We give in this paper a description of a new category related to shape category. We consider families of multi-valued functions between topological spaces which we call multi-nets. In a well-controlled way functions of a multi-net more and more resemble single-valued functions. We introduce a notion of homotopy for multi-nets and a composition of homotopy classes. The resultant homotopy category of multi-nets $\mathcal{H}M$ is naturally equivalent to the shape category provided we restrict to spaces which have ANRresolutions with onto projections. However, the homotopy category of multi-nets is interesting because it provides an intrinsic method of studying global properties of spaces. Our idea is to extend Borsuk's approach based on fundamental sequences to arbitrary topological spaces in analogy with Sanjurjo's description of shape category of compact metric spaces in terms of upper semi-continuous multi-valued functions.

We prove the following theorem: Let $T$ be superstable and let $A$ any set. Then there is no minimal model over $A$ which has an infinite set of indiscernibles over $A$.

Moore spaces are defined in proper homotopy theory. Some results on the existence and uniqueness of those spaces are proven. An example of two non properly equivalent Moore spaces is given.

We characterize reflexive modules over $QF- 3^{\prime}$ rings using a linear compactness condition relative to the Lambek torsion theory, and we also give a necessary and sufficient condition for a left $QF- 3^{\prime}$ maximal quotient ring to be right $QF- 3^{\prime}$.

This paper deals with the asymptotic energy distributions for large times of the solutions of elastic wave propagation problems in stratified media $R^{3}$. We construct asymptotic wave functions which approximate the solutions for large times and calculate the asymptotic energy of the solutions using these asymptotic wave functions. In particular, it is shown that the energy of Stoneley wave is asymptotically concentrated along the interface.

Let $A$ be a finite dimensional, basic and connected algebra over an algebraically closed field $k$. We define a notion of weakly separating family in the Auslander-Reiten quiver of $A$ which generalises the notion of a separating tubular family introduced by C. M. Ringel. Given an algebra $A$ having a weakly separating family $\mathcal{F}$ of stable tubes, we say that an algebra $B$ is a coil enlargement of $A$ using modules from $\mathcal{F}$ if $B$ is obtained from $A$ by an iteration of admissible operations performed either on a stable tube of $\mathcal{F}$, or on a coil obtained from a stable tube of $\mathcal{F}$ by means of the operations done so far. The purpose of this paper is to describe the module category of $B$. We also give a criterion for the tameness of $B$ if $A$ is a tame concealed algebra.

We prove that classical solutions of the dissipative wave equation $\epsilon u_{tt}+u_{t}-u_{xx}-(f(u_{x}))_{x}=0$ are globally defined in time, regardless of the size of the initial data, if $\epsilon$ is sufficiently small.