# Mathematica Moravica, Vol. 11 (2007)

On Fefferman's Non-existence Problems
Mathematica Moravica, Vol. 11 (2007), 1–7.

Abstract. Fefferman has propounded four open Cauchy problems A, B, C and D in Navier-Stokes equations. The problems A and B refer to sufficient condition for the existence of solutions, and the problems C and D to the ones for the non-existence of solutions. Here, we shall answer to the problems C and D.
Keywords. Navier-Stokes equations, non-existence problems, periodic function.

Common Fixed Point Theorems of Meir and Keeler Type for Weakly Compatible Maps
Mathematica Moravica, Vol. 11 (2007), 9–16.

Abstract. The subject of this paper is to prove a common fixed point theorem for four weakly compatible maps which extends and improves Theorem 1 of [2] and others by removing the assumption of continuity and relaxing the property of compatibility to weak compatibility. Also we give another extension of the same theorem.
Keywords. Weakly compatible maps, compatible and compatible maps of type (A), (B), (P) and (C), common fixed point theorems.

On Quaternary Associative Operations
Mathematica Moravica, Vol. 11 (2007), 17–25.

Abstract. Some properties of quaternary associative operations that have neutral sequences are investigated. A characterization of such operations by means of binary primitive operations is established.
Keywords. Quaternary operation, primitive operation, neutral sequence.

Common Fixed Point for Set and Single Valued Functions without Continuity and Compatibility
Mathematica Moravica, Vol. 11 (2007), 27–38.

Abstract. We prove some common fixed point theorems for set and single valued mappings without assuming continuity and compatibility. We show that completeness of the whole space is not necessary for existence of common fixed point. To prove the theorems we use a new noncompatible condition that is weak commutativity of type (KB). We also prove a fixed point theorem for sequence of set valued mappings.
Keywords. Common fixed point, noncompatible maps, weak commutativity of type (KB).

On Separation Axioms and Sequences
Mathematica Moravica, Vol. 11 (2007), 39–46.

Abstract. In 2003, Noiri has introduced the notion of $\beta$-$\theta$-open sets. By using these sets, the aim of this paper is to investigate the relation ships between separation axioms and sequences.
Keywords. Separation axioms, sequences, $\beta$-$\theta$-open sets.

An Alternative Moving Frame for Tubular Surface around the Spacelike Curve with a Spacelike Binormal in Minkowski 3-Space
Mathematica Moravica, Vol. 11 (2007), 47–54.

Abstract. We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike binormal in Minkowski 3-space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at points where the curve experiences high torsion. In our construction the tubular surface’s twist is bounded by the spacelike curve’s curvature and is independent of the spacelike curve’s torsion.
Keywords. Spacelike curve, canal surface, tubular surface, Minkowski 3-space.

Remark on Some Functional Equations
Mathematica Moravica, Vol. 11 (2007), 55–58.

Abstract. In this note we give a new and simple proofs of some D. Adamović’s results.
Keywords. Functional equation.

Intuitionistic Fuzzy Strong Precompactness in Coker's Sense
Mathematica Moravica, Vol. 11 (2007), 59–67.

Abstract. The concept of fuzzy strong precompactness in Coker’s space has been introduced and studied. Several preservation properties and some characterizations concerning fuzzy strong precompactness have been obtained.
Keywords. Intuitionistic fuzzy topology, intuitionistic fuzzy strong precompactness.

$L$-Ultrafilters, $L$-Sets and $LC$-Property
Mathematica Moravica, Vol. 11 (2007), 69–78.

Abstract. An $L$-filter base, $L$-filter, $L$-ultrafilter is a filter base, filter, ultra filter consisting exclusively of Lindelof sets. In this paper we consider $L$-filters (ultrafilters) and $LC$-property. A space $X$ is LC – space if every Lindelöf set in $X$ has the compact closure in $X$. A locally compact space $X$ is $LC$ – space if and only if every $L$-ultrafilter on $X$ converges. We also consider $L$-points, $L$-sets and $LC$-extensions.
Keywords. $L$-set, $LC$-space, $LC$-extensions.

Two Coincidence and Fixed Point Theorems for Hybrid Strict Contractions
Mathematica Moravica, Vol. 11 (2007), 79–83.

Abstract. In this paper two general coincidence and fixed point theorems for hybrid pairs satisfying an implicit relation are proved.
Keywords. Fixed point, coincidence point, property (E.A.), noncompatible mappings, implicit relations.

Best Approximations in Cone Metric Spaces
Mathematica Moravica, Vol. 11 (2007), 85–88.

Abstract. In this paper we introduced the fundamental elements of a new relativistic physics for the lower transversal spaces. The classical Einstein’s theory (physics) of the upper transversal spaces is linear via Lorentz’s transformations. On the other hand, the physics of the lower transversal spaces is nonlinear via the nonlinear transformations. Also, for the upper transversal spaces, as and Einstein’s physics, is essential that the "live" is finite but in the lower transversal spaces the “live” is infinite. This is a result (fact) of the deepest connection between new nonlinear physics and the geometry of the lower transversal spaces. The relativistic physics and the new nonlinear physics are essential different, but the equation for Energy in the form $E = mc^{2}$ is in the same in both physics.