# Nesterenko V. V.

### Points of joint continuity and large oscillations

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 791–800

For topological spaces $X$ and $Y$ and a metric space $Z$, we introduce a new class $N(X × Y,Z)$ of mappings $f:\; X × Y → Z$ containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping $f$ from this class and any countable-type set $B$ in $Y$, the set $C_B (f)$ of all points $x$ from $X$ such that $f$ is jointly continuous at any point of the set $\{x\} × B$ is residual in $X$: We also prove that if $X$ is a Baire space, $Y$ is a metrizable compact set, $Z$ is a metric space, and $f ∈ N(X×Y,Z)$, then, for any $ε > 0$, the projection of the set $D^{ε} (f)$ of all points $p ∈ X × Y$ at which the oscillation $ω_f (p) ≥ ε$ onto $X$ is a closed set nowhere dense in $X$.

### Ostrohrads'kyi Formalism for Singular Lagrangians with Higher Derivatives

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1034-1037

We generalize the Ostrohrads'kyi method for the construction of the Hamiltonian description of a nondegenerate (regular) variational problem of arbitrary order to the case of degenerate (singular) Lagrangians. These Lagrangians are of major interest in the contemporary theory of elementary particles. For simplicity, we consider the Hamiltonization of a variational problem defined by a singular second-order Lagrangian. Generalizing the Ostrohrads'kyi method, we derive equations of motion in the phase space. We determine a complete collection of constraints of the theory.

### Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1711-1714

We show that if *X*is a topological space, *Y*satisfies the second axiom of countability, and *Z*is a metrizable space, then, for every mapping *f*: *X*× *Y*→ *Z*that is horizontally quasicontinuous and continuous in the second variable, a set of points *x*∈ *X*such that *f*is continuous at every point from {*x*} × *Y*is residual in *X*. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.