The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.

A bijective correspondence between strong inclusions and compactifications in the setting of σ-frames is presented. The category of uniform σ-frames is defined and a description of the Samuel compactification is given. It is shown that the Samuel compactification of a uniform frame is completely determined by the σ-frame consisting of its uniform cozero part, and consequently, any compactificat...

In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined as a positive integer that does not belong to the numerical semigroup. The computation of gaps of numerical semigroups has been actively studied from the 19t...

A class of differential Riccati equations (DREs) is considered whereby the evolution of any solution can be identified with the propagation of a value function of a corresponding optimal control problem arising in L2-gain analysis. By exploiting the semigroup properties inherited from the attendant dynamic programming principle, a max-plus primal space fundamental solution semigroup of max-plus...

A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most finitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is...

A topological semigroup is a nonvoid, Hausdorff space, 5, together with a continuous, associative multiplication defined on 5. Here the term semigroup will always denote a topological semigroup. In [4], Wallace and Koch have shown that if the circle, B, is a semigroup such that B2 = B, then either B is a group, or the multiplication on B is of the trivial kind xy = x, or xy = y. In [6], Mostert...

We study algebraic and topological properties of the convolution semigroups of probability measures on topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P (G) over some topological group G if and only if S embeds into the semigroup exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maxi...

Let S be a semigroup. In certain cases we give some characterizations of extreme amenability of S and we show that in these cases extreme left amenability and extreme right amenability of S are equivalent. Also when S is a compact topological semigroup, we characterize extremely left amenable subalgebras of C(S), where C(S) is the space of all continuous bounded real valued functions on S

We discuss the Bohr compactification of a pseudofinite group, motivated by a question of Boris Zilber. Basically referring to results in the literature we point out that (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the “definable” Bohr compactification of any pseudofinite group G, relative to an ambient nonstandard model of set theory in which it...

In this paper, we introduce a generalized class of an Hv-semigroup obtained from an LA-semigroup H. This generalized Hv-structure is called an Hv-LA-semigroup. We provide several examples of Hv-LA-semigroups. Moreover, with the help of an example we obtain that each LA-semigroup endowed with an equivalence relation can induce an Hv-LA-semigroup. We also investigate isomorphism theorems with the...