PR
Keyword Search
▼キーワード検索
再生核研究所
再生核研究所声明
Calendar
Comments
Freepage List
2013.11.30
カテゴリ： カテゴリ未分類
再生核研究所声明１４２ アヴェイロの夢 （２０１３．１１．３０）
\documentclass{article}
\title{\Large\textbf{Announcement 142: An Aveiro Dream in Mathematics\\
}}
\author{ Institute of Reproducing Kernels, \\ Kawauchi-cho 5-1648-16, \\Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp}
\begin{document}
\maketitle
\begin{abstract}\noindent
\end{abstract}
\textbf{Keywords:} Linear operator, eigenfunction, initial value problem, integral transform, reproducing kernel, linear differential equation, partial differential equation, linear integral operator, Dirac delta function, Green function.
\section{Introduction}
In \cite{rassias}, we stated that mathematic is a set of relations and the fundamental relation
\begin{equation}
e^{\pi i} = -1
\end{equation}
among the basic numbers $-1, \pi, i, e$ is the best result in mathematics. There, we stated also that good results are fundamental, beautiful and give good impact to human beings. At the same time, in the paper we showed the basic relation between {\it analyticity of functions} and {\it nonlinear transforms}, by using the theory of reproducing kernels.
\section{God loves two}
On the universe, we stated repeatedely: The God loves two; and two is a fundamental concept on the universe, in our general, essayes of \cite{saito,Sa12} and we love the famous Pythagorean theorem with two powers. We established a general Pythagorean theory between the inputs and the outputs of a general linear system in the framework of Hilbert spaces in \cite{Sa83} and we established many entirely new type Pythagorean (isometric identity) theorems and we called it the fundamental theory in linear mappings. The theory establishes the basic fundamental relations among the inputs, the outputs and linear systems in the framework of Hilbert spaces \cite{Sa83}. The theory is the core of the theory of reproducing kernels, and the basic concepts Dirac's delta function and Green's function are the family of reproducing kernels.
\section{Aveiro Discretization Method }
\section{An Aveiro Dream in Mathematics}
By combining the very specialized research result of Professor M. M. Rodrigues and the Aveiro discretization method using the fundamental theory of linear mappings, we found the basic relations among linear operators, eigenfunctions, linear initial value problems, integral transforms and reproducing kernels.
Roughly speaking, when we know some eigenfunctions of a linear operator, we can consider the related partial differential equation and we can solve the associated initial value problem; in this method, we shall consider the reproducing kernel forms and related integral transforms ( linear mappings), and we can discuss the existence problem and construction problem of the initial value problem, and furthermore, we can consider the complete property of the solutions by using the theory of reproducing kernels.
From this general method, we find that we can consider many and many integral transforms and reproducing kernels in concrete forms from the known eigenfunctions.
We know a great tradition on concrete forms in Russia; many definite integrals, many eigenfunctions, many analytical solutions in differential and integral equations. Our theory will give a great impact on these topics.
\section{Research projects}
Our output results in the Aveiro Dream in Mathematics may be stated as follows:
1) Many concrete reproducing kernels may be calculated and the related reproducing kernel Hilbert spaces should be realized with concrete norms.
2) Eigenfunctions and the related initial value problems in partial differential and integral equations should be examined with their properties of the solutions.
3) Many new integral transforms and their properties; that is, isometric identities and inversion formulas should be established.
4) For the associated $t$ kernels and the related small reproducing kernels appeared in the general theory, we can consider the similar problems above.
From the great references \cite{poly02,poly03,poly08, gr, Pru3} containing the special function theory, we may consider expected new materials as the Aveiro Dream in Mathematics. We believe such materials in mathematics are definite values and fundamentals in mathematics.
\begin{thebibliography}{00}
\bibitem{cfrst}
L.P. Castro, H. Fujiwara, M.M. Rodrigues, S. Saitoh and V. K. Tuan,
\textit{Aveiro Discretization Method in Mathematics: A New Discretization Principle, MATHEMATICS WITHOUT BOUNDARIES: SURVEYS IN PURE MATHEMATICS}, Edited by Panos Pardalos and Themistocles M. Rassias (to appear). 52 pages.
\bibitem{cfqs} L.P. Castro, H. Fujiwara, T. Qian and S. Saitoh, \textit{How to catch smoothing properties and analyticity of functions by computers?,} MATHEMATICS WITHOUT BOUNDARIES: SURVEYS IN PURE MATHEMATICS, Edited by Panos Pardalos and Themistocles M. Rassias (to appear).
\bibitem{Erdlyi}
A. Erd\'elyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, 1954. \textit{Tables of Integral Transforms-Vol I, II}. Bateman Manuscript Project. California Institute of Technology, McGraw Hill, New York.
\bibitem{gr}
I. S. Gradshlein and I. M. Ryzhik,
\textit{Table of Integrals, Series, and Products}, 7th Edition, Elsever Inc. 2007.
\bibitem{Pru3}
A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, \textit{Integrals and Series, Volume 3: More special functions}, Gordon and Breach Publisher, New York, 1990.
\bibitem{poly02}
A. D. Polyanin, \textit{Handbook of Linear Partial Differential Equations for Engineers and Scientists}, CRC Press, 2002.
\bibitem{poly03}
A. D. Polyanin and V. F. Zaitsev, \textit{Handbook of Exact Solutions for Ordinary Differential Equations}, CRC Press, 2003.
\bibitem{poly08}A. D. Polyanin and A. V. Manzhirov, \textit{Handbook of Integral Equations}, CRC Press, 2008.
\bibitem{rassias}
T. M. Rassias,
Editor, Nonlinear Mathematical Analysis and Applications, HadronicPress,Palm Harbor,FL34682-1577,USA:ISBN1-57485-044-X,1998, pp.223-234: \textit{ Nonlinear transforms and analyticity of functions}, Saburou Saitoh.
\bibitem{saito}
S. Saito and Y. Saito, \textit{Yoakemae - Yocchan no Omoi - } (Predawn - Thoughts of Yotchan) in Japanese, Bungeisha, (2010).
\bibitem{Sa83} S.~Saitoh, \textit{Hilbert spaces induced by Hilbert space valued functions},
\newblock Proc. Amer. Math. Soc., {\bf 89} (1983), 74--78.
\bibitem{Sa97}
S. Saitoh,
\textit{Integral Transforms, Reproducing Kernels and their Applications},
\newblock Pitman Res. Notes in Math. Series {\bf 369},
\newblock Addison Wesley Longman, Harlow (1997).
\bibitem{Sa10}
S. Saitoh,
\textit{ Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces},
Amer. Math. Soc. Transl. Ser. 2, {\bf 230}, Amer. Math. Soc., Providence, RI, 2010.
\bibitem{Sa12}
S. Saitoh,
\textit{ What is mathematics - mathematics and human beings}, International Society for Mathematical Sciences, No. 81/2012.5, 7―15 (in Japanese).
\end{thebibliography}
\end{document}
\documentclass{article}
\title{\Large\textbf{Announcement 142: An Aveiro Dream in Mathematics\\
}}
\author{ Institute of Reproducing Kernels, \\ Kawauchi-cho 5-1648-16, \\Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp}
\begin{document}
\maketitle
\begin{abstract}\noindent
\end{abstract}
\textbf{Keywords:} Linear operator, eigenfunction, initial value problem, integral transform, reproducing kernel, linear differential equation, partial differential equation, linear integral operator, Dirac delta function, Green function.
\section{Introduction}
In \cite{rassias}, we stated that mathematic is a set of relations and the fundamental relation
\begin{equation}
e^{\pi i} = -1
\end{equation}
among the basic numbers $-1, \pi, i, e$ is the best result in mathematics. There, we stated also that good results are fundamental, beautiful and give good impact to human beings. At the same time, in the paper we showed the basic relation between {\it analyticity of functions} and {\it nonlinear transforms}, by using the theory of reproducing kernels.
\section{God loves two}
On the universe, we stated repeatedely: The God loves two; and two is a fundamental concept on the universe, in our general, essayes of \cite{saito,Sa12} and we love the famous Pythagorean theorem with two powers. We established a general Pythagorean theory between the inputs and the outputs of a general linear system in the framework of Hilbert spaces in \cite{Sa83} and we established many entirely new type Pythagorean (isometric identity) theorems and we called it the fundamental theory in linear mappings. The theory establishes the basic fundamental relations among the inputs, the outputs and linear systems in the framework of Hilbert spaces \cite{Sa83}. The theory is the core of the theory of reproducing kernels, and the basic concepts Dirac's delta function and Green's function are the family of reproducing kernels.
\section{Aveiro Discretization Method }
\section{An Aveiro Dream in Mathematics}
By combining the very specialized research result of Professor M. M. Rodrigues and the Aveiro discretization method using the fundamental theory of linear mappings, we found the basic relations among linear operators, eigenfunctions, linear initial value problems, integral transforms and reproducing kernels.
Roughly speaking, when we know some eigenfunctions of a linear operator, we can consider the related partial differential equation and we can solve the associated initial value problem; in this method, we shall consider the reproducing kernel forms and related integral transforms ( linear mappings), and we can discuss the existence problem and construction problem of the initial value problem, and furthermore, we can consider the complete property of the solutions by using the theory of reproducing kernels.
From this general method, we find that we can consider many and many integral transforms and reproducing kernels in concrete forms from the known eigenfunctions.
We know a great tradition on concrete forms in Russia; many definite integrals, many eigenfunctions, many analytical solutions in differential and integral equations. Our theory will give a great impact on these topics.
\section{Research projects}
Our output results in the Aveiro Dream in Mathematics may be stated as follows:
1) Many concrete reproducing kernels may be calculated and the related reproducing kernel Hilbert spaces should be realized with concrete norms.
2) Eigenfunctions and the related initial value problems in partial differential and integral equations should be examined with their properties of the solutions.
3) Many new integral transforms and their properties; that is, isometric identities and inversion formulas should be established.
4) For the associated $t$ kernels and the related small reproducing kernels appeared in the general theory, we can consider the similar problems above.
From the great references \cite{poly02,poly03,poly08, gr, Pru3} containing the special function theory, we may consider expected new materials as the Aveiro Dream in Mathematics. We believe such materials in mathematics are definite values and fundamentals in mathematics.
\begin{thebibliography}{00}
\bibitem{cfrst}
L.P. Castro, H. Fujiwara, M.M. Rodrigues, S. Saitoh and V. K. Tuan,
\textit{Aveiro Discretization Method in Mathematics: A New Discretization Principle, MATHEMATICS WITHOUT BOUNDARIES: SURVEYS IN PURE MATHEMATICS}, Edited by Panos Pardalos and Themistocles M. Rassias (to appear). 52 pages.
\bibitem{cfqs} L.P. Castro, H. Fujiwara, T. Qian and S. Saitoh, \textit{How to catch smoothing properties and analyticity of functions by computers?,} MATHEMATICS WITHOUT BOUNDARIES: SURVEYS IN PURE MATHEMATICS, Edited by Panos Pardalos and Themistocles M. Rassias (to appear).
\bibitem{Erdlyi}
A. Erd\'elyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, 1954. \textit{Tables of Integral Transforms-Vol I, II}. Bateman Manuscript Project. California Institute of Technology, McGraw Hill, New York.
\bibitem{gr}
I. S. Gradshlein and I. M. Ryzhik,
\textit{Table of Integrals, Series, and Products}, 7th Edition, Elsever Inc. 2007.
\bibitem{Pru3}
A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, \textit{Integrals and Series, Volume 3: More special functions}, Gordon and Breach Publisher, New York, 1990.
\bibitem{poly02}
A. D. Polyanin, \textit{Handbook of Linear Partial Differential Equations for Engineers and Scientists}, CRC Press, 2002.
\bibitem{poly03}
A. D. Polyanin and V. F. Zaitsev, \textit{Handbook of Exact Solutions for Ordinary Differential Equations}, CRC Press, 2003.
\bibitem{poly08}A. D. Polyanin and A. V. Manzhirov, \textit{Handbook of Integral Equations}, CRC Press, 2008.
\bibitem{rassias}
T. M. Rassias,
Editor, Nonlinear Mathematical Analysis and Applications, HadronicPress,Palm Harbor,FL34682-1577,USA:ISBN1-57485-044-X,1998, pp.223-234: \textit{ Nonlinear transforms and analyticity of functions}, Saburou Saitoh.
\bibitem{saito}
S. Saito and Y. Saito, \textit{Yoakemae - Yocchan no Omoi - } (Predawn - Thoughts of Yotchan) in Japanese, Bungeisha, (2010).
\bibitem{Sa83} S.~Saitoh, \textit{Hilbert spaces induced by Hilbert space valued functions},
\newblock Proc. Amer. Math. Soc., {\bf 89} (1983), 74--78.
\bibitem{Sa97}
S. Saitoh,
\textit{Integral Transforms, Reproducing Kernels and their Applications},
\newblock Pitman Res. Notes in Math. Series {\bf 369},
\newblock Addison Wesley Longman, Harlow (1997).
\bibitem{Sa10}
S. Saitoh,
\textit{ Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces},
Amer. Math. Soc. Transl. Ser. 2, {\bf 230}, Amer. Math. Soc., Providence, RI, 2010.
\bibitem{Sa12}
S. Saitoh,
\textit{ What is mathematics - mathematics and human beings}, International Society for Mathematical Sciences, No. 81/2012.5, 7―15 (in Japanese).
\end{thebibliography}
\end{document}
お気に入りの記事を「いいね！」で応援しよう
Last updated
2013.11.30 18:47:14
【毎日開催】
15記事にいいね！で1ポイント
© Rakuten Group, Inc.
