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Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half‐space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper, we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form S:={z∈핆|0<ℜ(z)

In this paper, we will express weighted monogenic functions as series composed of weighted monogenic polynomials. Firstly, the definition of p order homogeneous weighted monogenic polynomials is given. In order to obtain the basis of the set composed of the above polynomials, the hypercomplex variables are introduced. Secondly, we prove the relationship between the analytic as well as weighted monogenic functions and the p order homogeneous weighted monogenic polynomials. By the relationship, the Taylor expansion of the weighted monogenic functions at a certain point is given. Then, the uniform convergence of the Taylor expansion of $ E_{\omega }(x,\xi)=\displaystyle \frac{1}{det(B)^{\frac{1}{2}}\omega_{n}}\frac{1}{\rho^{n}}\sum_{i,j=1}^{n}\overline{\psi_{i}}A_{ij}(x_{j}-\xi_{j}) $ E ω (x , ξ) = 1 det (B) 1 2 ω n 1 ρ n ∑ i , j = 1 n ψ i ¯ A ij (x j − ξ j) on every compact subset of a certain domain is proved. From the above results, the uniform convergence of the Taylor expansion of arbitrary weighted monogenic function f on every compact subset of the above domain is further obtained, and the inverse theorem of Taylor expansion is also obtained. Finally, the uniqueness theorem is obtained by the Taylor expansion and the connectivity of Ω. [ABSTRACT FROM AUTHOR]

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity. [ABSTRACT FROM AUTHOR]

This article presents an innovative extension of the classical Almansi decomposition. Traditionally, the Almansi decomposition is employed to decompose polymonogenic functions into their monogenic counterparts, thereby elucidating the relationship between the Dirac operator and its successive iterations. Diverging from this conventional method, the new extension delves into the kernel of a polynomial related to the quaternionic Dirac operator, as opposed to focusing on the iterated operator. This is achieved by employing a polynomial-based partition of unity. The outcome of this approach is a distinctive decomposition that leverages the kernel of the Dirac–Helmholtz operators. [ABSTRACT FROM AUTHOR]

A function f$$ f $$ from a domain in ℝ3$$ {\mathrm{\mathbb{R}}}&amp;#x0005E;3 $$ to the quaternions is said to be inframonogenic if ∂‾f∂‾=0$$ \overline{\partial}f\overline{\partial}&amp;#x0003D;0 $$, where ∂‾=∂/∂x0+(∂/∂x1)e1+(∂/∂x2)e2$$ \overline{\partial}&amp;#x0003D;\partial /\partial {x}_0&amp;#x0002B;\left(\partial /\partial {x}_1\right){e}_1&amp;#x0002B;\left(\partial /\partial {x}_2\right){e}_2 $$. All inframonogenic functions are biharmonic. In the context of functions f=f0+f1e1+f2e2$$ f&amp;#x0003D;{f}_0&amp;#x0002B;{f}_1{e}_1&amp;#x0002B;{f}_2{e}_2 $$ taking values in the reduced quaternions, we show that the inframonogenic homogeneous polynomials of degree n$$ n $$ form a subspace of dimension 6n+3$$ 6n&amp;#x0002B;3 $$. We use the homogeneous polynomials to construct an explicit, computable orthogonal basis for the Hilbert space of square‐integrable inframonogenic functions defined in the ball in ℝ3$$ {\mathrm{\mathbb{R}}}&amp;#x0005E;3 $$. [ABSTRACT FROM AUTHOR]

In this paper, the hypercomplex Ruscheweyh derivative operator for special monogenic functions is defined. The representation in certain regions of such functions in terms of hypercomplex Ruscheweyh derivative bases of special monogenic polynomials (HRDBSMPs) are investigated. Precisely, we examine the approximation properties in different regions such as closed balls, open balls, closed regions surrounding closed balls, at the origin and for all entire special monogenic functions. Moreover, the order type and the Tρ-property for these bases are discussed. We also provide some interesting applications for some HRDBSMPs such as Bernoulli, Euler, and Bessel polynomials. The obtained results extend and enhance relevant results in the complex and Clifford setting. [ABSTRACT FROM AUTHOR]

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in R4$\mathbb {R}^4$, denoted by D$\mathcal {D}$. This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of Δ=DD¯$\Delta = \mathcal {D} {\overline{\mathcal D}}$. Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator D¯$ {\overline{\mathcal D}}$ and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of D2$ \mathcal {D}^2$. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors. [ABSTRACT FROM AUTHOR]

Edge detection is a technique in digital image processing that detects the contours of objects based on changes in brightness. Edges can be used to determine the size, orientation, and properties of the object of interest within an image. There are different techniques employed for edge detection, one of them being phase congruency, a recently developed but still relatively unknown technique due to its mathematical and computational complexity compared to more popular methods. Additionally, it requires the adjustment of a greater number of parameters than traditional techniques. Recently, a unique formulation was proposed for the mathematical description of phase congruency, leading to a better understanding of the technique. This formulation consists of three factors, including a quantification function, which, depending on its characteristics, allows for improved edge detection. However, a detailed study of the characteristics had not been conducted. Therefore, this article proposes the development of a generalized function for quantifying phase congruency, based on the family of functions that, according to a previous study, yielded the best results in edge detection. [ABSTRACT FROM AUTHOR]

In this paper we study the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e. the solutions of the sandwich equation ∂ x _ f ∂ x _ = 0 , where ∂ x _ stands for the Dirac operator in R m. The main steps in deriving our results are the establishment of some interior estimates for the first order derivatives of harmonic Hölder continuous functions and the proof of certain invariance property of the higher order Lipschitz class under the action of the Poisson integral. Using Mathematica we also implement an algorithm to find explicitly the solution of such a Dirichlet problem for a much wider class of partial differential equations in the ball of R 3 with polynomial boundary data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we study the continuity of Cauchy-type integral operator C Γ infra associated with inframonogenic functions and give the Plemelj formula. Secondly, we prove the properties of the Teodorescu operator related to the inframonogenic functions, including its boundness, continuity and differentiability. Finally, we give the related integral representation of Riemann boundary value problems for inframonogenic functions and generalized inframonogenic functions. [ABSTRACT FROM AUTHOR]

Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases. [ABSTRACT FROM AUTHOR]

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator in \mathbb {R}^4, denoted by \mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator \Delta in four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. \Delta = \mathcal {\overline {D}} \mathcal {D} to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of \mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the S-spectrum. [ABSTRACT FROM AUTHOR]

This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms. [ABSTRACT FROM AUTHOR]

This article is an editorial on the special issue "Current topics in applied hypercomplex analysis" dedicated to K. Gürlebeck on his 65th birthday. The special issue features papers from a workshop held in his honor in February 2020, which was delayed due to the start of the Corona pandemic. K. Gürlebeck is a renowned mathematician with expertise in areas such as quaternionic and hypercomplex analysis, numerical mathematics, parameter identification problems, fluid mechanics, and elasticity theory. The special issue showcases the advancements in these topics and pays tribute to K. Gürlebeck's contributions to the field. [Extracted from the article]

We consider a piecewise continuous biharmonic problem in an angle and the corresponding Schwartz-type boundary-value problem for monogenic functions in a commutative biharmonic algebra. These problems are reduced to a system of integral equations on the positive semiaxis. It is shown that, on each segment of this semiaxis, the set of solutions of this system coincides with the set of solutions of a certain system of Fredholm integral equations. [ABSTRACT FROM AUTHOR]

This paper shows a hypercomplex function theory emerging in the representation of paravector‐valued monogenic functions over the (m+1)$$ \left(m+1\right) $$‐dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well‐known Whittaker‐Cannon theorem to special monogenic functions defined in an open hyperball in ℝm+1$$ {\mathbb{R}}^{m+1} $$. More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three‐hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non‐Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non‐Cannon basis. [ABSTRACT FROM AUTHOR]

New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found [ABSTRACT FROM AUTHOR]

We consider special topological vector spaces with a commutative multiplication for some of elements of the spaces and monogenic functions taking values in these spaces. Monogenic functions are understood as continuous and differentiable in the sense of Gâteaux functions. We describe relations between the mentioned monogenic functions and harmonic vectors in the three-dimensional real space and establish sufficient conditions for infinite monogeneity of functions. Unlike the classical complex analysis, it is done in the case where the validity of the Cauchy integral formula for monogenic functions remains an open problem. [ABSTRACT FROM AUTHOR]

We prove that any commutative and associative algebra 픹* of the second rank with unit over the field of complex numbers ℂ contains bases {e1, e2} for which 픹*-valued "analytic" functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differentional equation of the fourth order with complex coefficients whose characteristic equation has a single multiple root and the remaining roots are simple. We present a complete description of the set of all triples (픹*, {e1, e2}, Φ). [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

Solutions of the sandwich equation ϕ ∂ ̲ [ f ] ψ ∂ ̲ = 0 , where ϕ ∂ ̲ stands for the Dirac operator with respect to a structural set ϕ , are referred to as (ϕ , ψ) -inframonogenic functions and capture the standard inframonogenic ones as special case. We derive a new integral representation formula for such functions as well as for multidimensional Ahlfors–Beurling transforms closely connected to the use of two different orthogonal basis in R m . Moreover, we also establish sufficient conditions for the solvability of a jump problem for the system ϕ ∂ ̲ [ f ] ψ ∂ ̲ = 0 in domains with fractal boundary. [ABSTRACT FROM AUTHOR]

The statement that any two-dimensional algebra 픹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 픹*-valued "analytic" functions Φ(xe1 + ye2) (x, y are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples (픹*; {e1; e2}; Φ) is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these "analytic" functions. [ABSTRACT FROM AUTHOR]

Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics. We introduce a new type of generalized Clifford algebra such that all components of a monogenic function are solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations within the framework of Clifford analysis. We prove some Cauchy integral representation formulas for monogenic functions in these cases. [ABSTRACT FROM AUTHOR]

We define a very general notion of regularity for functions taking values in an alternative real ⁎-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail. [ABSTRACT FROM AUTHOR]

The aim of the present work is to weaken the conditions of monogeneity for functions taking values in a given three-dimensional commutative algebra over the field of complex numbers. The monogeneity of a function is understood as a combination of its continuity with the existence of Gâteaux derivative. [ABSTRACT FROM AUTHOR]

A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]

A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S-functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra Rn. The methodology can be generalized, for example, to handle the case of noncommuting matrix variables. [ABSTRACT FROM AUTHOR]

A synthetic aperture radar (SAR) target recognition method based on image blocking and matching is proposed. The test SAR image is first separated into four blocks, which are analyzed and matched separately. For each block, the monogenic signal is employed to describe its time-frequency distribution and local details with a feature vector. The sparse representation-based classification (SRC) is used to classify the four monogenic feature vectors and produce the reconstruction error vectors. Afterwards, a random weight matrix with a rich set of weight vectors is used to linearly fuse the feature vectors and all the results are analyzed in a statistical way. Finally, a decision value is designed based on the statistical analysis to determine the target label. The proposed method is tested on the moving and stationary target acquisition and recognition (MSTAR) dataset and the results confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

Among all two-dimensional algebras of the second rank with unit e over the field of complex numbers ℂ, we find a semisimple algebra 𝔹0 := {c1e + c2𝜔 : ck 𝜖 ℂ, k = 1, 2}, 𝜔2 = e, containing bases {e1, e2} such that the 𝔹0-valued "analytic" functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differential equation of the fourth order, which has only simple nonzero characteristics. The set of pairs ({e1, e2},Φ) is described in the explicit form. [ABSTRACT FROM AUTHOR]

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G := ∥ x → ∥ 2 ∂ 0 + x → ∑ k = 1 3 x k ∂ k , according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory. 2019;13(6):2527–2539; Colombo F, González-Cervantes JO, Sabadini I. A non-constant coefficients differential operator associated to slice monogenic functions. Trans Am Math Soc. 2013;365:303–318]. This association allows to show a behavior of the theory of slice regular functions similar to the well known theories of the hypercomplex analysis. [ABSTRACT FROM AUTHOR]

Examining the asymptotic growth behavior of holomorphic and meromorphic functions has significant importance in complex analysis. Estimations of upper bounds for the rate of growth of entire functions are mostly guaranteed. However, the analog estimations for lower bounds of the rate of growth are not always attainable. In this paper, we give the lower rate of growth of the generalized Hadamard product of two entire axially monogenic functions. It has also been shown that the product entire axially monogenic function is of regular growth or perfectly regular growth when its constituent entire axially monogenic functions possess these properties. The investigation of both upper and lower bounds by means of linear transmutation is also provided. Furthermore, some applications related to approximation theory are outlined. [ABSTRACT FROM AUTHOR]

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations. [ABSTRACT FROM AUTHOR]

The main objective of this work is to interpret aeromagnetic data of the central In Ouzzal terrane and adjacent zone, southern Algeria using the Riesz Transform (RT) method. In this paper, we developed the RT method using the concepts of monogenic functions with Cauchy Riemann conditions. The result, obtained by calculation, led us to the new expression of the Riesz Analytic Signal Amplitude (RASA) and Riesz Local Phase (RLP), respectively. Tests on a synthetic magnetic model showed that the RASA and RLP had a better performance in delineating the geological contacts that were not seen in the original data. The advantage of this RT method is that it is less sensitive to noise. By applying the Euler deconvolution method using the RASA to the aeromagnetic data of the study area, we obtained the West Ouzzalian Fault (WOF), which is below the Paleozoic cover and has a depth of 0.8 km. At a distance of 30 km from the WOF, we find the East Ouzzalian Fault (EOF) with a depth of about 5 km. According to the RLP, the dip of the WOF, the EOF and the Adrar Fault are vertical. The fault systems located inside of the central In Ouzzal terrane are, generally, inclined towards the west. [ABSTRACT FROM AUTHOR]

In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially monogenic functions, by making use of the Funk–Hecke theorem. Then we present the explicit form of the general Cauchy–Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann–Radon transform for Cauchy–Kowalewski extension. [ABSTRACT FROM AUTHOR]

We consider a class of plane orthotropic deformations of the form εx = σx + a12σy, γxy = 2(p − a12)Txy, εy = a12σx + σy, where σx, Txy, σy and ε x γ xy 2 , ε Y are components of the stress tensor and the deformation tensor, respectively, real parameters p and a12 satisfy the inequalities: -1 < p < 1, -1 < a12 < p. A class of solutions of the Lamé equilibrium system for displacements is built in the form of linear combinations of components of "analytic" functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers. [ABSTRACT FROM AUTHOR]

In this paper, we consider functions satisfying the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. Such functions are referred as inframonogenic and represent an extension of the monogenic functions, ie, null solutions of ∂x_. In particular, for odd m, we prove that a C2‐function is both inframonogenic and harmonic in Ω⊂Rm if and only if it can be represented in Ω as f=f1+f2+f3x_+x_f4,where f1 and f2 are, respectively, left and right monogenic functions in Ω, while f3 and f4 are two‐sided monogenic functions there. Finally, in deriving some applications of our results, we have made use of the deep connection between the class of inframonogenic vector fields and the universal solutions of the Lamé‐Navier system in R3. [ABSTRACT FROM AUTHOR]

The article presents a monogenic exponential function for the hypercomplex function theory of quaternion-valued monogenic functions in a reduced quaternionic variable, which has essential properties analogous to the complex exponential function ,. In particular, it is shown that the Taylor series of the monogenic exponential function based on a complete orthogonal system of monogenic Appell polynomials also generalizes the corresponding complex series representation in a canonical manner. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract In this paper we first prove an important formula for the fractional Laplacian, and then we use it to invert the Fueter mapping theorem for axially monogenic functions of degree k. In fact, we prove that for every axially monogenic function of degree k f (x) = [ A (x 0 , | x _ |) + x _ | x _ | B (x 0 , | x _ |) ] P k (x _) , x ∈ R n + 1 , there exists a holomorphic intrinsic function f k in C such that f (x) = τ k (f k) (x) : = (− Δ) k + (n − 1) / 2 ( f → k (x) P k (x _)) , where n can be any positive integer, k can be any non-negative integer, f → k is the slice monogenic function induced by f k , and P k (x _) is an inner spherical monogenic polynomial of degree k. Using the maps τ k , k = 0 , 1 , 2 , ... , we obtain a decomposition of a monogenic function for any value of the dimension n. [ABSTRACT FROM AUTHOR]

BACKGROUNDIn the Joslin Medalist Study (Medalists), we determined whether significant associations exist between β cell function and pathology and clinical characteristics.METHODSIndividuals with type 1 diabetes (T1D) for 50 or more years underwent evaluation including HLA analysis, basal and longitudinal autoantibody (AAb) status, and β cell function by a mixed-meal tolerance test (MMTT) and a hyperglycemia/arginine clamp procedure. Postmortem analysis of pancreases from 68 Medalists was performed. Monogenic diabetes genes were screened for the entire cohort.RESULTSOf the 1019 Medalists, 32.4% retained detectable C-peptide levels (>0.05 ng/mL, median: 0.21 ng/mL). In those who underwent a MMTT (n = 516), 5.8% responded with a doubling of baseline C-peptide levels. Longitudinally (n = 181, median: 4 years), C-peptide levels increased in 12.2% (n = 22) and decreased in 37% (n = 67) of the Medalists. Among those with repeated MMTTs, 5.4% (3 of 56) and 16.1% (9 of 56) had waxing and waning responses, respectively. Thirty Medalists with baseline C-peptide levels of 0.1 ng/mL or higher underwent the clamp procedure, with HLA-/AAb- and HLA+/AAb- Medalists being most responsive. Postmortem examination of pancreases from 68 Medalists showed that all had scattered insulin-positive cells; 59 additionally had few insulin-positive cells within a few islets; and 14 additionally had lobes with multiple islets with numerous insulin-positive cells. Genetic analysis revealed that 280 Medalists (27.5%) had monogenic diabetes variants; in 80 (7.9%) of these Medalists, the variants were classified as "likely pathogenic" (rare exome variant ensemble learner [REVEL] >0.75).CONCLUSIONAll Medalists retained insulin-positive β cells, with many responding to metabolic stimuli even after 50 years of T1D. The Medalists were heterogeneous with respect to β cell function, and many with HLA+ diabetes risk alleles also had monogenic diabetes variants, indicating the importance of genetic testing for clinically diagnosed T1D.FUNDINGFunding for this work was provided by the Dianne Nunnally Hoppes Fund; the Beatson Pledge Fund; the NIH, National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK); and the American Diabetes Association (ADA). [ABSTRACT FROM AUTHOR]