The Bolzano-Poincaré Type Theorems

In 1883–1884, Henri Poincaré announced the result about the structure of the set of zeros of function , or alternatively the existence of solutions of the equation . In the case the Poincaré Theorem is well known Bolzano Theorem. In 1940 Miranda rediscovered the Poincaré Theorem. Except for few isolated results it is essentially a non-algorithmic theory. The aim of this article is to introduce an algorithmical proof of the Theorem “On the existence of a chain” and for an algorithmical proof of the Bolzano-Poincaré Theorem and to show the equivalence of Poincaré, Brouwer and “On the existence of a chain” theorems.

Strong Poincaré recurrence theorem in MV-algebras

The classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.

Mean Circulation and Structures of Tilted Ocean Deep Convection

Convection in a homogeneous ocean is investigated by numerically integrating the three-dimensional Boussinesq equations in a tilted, rotating frame ( f–F plane) subject to a negative buoyancy flux (cooling) at the surface. The study focuses on determining the influence of the angle (tilt) between the axis of rotation and gravity on the convection process. To this end the following two essential parameters are varied: (i) the magnitude of the surface heat flux, and (ii) the angle (tilt) between the axis of rotation and gravity. The range of the parameters investigated is a subset of typical open-ocean deep convection events. It is demonstrated that when gravity and rotation vector are tilted with respect to each other (i) the Taylor–Proudman–Poincaré theorem leaves an imprint in the convective structures, (ii) a horizontal mean circulation is established, and (iii) the second-order moments involving horizontal velocity components are considerably increased. Tilted rotation thus leaves a substantial imprint in the dynamics of ocean convection.

INNER STRUCTURE OF GAUSS–BONNET–CHERN THEOREM AND THE MORSE THEORY

We define a new one-form HA based on the second fundamental tensor [Formula: see text], the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.