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A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called $$\alpha $$ α -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ α -hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case $$\alpha =n-2$$ α = n - 2 , the $$n-2$$ n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$ α -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

In this paper we introduce the modified Dirac operators $$\mathcal {M}_{\mathbf {x}}^\kappa f{:}= \,\partial _{\mathbf {x}}f-\frac{\kappa }{x_m}e_m\cdot f$$ and $$\mathcal {M}_{\mathbf {y}}^\tau f{:}= \,\partial _{\mathbf {y}}f-\frac{\tau }{y_m}e_m \cdot f$$ , where $$f:\Omega \subset \mathbb {R}_+^{m}\times \mathbb {R}_+^{m}\rightarrow {{\mathrm{\mathcal {C}\ell }}}_{0,m}$$ is differentiable function, and $${{\mathrm{\mathcal {C}\ell }}}_{0,m}$$ is the Clifford algebra generated by the basis vectors of $$\mathbb {R}^m$$ . We look for solutions $$f(\mathbf {x},\mathbf {y}) =f(\underline{x},x_m,\underline{y},y_m)$$ of the system $$\mathcal {M}_{\mathbf {x}}^\kappa f(\mathbf {x,y})= \mathcal {M}_{\mathbf {y}}^\tau f(\mathbf {x,y})=0$$ , where the first and third variables are invariant under rotations. These functions are called $$(\kappa ,\tau )$$ -hypermonogenic functions. We discuss about axially symmetric functions with respect to the symmetric group $$SO(m-1)$$ . Some examples of axially symmetric $$(\kappa ,\tau )$$ -hypermonogenic functions generated by homogeneous functions and hypergeometric functions are presented.

Complex holomorphic functions are defined using a complex derivative. In higher dimensions the meaningful generalization of complex derivative is not straight forward. Sudbery defined a derivative for quaternion regular functions using differential forms. Gurlebeck and Malonek generalized that for monogenic functions. In this paper we find similar characterizations for k-hypermonogenic functions which are holomorphic functions based on the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}}{x_{2}^{2k}}. \end{aligned}$$ When \(k=0\), we obtain the hypercomplex derivative by Gurlebeck and Malonek. Just like in the complex case derivative of k-hypermonogenic is the usual partial derivative with respect to the first coordinate.

We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called $$\alpha $$α-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric $$\begin{aligned} ds_{\alpha }^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\alpha }} \end{aligned}$$dsα2=dx02+dx12+dx22+dx32x3αin the upper half space $${\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}$$R+4={x0,x1,x2,x3∈R4:x3>0}. If $$\alpha =2$$α=2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function $$x^{m}\,(m\in {\mathbb {Z}})$$xm(m∈Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental $$\alpha $$α-hyperbolic harmonic functions, depending only on the hyperbolic distance and $$x_{3}$$x3, we verify a Cauchy type integral formula for conjugate gradient of $$\alpha $$α-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued $$\alpha $$α-hypermonogenic in the Clifford algebra $${{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}$$Cℓ0,3.

We are studying hyperbolic function theory in the skew-field of quaternions. This theory is connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$\displaystyle ds_{k}^{2}=\frac {dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{k}} $$ in the upper half space \(\mathbb {R}_{+}^{4}=\{ \left ( x_{0},x_{1},x_{2},x_{3}\right )\in \mathbb {R}^4 : x_{3}>0\} \). In the case k = 2, the metric is the hyperbolic metric of the Poincare upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function \(x^{m}\,(m\in \mathbb {Z})\), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. We find fundamental k-hyperbolic harmonic functions depending only on the hyperbolic distance and x3. Using these functions we are able to verify a Cauchy type integral formula. Earlier these results have been verified for quaternionic functions depending only on reduced variables \(\left ( x_{0},x_{1},x_{2}\right )\). Our functions are depending on four variables.

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

We present how the education of subject teachers is organized in mathematics, science and computer science in Tampere. It is based on the idea that both engineering students and students from mathematics and science may choose to become a subject teacher. Students are accepted either to the master’s degree program in Science and Engineering of Tampere University of Technology or the master’s program of Mathematics and Statistics of University of Tampere. Students from different universities are giving opportunities to learn from each other. They study physics and chemistry in Tampere University of Technology and do pedagogical studies in University of Tampere. Both universities have also developed special motivating courses based on the didactical research to their students. In mathematics, there is a joined course for the second or third year students motivating towards teaching carrier. In both universities there are possibilities to do the master or bachelor thesis in didactics of mathematics or science. Both universities have an important role in education of subject teachers in Finland. Tampere University of Technology is providing opportunities during studies to cooperate between schools and industry. It gives ideas how science and mathematics are applied in the modern society. University of Tampere also trains primary school teachers with specialization in mathematics.

We study a hyperbolic version of holomorphic functions to higher dimensions. In this frame work, a generalization of holomorphic functions are called \(k\)-hypermonogenic functions. These functions are depending on several real variables and their values are in a Clifford algebra. They are defined in terms of hyperbolic Dirac operators. They are connected to harmonic functions with respect to the Riemannian metric \(ds^{2}_k=x_{n}^{2k/\left( 1-n\right) }\left( dx_{0}^{2}+\cdots +dx_{n}^{2}\right) \) in the same way as the usual harmonic function to holomorphic functions. We present the mean value property for \(k\)-hypermonogenic functions and related results. Earlier the mean value properties has been proved for hypermonogenic functions. The key tools are the invariance properties of the hyperbolic metric.

In this paper we consider the solutions of the equation $${\mathcal {M}}_\kappa f=0$$ , where $${\mathcal {M}}_\kappa $$ is the so called modifier Dirac operator acting on functions $$f$$ defined in the upper half-space and taking values in the Clifford algebra. We look for solutions $$f(\underline{x},x_{n})$$ where the first variable is invariant under rotations. A special type of solution is generated by the so called spherical monogenic functions. These solutions may be characterize by a Vekua-type system and this system may be solved using Bessel functions. We will see that the solution of the equation $${\mathcal {M}}_\kappa f=0$$ in this case will be a product of Bessel functions.

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We consider the modified Cauchy- Riemann operator M-k = Sigma(n)(i=0)=0(ei partial derivative xi) + k/xn Q' in the universal Clifford algebra Cl-0,Cl-n with the basis e1, ... ,en. The null- solutions of this operator are called k-hypermonogenic functions. We calculate the k- hyperbolic harmonic fundamental solutions, i. e. solutions to M-k(M)over bar(k)f = 0, and use these solutions to find k-hypermonogenic kernels for a Cauchy-type integral formula in the upper half-space.

In this paper we study the solutions of the equation $$\Delta w- \frac{\alpha}{y}\partial_{y}w = 0,$$ where w is a complex valued function. This equation is related to the generalized axially symmetric potential theory which has been studied notably by Weinstein, see [12]. We have researched this equation earlier in higher dimensions in connection with the hyperbolic function theory. In this paper will see how this equation is related to the generalized analytic functions in the hyperbolic upper half-plane. We also study harmonic differential forms in the hyperbolic plane and using these we obtain special type of solutions for the preceding equation.

We study the Weinstein equation $$\Delta u - \frac{k}{{x}_{2}} \frac{\partial}{\partial{x}_{2}} + \frac{l}{x^{2}_{2}}u = 0$$ , on the upper half space \({\mathbb{R}^3_{+} = \{ (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} | x_2 > 0\}}\) in case \({4l \leq (k + 1)^{2}}\) . If l = 0 then the operator \({x^{2k}_{2} (\Delta - \frac{k}{x_{2}} \frac{\partial}{\partial{x}_{2}})}\) is the Laplace- Beltrami operator of the Riemannian metric \({ds^2 = x^{-2k}_{2} (\sum^{2}_{i = 0} dx^{2}_{i})}\) . The general case \({\mathbb{R}^{n}_{+}}\) has been studied earlier by the authors, but the results are improved in case \({\mathbb{R}^3_{+}}\) . If k = 1 then the Riemannian metric is the hyperbolic distance of Poincare upper half-space. The Weinstein equation is connected to the axially symmetric potentials. We compute solutions of the Weinstein equation depending only on the hyperbolic distance and x2. The solutions of the Weinstein equation form a socalled Brelot harmonic space and therefore it is known that they satisfy the mean value properties with respect to the harmonic measure. However, without using the theory of Brelot harmonic spaces, we present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. Earlier these results were proved only for k = 1 and l = 0 or k = 1 and l = 1. We also compute the fundamental solutions. The main tools are the hyperbolic metric and its invariance properties. In the consecutive papers, these results are applied to find explicit kernels for k-hypermonogenic functions that are higher dimensional generalizations of complex holomorphic functions.

We research a function theory in higher dimensions based on the hyperbolic metric . The complex numbers are extended by the Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . In 1992, H. Leutwiler noticed that the power function (x 0 + x 1 e 1 + ··· + x n e n ) m is the generalized conjugate gradient of the functions . In the complex field (n = 1) this function h is harmonic in the usual sense, but in the higher dimensional case it is harmonic with respect to the Laplace–Beltrami operator with respect to the Riemannian hyperbolic metric . He started to study these type of functions, called H-solutions, that include positive and negative powers and elementary functions. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined by H. Leutwiler and the first author in 2000. The integral formula has been proved by the first author. In this article, we present a simple way to find hyperbolic harmonic functions depending on the hyperbolic ...

We consider the hyperbolic generalization of the classical complex analysis to higher dimensions based on the hyperbolic metric . The field of complex numbers is generalized by the associative Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . H. Leutwiler has noticed around 1992 that the power function (x 0 + x 1 e 1 + ··· x n e n ) m is the conjugate gradient of a hyperbolic harmonic function. He started to study these types of functions that include positive and negative powers and elementary functions, defined similarly as in classical complex analysis. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined in terms of the modified Dirac operator introduced by H. Leutwiler and the author in 2000. The integral formula for hypermonogenic functions has been proved by the author. In this article we compute the same kernels using the hyperbolic distance. The kernel is surprisingly the shifted Euclidean Cauchy kernel. Using this we ...

In this paper we consider solutions of the Weinstein equation $$\begin{aligned} \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}+\frac{\ell }{ x_{n}^{2}}u=0, \end{aligned}$$ on some open subset $$\Omega \subset \mathbb R ^{n}\cap \{x_{n}>0\}$$ subject to the conditions $$4\ell \le (k+1)^{2}$$ . If $$l=0$$ , the operator $$x_{n}^{2k/n-2}\left( \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}\right) $$ is the Laplace–Beltrami operator with respect to the Riemannian metric $$ds^{2}=x_{n}^{-2k/n-2}\left( \sum _{i=1}^{n}dx_{i} ^{2}\right) $$ . In case $$k=n-2$$ the Riemannian metric is the hyperbolic distance of Poincare upper half space. The Weinstein equation is connected to the axially symmetric potentials. The solutions of of the Weinstein equation form a so-called Brelot harmonic space and therefore it is known they satisfy the mean value properties with respect to the harmonic measure. We present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. The key idea is to transform the solutions to the eigenfunctions of the Laplace–Beltrami operator in the Poincare upper half-space model.

In complex function theory holomorphic functions are conjugate gradient of real harmonic functions. We may build function theories in higher dimensions based on this idea if we generalize harmonic functions and define the conjugate gradient operator. We study this type of function theory in \( \mathbb{R}^3 \) connected to harmonic functions with respect to the Laplace–Beltrami operator of the Riemannian metric \( ds^{2} = x_2^{-2k} \left(\sum \nolimits ^{2}_{i=0} dx^{2}_{i}\right) \). The domain of the definition of our functions is in \( \mathbb{R}^3 \) and the image space is the associative algebra of quaternions \( \mathbb{H} \) generated by 1, e1, e2 and e12 = e1e2 satisfying the relation e i e j + e j e i = –2δ ij , i, j = 1, 2. The complex field \( \mathbb{C} \) is identified by the set \( \{ x0 + x1e1 \, | \, x0,x1 \, \varepsilon \, \mathbb{B} \} \). The conjugate gradient is defined in terms of modified Dirac operator, introduced by \( M_{k}f = Df + kx_{2}^{-1}\overline{Qf} \), where Qf is given by the decomposition f (x) = Pf (x) + Qf (x) e2 with Pf (x) and Qf (x) in \( \mathbb{C} \) and \( \overline{Qf} \) is the usual complex conjugation.

Yläkoulun matematiikan opetuksen haasteena on sytyttää oppilaiden innostus ja auttaa heitä ymmärtämään matematiikan kokonaiskuva. Nykyisiä matematiikan opetusmenetelmiä pidetään usein teoreettisina ja ne opettavat vain osan tarvittavista taidoista. Projektiopiskelu tai -oppiminen saattaa olla vastaus näihin haasteisiin. Projektioppiminen on tapa järjestää opetus muotoon, jossa keskitytään ympäröivään maailmaan. Tavoitteena on liittää opiskeluaiheet osaksi opiskelijoiden jokapäiväistä elämää ja tulevaa työuraa. Matematiikkaa ei opeteta vain jatko-opintoja varten, vaan sitä tarvitaan muutenkin elämässä. Tutkimukset osoittavat, että projektioppiminen kehittää paitsi matemaattisten sisältöjen osaamista, myös työnantajien arvostamia meta-taitoja, kuten ongelmanratkaisutaitoja, spontaaniutta, vuorovaikutustaitoja ja yhteistyötaitoja, pitkäjänteisyyttä sekä tietotekniikkataitoja. Toisaalta projektit ovat usein toiminnallisia, mikä vaikuttaa positiivisesti tunnelmaan ja työrauhaan luokassa. Projektit, jotka tehdään yhteistyössä yritysten kanssa, lisäävät myös opiskelijoiden tietoa paikallisesta elinkeinoelämästä. Ulkomaiset tutkimukset projektioppimisesta osoittavat, että opiskelutavan käyttöönotossa on tiettyjä haasteita. Suomessa projektioppimista on sovellettu vähän yläkoulun matematiikan opetuksessa, joten sen käyttöönottoon tarvitaan lisätukea. Vuoden 2015 alussa käynnistetyn Projektioppiminen-kehittämishankkeen tavoitteena on lisätä opiskelijoiden innostusta matematiikkaan ja auttaa heitä ymmärtämään opiskeltavien asioiden laajempia yhteyksiä. Hankkeen aikana luodaan projektipankki yläkoulun matematiikan opettajien käyttöön ja testataan pankin projekteja käytännössä useissa eri peruskouluissa.

In this paper we study a mean-value property for solutions of the eigenvalue equa- tion of the Laplace-Beltrami operator ¢lbh = i(n i 1)h with respect to the volume and the surface integrals on the Poincare upper-half space R n+1 + = f(x0;:::;xn) 2 R n+1 : xn > 0g with the Riemannian metric ds 2 = dx 2 0+dx 2 1+¢¢¢+dx 2 n x2 . 1. Preliminaries In this section we recall the Laplace-Beltrami operator in the Poincare upper-half space and formulate its connections with the so called hypermonogenic functions. Let us denote R n+1 + = f(x0;x1;:::;xn) 2 R n+1 : xn > 0g. The Poincare half-space is the Riemannian manifold (R n+1 + ;ds 2 ), where the Riemannian metric is

The theory of monogenic functions or regular functions is based on Euclidean metric. We consider a function theory in higher dimensions based on hyperbolic metric. The advantage of this theory is that positive and negative powers of hyper complex variables are included to the theory, which is not in the monogenic case. Hence elementary functions can be defined similarly as in classical complex analysis.

In this paper we study geometric algebra valued null solutions of the equation $$D_{\ell}f- {k \over x_{0}}Q_{0}f=0$$ on the upper half \({\rm R}^{n+1}\cap \lbrace x_{0}>0\rbrace\), where D l is the Dirac operator and Q 0 is a projection-type mapping. Null solutions are called hypergenic functions. We will also study their local properties and integral representations.

In the first part of this article, we give the definition of bihypermonogenic functions in Clifford analysis. Using the idea of quasi-permutation, introduced by Sha Huang [Quasi-permutations and generalized regular functions in real Clifford analysis, J. Sys. Sci. and Math. Sci 18 (1998), pp. 380–384], we state an equivalent condition for bihypermonogenicity. In the second part, we discuss the Cauchy integral formula and Plemelj formula for the bihypermonogenic functions in real Clifford analysis.

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra \({\mathcal {C}}\ell_{n+1, 0}\). The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra \({\mathcal {C}}\ell_{n+1, 0}\) is generated by unit vectors \(\{e_{i}\}^n_{i=0}\) with positive squares e2i = + 1. The hyperbolic Dirac operator is of the form \(H_{k}f = Df - \frac{k} {x_{0}}Q_{0}f\) where Q0f is represented by the composition \(f = P_{0}f +e_{0}Q_{0}f\). If \(f : \Omega \rightarrow {\mathcal {C}}\ell_{n+1,0}\) is a solution of Hkf = 0, then f is called k-hypergenic in Ω, where \(\Omega \subset {\mathbb{R}}^{n+1}\) is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on \({\mathcal {C}}\ell_{n+1, 0}\).

A new Cauchy-type formula for hypermonogenic functions is derived. Hypermonogenic functions, introduced in [6], are a generalization of holomorphic functions to several dimensions. The power function xm is hypermonogenic.

The hyperbolic version of the standard Clifford analysis will be considered. In this modification the power function x m becomes a solution. In more details, the Dirac operator \(Df = \sum^n_{i=0} e_i \frac{\partial f} {\partial x_i}\) with e 0 = 1, defined with respect to the Clifford algebra Cl n , is replaced by the operator \(M_kf(x) = Df (x) + \frac{k}{x_n} Q^{\prime}f(x)\), where ′ denotes the main involution in Cl n and Qf is given by the unique decomposition f(x) = Pf(x) + Qf(x)e n with Pf(x),Qf(x) ∈Cl n-1. The operator M k (k ∈R) will mainly be considered for k = 0, k = n − 1 and k = 1 − n. In case k = 0 the equation Mkf = 0 yields the well-known monogenic functions, in case k = n − 1 one obtains the so-called hypermonogenic functions introduced in [5]. Besides M k we also study the operator \(\overline{M_k}M_k = M_k\overline{M_k}\), a natural generalization of the Laplace operator Δ. Solutions of the equation \(\overline{M_{n-1}}M_{n-1}f =0\) are called hyperbolic harmonic functions. The main goal of this article is to give integral representations for hypermonogenic and hyperbolic harmonic functions in the upper half space \({\mathbb{R}}^{n+1}_+\).

The aim of this article is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function x m is included. The leading idea is that the power function is the conjugate gradient of a harmonic function, defined with respect to the hyperbolic metric of the upper half space. In this paper we give a new approach to this hyperbolic function theory and survey some of its results.

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Qp-spaces.

We consider solutions of the equation yΔh (x,y)−k∂h∂y=0 in the plane. These functions already have been investigated by Weinstein around 1950 in connection of generalized axially symmetric potential theory. We have found several results concerning these type of functions, called k-hyperbolic harmonic functions, in higher dimensions. In this paper, we show in the plane case that it is possible to compute the explicit fundamental solutions in terms of the hyperbolic metric. These results may be used to find fundamental solutions in all even dimensional spaces. The key tools are the transformation properties of hyperbolic metric of the Poincare upper half space model.

Let Cl n be the (universal) Clifford algebra generated by e 1, …, en , satisfying . The Dirac operator in Cl n is defined by , where e 0=1. The modified Dirac operator is introduced in (Eriksson-Bique and Leutwiler 2000, Hypermonogenic functions. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2 (Boston: Birkhauser), pp. 287–302) by Mf=Df+k(Q′f/x n ), where ′ denotes the main involution in Cl n and Qf is given by the decomposition with . A continuously differentiable function f:Ω→ Cl n is called hypermonogenic in an open subset Ω of , if , for all x ∈Ω. Paravector-valued hypermonogenic functions are called H-solutions, see (Leutwiler, 1992, Modified Clifford analysis, Complex Variables, 17, 153–171). The power function is an H-solution. We give a Cauchy-type formula for H-solutions. Furthermore we derive the equation for the restriction g of the hypermonogenic function f to the hyperplane xn =0. This equation has been considered by Laville, Lehman and Ramadanoff, see (2004, Anal...

Bagdad – Mathematics from here to eternity is an exhibition produced by Swedish Navet Sicence Center that was held in Museum Centre Vapriikki in Tampere 11 April–31 May 2013. The Bagdad exhibition was a fascinating journey to the history of mathematics guided by few historical mathematicians. Al-Khwarizmi, Hypatia, Sofia Kovalevskaya, Archimedes, Brahmagupta and other mathematical masters led the audience to the world of mathematics through drama, games and different problems to be solved. Each exhibition session lasted an hour and a half and included introduction, problems to be solved together and in groups, and individual activities. The introduction was in the form of drama and guided the audience to the mathematical problems. After the introduction, a mathematical problem was solved in groups. After that the audience had the opportunity to explore the tent and games and problems inside of it. At the end, the audience pondered and solved a mathematical problem together.

We considerC 2-solutionsf=u+iv+jw of the system $$t\left( {\frac{{\partial u}}{{\partial x}} - \frac{{\partial v}}{{\partial y}} - \frac{{\partial w}}{{\partial t}}} \right) + w = 0$$ $$\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}, \frac{{\partial u}}{{\partial t}} = - \frac{{\partial w}}{{\partial x}},\frac{{\partial v}}{{\partial t}} = \frac{{\partial w}}{{\partial y}}$$ calledH-solutions introduced by H. Leutwiler. Iff is anH-solution in ω, thenf | Ω∩ℂ is holomorphic. SinceH-solutions are real analytic, a non-zeroH-solution cannot vanish in an open subdomain of ℝ3. Our object is, by the way of examples, to show that there are many kinds of null-sets ofH-solutions in ℝ3. This is in sharp contrast to a holomorphic functionf in ℂ, where the setf −1 ({0}) consists of discrete points only unlessf≡0.

We consider two generalizations of Cauchy-Riemann equations to the n-dimensional Euclidean space. The first one, obtained by the Dirac operator, is given by . The latter system can be considered as a non-euclidean version of the former one. It is known that x m satisfies the second system, but not the first one. We study elementary homogeneous polynomial solutions and the left module generated by them.

The function theory in \( {\Bbb R}^3 \), introduced in Complex Variables 20 (1992) by the second author, will be further developed. It has the property that its underlying generalized Cauchy-Riemann system is fulfilled by the power-function \( z\rightarrow z^n, z = x + iy + jt, n\in {\Bbb N} \). In this paper we study the basic polynomials \( L_n^k:z\rightarrow {1\over k!} {{\partial^k z^{n+k}} \over \partial y^k} \). Explicit formulas for the polynomials L n k are deduced. For k = 0,1, . . . , n the real parts Re L n k yield an explicit basis for the vector space of hyperbolic harmonic, homogeneous polynomials of degree n, hyperbolic harmonicity being defined by the Poincare metric ds 2 = t -2 (dx 2 + dy 2 + dt 2).

We consider multiply hyperharmonic functions on the product space of two harmonic spaces in the sense of Constantinescu and Cornea. Earlier multiply superharmonic and harmonic functions have been studied in Brelot spaces notably by GowriSankaran. Important examples of Brelot spaces are solutions of elliptic differential equations. The theory of general harmonic spaces covers in addition to Brelot spaces also solution of parabolic differential equations. A locally lower bounded function is multiply hyperharmonic on the product space of two harmonic spaces if it is a hyperharmonic function in each variable for every fixed value of the other. We prove similar results as in Brelot spaces, but our approach is different. We study sheaf properties of multiply hyperharmonic functions. Our main theorem states that multiply hyperharmonic functions are lower semicontinuous and satisfy the axiom of completeness with respect to products of relatively compact sets. We also study nearly multiply hyperharmonic functions.

This paper derives formulas for least-squares transformations between point-sets in ℝ d . We consider affine transformations, with optional constraints for linearity, scaling, and orientation. We base the derivations hierarchically on reductions, and use trace manipulation to achieve short derivations. For the unconstrained problems, we provide a new formula which maximizes the orthogonality of the transform matrix.

In this paper we consider harmonic differential forms and Clifford multi-vector functions on the hyperbolic upper half-space. We see how the operators and their solutions are related and present a Moisil-Theodorescu-type system related to the harmonic multi-vectors.

In this paper we study a mean-value property for solutions of the Laplace-Beltrami equation $$x^{2}_{n} \Delta h - (n-1) x_n \frac{\partial h}{\partial x_n} = 0$$ (Equation 1 ) with respect to the volume and the surface integral on the Poincare upper-half space \(\mathbb{R}^{n+1}_{+} = \{(x_0,...,x_n)\ \in \mathbb{R}^{n+1} : x_n > 0\}\) with the Riemannian metric \(g = \frac{dx^{2}_{0} + dx^{2}_{1} +...+ dx^{2}_{n}}{x^{2}_{n}}\). We also compute the Cauchy type kernels in terms of the hyperbolic metric.

In this paper we present a new integral formulas for hypermonogenic functions where the kernels are also hypermonogenic functions. We also introduce dual k-hypermonogenic functions. If k = 0, then k-hypermonogenic functions are monogenic functions and their dual functions are also monogenic. If k is nonzero the only function that is k-hypermonogenic function and dual hypermogenic is zero function.

Let X be a harmonic space in the sense of C. Constantinescu and A. Cornea. We show that, for any subset E of X, a positive superharmonic function u on X has a representation u = p + h, where p is the greatest specific minorant of u satisfying . This result is a generalization of a theorem of M. Brelot. We also state some characterizations of extremal superharmonic functions.

We study k-hyperbolic harmonic functionsin the upper half space . The operator is the Laplace-Beltrami operator with respect to the Riemannian metric . In case k = n — 1 the Riemannian metric is the hyperbolic distance of Poincare upper half space. The proposed functions are connected to the axially symmetric potentials studied notably by Weinstein, Huber and Leutwiler. We present the fundamental solution in case n is even using the hyperbolic metric. The main tool is the transformation of k-hyperbolic harmonic functions to eigenfunctions of the hyperbolic Laplace operator.