By Antje Düsterhöft, Meike Klettke, Klaus-Dieter Schewe
This Festschrift, released in honor of Bernhard Thalheim at the social gathering of his sixtieth birthday provides 20 articles by way of colleagues from around the globe with whom Bernhard Thalheim had cooperation in quite a few respects; additionally incorporated is a systematic biography contributed via the amount editors. The 20 contributions replicate the breadth and the intensity of the paintings of Bernhard Thalheim in conceptual modeling and database concept in the course of his clinical occupation spanning greater than 35 years of lively learn. particularly, ten articles are concentrating on themes like database dependency thought, object-oriented databases, triggers, summary kingdom machines, database and knowledge structures layout, net semantics, and company processes.
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Additional resources for Conceptual Modelling and Its Theoretical Foundations: Essays Dedicated to Bernhard Thalheim on the Occasion of His 60th Birthday
Random tuples (or records), respectively. One of basic database operations, join, combines two tables into a third one by matching values for given columns (attributes). , names) from the tables. , Kiefer et al. 2005, Copas and Hilton, 1990, and references therein). , Seleznjev and Thalheim (2008). A set of attributes A in a table T is called an -key (test) if there are no -close sub-tuples tA (j), j = 1, . . , m. , Thalheim (2000). By joining a table with itself (self-join) we identify also -keys and key-properties for a set of attributes for a random table (Seleznjev and Thalheim, 2003, Leonenko and Seleznjev, 2010).
Proof. We have to prove that (i) and (ii) imply both (6) and (7). Let us start with some elementary lemmas. The ﬁrst two of them prove that if a sequence of non-negative numbers is given with a ﬁxed sum and an upper bound c is given on them then the sum of their squares is maximized for a choice with (one exception) all members = c or 0. We give the proof for sake of completeness. Lemma 1. Let the real numbers 0 ≤ a, b, c satisfy the inequalities b ≤ a ≤ c ≤ a + b. Then (10) a2 + b2 ≤ c2 + (a + b − c)2 holds.
Applying the notions introduced above to the comparison of the special situation of Proposition 1 with the more general case, we see the following. Each of the attributes a and b of the special situation is 1-prime and forms a singleton ∼mk -equivalence class; correspondingly, in the general situation each of the sets K1 \ E and K2 \ E consists of 1-prime attributes and forms an ∼mk -equivalence class. Summarizing and formalizing the arguments given so far, the general situation of schemes having exactly two minimal keys is fully characterized by the following Theorem 1 and visualized in Figure 1.
Conceptual Modelling and Its Theoretical Foundations: Essays Dedicated to Bernhard Thalheim on the Occasion of His 60th Birthday by Antje Düsterhöft, Meike Klettke, Klaus-Dieter Schewe