An effort to initiate the subject of the title: the basic tool is the study of the abstract closed interval equipped with certain equational structures. The title is wishful thinking; there ought to be a subject that deserves the name “algebraic real analysis.” Herein is a possible beginning. For reasons that can easily be considered abstruse we were led to the belief that the closed interval—n...

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ k[X ] such that f | Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if (1) there is a nonempty open subset U ⊆ X consisting of clo...

Text appearing between brackets like This ] is extracted from the rst printing; text appearing between pararentheses like f This g is to be inserted for the second printing. Chapter 1 Page 18, Replace the value of a closed-term is always a closed abstraction ] by f M] ] is a closed abstraction if it is deened g.

In the present paper, we introduce soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. A sufficient condition for a soft g-closed set to be a soft closed is also presented. Moreover, the union and intersection of two soft g-closed sets are discussed. Finally, the new soft separation axiom, namely soft 1 2 T space is ...

A closed subgroup H of the affine, algebraic group G is called observable if G/H is a quasi-affine algebraic variety. In this paper we define the notion of an observable subgroup of the affine, algebraic monoid M . We prove that a subgroup H of G is observable in M if and only if H is closed in M and there are “enough” H-semiinvariant functions in k[M ]. We show also that a closed, normal subgr...

Introduction Let G be a connected reductive group over an algebraically closed eld k; let B G be a Borel subgroup and K G a closed subgroup. Assume that K is a spherical subgroup of G, that is, the number of K-orbits in the ag variety G=B is nite; equivalently, the set KnG=B of (K; B)-double cosets in G is nite. Then the following problems arise naturally.

Let f, g : X → G/K be maps from a closed connected orientable manifold X to an orientable coset space M = G/K where G is a compact connected Lie group, K a closed subgroup and dimX = dimM . In this paper, we show that if L(f, g) = 0 then N(f, g) = 0; if L(f, g) 6= 0 then N(f, g) = R(f, g) where L(f, g), N(f, g), and R(f, g) denote the Lefschetz, Nielsen, and Reidemeister coincidence numbers of ...

For an assignment of numbers to the vertices of a graph, let S[u] be the sum of the labels of all the vertices in the closed neighborhood of u, for a vertex u. Such an assignment is called closed distinguishing if S[u] 6= S[v] for any two adjacent vertices u and v unless the closed neighborhoods of u and v coincide. In this note we investigate dis[G], the smallest integer k such that there is a...