An Elementary Transition to Abstract Mathematics

2019 ◽  
Author(s):  
Gove Effinger ◽  
Gary L. Mullen
1992 ◽  
Vol 96 (1) ◽  
pp. 3-33 ◽  
Author(s):  
M. Nielsen ◽  
G. Rozenberg ◽  
P.S. Thiagarajan

2015 ◽  
Vol 105 ◽  
pp. 5-21 ◽  
Author(s):  
Yuichi Mitsutake ◽  
Suhaimi Illias ◽  
Koutarou Tsubaki ◽  
Mohammad Nasim Hasan ◽  
Masanori Monde

1992 ◽  
Vol 29 (6-7) ◽  
pp. 555-578 ◽  
Author(s):  
M. Nielsen ◽  
G. Rozenberg ◽  
P. S. Thiagarajan

1992 ◽  
Vol 21 (399) ◽  
Author(s):  
Madhavan Mukund

<p>Labelled transition systems can be extended to faithfully model concurrency by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step, We can characterize a subclass of these <em>step transition systems</em>, called PN-transition systems, which describe the behaviour of Petri nets.</p><p>This correspondence is formally described in terms of a coreflection between a category of <em>PN</em>-transition systems and a category of Petri nets.</p><p>In this paper, we show that we can define subcategories of <em>PN</em>-transition systems whose objects are <em> safe PN-transition systems and elementary PN-transition systems</em> such that there is a coreflection between these subcategories and subcategories of our category of Petri nets corresponding to safe nets and elementary net systems.</p><p>We also prove that our category of elementary <em>PN</em>-transition systems is equivalent to the category of (sequential) <em> elementary transition systems</em> defined by Nielsen, Rozenberg and Thiagarajan, thereby establishing that the concurrent behaviour of an elementary net system can be completely recovered from a description of its sequential behaviour. Finally, we establish a coreflection between our category of safe <em>PN</em>-transition system and a subcategory of <em>asynchronous transition systems</em> which has been shown by Winskel and Nielsen to be closely linked to safe nets.</p>


1964 ◽  
Vol 17 (8) ◽  
pp. 833 ◽  
Author(s):  
JC Sheldon

It is proposed that the valence number, n, for bonding in the elemental state may be estimated by n = rΔHf�/C, where r is half the interatomic distance, -ΔHf� is the heat of formation of the elemental state, and C is a constant and approximately 28 and 40 Ǻ kcal g-atom-1 for non-metals and metals respectively. It is shown that the valence states of the elementary transition metals, as given by n, correspond closely to the most stable oxidation states (MSOS) displayed in binary transition metal fluorides, chlorides, and related compounds. It is concluded that the trend of the MSOS of transition metal binary compounds results mainly from each metal possessing a preferred valence state, which is strongly determined by the availability of the (n-l)d orbitals for bonding as judged by their size relative to that of the ns and p orbitals.


1990 ◽  
Vol 19 (310) ◽  
Author(s):  
Mogens Nielsen ◽  
Grzegorz Rozenberg ◽  
P. S. Thiagarajan

<p>Transition systems are a simple and powerful formalism for explaining the operational behaviour of models of concurrency. They provide a common framework for investigating the interrelationships between different approaches to the study of distributed systems. Hence an important question to be answered is: which subclass of transition systems corresponds to a particular model of distributed systems? In this paper we provide an answer to this question for elementary net systems.</p>


1991 ◽  
Vol 20 (346) ◽  
Author(s):  
Mogens Nielsen ◽  
Grzegorz Rozenberg ◽  
P. S. Thiagarajan

The model of Elementary Transition Systems has been introduced by the authors as an abstraction of Elementary Net Systems - with a formal embedding in terms of a categorical coreflection, keeping behavioural information like causality, concurrency and conflict, but forgetting the concrete programming of a particular behaviour over an event set using conditions. In this paper we give one example of the advantages of ETS over ENS, - the definition of local state refinement. We show that the well known problems in understanding within nets the simple notion of syntactic substitution of conditions by (sub) nets behaviourally, - these problems seem to disappear when moving to the more abstract level of ETS. Formally, we show that the ETS-version of condition-substitution does satisfy nice and natural properties, e.g., projection and compositionality results w.r.t. a standard notion of transition system morphisms.


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