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The two-parameter class of Schröder inversions

J. Schröder — 2013

Commentationes Mathematicae Universitatis Carolinae

Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.

Natural sinks on Y β

J. Schröder — 1992

Commentationes Mathematicae Universitatis Carolinae

Let ( e β : 𝐐 Y β ) β Ord be the large source of epimorphisms in the category Ury of Urysohn spaces constructed in [2]. A sink ( g β : Y β X ) β Ord is called natural, if g β e β = g β ' e β ' for all β , β ' Ord . In this paper natural sinks are characterized. As a result it is shown that Ury permits no ( E p i , ) -factorization structure for arbitrary (large) sources.

Filling boxes densely and disjointly

J. Schröder — 2003

Commentationes Mathematicae Universitatis Carolinae

We effectively construct in the Hilbert cube = [ 0 , 1 ] ω two sets V , W with the following properties: (a) V W = , (b) V W is discrete-dense, i.e. dense in [ 0 , 1 ] D ω , where [ 0 , 1 ] D denotes the unit interval equipped with the discrete topology, (c) V , W are open in . In fact, V = V i , W = W i , where V i = 0 2 i - 1 - 1 V i j , W i = 0 2 i - 1 - 1 W i j . V i j , W i j are basic open sets and ( 0 , 0 , 0 , ... ) V i j , ( 1 , 1 , 1 , ... ) W i j , (d) V i W i , i is point symmetric about ( 1 / 2 , 1 / 2 , 1 / 2 , ... ) . Instead of [ 0 , 1 ] we could have taken any T 4 -space or a digital interval, where the resolution (number of points) increases with i .

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