MODIFIED WHYBURN SEMIGROUPS

. Let f: X Y be a continuous semigroup homomorphism. Conditions are given which will ensure that the semigroup X u Y .is a topological semlgroup, when the modified Whyburn topology is placed on X u Y.


I. INTRODUCTION.
Let (X,m I) and (Y,m 2) be semigroups and let f: X /Y be a semigroup homomorphism. An associative multiplication m may be defined on the disjoint union of X and Y as follows: m is m on X, m 2 on Y and m2(f(x),y) if x e X and y e Y. If we assume that X and Y are Hausdorff semlgroups and that f is continuous, then m is continuous in the disjoint union (or direct sum) topology. Let (X u Y,m) denote this Hausdorff semigroup.
Let Z denote the disjoint union of X and Y with Whyburn's unified topology [I]; i.e., V is open in Z iff V n X and V n Y are open in X and Y, -I respectively, and for any compact K in V n Y, f (K) V is compact. If X is locally compact, then Z is Hausdorff, and if Y is also locally compact, so is Z.
If f is a compact map, then Z and X v y are the same. If X and Y are locally compact, Hausdorff semlgroups, (Z,m) is a locally compact Hausdorff semlgroup provided m is a compact map [2].
In this paper we consider the modified Whyburn topology which is coarser than the disjoint union topology, but finer than the Whyburn topology and ask what conditions will insure that m will be continuous.

MAIN RESULTS.
Let W denote the disjoint union of X and Y with the modified Whyburn topology; V is open in W iff V n X and V n Y are open in X and Y, -I respectively, and f (y) V is compact for every y in V Y. The following notions and facts are due to Stallings [3]. A subset A of X is fiber compact -I relative to f: X +Y iff A is closed in X and A n f (x) is compact for every y e Y, and X is locally fiber compact iff every point in X has a neighborhood with a fiber compact closure. Fiber compact subsets of X are closed in W and W is Hausdorff if X is locally fiber compact. If Y is first countable, then Z and W B.B. REYNOLDS and V. SCHNEIDER are the same iff f is closed.
The proof given in [2] that m is a continuous operation on Z did not use the -i assumption that m (K) is compact for every compact K in X, but used an equivalent condition instead. The appropriate generalization of that condition for W is: CONDITION I. For every fiber compact K in X, there is a fiber compact K 2 in X such that for all x, y g X, if ml(x,y E K I, then x g K 2 and y K 2. (ml-l(K)), i 1,2 are fiber compact for each This condition is equivalent to: Pi fiber compact K in X, where Pl and P2 are the projections on X X. Thus A is open and must equal X since X is connected. All of this yields m2(Y,y) z. Let t',y' e Y and let z' m2(t',y'). The argument above will give that m2(t',Y) z'. Hence z z' and Y has the zero multiplication.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009