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Register a new userIntroduction into Calculus over Banach algebra
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Authors
Aleks Kleyn
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Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d f(x)}{d x}:Arightarrow B$$ is linear map and $o:Arightarrow B$ is such continuous map that $$lim_{arightarrow 0}frac{|o(a)|_B}{|a|_A}=0$$ Linear map $displaystylefrac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $omegainmathcal{LA}(D;Arightarrow B)$ is defined by map $g:Arightarrow Botimes B$, $omega=gcirc dx$. If the map $g$, is derivative of the map $f:Arightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=int g(x)circ dx=intomega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$int_a^bomega=int_{gamma}omega$$ for any path $gamma$ from $a$ to $b$.
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In the book, I considered differential equations of order $1$ over Banach $D$Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential equations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.
Let $A$ be Banach algebra over commutative ring $D$. The map $f:Arightarrow A $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=partial f(x)circ a+o(a)$$ where the Gateaux derivative $partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ lim_{arightarrow 0}frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $partial^{n-1} f(x)$ of order $n-1$, we define $$ partial^n f(x)circ(a_1otimes...otimes a_n) =partial(partial^{n-1} f(x)circ(a_1otimes...otimes a_{n-1}))circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=sum_{n=0}^{infty}(n!)^{-1}partial^n f(x_0)circ(x-x_0)^n $$
In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.
With the theory of general relativity, Einstein abolished the interpretation of gravitation as a force and associated it to the curvature of spacetime. Tensorial calculus and differential geometry are the mathematical resources necessary to study the spacetime manifold in the context of Einsteins theory. In 1961, Tullio Regge published a work on which he uses the old idea of triangulation of surfaces aiming the description of curvature, and, therefore, gravitation, through the use of discrete calculus. In this paper, we approach Regge Calculus pedagogically, as well as the main results towards a discretized version of Einsteins theory of gravitation.
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general point of view I started the book from consideration of Cartesian product of representations. Polymorphism of representations is a map of Cartesian product of representations which is a morphism of representations with respect to each separate independent variable. Reduced morphism of representations allows us to simplify the study of morphisms of representations. However a representation has to satisfy specific requirements for existence of reduced polymomorphism of representations. It is possible that Abelian group is only $Omega$-algebra, such that representation in this algebra admits polymorphism of representations. However, today, this statement has not been proved. Multiplicative $Omega$-group is $Omega$-algebra in which product is defined. The definition of tensor product of representations of Abelian multiplicative $Omega$-group is based on properties of reduced polymorphism of representations of Abelian multiplicative $Omega$-group. Since an algebra is a module in which the product is defined, then we can use this theory to study linear map of algebra. For instance, we can study the set of linear transformations of $D$-algebra $A$ as representation of algebra $Aotimes A$ in algebra $A$.
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