The simplest singularities of smooth mappings are fold singularities. We say that a mapping f is a fold mapping if every singular point of f is of the fold type. We prove 1 that for a closed orientable 4-manifold M4 the following conditions are equivalent: (1) M4 admits a fold mapping into R3; (2) for every orientable 3-manifold N3, every homotopy class of mappings of M4 into N3 contains a fold...

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

Recently, [Doyle and McMullen 1989] devised an iterative solution to the fifth degree polynomial. At the method’s core is a rational mapping f of CP with the icosahedral symmetry of a general quintic. Algebraically, this means that f commutes with a group of Möbius transformations that is isomorphic to the alternating groupA5. Moreover, thisA5-equivariant posseses nice dynamics: for almost any ...

Recently, P. Doyle and C. McMullen devised an iterative solution to the fifth degree polynomial [4]. At the method’s core is a rational mapping f of CP with the icosahedral (A5) symmetry of a general quintic. Moreover, this A5-equivariant posseses nice dynamics: for almost any initial point a0 ∈ CP, the sequence of iterates f(a0) converges to one of the periodic cycles that comprise an icosahed...

There has been recent interest in finding combinatorial models for groups, i.e. finding a simplicial complex whose automorphism group is a given group. For example, CharneyDavis showed that the Coxeter Diagram is a model for Out(W ) (for certain Coxeter groups W) [3], and Bridson-Vogtmann showed that a spine of Outer Space is a model for Out(Fn) [1]. In this paper, we prove that the complex of ...

Considering a mapping g holomorphic on a neighbourhood of a rational convex set K ⊂ C, and range into the complex projective space CP, the main objective of this paper is to show that we can uniformly approximate g on K by rational mappings defined from C into CP. We only need to ask that the second Čech cohomology group Ȟ(K, Z) vanishes.

In this paper we construct and study a new 15-vertex triangulationX of the complex projective plane CP. The automorphism group of X is isomorphic to S4 ×S3. We prove that the triangulation X is the minimal by the number of vertices triangulation of CP admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for simplices of X and show that the automorphism...

Atlantic herring (Clupea harengus) is an important commercial fish and shows to be more and more demanded for human consumption. Therefore, it is very important to find good methods for monitoring the freshness of the fish in order to keep it in the best quality for human consumption. In this study, the fish was stored in ice up to 2 weeks. Quality changes during storage were assessed by the Qu...

Many numerical problems and methods for their approximate solution can be brought into the following abstract form: We are given a bounded linear operator S E L(X, Y) between Banach spaces X and Y, the “solution operator.” That is, for x E X, Sx is the true solution of the problem. We assume that we have only partial information on the “datum” x E X, which is given by a mapping N: X + R”, the “...