1. Preliminaries. We are concerned with involutions without fixed points, together with equivariant maps connecting such involutions. An involution T is a homeomorphism of period 2 of a Hausdorff space X onto itself; that is, T(x) = x for all x £ X . There is associated with an involution T on X the orbit space X/T, obtained by identifying x with T(x) for all x G Z . Denote by v\ X—+X/T the dec...

In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p. 179–183; 1.1–1.12]. The duality is based on the concept of the Galois connection. Let S, T be complete lattices. Note that there exists a connection between S and T which is Galois. One can prove the following proposition (1) Let S, T , S , T be non empty relati...

If a graph GM is embedded into a closed surface S such that S\GM is a collection of disjoint open discs, then M = (GM , S) is called a map. A zigzag in a map M is a closed path which alternates choosing, at each star of a vertex, the leftmost and the rightmost possibilities for its next edge. If a map has a single zigzag we show that the cyclic ordering of the edges along it induces linear tran...

We prove an extension of Waldhausen's theorem [5] conjectured by Hempel in [3]. We prove the following extension of Waldhausen's theorem [5]: Theorem 1. Let M, N be P2-irreducible 3-manifolds. Suppose that M is compact, sufficiently large and f: (M, dM) -^ (N, dN) is a continuous map inducing an injection fM: trx(M)^>irx(N). Then, there is a proper homotopy f: (M, 3M) —» (N, dN) such that f0 = ...

Extracellular signal-regulated kinases (ERKs), also known as mitogen-activated protein (MAP) kinases, are rapidly phosphorylated and activated in response to a number of external factors which promote growth and differentiation (T. G. Boulton, S. H. Nye, D. J. Robbins, N. Y. Ip, E. Radziejewska, S. D. Morgenbesser, R. A. DePinho, N. Panayotatos, M. H. Cobb, and G. D. Yancopoulos, Cell, 65: 663-...

Due to its fast growth rate and little nursery requirements, willow is largely planted as short rotation coppice species for bioenergy resources in temperate regions. Willow coppices can be easily established by vegetative propagation of cuttings and can remain productive for up to 25 years. In China, diploid species S. leucopithecia and S. erioclada L. are two promising willow species that can...

The main idea of this chapter is to try to measure to which extent a surface S is different from a plane, in other words, how “curved” is a surface. The idea of doing this is by assigning to each point P on S a unit normal vector N(P ) (that is, a vector perpendicular to the tangent plane at P ). We are measuring to which extent is the map from S to R given by P 7→ N(P ) (called the Gauss map) ...

The purpose of this study was to determine the viability of Mycobacterium avium subsp. paratuberculosis (MAP), Escherichia coli (E. coli), and Salmonella Enteritidis (S. Enteritidis) during preparation and refrigerated storage of yogurt. Three yogurts were prepared using pasteurized commercial milk. Each yogurt was artificially contaminated with (1) MAP, (2) E. coli + S. Enteritidis, and (3) MA...

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y ...

(1) π∗(S) −→ π∗(tmf ) −→ MF∗ that we now describe. Both maps are surprisingly close to being isomorphisms (even though π∗(S) and MF∗ have nothing to do with each other). The first map (1) is the Hurewitz homomorphism: being a ring spectrum, tmf admits a unit map from the sphere spectrum S. This induces a map in homotopy π∗(S) → π∗(tmf ), which is an isomorphism on π0. The only torsion in π∗(tmf...