to 2 is the Irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers whose Irrationality exponent is not computable.
