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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.

Filling boxes densely and disjointly

J. Schröder — 2003

Commentationes Mathematicae Universitatis Carolinae

We effectively construct in the Hilbert cube $ℍ = {[0, 1]}^{ω}$ two sets $V, W \subset ℍ$ with the following properties: (a) $V \cap W = \emptyset$ , (b) $V \cup W$ 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 $ℍ$ . In fact, $V = ⋃_{ℕ} V_{i}$ , $W = ⋃_{ℕ} W_{i}$ , 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 $(0, 0, 0, ...) \in V_{i j}$ , $(1, 1, 1, ...) \in W_{i j}$ , (d) $V_{i} \cup W_{i}$ , $i \in ℕ$ 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 $i$ .

Delannoy and tetrahedral numbers

J. Schröder — 2007

Commentationes Mathematicae Universitatis Carolinae

We establish an identity between Delannoy numbers and tetrahedral numbers of arbitrary dimension.

Natural sinks on $Y_{β}$

J. Schröder — 1992

Commentationes Mathematicae Universitatis Carolinae

Let ${(e_{β} : 𝐐 \to Y_{β})}_{β \in Ord}$ be the large source of epimorphisms in the category $Ury$ of Urysohn spaces constructed in [2]. A sink ${(g_{β} : Y_{β} \to X)}_{β \in Ord}$ is called natural, if $g_{β} \circ e_{β} = g_{β^{'}} \circ e_{β^{'}}$ for all $β, β^{'} \in Ord$ . In this paper natural sinks are characterized. As a result it is shown that $Ury$ permits no $(E p i, ℳ)$ -factorization structure for arbitrary (large) sources.

On sub-, pseudo- and quasimaximal spaces

J. Schröder — 1998

Commentationes Mathematicae Universitatis Carolinae

The structure of sub-, pseudo- and quasimaximal spaces is investigated. A method of constructing non-trivial quasimaximal spaces is presented.

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. -Solving Difference Equations for Boundary Value Problems by Reduction Methods II.

J. Schröder; U. Trottenberg; H. Reutersberg — 1976

Numerische Mathematik

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Operator-Ungleichungen und ihre numerische Anwendung bei Randwertaufgaben.

Fehlerabschätzung mit Rechenanlagen bei gewöhnlichen Differentialgleichungen erster Ordnung .

Verbesserung einer Fehlerabschätzung für gewöhnliche Differentialgleichungen erster Ordnung.

Upper and Lower Bounds for Solutions of Generalized Two-Point Boundary Value Problems.

Proving Inverse-Positivity of Linear Operators by Reduction.

Beiträge zur Darstellung der Möbiusschen Funktion.

The two-parameter class of Schröder inversions

Filling boxes densely and disjointly

Delannoy and tetrahedral numbers

Natural sinks on $Y_{β}$

On sub-, pseudo- and quasimaximal spaces

Einschließen der Lösungen von Randwertaufgaben.

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I.

Wallman's method

The Uniqueness of the Transfer Function of Linear System from Input-Output Observations.

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. -Solving Difference Equations for Boundary Value Problems by Reduction Methods II.