Zeta function
Zeta functions are one of the most important concepts in mathematics, whose origin is Riemann's zeta function. In statistical learning theory, Gel'fand zeta function plays the central role. In 1954, at the international congress of mathematicians (ICM1954), Gel'fand conjectured that his zeta function can be analytically continued to the unique meromorphic function on the entire complex plane.
Zeta functions are one of the most important concepts in mathematics, whose origin is Riemann's zeta function. In statistical learning theory, Gel'fand zeta function plays the central role. In 1954, at the international congress of mathematicians (ICM1954), Gel'fand conjectured that his zeta function can be analytically continued to the unique meromorphic function on the entire complex plane.
In 1970, Atiyah showed that Gel'fand conjecture can be proved by Hironaka's resolution of singularities. In 1972, Bernstein and Sato independently proved the existence of b-function. For an arbitrary analytic function K(w), there exist both a differential operator D and a polynomial b(z) which satisfy this equation. If you are a mathematician, please see also Igusa zeta function and local zeta function.
In 1970, Atiyah showed that Gel'fand conjecture can be proved by Hironaka's resolution of singularities. In 1972, Bernstein and Sato independently proved the existence of b-function. For an arbitrary analytic function K(w), there exist both a differential operator D and a polynomial b(z) which satisfy this equation. If you are a mathematician, please see also Igusa zeta function and local zeta function.
By the existence of b-function, meromorphic property of Gel'fand zeta function can be proved. Its poles are all real, rational, and negative numbers. In 1997, Oaku constructed the algorithm how to find the b-function for an arbitrary polynomial using D-module Theory. B-function is one of the important concepts in algebraic analysis which was established in Research Institute of Mathematical Science (RIMS), Kyoto University.
By the existence of b-function, meromorphic property of Gel'fand zeta function can be proved. Its poles are all real, rational, and negative numbers. In 1997, Oaku constructed the algorithm how to find the b-function for an arbitrary polynomial using D-module Theory. B-function is one of the important concepts in algebraic analysis which was established in Research Institute of Mathematical Science (RIMS), Kyoto University.
Let us illustrate a simple example. The function eq.(1) is defined in the region Re(z)>-1/2, whereas eq.(2) can be defined for all z that is not (-1/2). Thus eq.(2) can be understood as the analytic continuation of eq.(1).
Let us illustrate a simple example. The function eq.(1) is defined in the region Re(z)>-1/2, whereas eq.(2) can be defined for all z that is not (-1/2). Thus eq.(2) can be understood as the analytic continuation of eq.(1).
The same method can be applied to the general case, where phi(w) is a C-infinity class function with a compact support.
The same method can be applied to the general case, where phi(w) is a C-infinity class function with a compact support.
Also the same method can be generalized for normal crossing cases. By Hironaka's theorem, an arbitrary analytic function can be made normal crossing in local coordinate. Bernstein-Sato polynomial gives the other proof of analytic continuation of the zeta function.
Also the same method can be generalized for normal crossing cases. By Hironaka's theorem, an arbitrary analytic function can be made normal crossing in local coordinate. Bernstein-Sato polynomial gives the other proof of analytic continuation of the zeta function.
The Schwartz distribution w^z can be expanded by using delta function. This is a kind of Taylor expansion in the space of Schwartz distributions. In this example, the expressions on the left and right sides appear to be completely different, but they are equal to each other as Schwartz distributions. This mathematical structure plays the important role not only in mathematics but also in statistical learning theory.
The Schwartz distribution w^z can be expanded by using delta function. This is a kind of Taylor expansion in the space of Schwartz distributions. In this example, the expressions on the left and right sides appear to be completely different, but they are equal to each other as Schwartz distributions. This mathematical structure plays the important role not only in mathematics but also in statistical learning theory.
These equations describe the dream journey back from algebraic geometry to statistical learning theory. The Gelfand zeta function is a guidepost that illuminates the way to travel through the wonderland. Hironaka's theorem is mathematically connected to statistical learning theory. This miracle path was firstly found by Professor M.F. Atiyah.
These equations describe the dream journey back from algebraic geometry to statistical learning theory. The Gelfand zeta function is a guidepost that illuminates the way to travel through the wonderland. Hironaka's theorem is mathematically connected to statistical learning theory. This miracle path was firstly found by Professor M.F. Atiyah.
M.F. Atiyah, Resolution of singularities and division of distributions. communications on pure and applied mathematics, pp.145-150, 1970.
This is an example.
The inverse Mellin transform of the meromorphic function is equal to the product of t and log(t). This relation is universally employed in the zeta function theory.