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In this page, we introduce Gelfand zeta function, Bernstein-Sato polynomial, algebraic analysis, and their connections to statistical learning theory.
In this page, we introduce Gelfand zeta function, Bernstein-Sato polynomial, algebraic analysis, and their connections to statistical learning theory.
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The 2025 Abel Prize is awarded to Professor Masaki Kashiwara in RIMS Kyoto University for his contributions to algebraic analysis. Congratulations ! One result among Professor Kashiwara's many achievements is the rationality of the roots of the Bernstein-Sato polynomial, which are essentially the same as the real log canonical thresholds that play an important role in singular learning theory. Professor Kashiwara's mathematics has relations to many mathematical fields, which also connects to today's artificial intelligence studies.
What algebraic analysis is :
When we talk about "analysis," it is generally understood as the branch of mathematics that rigorously deals with limit operations such as differentiation and integration, and this understanding is not incorrect. However, "algebraic analysis" studies the algebraic properties that appear in analysis, independently of those limit operations. For example, if we define X as the operation of "multiplying by x" and ∂ as the operation of "differentiating with respect to x," the commutation relation ∂X−X∂=1 holds. Algebraic analysis investigates the properties of sets that satisfy such algebraic relations. The module over a ring of differential operators is called a D-module. Algebraic analysis is sometimes referred to as the "representation theory of D-modules." Research topics include questions like "What algebraic characteristics do solvable nonlinear differential equations have?" or "Does there exist a differential operator that has the complex power f(x)^z of a polynomial f(x) as an eigenvector?" Addressing these questions is known to extend mathematical world and to provide solutions to many unresolved problems in mathematics. In this subpage, we introduce the relation between algebraic analysis and statistical learning theory.
Family of Zeta functions is one of the most important concepts in mathematics, whose origin is Riemann zeta function. The Reimann zeta function is defined by the unique analytic continuation of the left equation (Re(z)>0) onto the entire complex plain as a meromorphic function. It is trivial that zeta is equal to zero at z=-2,-4,-6,...,. Riemann Hypothesis is that the real part of nontrivial zeros of the zeta function is equal to 1/2.
Family of Zeta functions is one of the most important concepts in mathematics, whose origin is Riemann zeta function. The Reimann zeta function is defined by the unique analytic continuation of the left equation (Re(z)>0) onto the entire complex plain as a meromorphic function. It is trivial that zeta is equal to zero at z=-2,-4,-6,...,. Riemann Hypothesis is that the real part of nontrivial zeros of the zeta function is equal to 1/2.
In statistical learning theory, Gelfand zeta function plays the central role, whose definition is given in the left figure (Re(z)>0). In 1954, at the international congress of mathematicians (ICM1954), Professor Gelfand conjectured that his zeta function can be analytically continued to the unique meromorphic function on the entire complex plane.
In statistical learning theory, Gelfand zeta function plays the central role, whose definition is given in the left figure (Re(z)>0). In 1954, at the international congress of mathematicians (ICM1954), Professor Gelfand 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 [1] and Sato [2] independently proved the existence of the b-function, which is called Bernstein-Sato polynomial. 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 [3] and local zeta function. I would like to thank many mathematicians' results on which singular learning theory could be constructed.
In 1970, Atiyah showed that Gel'fand conjecture can be proved by Hironaka's resolution of singularities. In 1972, Bernstein [1] and Sato [2] independently proved the existence of the b-function, which is called Bernstein-Sato polynomial. 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 [3] and local zeta function. I would like to thank many mathematicians' results on which singular learning theory could be constructed.
[1] J.Bernstein, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients. Functional Analysis and Its Applications. vol.5, pp. 89–101, 1971.
[2] M.Sato, T. Shintani, On zeta functions associated with prehomogeneous vector spaces". Annals of Mathematics. vol.100, pp. 131–170. 1974.
[3] J. Igusa. Complex powers and asymptotic expansions. I. Functions of certain types", Journal für die reine und angewandte Mathematik, vol.0268-0269, pp.110-130, 1974.
By the existence of b-function, meromorphic property of Gel'fand zeta function can be proved. In 1976, Kashiwara proved that its poles are all real, rational, and negative numbers by using Hironaka resolution theorem [4]. In 1997, Oaku constructed the algorithm how to find the b-function for an arbitrary polynomial based on D-module Theory [5]. In 2007, Saito showed the relation between the root of b-function and real log canonical threshold [6]. 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. In 1976, Kashiwara proved that its poles are all real, rational, and negative numbers by using Hironaka resolution theorem [4]. In 1997, Oaku constructed the algorithm how to find the b-function for an arbitrary polynomial based on D-module Theory [5]. In 2007, Saito showed the relation between the root of b-function and real log canonical threshold [6]. B-function is one of the important concepts in algebraic analysis which was established in Research Institute of Mathematical Science (RIMS), Kyoto University.
[4] M. Kashiwara, B-functions and Holonomic systems. Inventiones Mathematicae, Vol.38, pp.33-53, 1976.
[5] T. Oaku, Algorithms for the b-function and D-modules associated with a polynomial. Journal of Pure and Applied Algebra. Vol.117-118, pp.495-518, 1997.
[6] M. Saito, On real log canonical thresholds. arXiv:0707.2308, 2007.
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). In Re(z)<-1/2, the function (w^2)^z can be understood as a Schwartz distribution or a Sato hyperfunction. Note that
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). In Re(z)<-1/2, the function (w^2)^z can be understood as a Schwartz distribution or a Sato hyperfunction. Note that
(d^2/dw^2)(w^(2z+2))=(2z+2)(2z+1) w^(2z).
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.
By using Hironaka resolution map, any analytic function can be made a normal crossing function in any local neighborhood.
This is an example of a zeta function.
The integration of a normal crossing function can be concretely calculated, which is equal to a meromorphic function whose largest pole is the minus real log canonical threshold.
This is an example of Mellin transform.
The inverse Mellin transform of the meromorphic function is equal to a product of t and log(t). This relation is universally employed in the zeta function theory.
This is an example of Laplace transform.
When n tends to infinity, the main term of the Laplace transform is equal to
( log n)^(m-1)/n^{lambda} . This result will be continued to the marginal likelihood using empirical process theory.
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