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Introduction to Liouville Numbers

Adam Grabowski, Artur Korniłowicz
2017 Formalized Mathematics  
The aim is to show that all Liouville numbers are transcendental.  ...  A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational.  ...  It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number.  ... 
doi:10.1515/forma-2017-0003 fatcat:w4nyxx4nubgghbjict34tcyfxu

On the k-Riemann-Liouville fractional derivative

L. G. Romero, L. L. Luque, G. A. Dorrego, R. A. Cerutti
2013 International Journal of Contemporary Mathematical Sciences  
A new transform Elzaki transform of k-Wright function and r-Riemann-Liouville fractional integral are also obtained. Corollaries of the main theorems have also been derived.  ...  __________________________________________________________________________________ Abstract: In this paper some certain results of r-Riemann-Liouville fractional integration and differentiation of k-Wright  ...  Let be a real number such that The r-Riemann-Liouville fractional derivative was introduced as (cf.  ... 
doi:10.12988/ijcms.2013.13004 fatcat:fgcjer2okbh3dc76hks27mvv7u

Page 330 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 27, Issue [page]

1921 American Mathematical Society. Bulletin of the American Mathematical Society  
330 AN ARITHMETIC FORMULA OF LIOUVILLE. [Apr., ON A GENERAL ARITHMETIC FORMULA OF LIOUVILLE. BY PROFESSOR E. T. BELL. (Read before the American Mathematical Society April 9, 1921.) 1. Introduction.  ...  In his History of the Theory of Numbers (vol. 2, chap. 11), Dickson gives a resumé of the present state of the celebrated general formulas of Liouville in the theory of numbers, and remarks that the only  ... 

Numerical Computation Of Sturm-Liouville Problem With Robin Boundary Condition

Theddeus T. Akano, Omotayo A. Fakinlede
2015 Zenodo  
The modelling of physical phenomena, such as the earth's free oscillations, the vibration of strings, the interaction of atomic particles, or the steady state flow in a bar give rise to Sturm- Liouville  ...  Numerical solution of classical Sturm–Liouville problem is presented. The results show an agreement with the exact solution. High results precision is achieved with higher number of elements.  ...  INTRODUCTION TURM-LIOUVILLE boundary value problem or eigenvalue problem is an important theory, which essentially is an extension of the spectral theorem from discretised vector spaces into continuous  ... 
doi:10.5281/zenodo.1110336 fatcat:s6k6jvx4z5eepoz5gqlurklh2y

THE STOYANOVSKY–RIBAULT–TESCHNER MAP AND STRING SCATTERING AMPLITUDES

GASTON GIRIBET, YU NAKAYAMA
2006 International Journal of Modern Physics A  
Furthermore, we discuss the scattering amplitudes violating the winding number conservation in those backgrounds and provide a formula connecting such amplitudes with observables in Liouville field theory  ...  For instance, we analyze how the divergences corresponding to worldsheet instantons in AdS_3 can be understood as arising from the insertion of the dual screening operator in the Liouville theory side.  ...  Acknowledgements Gaston Giribet thanks Simeon Hellerman, Sameer Murthy and Ari Pakman for useful conversations on sine-Liouville field theory and related topics.  ... 
doi:10.1142/s0217751x06031697 fatcat:3fy74nnjvzdbjifvwmcfnx7dxe

On a question proposed by K. Mahler concerning Liouville numbers [article]

Jean Lelis, Diego Marques, Josimar Ramirez
2018 arXiv   pre-print
In this work, we provide an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.  ...  In 1906, Maillet proved that given a non-constant rational function f, with rational coefficients, if ξ is a Liouville number, then so is f(ξ).  ...  Thus, in order to prove that F (ξ) is a Liouville number, it suffices to prove that |F (ξ) − γ n | > 0 for infinitely many integers n.  ... 
arXiv:1603.04111v3 fatcat:bjae2iragzfv5gxzznirzmxa5y

On the Arithmetic Behavior of Liouville Numbers under Rational Maps [article]

Ana Paula Chaves, Diego Marques, Pavel Trojovský
2019 arXiv   pre-print
In 1972, Alniaçik proved that every strong Liouville number is mapped into the set of U_m-numbers, for any non-constant rational function with coefficients belonging to an m-degree number field.  ...  In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain  ...  Introduction The beginning of the transcendental number theory happened in 1844, when Liouville [11] showed that algebraic numbers are not "well-approximated" by rationals.  ... 
arXiv:1910.14190v1 fatcat:6fdzfalsfnhklpsaussncltigm

Normality and Finite-state Dimension of Liouville numbers [article]

Satyadev Nandakumar, Santhosh Kumar Vangapelli
2014 arXiv   pre-print
Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers.  ...  In the second number-theoretic construction, we use an arithmetic property of numbers - the existence of primitive roots - to construct Liouville numbers normal in finitely many bases, assuming a Generalized  ...  Acknowledgements The first author would like to gratefully acknowledge the help of David Kandathil and Sujith Vijay, for their suggestions during early versions of this work and Mrinal Ghosh, Aurko Roy  ... 
arXiv:1204.4104v2 fatcat:ip5u7lyuevdojoegq2xxdtpxia

Non-Liouville numbers and a Theorem of Hörmander

Edith Kregelius Petersen, G.H Meisters
1978 Journal of Functional Analysis  
Hence Lemma 2 applies to show that a//3 is not a Liouville number.  ...  A and B in b', then a//I is not a Liouville number.  ... 
doi:10.1016/0022-1236(78)90002-2 fatcat:kfah675wizhwrpyebjl4ol3v7e

An irrationality measure for Liouville numbers and conditional measures for Euler's constant [article]

Jonathan Sondow
2003 arXiv   pre-print
The irrationality exponent μ(t) of an irrational number t, defined using the irrationality measure 1/q^μ, distinguishes among non-Liouville numbers and is infinite for Liouville numbers.  ...  Using the irrationality measure 1/β^q, we define the "irrationality base" β(t), which distinguishes among Liouville numbers and is 1 for non-Liouville numbers. We give some properties and examples.  ...  In particular, a super Liouville number is also a Liouville number. Proof.  ... 
arXiv:math/0307308v1 fatcat:bnjxra7hxvdu3b3ygsa6s7uo4y

(0, 2) CHIRAL LIOUVILLE FIELD THEORY

YU NAKAYAMA
2013 Modern Physics Letters A  
existence proof of the (0,2) heterotic supercurrent supermultiplets in (1+1) dimensional quantum field theories which are consistent with the warped superconformal algebra, we construct the (0,2) chiral Liouville  ...  Introduction of Liouville potential One feature of the construction of the (0, 2) chiral Liouville field theory in the last section was there is one-parameter freedom to choose the energy-momentum tensor  ...  The introduction of the Liouville potential term will be discussed in section 4 with the addition of a Fermi-multiplet.  ... 
doi:10.1142/s0217732313501782 fatcat:ji2piteshbeldcfgpbfz7tx4ve

Transcendence Measures for some U_m-numbers related to Liouville's constant [article]

Ana Paula Chaves, Diego Marques
2009 arXiv   pre-print
In this note, we shall prove that the sum and the product of an algebraic number α by the Liouville constant L=∑_j=1^∞10^-j! is a U-number with type equals to the degree of α (with respect to Q).  ...  Acknowledgement The second author thanks to the Department of Mathematics of Universidade Federal do Ceará for its hospitality. We also thank Yann Bugeaud for nice discussions on the subject.  ...  Introduction A real number ξ is called a Liouville number, if for any positive real number w there exist infinitely many rational numbers p/q, with q ≥ 1, such that 0 < ξ − p q < 1 q w .  ... 
arXiv:0910.3715v1 fatcat:3nwesis5vrbyvf4oabhh656uja

Bulk one-point function on disk in one-matrix model

Alexander Belavin, Chaiho Rim
2010 Physics Letters B  
Using the recently found so-called resonance transformation from the KdV to the Liouville frame, we obtain an explicit expression for the bulk one-point function.  ...  We consider bulk correlation numbers on disk in one-matrix model.  ...  Aknowledgements Authors are indebted to V. Belavin  ... 
doi:10.1016/j.physletb.2010.03.020 fatcat:u5ybxyjx6vgjfmtwtmsxc22tcm

On the transcendence of some power series

Gül Karadeniz Gözeri, Ayten Pekin, Adem Kılıçman
2013 Advances in Difference Equations  
It is proved that these values are either a Liouville number or a rational number under certain conditions.  ...  In this study, we consider some power series with rational coefficients and investigate transcendence of the values of these series for Liouville number arguments.  ...  The first author acknowledges that this work was partially supported by Scientific Research Projects Coordination Unit of Istanbul University under project number 3414.  ... 
doi:10.1186/1687-1847-2013-17 fatcat:tlgg2jsuujbjlbvgscqg6zfrh4

An extended liouville equation for variable particle number systems [article]

Michael H. Peters
1998 arXiv   pre-print
It is well-known that the Liouville equation of statistical mechanics is restricted to systems where the total number of particles (N) is fixed.  ...  In general, the extended Liouville equation can be used to study nonequilibrium systems where the total number of particles can vary.  ...  Acknowledgements I want to thank Professor I. Bitsanis for suggesting this problem to me.  ... 
arXiv:physics/9809039v2 fatcat:y4mo4a7rbzeybl4zymz3tg6bha
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