Tanaka, Takaaki

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Associate Professor

E-mail Address

E-mail address

Related Websites

External Links

Career 【 Display / hide

  • 1996.04
    -
    1998.03

    日本学術振興会 ,特別研究員

  • 1998.04
    -
    2007.03

    慶應義塾大学(理工学部),助手

  • 2001.02
    -
    2002.03

    フンボルト財団(ドイツ連邦共和国,ケルン大学) ,奨学研究員

  • 2006.04
    -
    2009.03

    慶應義塾大学(経済学部),非常勤講師

  • 2007.04
    -
    2010.03

    慶應義塾大学(理工学部),助教

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Academic Background 【 Display / hide

  • 1993.03

    Keio University, Faculty of Science and Engineering, 数理科学科

    University, Graduated

  • 1995.03

    Keio University, Graduate School, Division of Science and Engineeri, 数理科学専攻

    Graduate School, Completed, Master's course

  • 1998.03

    Keio University, Graduate School, Division of Science and Engineeri, 数理科学専攻

    Graduate School, Completed, Doctoral course

Academic Degrees 【 Display / hide

  • 博士(理学), Keio University, Coursework, 1998.03

Licenses and Qualifications 【 Display / hide

  • 気象予報士, 1999.10

 

Research Areas 【 Display / hide

  • Algebra (Analytic Number Theory)

Research Keywords 【 Display / hide

  • algebraic independence

  • transcendental numbers

 

Papers 【 Display / hide

  • Algebraic independence of the values of a certain map defined on the set of orbits of the action of Klein four-group

    TANAKA TAKA-AKI

    RIMS Kôkyûroku (RIMS)  2131   177 - 187 2019.10

    Research paper (international conference proceedings), Single Work, Except for reviews

     View Summary

    In the previous work the author constructed a function of two variables taking algebraically independent values at any distinct points belonging to the cartesian product of the set of nonzero algebraic numbers and that of nonzero algebraic numbers inside the unit circle. In this paper the author constructs a balanced analogue of the two-variable function stated above, which is defined on a wider domain, the cartesian product of the set of complex numbers and that of complex numbers except the unit circle. This balanced function can also be regarded as a map defined on the set of orbits of the action of Klein four-group on the domain of definition of the balanced function and the latter map takes algebraically independent values at any distinct orbits represented by algebraic points.

  • Algebraic independence of the values of the Hecke-Mahler series and its derivatives at algebraic numbers

    Tanaka, Taka-aki, Tanuma, Yusuke

    Int. J. Number Theory (International Journal of Number Theory)  14 ( 9 ) 2369 - 2384 2018.10

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  17930421

     View Summary

    Hecke-Mahler series is the generating function of the sequence {[nω]} of the integral parts [nω] of the positive integer multiple nω for ω real. In this paper, we prove that the Hecke-Mahler series has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition, including the case where ω is in addition an algebraic integer. Using Mahler's method, we reduce the algebraic independence of the values of the Hecke-Mahler series to the linear independence of the Hecke-Mahler series themselves modulo the rational function field. For proving this linear independence, we observe the distribution of the sequence of points whose components consist of the fractional parts of the components of scalar
    multiple of a real vector.

  • Algebraic independence of the values of functions satisfying Mahler type functional equations under the transformation represented by a power relatively prime to the characteristic of the base field

    Goto, Akinari, Tanaka Taka-aki

    J. Number Theory (Journal of Number Theory)  184   384 - 410 2018.03

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  0022314X

     View Summary

    We give positive characteristic analogues of complex entire functions having remarkable property that their values as well as their derivatives of any order at any nonzero algebraic numbers are algebraically independent. These results are obtained by establishing a criterion for the algebraic independence of the values of Mahler functions as well as that of the algebraic independence of the Mahler functions themselves over any function fields of positive characteristic.

  • Explicite algebraic dependence formulae of infinite products related with Fibonacci and Lucas numbers

    Kaneko, H., Kurosawa, T., Tachiya, Y., and Tanaka Taka-aki

    Acta Arithmetica 168 ( 2 ) 161 - 186 2015.04

    Research paper (scientific journal), Joint Work, Accepted

  • Arithmetic properties of solutions of certain functional equations with transformations represented by matrices including a negative entry

    Tanaka, Taka-aki

    Tokyo J. Math. 37 ( 1 ) 211 - 223 2014.06

    Research paper (scientific journal), Single Work, Accepted

     View Summary

    Mahler関数において変数の変換を表す行列は非負行列に限る、という従来の制約の外にある多変数Mahler関数の新しいクラスについてその値の代数的独立性を示し、2項回帰数列で生成されるLaurent級数の値の代数的独立性を証明した。

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Papers, etc., Registered in KOARA 【 Display / hide

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Presentations 【 Display / hide

  • Algebraic independence properties of a certain map defined on the set of orbits of the action of Klein four-group

    TANAKA TAKA-AKI

    Keio-Yonsei Number Theory Workshop (Keio Univ.) , 2018.12, Oral Presentation(general)

     View Summary

    In the previous work the speaker constructed a function of two variables taking algebraically independent values at any distinct points belonging to the cartesian product of the set of nonzero algebraic numbers and that of nonzero algebraic numbers inside the unit circle. In this talk the speaker
    introduces a balanced analogue of the two-variable function stated above,
    which is defined on a wider domain, the cartesian product of the set of
    complex numbers and that of complex numbers except the unit circle.
    The balanced function can also be regarded as a map defined on the set of orbits of the action of Klein four-group on the domain of definition of the balanced function and the latter map takes algebraically independent values at any distinct orbits represented by algebraic points.

  • Algebraic independence of the values of a certain map defined on the set of orbits of the action of Klein four-group

    TANAKA TAKA-AKI

    Analytic Number Theory and Related Topics (Kyoto) , 2018.10, Oral Presentation(guest/special), Research Institute for Mathematical Sciences

     View Summary

    先行研究において講演者は、0以外の代数的数全体と単位円内の0でない代数的数全体の直積集合に属する任意の相異なる点における値がすべて代数的独立となる2変数関数を構成した。 本講演では、上記の2変数関数にある種のバランス化を施し、定義域を複素数全体と単位円周を除く複素数全体の直積集合に拡張した関数を紹介する。この関数は定義域に対するクラインの4元群の作用に関する軌道全体の集合上で定義された写像とも見做すことができ、この写像は代数点に代表される任意の相異なる軌道における値がすべて代数的独立となる。

  • On the functions having `perfect' algebraic independence property at algebraic numbers

    Taka-aki Tanaka

    Diophantine Analysis and Related Fields 2017 (Tokyo) , 2017.01, Oral Presentation(general), Nihon Univ.

  • 有限個の素数pに対するQpとRの`共通部分'に属する超越数から成る代数的独立な無限集合について

    Taka-aki Tanaka, Miho Nakashima

    日本数学会年会 (Tsukuba Univ.) , 2016.03, Oral Presentation(general)

  • Algebraic independence of the values of Mahler functions in a certain case of positive characteristic

    AKINARI GOTO, TAKA-AKI TANAKA

    Diophantine Analysis and Related Fields 2015 (桐生市市民文化会館) , 2015.03, Oral Presentation(general), Gunma University

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Research Projects of Competitive Funds, etc. 【 Display / hide

  • 完全代数的独立性の拡張と高次元化

    2020.04
    -
    2024.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, 田中 孝明, Grant-in-Aid for Scientific Research (C), Principal Investigator

  • 非アルキメデス付値体における完全代数的独立性

    2015.04
    -
    2019.03

    日本学術振興会, JSPS

  • Mahler関数の特殊値による超越数の構造解明

    2010.04
    -
    2013.03

    日本学術振興会, JSPS, Principal Investigator

  • p進数体及び関数体上のMahler関数の超越性と代数的独立性及びその応用

    2006.04
    -
    2009.03

    JSPS, 科学研究費補助金, Principal Investigator

 

Courses Taught 【 Display / hide

  • TOPICS IN ALGEBRA A

    2020

  • MATHEMATICS 4B

    2020

  • MATHEMATICS 4A

    2020

  • INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

    2020

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 2

    2020

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Courses Previously Taught 【 Display / hide

  • 代数学特論C

    慶應義塾大学大学院理工学研究科, 2018

  • 代数学特論A

    慶應義塾大学大学院理工学研究科, 2018

  • 代数学第1同演習

    慶應義塾大学理工学部, 2018

  • 代数学基礎同演習

    慶應義塾大学理工学部, 2018

  • 関数論第1同演習

    慶應義塾大学理工学部, 2018

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Social Activities 【 Display / hide

  • 日本気象予報士会

    1999.10
    -
    Present

Memberships in Academic Societies 【 Display / hide

  • 日本数学会, 

    1995.01
    -
    Present