2000 Mathematics Subject Classification: 44A40, 44A35A direct algebraic construction of a family of operational calculi for the Euler differential operator δ = t d/dt is proposed. It extends the Mikusiński's approach to the Heaviside operational calculus for the case when the classical Duhamel convolution is replaced by the convolution ...

The one-dimensional steady-state convection-diffusion problem for the unknown temperature $y\left(x\right)$ of a medium entering the interval $(a,b)$ with the temperature $y_{min}$ and flowing with a positive velocity $v\left(x\right)$ is studied. The medium is being heated with an intensity corresponding to $y_{max}-y\left(x\right)$ for a constant $y_{max}>y_{min}$. We are looking for a velocity $v\left(x\right)$ with a given average such that the outflow temperature $y\left(b\right)$ is maximal and discuss the influence of the boundary condition at the point $b$ on the “maximizing” function $v\left(x\right)$.

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}+q\left(t\right)y=0(*)$$ are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}-\frac{{\mu }_{n,\alpha }}{t^{2n-\alpha }}y=0,$$ where ${\mu }_{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.