Related papers: Introduction into Calculus over Banach algebra

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear…

A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…

A linear Diophantine equation $ax + by = n$ is solvable if and only if gcd$(a; b)$ divides $n$. A graph $G$ of order $n$ is called Diophantine if there exists a labeling function $f$ of vertices such that gcd$(f(u); f(v))$ divides $n$ for…

Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which…

We explicitly construct a finite set of separating invariants for the basic $\Ga$-actions. These are the finite dimensional indecomposable rational linear representations of the additive group $\Ga$ of a field of characteristic zero, and…

This paper introduces DD calculus and describes the basic calculus concepts of derivative and integral in a direct and non-traditional way, without limit definition: Derivative is computed from the point-slope equation of a tangent line and…

Let $T:A\to X$ be a bounded linear operator, where $A$ is a C$^*$-algebra, and $X$ denotes an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(a)$ $T$ is anti-derivable at zero (i.e. $ab =0$ in $A$…

A linear mapping $\phi$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B=A\phi(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$, and $\phi$ is called a derivable mapping at…

We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous…

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At…

We show that the derivatives in the sense of Fr\'echet and G\^ateaux can be viewed as derivatives oriented towards a star convex set with the origin as center. The resulting oriented differential calculus extends the mean value theorem, the…

Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets…

Let $n\in \Bbb N-\{1\},$ and let $A$ be a Banach algebra. An additive map $D: A\to A$ is called n-Jordan derivation if $$D(a^n)=D(a)a^{n-1}+aD(a)a^{n-2}+...+a^{n-2}D(a)a+a^{n-1}D(a),$$ for all $a \in {A}$. Using fixed point methods, we…

Let $A$ and $B$ be unital complex Banach algebras having no quotients isomorphic to $\mathbb{C}$ or $M_2(\mathbb{C})$. Assume additionally that $B$ is semisimple. If a surjective additive mapping $\Phi\colon A\to B$ satisfies…

Let $F: \mathbb{L}^2(\Omega, \mathbb{R}) \to \mathbb{R}$ be a law invariant and continuously Fr\'echet differentiable mapping. Based on Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \bea…

Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of…

Let $A$ be a Banach algebra. The flip on $A \otimes A^\op$ is defined through $A \otimes A^\op \ni a \tensor b \mapsto b \tensor a$. If $A$ is ultraprime, $\El(A)$, the algebra of all elementary operators on $A$, can be algebraically…

We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the…

Let $ \mathcal D$ be a dense linear manifold in a Hilbert space $\mathcal H$ and let $L^+(\mathcal D)$ be the *-algebra of all linear operators $A$ such that $A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$. Denote by…

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\rm{Pic}^0(A)$ injects into $\rm{Pic}^0(T)$ and let $L$ be a line bundle on $X$. Denote by…