After introducing Berezin integral for polynomials of odd variables, we develop the elementary in... more After introducing Berezin integral for polynomials of odd variables, we develop the elementary integral calculus based on supersmooth functions on the superspace R^m|n. Here, R is the Fréchet-Grassmann algebra with countably infinite Grassmann generators, which plays the role of real number field R. As is well-known that the formula of change of variables under integral sign is indispensable not only to treat PDE applying funtional analytic method but also to introduce analysis on supermanifolds. But, if we define naively the integral for supersmooth functions, there exists discrepancy which should be ameliorated. Here, we extend the contour integral modifying the parameter space introduced basically by de Witt, Rogers and Vladimirov and Volovich
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of tha...
After introducing Berezin integral for polynomials of odd variables, we develop the elementary in... more After introducing Berezin integral for polynomials of odd variables, we develop the elementary integral calculus based on supersmooth functions on the superspace ${\mathfrak{R}}^{m|n}$. Here, ${\mathfrak{R}}$ is the Fr\'echet-Grassmann algebra with countably infinite Grassmann generators, which plays the role of real number field ${\mathbb{R}}$. As is well-known that the formula of change of variables under integral sign is indispensable not only to treat PDE applying funtional analytic method but also to introduce analysis on supermanifolds. But, if we define naively the integral for supersmooth functions, there exists discrepancy which should be ameliorated. Here, we extend the contour integral modifying the parameter space introduced basically by de Witt, Rogers and Vladimirov and Volovich
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of tha...
In order to treat "boson" and "fermion" on equal footing, the s0-called even (bosonic) and odd (f... more In order to treat "boson" and "fermion" on equal footing, the s0-called even (bosonic) and odd (ferminionic) variables are introduced formally in physics literature. To make rigorous such new variables, we introduced the new algebras called Frechet-Grassmann
In this note, we construct, strong and classical solutions of the Hopf equation, a statistical ve... more In this note, we construct, strong and classical solutions of the Hopf equation, a statistical version of the Navier-Stokes equation on a compact Riemannian manifold with or without boundary. Our points are to regard the Hopf equation as a given Functional Derivative Equation (F.D.E. for short) of second order, to derive the Navier-Stokes equation as the characteristic equation of it and to give an exact meaning to the 'trace* of the second order functional derivatives which appear in the Hopf equation. To construct a solution of the Hopf-Foias equation with the energy inequality of strong form, we apply Foia §'s argument with slight modifications instead of using Prokhorov's compactness argument.
Preparing the Fréchet-Grassmann (FG-)algebra R composed with countably infinite Grassmann generat... more Preparing the Fréchet-Grassmann (FG-)algebra R composed with countably infinite Grassmann generators, we introduce the superspace R m|n. After defining Grassmann continuation of smooth functions on R m to those on R m|0 , we introduce a class of functions which are called supersmooth (alias superfields) and are regarded as one of those with countably infinite independent variables. In this paper, we characterize such supersmooth functions in Gâteaux (but not necessarily Fréchet) differentiable category in Fréchet but not in Banach space. This type of arguments for G ∞-functions is done in the Banach-Grassmann (BG-)algebra, but we find it rather natural to work within FG-algebra when we treat systems of PDE such as Dirac equation. Though we took this point of view in our previous works, but is insufficiently managed.
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of t...
After introducing Berezin integral for polynomials of odd variables, we develop the elementary in... more After introducing Berezin integral for polynomials of odd variables, we develop the elementary integral calculus based on supersmooth functions on the superspace R^m|n. Here, R is the Fréchet-Grassmann algebra with countably infinite Grassmann generators, which plays the role of real number field R. As is well-known that the formula of change of variables under integral sign is indispensable not only to treat PDE applying funtional analytic method but also to introduce analysis on supermanifolds. But, if we define naively the integral for supersmooth functions, there exists discrepancy which should be ameliorated. Here, we extend the contour integral modifying the parameter space introduced basically by de Witt, Rogers and Vladimirov and Volovich
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of tha...
After introducing Berezin integral for polynomials of odd variables, we develop the elementary in... more After introducing Berezin integral for polynomials of odd variables, we develop the elementary integral calculus based on supersmooth functions on the superspace ${\mathfrak{R}}^{m|n}$. Here, ${\mathfrak{R}}$ is the Fr\'echet-Grassmann algebra with countably infinite Grassmann generators, which plays the role of real number field ${\mathbb{R}}$. As is well-known that the formula of change of variables under integral sign is indispensable not only to treat PDE applying funtional analytic method but also to introduce analysis on supermanifolds. But, if we define naively the integral for supersmooth functions, there exists discrepancy which should be ameliorated. Here, we extend the contour integral modifying the parameter space introduced basically by de Witt, Rogers and Vladimirov and Volovich
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of tha...
In order to treat "boson" and "fermion" on equal footing, the s0-called even (bosonic) and odd (f... more In order to treat "boson" and "fermion" on equal footing, the s0-called even (bosonic) and odd (ferminionic) variables are introduced formally in physics literature. To make rigorous such new variables, we introduced the new algebras called Frechet-Grassmann
In this note, we construct, strong and classical solutions of the Hopf equation, a statistical ve... more In this note, we construct, strong and classical solutions of the Hopf equation, a statistical version of the Navier-Stokes equation on a compact Riemannian manifold with or without boundary. Our points are to regard the Hopf equation as a given Functional Derivative Equation (F.D.E. for short) of second order, to derive the Navier-Stokes equation as the characteristic equation of it and to give an exact meaning to the 'trace* of the second order functional derivatives which appear in the Hopf equation. To construct a solution of the Hopf-Foias equation with the energy inequality of strong form, we apply Foia §'s argument with slight modifications instead of using Prokhorov's compactness argument.
Preparing the Fréchet-Grassmann (FG-)algebra R composed with countably infinite Grassmann generat... more Preparing the Fréchet-Grassmann (FG-)algebra R composed with countably infinite Grassmann generators, we introduce the superspace R m|n. After defining Grassmann continuation of smooth functions on R m to those on R m|0 , we introduce a class of functions which are called supersmooth (alias superfields) and are regarded as one of those with countably infinite independent variables. In this paper, we characterize such supersmooth functions in Gâteaux (but not necessarily Fréchet) differentiable category in Fréchet but not in Banach space. This type of arguments for G ∞-functions is done in the Banach-Grassmann (BG-)algebra, but we find it rather natural to work within FG-algebra when we treat systems of PDE such as Dirac equation. Though we took this point of view in our previous works, but is insufficiently managed.
By taking the Weyl equation with external electro-magnetic potentials as the simplest representat... more By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of t...
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