In this paper we study Doob's transform of fractional Brownian motion (FBM). It is well known that Doob's transform of standard Brownian motion is identical in law with the Ornstein-Uhlenbeck diffusion defined as the solution of the (stochastic) Langevin equation where the driving process is a Brownian motion. It is also known that Doob's transform of FBM and the process obtained from the Langevin equation with FBM as the driving process are different. However, also the first one of these can be described as a solution of a Langevin equation but now with some other driving process than FBM. We are mainly interested in the properties of this new driving process denoted Y (1) . We also study the solution of the Langevin equation with Y (1) as the driving process. Moreover, we show that the covariance of Y (1) grows linearly; hence, in this respect Y (1) is more like a standard Brownian motion than a FBM. In fact, it is proved that a properly scaled version of Y (1) converges weakly to Brownian motion.

Artikkelissa tarkastellaan evoluutioyhtälöpohjaisen jännitysväsymismallin stokastista laajennusta. Esitetty malli on muodostettu yleisten kontinuumimekaniikan periaatteiden mukaisesti ja on siten luonnostaan moniakselinen ja käsittelee kaikki jännityskomponentit ekvivalentilla tavalla. Malli soveltuu myos mielivaltaiselle kuormitushistorialle. Esimerkkinä tarkastellaan yksinkertaista valkoisella kohinalla häirityn säännollisen kuormituksen aiheuttaman elinikäaennusteen jakaumaa.

Summary. A very general continuum based approach to model both low-and high cycle fatigue behaviour is described. The approach allows for both isotropic and anisotropic properties under very general random multiaxial loading histories.

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.

MathCheck is a relatively new online tool that gives students feedback on their solutions to elementary university mathematics and theoretical computer science exercises. MathCheck was designed with constructivism learning theory in mind and it differs from other online tools as it checks the solutions step by step and shows a counter-example if the step is incorrect. It has been in student use since the autumn of 2015 and under design-based research from the first online day. The main research questions of this study are the following. 1) How can the usage of MathCheck support the aspects of conceptual understanding and procedural fluency of constructivism learning? 2) How can MathCheck empower both students and teachers in the education of mathematics? This paper presents the results of five pedagogical experiments considering both students’ and teachers’ point of views. In each experiment, the students have suggested improvements, which have affected the further development of MathCheck. In general, both students and teachers have given positive feedback on MathCheck. MathCheck seems to support learning better than tools that only provide the “incorrect”/“correct” verdict after checking the answer. MathCheck is suitable for independent studying as well as an addition to traditional lectures. In the best case, it can reduce teachers’ workload during courses.