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The Theory of Analytic Functions in Normed Abelian Vector Rings
... У роботі Е. Р. Лорха [23] доведено інтегральну теорему Коші та інтегральну формулу Коші, теореми Тейлора та Морера для функцій, диференційовних у сенсі Лорха в довільній опуклій області комутативної банахової алгебри. Умову опуклості області в цих результатах було знято Е.К. ...
... Другий крок в доведенні теореми Коші для гіперкомплексного криволінійного інтеграла здійснюється у випадку, коли область Ω Ă E 3 є опуклою і γ -довільна замкнена спрямлювана жорданова крива в області Ω. У цьому випадку рівність (3.2) для кожної моногенної функції Φ : Ω Ñ A m n може бути доведена класичним способом так, як це зробив Е. Р. Лорх [23] в опуклій області усієї алгебри. ...
... Отже, інтеграл J 1 прямує до нуля, коли ε Ñ 0. Нарешті, переходячи до границі в рівності (3.16) при ε Ñ 0, отримуємо рівність (3.14). □ На відміну від подібних результатів Е. Лорха [23] і Е. Блюма [5], функція Φ : Ω Ñ A m n в теоремі 3.9 задана тільки в області Ω підпростору E 3 , а не в області з усієї алгебри. Більш того, зауважимо, що інтегральна формула Коші, встановлена у роботах [5,23], не застосовна до моногенної функції Φ : Ω Ñ A m n , оскільки в ній інтегрування здійснюється вздовж кривої, на якій функція Φ, взагалі кажучи, не визначена. ...
For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.
... Ketchum's work stands out, which perhaps has not been understood or has been misinterpreted, because he works with algebras in a general way, which makes it seem a bit complicated to construct solutions of PDEs by the method he proposes. The theory of analytic functions over algebras has been developed since the end of the 19th century, also see [10][11][12][13][14], and [15]. ...
... If = and ∶ R → R is the identity transformation ( ) = , the A-differentiability will be called A-differentiability and the A-derivative of will be denoted by ′ . This last differentiability is known as Lorch differentiability, see [12]. Differentiability related to commutative and noncommutative algebras is considered in [10]. ...
... Thus, a type of calculus on algebras and their corresponding differential equations has been introduced. The A-differentiability is an extension of differentiability in the sense of Lorch [12]. ...
In this paper we introduce the -\emph{differentiability} for functions , where U is an open set, is the linear space endowed with a unital associative commutative algebra product, and is a differentiable function in the usual sense. We call it \emph{pre-twisted differentiability}. With respect to the -differentiability we introduce: (a) a type Cauchy--Riemann equations, which serve as -differentiability criteria, (b) a Cauchy-integral theorem, and (c) \emph{-differential equations}, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the -differentiable functions define a complete solutions for the PDEs of the form , which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of -differential equations define solutions for linear and nonlinear PDE systems.
... The corresponding ideas were given by Ketchum 1 where differentiability in the sense of Lorch with respect to associative commutative algebras with unit e is used; see other studies. [2][3][4] Instead, we propose the use of pre-twisted differentiability with respect to a family of algebras with unit e = e 1 ; see preprint arXiv 1805.10524. We obtain a complete solution of each PDE of the family ...
... Theorem 3.1. Consider the PDE (2) and the vector field given in (5). Suppose that Ac 2 + Bcd + Cd 2 ≠ 0. Thus, for the algebra A = A 1 (p 1 , p 2 ) with parameters p 1 and p 2 given by ...
... the components f and g of the exponential function (6) defined with respect to A, are solutions of the PDE (2). ...
The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form A u x x + B u x y + C u y y = 0, and for each affine planar vector field φ, we give an algebra 𝔸 with unit e = e1, with respect to which the components of all functions of the form L ∘ φ are all the solutions for this PDE, where L is differentiable in the sense of Lorch with respect to 𝔸. Solutions are also constructed for the following equations: A u x x + B u x y + C u y y + D u x + E u y + F u = 0, 3 rd-order PDEs, and 4 th-order PDEs; among these are the bi-harmonic, the bi-wave, and the bi-telegraph equations.
... The class of Lorch-analytic mappings (cf. Definition 2.1) was introduced by E. R. Lorch in [14]. It was shown in [17] that H L (E, E) is a closed subalgebra of H b (E, E). ...
... In this note, Section 3, we investigate this set in the case where E is a general Banach algebra E, and we show that it is spaceable, but not residual in H b (E, E). For background on Lorch-analytic mappings, we refer to [14,17]. ...
... E. R. Lorch in [14] introduced a definition of analytic mappings (see Definition 3.1), that have for their domains and ranges a complex commutative Banach algebra with identity. ...
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
... The Cauchy integral theorem is satisfied on algebras, for a proof by using analytic arguments, see [8] and [10]; [9] and [14] for proofs by using conservative vector fields. Thus, the fundamental theorem of calculus on algebras is satisfied. ...
... Thus, the fundamental theorem of calculus on algebras is satisfied. The theory of analytic functions over algebras extends other results of the classical complex function theory; see [3], [8], [10], [15], [18], and [19]. Differentiability with respect to finite dimensional associative algebras over R (not necessarily commutative) has been recently reconsidered in [4] under the name of A-Calculus. ...
... The structure constants of A 2 1,2 are c iii = 1 for i = 1, 2 and c ijk = 0 in another case. By substituting these constants in equations (8), we obtain (10). Thus, the proof is finished. ...
Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics under which they are geodesible. Furthermore, for each of these vector fields F we give two first integrals and such that the integral curves of F are locally defined by the intersections of the level surfaces of and .
... Generalizing such an approach to the case of mappings given in a domain of an arbitrary commutative associative Banach algebra, E. R. Lorch [37] introduced a derivative which is also understood as a function given in the same domain. ...
... The function Φ : Ω −→ A is called differentiable in the sense of Lorch (cf. [37]) in a domain Ω ⊂ E 3 , if, for every ζ ∈ Ω, there exists an element Φ ′ L (ζ) ∈ A such that, for any ε > 0, there exists δ > 0 such that, for all h ∈ E 3 with h < δ, the following inequality is fulfilled: ...
... Some properties similar to the properties of analytic functions of a complex variable are established for the functions differentiable in the sense of Lorch [37] in an arbitrary convex domain of a commutative Banach algebra. In particular, the integral Cauchy theorem, integral Cauchy formula, Taylor expansion, and Morera theorem were proved in [37] in such a way as for analytic functions of a complex variable. ...
The methods involving the functions analytic in a complex plane for plane potential fields inspire the search for the analogous efficient methods for solving the spatial and multidimensional problems of mathematical physics. Many such methods are based on the mappings of hypercomplex algebras. The essence of the algebraic-analytic approach to elliptic equations of mathematical physics consists in the finding of a commutative Banach algebra such that the differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. The use of differentiable functions given in commutative Banach algebras combines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs. The paper contains the review of results reflecting the formation and the development of the mentioned approach.
... 2 On the distribution and non-existence of limit cycles A useful relationship between limit cycles and iifs is established in the following theorem (see [10]), that relates limit cycles with the zero level curve of an iif. ...
... Theorem 2.1. ( [10], th. 9) Let V : U → R be an inverse integrating factor of the system (1). ...
... Example 2.16. Saturated incidence rate of the form f (S, I) = βSI/(1 + κS), where β > 0 and κ ≥ 0. This saturation incidence was proposed by May and Anderson [10]. ...
... [1,2]). В теореме 1.8 из [2] показано, что в A 2 гармоническими являются базисы {e 1 , e 2 , e 3 }, разложения которых по базису {I 1 , I 2 , ρ} имеют вид e 1 = I 1 + I 2 , e 2 = n 1 I 1 + n 2 I 2 + n 3 ρ , e 3 = m 1 I 1 + m 2 I 2 + m 3 ρ , (3) где n k , m k при k = 1, 2, 3 -комплексные числа, удовлетворяющие системе 1 + n 2 1 + m 2 1 = 0 , 1 + n 2 2 + m 2 2 = 0 , n 2 n 3 + m 2 m 3 = 0, ...
... При этом умножением элементов гармонических базисов вида (3) на произвольные обратимые элементы алгебры могут быть получены все гармонические базисы в алгебре A 2 [2, c. 35]. Выделим в алгебре A 2 линейную оболочку E 3 := {ζ = xe 1 + ye 2 + +ze 3 : x, y, z ∈ R} над полем действительных чисел R, порожденную векторами e 1 , e 2 , e 3 . ...
... В работе [3] установлена разложимость в степенной ряд функций, дифференцируемых по Лорху в выпуклой области произвольной коммутативной ассоциативной банаховой алгебры. В работе [4] в указанном результате из [3] снято условие выпуклости области определения заданных функций. ...
В тривимiрнiй гармонiчнiй алгебрi з одновимірним радикалом одержано тейлорiвськi та лоранiвськi розклади моногенних функцiй, класифiковано їх особливостi та встановлено мономорфізм між
алгебрами моногенних функцій при переході від одного гармонічного базиса до іншого.
Taylor’s and Laurent’s expansions of monogenic functions taking values in a three-dimensional harmonic algebra with one-dimensional radical are obtained, singularities of these functions are classified and a monomorphism between algebras of monogenic functions at transition from a harmonic basis to another
one is established.
... In the paper [15] for functions differentiable in the sense of Lorch in an arbitrary convex domain of a commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the integral Cauchy theorem and the integral Cauchy formula and the Morera theorem) are established. The convexity of the domain in the mentioned results from [15] is withdrawn by E. K. Blum [16]. ...
... In the paper [15] for functions differentiable in the sense of Lorch in an arbitrary convex domain of a commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the integral Cauchy theorem and the integral Cauchy formula and the Morera theorem) are established. The convexity of the domain in the mentioned results from [15] is withdrawn by E. K. Blum [16]. ...
... Let us note that a priori the differentiability of the function Φ in the sense of Gateaux is a restriction weaker than the differentiability of this function in the sense of Lorch. Moreover, note that the integral Cauchy formula established in the papers [15,16] is not applicable to a monogenic function Φ : Ω → A n because it deals with an integration along a curve on which the function Φ is not given, generally speaking. ...
We obtained a constructive description of monogenic functions taking values in a finite-dimensional semi-simple commutative algebra by means of analytic functions of the complex variable. We proved that the mentioned monogenic functions have the Gateaux derivatives of all orders. We have proved analogs of classical integral theorems of the theory of analytic functions of the complex variable: the Cauchy integral theorems for surface integral and curvilinear integral, the Morera theorem and the Cauchy integral formula.
... To obtain the equality (14) in the case k − 1 > m > 0, primarily, we substitute an expression P k−1,m of the form (15) into the equality (15) and get m (y, z) . Now, the last equality coincides with (14) in the case k − 2 = m > 0 . ...
... To obtain the equality (14) in the case k − 1 > m > 0, primarily, we substitute an expression P k−1,m of the form (15) into the equality (15) and get m (y, z) . Now, the last equality coincides with (14) in the case k − 2 = m > 0 . ...
... Finally, continuing similar operations, after k − m − 1 steps we get the equality (14). ...
We obtain a constructive description of monogenic functions taking
values in a �nite-dimensional commutative algebra with unit and radical of maximal
dimensionality by means of holomorphic functions of the complex variable. We
prove that the mentioned monogenic functions have the Gateaux derivatives of all
orders, and analogues of classical theorems of the complex analysis hold for them:
the Cauchy integral theorem and the Cauchy integral formula, the Taylor expansion
and the Morera theorem.
... In the paper of E. R. Lorch [1] for functions differentiable in the sense of Lorch in an arbitrary convex domain of commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the curvilinear integral Cauchy theorem and the integral Cauchy formula, the Taylor expansion and the Morera theorem) are established. E. K. Blum [2] withdrew a convexity condition of a domain in the mentioned results from [1]. ...
... In the paper of E. R. Lorch [1] for functions differentiable in the sense of Lorch in an arbitrary convex domain of commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the curvilinear integral Cauchy theorem and the integral Cauchy formula, the Taylor expansion and the Morera theorem) are established. E. K. Blum [2] withdrew a convexity condition of a domain in the mentioned results from [1]. ...
... Also we assume that a monogenic function is given in a domain of three-dimensional subspace of an arbitrary commutative associative algebra with unit over the field of complex numbers. In this situation the results established in the papers [1,2] is not applicable for a mentioned monogenic function, because it deals with an integration along a curve on which the function is not given, generally speaking. ...
We consider an arbitrary finite-dimensional commutative associative algebra,
, with unit over the field of complex number with m
idempotents. Let be elements of which are
linearly independent over the field of real numbers. We consider monogenic
(i.e. continuous and differentiable in the sense of Gateaux) functions of the
variable , where x,y,z are real. For mentioned monogenic
function we prove curvilinear analogues of the Cauchy integral theorem, the
Morera theorem and the Cauchy integral formula.
... In the paper of E. R. Lorch [1] for functions differentiable in the sense of Lorch in an arbitrary convex domain of commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the curvilinear integral Cauchy theorem and the integral Cauchy formula, the Taylor expansion and the Morera theorem) are established. E. K. Blum [2] withdrew a convexity condition of a domain in the mentioned results from [1]. ...
... In the paper of E. R. Lorch [1] for functions differentiable in the sense of Lorch in an arbitrary convex domain of commutative associative Banach algebra, some properties similar to properties of holomorphic functions of complex variable (in particular, the curvilinear integral Cauchy theorem and the integral Cauchy formula, the Taylor expansion and the Morera theorem) are established. E. K. Blum [2] withdrew a convexity condition of a domain in the mentioned results from [1]. ...
... Also we assume that a monogenic function is given in a domain of three-dimensional subspace of an arbitrary commutative associative algebra with unit over the field of complex numbers. In this situation the results established in the papers [1,2] is not applicable for a mentioned monogenic function, because it deals with an integration along a curve on which the function is not given, generally speaking. ...
Let be an arbitrary n-dimensional commutative associative
algebra over the field of complex numbers with m idempotents. Let
with be elements of
which are linearly independent over the field of real numbers. We consider
monogenic (i.e. continuous and differentiable in the sense of Gateaux)
functions of the variable , where
are real, and we prove curvilinear analogues of the Cauchy integral theorem,
the Morera theorem and the Cauchy integral formula in k-dimensional () real subset of the algebra . The present article is
generalized of the author's paper [1], where mentioned results are obtained for
k=3.
... Generalizing such an approach to the case of functions given in a domain of an arbitrary commutative Banach algebra, E.R. Lorch [7] introduced a derivative, which is also understood as a function given in the same domain. ...
... Here, Φ ′ L (ζ) is called the Lorch derivative of the function Φ at the point ζ (cf. E.R. Lorch [7]). ...
We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.
... In the case K = C, Ketchum [2] was the first to establish the basic analogues of Cauchy's and Morera's Integral Theorems, Cauchy's Integral Formula, A-analyticity, and others. This in turn was later generalized to infinite-dimensional A by Lorch [6] and Blum [7]. ...
... 4 For example, idA is not Fueter-regular. 5 Finite-dimensional commutative local (Artinian) algebras (A, m) with residue field A/m ∼ = R. 6 As opposed to the complex-analytic families in Kodaira-Spencer deformation theory. of smooth manifolds over Weil algebras. In the same vein, A-differentiability involves a (non-canonical) choice of an A-structure on a vector space V and is thus intimately related to the representation theory of A. This point of view too is out of the scope of the present article, which only aims to give an overview of the core facts that underpin the local theory. ...
To a morphism of finite-dimensional commutative associative unital Banach -algebras one can associate a sheaf on the underlying topological space of consisting of -valued differentiable functions f with -linear differential Df. It turns out that this class of functions exhibits a theory very similar to the classical complex analysis of one variable. In this article we give only an overview of some new results concerning the fundamentals of the corresponding local theory while at the same time also strengthen various already existing results scattered throughout the literature.
... The theory of analytic functions in algebras was started by Sheffers [12] at the end of nineteenth century. Other notable works are [3,6,8,9,11,13]. The corresponding differentiability is known as Lorch differentiability which is associated to algebras (in all this work algebra will be an ℝ-algebra associative commutative with unit), so we call it -differentiability, see Section 2.2. ...
... The -differentiability of vector fields is the same definition as the differentiability in the sense of Lorch with respect to , see [8]. ...
We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 𝔸 {\mathbb{A}} for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 𝔸 {\mathbb{A}} , which is also known as the Lorch derivative with respect to 𝔸 {\mathbb{A}} , and recently simply called 𝔸 {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.
... Ketchum's work stands out, which perhaps has not been understood or has been misinterpreted, because he works with algebras in a general way, which makes it seem a bit complicated to construct solutions of PDEs by the method he proposes. The theory of analytic functions over algebras has been developed since the end of the 19th century, also see [3], [6], [13], [14], [25], and [30]. ...
... If k = n and ϕ : R n → R n is the identity transformation ϕ(x) = x, the ϕA-differentiability will be called A-differentiability and the A-derivative of f will be denoted by f . This last differentiability is known as Lorch differentiability, see [13]. Differentiability related to commutative and noncommutative algebras is considered in [3]. ...
... Узагальнюючи такий пiдхiд на випадок довiльної комутативної банахової алгебри, Е.Р. Лорх [32] ввiв сильну похiдну функцiї, яка також розумiється як функцiя, визначена в тiй же областi, що i сама функцiя. ...
The aim of this work is to weaken the conditions of monogenity for functions that take values in subspaces of one concrete three-dimensional commutative algebras over the field of complex numbers. The monogenity of the function understood as a combination of its continuity with the existence of a Gato derivative.
... Definition 7.1. We say that the function f : D → A, D ⊂ A 3 , of three hypercomplex variables φ, ψ, η ∈ A has a partial derivative with respect to the variable φ in the domain D, if f is differentiable with respect to this variable in the sense of Lorch [6] at each point of the domain D, i.e. if there exists an element of algebra f φ such that for an arbitrary ε > 0 there exists δ > 0 such that for all h 1 ∈ A with ∥h 1 ∥ < δ the inequality ...
The work develops Humbert's method of solving PDEs. It applies method for constructing solutions of the three-dimensional Laplace, Helmholtz, and Poisson equations in the form of components of holomorphic functions of several variables (complex and hypercomplex). For more information see https://ejde.math.txstate.edu/Volumes/2024/71/abstr.html
... From representation (3.5), it follows that under the conditions of Theorem 3.1, each function Φ monogenic in the domain Ω is differentiable in a strong sense, in particular, in the sense of Lorch [25]. ...
In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras.
... It follows from (2.4) that under the assumptions of Theorem 2.2 each monogenic in the domain Ω function Φ is differentiable in a strong sense, and in particular, in the sense of Lorch [8]. The representation of the monogenic function Φ in the form (2.4) is unique. ...
In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.
... Для функцiї, заданої в областi скiнченновимiрної комутативної асоцiативної алгебри, Г. Шефферс [30] розглядав похiдну, яка розумiється як функцiя, визначена в тiй самiй областi. Узагальнюючи такий пiдхiд на випадок довiльної комутативної банахової алгебри, Е. Р. Лорх [31] увiв сильну похiдну функцiї, яка також розумiється як функцiя, визначена в тiй же областi, що i сама функцiя. ...
УДК 517.54 Послаблено умови моногенності функцій зі значеннями в певній тривимірній комутативній алгебрі над полем комплексних чисел.Під моногенністю мається на увазі неперервність та існування похідної Гато.
... The pre-twisted differentiability is defined in [5], this definition is closely related with the differentiability in the sense of Lorch, see [6]. Let A be an algebra and ϕ a differentiable planar vector field in the usual sense. ...
The components of complex differentiable functions define solutions for the Laplace’s equation. In this paper we generalize this result; for each PDE of the form we give an affine planar vector field and an associative and commutative 2D algebra with unit , with respect to which the components of all functions of the form define solutions for this PDE, where is differentiable in the sense of Lorch with respect to . In the same way, for each PDE of the form , the components of the exponential function defined with respect to , define solutions for this PDE. In the case of PDEs of the form , sine, cosine, hyperbolic sine, and hyperbolic cosine functions can be used instead of the exponential function. Also, solutions for two dependent variables order PDEs and a order PDE are constructed.
... Узагальнюючи такий пiдхiд на випадок довiльної комутативної банахової алгебри, Е.Р. Лорх [27] ввiв сильну похiдну функцiї, яка також розумiється як функцiя, визначена в тiй же областi, що i сама функцiя. ...
The aim of this work is to weaken the conditions of monogenity for functions that take values in one concrete three-dimensional commutative algebras over the field of complex numbers. The monogenity of the function understood as a combination of its continuity with the existence of a Gato derivative.
... (4) When A is a commutative Banach algebra, Lorch ( [2]) gave a definition that an A-valued function on an open set of A is A-holomorphic. With the norm max j=1,2,··· ,n |z j |, C n is a locally compact one. ...
... (4) When A is a commutative Banach algebra, Lorch ( [6]) gave a definition that an A-valued function on an open set of A is A-holomorphic. With the norm max j=1,2,··· ,n |z j |, C n is a locally compact one. ...
... An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the nite direct product C n of the complex number eld. On the other hand, when A is a commutative Banach algebra, Lorch [7] gave a de nition that an A-valued function on an open set of A is holomorphic. The de nition of a holomorphic function by Lorch can be straightforwardly generalized to a Fréchet di erentiable mapping from an open set of a Banach A-module to a Banach A-module. ...
Let Γ ( M ) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X . In this paper, we introduce a C ( X )-manifold structure on Γ ( M ). Especially, if X is contractible, then Γ ( M ) is a finite-dimensional C ( X )-manifold. Here, C ( X ) denotes the Banach algebra of all complex-valued continuous functions on X .
... (4) When A is a commutative Banach algebra, Lorch ( [6]) gave a definition that an A-valued function on an open set of A is A-holomorphic. With the norm max j=1,2,··· ,n |z j |, C n is a locally compact one. ...
Cartan-Thullen theorem states that for any open set of ,
the following conditions are equivalent:
(a) it is a domain of existence,
(b) it is a domain of holomorphy
and (c) it is holomorphically convex.
On the other hand, when
is a -valued function
on an open set U
of ,
f is said to be -analytic,
if f is complex analytic and for any i and j,
implies .
Here, holds.
We note that a -analytic mapping
and a -analytic manifold can be easily defined.
In this paper, we show an analogue of Cartan-Thullen theorem
for a -analytic function.
For n=1, it gives Cartan-Thullen theorem itself.
Our proof is almost the same as Cartan-Thullen theorem.
Thus, our generalization seems to be natural.
On the other hand, our result is partial,
because we do not answer the following question.
That is, does a connected open -holomorphically convex set U
exist such that U is not the direct product
of any holomorphically convex sets
and ?
As a corollary of our generalization, we only give a little partial answer.
Also, f is said to be -triangular,
if f is complex analytic and for any i and j,
implies .
Kasuya suggested that a -analytic manifold
and a -triangular manifold might, for example,
be related to a holomorphic web and a holomorphic foliation.
... (7) If A is infinite-dimensional, using Definition 3.0.1 is in fact the only sensible approach (also used in [Lor43]) as one can no longer expect A × to remain dense in A [see DF02]. ...
We study in detail the one-variable local theory of functions holomorphic over a finite-dimensional commutative associative unital -algebra , showing that it shares a multitude of features with the classical one-variable Complex Analysis, including the validity of the Jacobian conjecture for -holomorphic regular maps and a generalized Homological Cauchy's Integral Formula. In fact, in doing so we replace by a morphism in the category of finite-dimensional commutative associative unital -algebras in a natural manner, paving a way to establishing an appropriate category of Funktionentheorien (ger. function theories). We also treat the very instructive case of non-unital finite-dimensional commutative associative -algebras as far as it serves above agenda.
... Namely, it is shown in [5] that the class of H -analytic functions coincides with the class of Scheffers differentiable functions [20] (d f = f (η)dη). It follows from the results of paper [21] that the class of H -analytic functions coincides with the classes of functions differentiable in the sense of Gateaux and in the sense of Lorch [11]. ...
We consider the concept of the Hausdorff analyticity for functions ranged in real algebras and the corresponding notion of the Hausdorff derivative. Both apply to the real algebra H of Hamilton’s quaternions. The main aim of the work is to compare them with the well-known classes of H-valued functions which have their own definitions of the derivative.
... Hamilton (1866), Moisil andTheodoresco (1931), Fueter (1935), Sudbery (1979), Gürlebeck and Sprössig (1997), Kravchenko and Shapiro (1996), Leutwiler (1992), Ryan (1998), Colombo et al. (2011) and many other developed methods which are based on mappings of noncommutative algebras. P. Ketchum (1928Ketchum ( , 1929, Ringleb (1933), Sobrero (1934), Lorch (1943), Wagner (1948, Ward (1953), Riley (1953), Blum (1955), Roşculeţ (1955), Kunz (1971), Edenhofer (1976), Snyder (1982), I. Mel'nichenko (1975, 1986, Kovalev andMel'nichenko (1981), Mel'nichenko andPlaksa (2008) and many other developed methods which are based on mappings of commutative algebras. ...
We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid vector fields. We establish relations between axial-symmetric potential solenoid fields and principal extensions of complex analytic functions into a special topological vector space containing an infinite-dimensional commutative Banach algebra. In such a way we substantiate a method for explicit constructing axial-symmetric potentials and Stokes flow functions by means of components of the mentioned principal extensions and establish integral expressions for axial-symmetric potentials and Stokes flow functions in an arbitrary simply connected domain symmetric with respect to an axis. The obtained integral expression of Stokes flow function is applied for solving boundary problem about a streamline of the ideal incompressible fluid along an axial-symmetric body. We obtain criteria of solvability of the problem by means distributions of sources and dipoles on the axis of symmetry and construct unknown solutions using multipoles together with dipoles distributed on the axis.
... If k = n and ϕ : R n → R n is the identity transformation ϕ(x) = x, the ϕA-differentiability will be called A-differentiability and the A-derivative of f will be denoted by f ′ . This last differentiability is known as Lorch differenciability, see [9]. Differentiability related with commutative and noncommutative algebras is considered in [4]. ...
We introduce the \emph{-differentiability} for functions where is the linear space endowed with an algebra product which is unital, associative, commutative, U is an open set, and is a differentiable function in the usual sense. We also introduce the corresponding generalized Cauchy-Riemann equations (-CREs), the Cauchy-integral theorem, and the \emph{-differential equations}. The four-dimensional vector fields associated with triangular billiards are -differentiable. The \emph{-differential equations} can be used for constructing exact solutions of partial differential equations like the three-dimensional heat equation.
... Thus, ∂G ln Φ(ζ) = D1(∂G) F 1 (ξ 1 )I 1 + D2(∂G) F 2 (ξ 2 )I 2 and, in view of the principle of argument for analytic functions of a complex variable (see, e.g., [18, sec. 10]), equality (14) is transformed into (13). ...
We consider monogenic functions taking values in a three-dimensional commutative algebra A 2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A 2 into A 2 . It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A 2 , and, in general case, is a hypercomplex number.
... Rosenfeld showed there were very few differentiable functions for a large class of simple algebras, in contrast, we show there are algebras with nilpotent elements which support many differentiable functions 2 . Lorch's work [30] is also interesting. Lorch's concept of differentiation over an algebra was foundational for the recent papers [16] and [17] where it is shown how to solve certain ordinary differential equations via an algebra substitution. ...
Let denote a real, n-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over . An -differentiable function is one for which the differential is right--linear. We discuss the basis-dependent correspondence between right--linear maps and the regular representation of real matrices in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of gives generalized -CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra. The Wirtinger calculus is shown to generalize. We find the -CReqns are equivalent to the condition that the partial derivatives in all conjugate variables vanish. Our construction modifies that given by Alvarez-Parrilla, Fr\'ias-Armenta, L\'opez-Gonz\'alez and Yee-Romero in a 2012 paper. We also discuss how this conjugate technology gives us a method to convert real PDEs into differential equations over . Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. Taylor's Theorem and a Tableau for -differentiable function are derived. We prove many of the usual theorems of integral calculus including Cauchy's Integral Theorem for and the Fundamental Theorems of Calculus part I and II. Certainly we do not claim originality in some of what we present, however, we hope this paper adds something useful to the existing literature.
... Surprisingly however, it does not seem that there is any literature available that studies the properties of the logarithm for completely general commutative associative algebras of finite dimension. The closest to this might be [5], where the properties of the logarithm are deduced for algebras that take C as their base field. ...
Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for any multiplicative unital nil algebra the exponential function is injective, and hence the algebra has a unique logarithm on the image of the exponential. We extend this result to show that for a large class of algebras, the logarithms behave incredibly similarly to the logarithms over the real and complex numbers depending on if they are "Type-R" or "Type-C" algebras.
... In 1943 E. R. Lorch introduced an analytic theory for mappings whose domain and range lie in a commutative Banach algebra with identity [3]. Let A be such an algebra, and let D be an open connected subset of A. A mapping O : D-^-A is L-analytic, that is analytic in the sense of Lorch if in a neighborhood of each g e D we have a power series expansion ...
It is shown that two classes of function transformations coincide when the transformations take place within the disk algebra. The first class is thatof the ʟ-Analytic mappings. These are the ones given locally by power series:'formula presented'. The second class is that of locally pointwise mappings. A mapping fulfil is pointwise if it has the form 'formula presented'. It is a by-product of the disk algebra investigation that if a set Х has certain topological properties, then every locally pointwise mapping in ᴄ(Х) is continuous.
... The theory of analytic functions on algebras is based on Lorch analyticity; see [1][2][3][4][5]. Results of classical function theory have been extended to finite dimensional associative commutative unital algebras: ...
Given a planar system of nonautonomous ordinary differential equations,
d
w
/
d
t
=
F
(
t
,
w
)
, conditions are given for the existence of an associative commutative unital algebra
A
with unit
e
and a function
H
:
Ω
⊂
R
2
×
R
2
→
R
2
on an open set
Ω
such that
F
(
t
,
w
)
=
H
(
t
e
,
w
)
and the maps
H
1
(
τ
)
=
H
(
τ
,
ξ
)
and
H
2
(
ξ
)
=
H
(
τ
,
ξ
)
are Lorch differentiable with respect to
A
for all
(
τ
,
ξ
)
∈
Ω
, where
τ
and
ξ
represent variables in
A
. Under these conditions the solutions
ξ
(
τ
)
of the differential equation
d
ξ
/
d
τ
=
H
(
τ
,
ξ
)
over
A
define solutions
(
x
(
t
)
,
y
(
t
)
)
=
ξ
(
t
e
)
of the planar system.
In the paper [1] we consider a new class, so-called, G-monogenic (differentiable in the sense of Gateaux) quaternionic mappings. In the present paper we introduce quaternionic H-monogenic (differentiable in the sense of Hausdorff) mappings and establish the relation between G-monogenic and H-monogenic mappings. The equivalence of different definitions of G-monogenic mapping is proved.
An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of A is holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product of A. Therefore, a manifold modeled on the finite direct product of A (an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n=1, that is, Riemann surfaces).
By the way, if X is a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (furthermore, a commutative C*-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family M of compact complex manifolds (a topological deformation M of compact complex analytic structures) on X is denoted by G(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on X is introduced into G(M). Therefore, especially, if X is contractible, then G(M) is a finite-dimensional C(X)-manifold.
------------
(The published version is in Complex Manifolds, 6 (2019), 228--264.:) https://doi.org/10.1515/coma-2019-0012
An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of A is holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product of A. Therefore, a manifold modeled on the finite direct product of A (an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n=1, that is, Riemann surfaces).
By the way, if X is a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (furthermore, a commutative C*-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family M of compact complex manifolds (a topological deformation M of compact complex analytic structures) on X is denoted by G(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on X is introduced into G(M). Therefore, especially, if X is contractible, then G(M) is a finite-dimensional C(X)-manifold.
------------
(The published version is in Complex Manifolds, 6 (2019), 228--264.:) https://doi.org/10.1515/coma-2019-0012
and are simple examples of a real commutative Banach algebra. and are simple examples of a commutative -algebra. Here, we consider , which is the set of all real-valued continuous functions on the bounded closed interval [0,1]. The interval is a contractible compact Hausdorff space. The direct product space is a real Banach space and a free -module. A -mapping from an open set of to
is said to be -smooth, if its Frechet derivatives are -linear. Then, we can define a concept of an n-dimensional smooth -manifold.
In this memo, we see existence of a connected metrizable 1-dimensional smooth -manifold that cannot be embedded in . In Section 1, as a primitive observation, we see that embeddability in the Cartesian space as a smooth -submanifold
implies existence of an analog of a bump function. Then, in Section 2, we construct an example where no such analog exists.
The notion of analytically equivalent domains can be extended from the complex plane to commutative Banach algebras with identity. In C ( X ) C(X) a domain equivalent to the unit ball must have a boundary that is in a certain sense continuous. This paper shows that in the disk algebra “continuous” must be replaced with “analytic.” These results set limits in the classical Riemann mapping theorem on how smoothly the mapping can respond to changes in the domain being mapped.
In the paper [1] we consider a new class, so-called, G-monogenic (differentiable in the sense of Gateaux) quaternionic mappings. In the present paper we introduce quaternionic H-monogenic (differentiable in the sense of Hausdorff) mappings and establish the relation between G-monogenic and H-monogenic mappings. The equivalence of different definitions of G-monogenic mapping is proved.
The inverse function theorem for analytic functions has a generalization to C(X) which fails in the disk algebra.
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