We study otopy classes of equivariant local maps and prove a Hopf type theorem for such maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group.

We prove the Hopf theorem for gradient local vector fields on manifolds, i.e., we show that there is a natural bijection between the set of gradient otopy classes of gradient local vector fields and the integers if the manifold is connected Riemannian without boundary.

We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of Z.

Rozdział w monografii dotyczy działalności projektowej Barbary Hoff, od 1956 roku kiedy w "Przekroju" prezentowała pojedyncze projekty sukienek, poprzez pierwszą "kolekcję masową" w 1967 roku aż do powstania marki Hoffland i stoiska " z kolekcją własną Hoff" w Juniorze w Domach towarowych Centrum w warszawie w 1977 roku

We study a mathematical model of gene transcription and protein synthesis with negative feedback. We consider a system of equations taking into account the formation of dimers (i.e., complex formed by two protein monomers), the way in which dimers bind to DNA and time delay in translation process. For the model consisting of three ordinary differential equations with time delay, we derive conditions for stability of the positive...