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Introduction to Liouville Numbers

2017
*
Formalized Mathematics
*

The aim is

doi:10.1515/forma-2017-0003
fatcat:w4nyxx4nubgghbjict34tcyfxu
*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*. ...##
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On the k-Riemann-Liouville fractional derivative

2013
*
International Journal of Contemporary Mathematical Sciences
*

A new transform Elzaki transform of k-Wright function and r-Riemann-

doi:10.12988/ijcms.2013.13004
fatcat:fgcjer2okbh3dc76hks27mvv7u
*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. ...##
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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 ...##
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Numerical Computation Of Sturm-Liouville Problem With Robin Boundary Condition

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

doi:10.5281/zenodo.1110336
fatcat:s6k6jvx4z5eepoz5gqlurklh2y
*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 ...##
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THE STOYANOVSKY–RIBAULT–TESCHNER MAP AND STRING SCATTERING AMPLITUDES

2006
*
International Journal of Modern Physics A
*

Furthermore, we discuss the scattering amplitudes violating the winding

doi:10.1142/s0217751x06031697
fatcat:3fy74nnjvzdbjifvwmcfnx7dxe
*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. ...##
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On a question proposed by K. Mahler concerning Liouville numbers
[article]

2018
*
arXiv
*
pre-print

In this work, we provide an uncountable subset of

arXiv:1603.04111v3
fatcat:bjae2iragzfv5gxzznirzmxa5y
*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. ...##
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On the Arithmetic Behavior of Liouville Numbers under Rational Maps
[article]

2019
*
arXiv
*
pre-print

In 1972, Alniaçik proved that every strong

arXiv:1910.14190v1
fatcat:6fdzfalsfnhklpsaussncltigm
*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. ...##
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Normality and Finite-state Dimension of Liouville numbers
[article]

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 ...

##
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Non-Liouville numbers and a Theorem of Hörmander

1978
*
Journal of Functional Analysis
*

Hence Lemma 2 applies

doi:10.1016/0022-1236(78)90002-2
fatcat:kfah675wizhwrpyebjl4ol3v7e
*to*show that a//3 is not a*Liouville**number*. ... A and B in b', then a//I is not a*Liouville**number*. ...##
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An irrationality measure for Liouville numbers and conditional measures for Euler's constant
[article]

2003
*
arXiv
*
pre-print

The irrationality exponent μ(t) of an irrational

arXiv:math/0307308v1
fatcat:bnjxra7hxvdu3b3ygsa6s7uo4y
*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. ...##
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(0, 2) CHIRAL LIOUVILLE FIELD THEORY

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

doi:10.1142/s0217732313501782
fatcat:ji2piteshbeldcfgpbfz7tx4ve
*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. ...##
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Transcendence Measures for some U_m-numbers related to Liouville's constant
[article]

2009
*
arXiv
*
pre-print

In this note, we shall prove that the sum and the product of an algebraic

arXiv:0910.3715v1
fatcat:3nwesis5vrbyvf4oabhh656uja
*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 . ...##
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Bulk one-point function on disk in one-matrix model

2010
*
Physics Letters B
*

Using the recently found so-called resonance transformation from the KdV

doi:10.1016/j.physletb.2010.03.020
fatcat:u5ybxyjx6vgjfmtwtmsxc22tcm
*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 ...##
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On the transcendence of some power series

2013
*
Advances in Difference Equations
*

It is proved that these values are either a

doi:10.1186/1687-1847-2013-17
fatcat:tlgg2jsuujbjlbvgscqg6zfrh4
*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. ...##
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An extended liouville equation for variable particle number systems
[article]

1998
*
arXiv
*
pre-print

It is well-known that the

arXiv:physics/9809039v2
fatcat:y4mo4a7rbzeybl4zymz3tg6bha
*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. ...
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