arXiv subjects
Publications and source records attributed to V. K. Maslyuchenko.
It is obtained necessary and sufficient conditions of dependence on $\aleph$ coordinates for functions of several variables, each of which is a product of metrizable factors. The set of discontinuity points of such functions is characterized in the case, when each variable is a product of separable metrizable spaces.
It is shown that a set in product of $n$ metrizable spaces is the discontinuity points set of some separately continuous function if and only if this set can be represented as the union of a sequence of $F_σ$-sets which are locally projectively meager.
It is proved that a differentiable with respect to each variable function $f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a function $φ:\mathbb R\to\mathbb R$ such that $f(x,y)=φ(x-y)$. This gives a positive answer to a question of R.~Baire. Besides, we use this result to solving analogous partial differential equations in abstract spaces and partial differential equations of higher-order.