December 1999 - Categories - categories.org.au

Real midpoints
by Peter Freyd 19 Jan '00

19 Jan '00

It could well be that Vaughan and I are defining the midpoint structure in the same way. Here's how I described it (using the conventions from my last posting). Let F:I --> I v I be a final coalgebra. We will denote the top of I as T and the bottom as B. Construct the "halving" map, h:I --> I, (on [-1,1] it will send x to x/2) as: T v F v B F'v F' F' I --> 1 v I v 1 ------> I v I v I v I ---> I v I --> I where F' denotes the inverse of F, and, by a little overloading, T and B denote the maps constantly equal to T and B. The leftmost map records the fact that the terminator is a unit for the ordered-wedge functor. Let g be the endo-function on I x I defined recursively by: g<x,y> = if dx = T and dy = T then <x,y> else if dx = T and uy = B then h(g(dx,uy>) else if ux = B and dy = T then h(g<ux,dy>) else if ux = B and uy = B then <x,y>. The values of g lie in the first and third quadrants, that is, those points such that either dx = dy = T or ux = uy = B. The two maps g d x d g u x u I x I --> I x I --> I x I and I x I --> I x I --> I x I give a coalgebra structure on I x I. The midpoint operation may be defined as the induced map to the final coalgebra.

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Pre-calculus
by Peter Freyd 31 Dec '99

31 Dec '99

Starting with the closed interval, regarded as an ordered midpoint algebra with duality, what's the best way to get the reals? My previous answer was first to reflect the midpoint algebra in the subcategory of abelian groups and then define the multiplication via monotonic endomorphisms. I think there's a more natural way. First, a re-cap of the definitions. A midpoint operation is a binary operation, with values here denoted as x|y, satisfying: idempotence, x|x = x; commutativity, x|y = y|x; middle-two-interchange, (u|v)|(x|y) = (u|x)|(v|y); and cancelation, x|y = x|z => y = z. The closed interval is a totally ordered midpoint algebra with anti-involution, with values here denoted as -x. It follows that there's a unique element of the form (-x)|x. We'll call it the center. The middle-two-interchange law is just what is needed to see that the monoid of midpoint-endomorphisms is itself a midpoint algebra (where the midpoint of f and g is defined pointwise: (f|g)(x) = (fx)|(gx)). If f and g are both order-preserving or both order- reversing then it's easy to see that so is f|g. But we'll want all the _monotonic_ endomorphisms (the OED defines "monotonic" as "varying in such a way that it either never increases or never decreases") and there's no apparent reason to expect that monotonicity is preserved by midpointing. The wonderful fact, though, is: Midpoint-preserving functions between intervals are monotonic. (The axiom of choice yields 2^{2^N} counterexamples if the interval is replaced by the reals.) It follows that a midpoint-preserving function is determined by its values on any two points. The monoid of midpoint- and duality- preserving endo-functions on a closed interval is naturally isomorphic (via evaluation at Top) to the closed interval. For the entire reals use the stalk at the center of the germs of such functions. PS For a proof suppose that f:[-1,1] --> [-K,K] preserves midpoints and duality. For any x:[-1,1] and dyadic rational q such that qx:[-1,1], we may uniquely identify qx using just midpoints and duality, hence f(qx) = qf(x). Suppose that f is not monotonic. We wish to reach a contradiction. Let u < v and x < y in [-1,1] be such that fu < fv and fx > fy. Define a = (-u)|v and b = (-x)|y to obtain a,b > 0 and fb < 0 < fa. Let q > 0 be a dyadic rational such that a/b - (fa)/(bK) < q < a/b and define c = (-qb)|a. Let r be a dyadic rational such that (2K)/(fa) < r < 1/c. Then rc is in [-1,1] but f(rc) can not be in [-K,K] (because fc > (fa)/2). The theorem I stated doesn't say anything about preserving duality. So let g be a midpoint-preserving function between intervals. Let a be the dual of the value of g at the center. Then the function \x.a|(gx) preserves the center and continues to preserve midpoints. Since -x may be uniquely identified by (-x)|x = 0 the preservation of the center implies preservation of the duality and \x.a|(gx) is monotonic. Since \y.a|y not only preserves but reflects the order, we know that g is monotonic.

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Real coalgebra
by Peter Freyd 24 Dec '99

24 Dec '99

I've been looking at the Proceedings of the Second Workshop on Coalgebraic Methods in Computer Science (CMCS'99), Electronic Notes in Theoretical Computer Science, Volume 19, to be found at www.elsevier.nl:80/cas/tree/store/tcs/free/noncas/pc/menu.htm There's a nice paper by Dusko Pavlovic and Vaughan Pratt. It's entitled On Coalgebra of Real Numbers and it has turned me on. Their abstract begins: We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor X x omega, ordinal product with omega. This makes an attractive analogy with the definition of the ordinal omega itself as the initial algebra of the functor 1;X, prepend unity, with both definitions made in the category of posets. I thought of using another functor. And damned if it isn't just what I should have had for my CTCS talk last September at Edinburgh. In the category of posets with top and bottom consider the binary functor, X v Y, obtained by starting with the disjoint union X;Y, with everything in X ordered below Y, and then identifying the top of X with the bottom of Y. The functor X v X preserves the terminator (the one-element poset) hence the terminator is its final coalgebra. So restrict to the category of posets with _distinct_ top and bottom. The functor X v X again has a final coalgebra: this time it's the closed interval, I. (For Dusko and Vaughan's functor it's the half-open interval. For yet another functor, one that ought to have the open interval as its final coalgebra, but doesn't, see PS below.) If X v Y is described as the subobject of the product X x Y consisting of those pairs <x,y> such that either x is top or y is bottom, then a coalgebra structure for X v X can be described as a pair of endo-functions d,u such that for each x either d(x) is top or u(x) is bottom. On the interval [a,b] the final-coalgebra structure is understood to be given by d(x) = min (b, 2x - a) and u(x) = max (a, 2x - b). In fact, we didn't need to start in the category of posets. It would have sufficed to work in the category of sets with distinct top and bottom (see PPS below for a proof). The final coalgebra is still the closed interval and, yes, the ordering is implicit: it is the most inclusive relation preserved by d and u that avoids placing top below bottom. (It can also be obtained by constructing either of the two lattice operations on I as the unique coalgebra map I x I --> I for an appropriately chosen coalgebra structure on I x I. A similar construction works for the Pavlovic-Pratt coalgebra.) Indeed, all of the structure of the closed interval is definable from this coalgebra definition. Go back to the category of posets with distinct top and bottom. It is routine to verify that X v X commutes with the opposite-poset functor hence -- using the fact that that functor is an equivalence -- its final coalgebra will be invariant under that functor. That is, there is an isomorphism from I to its opposite. It takes more work, but not an infinite amount, to construct a coalgebra structure on I x I so that the induced binary operation I x I --> I is the midpoint operation, the values of which will be denoted here as x|y. It is pretty much characterized by the equations: idempotence, x|x = x; commutativity, x|y = y|x; and middle- two-interchange, (u|v)|(x|y) = (u|x)|(v|y); together with cancelation: a|x = a|y => x = y. If one chooses a zero, then one may prove that there's an ambient abelian group with unique division by 2, such that the given midpoint algebra is a subset closed under the operation x|y = (x + y)/2. (There must be an existent reference for this.) Actually we want that ambient abelian group for I; it's none other than the reals. The order-duality makes its construction easier than in the general case. So let's use 1 to denote the top of the final coalgebra, I, -1 for the bottom, and 0 for their midpoint, -1|1. Let h:I -> I denote the "halving" map that sends x to 0|x. Note that h is an endomorphism with respect to 0, the ordering, the duality, the midpoint structure (and not, of course, the top and bottom). For a moment enter the category of endomorphisms, and reflect the object <I, h> into the full subcategory of automorphisms. (The reflection may be explicitly constructed as the colimit of the diagram h h h I --> I --> I -->...) The result is a poset-with-0-and-duality-and- midpoint-operation we'll denote as R. The midpoint operation respects the order and commutes with duality. 0 is self-dual. By making h an automorphism we know that for all y there is a unique x such that 0|x = y. On R define a + b by the equation 0|(a + b) = a|b and verify 0 + b = b = b + 0 and (a + b) + (c + d) = (a + c) + (b + d). Use the duality to define -a and verify (-a) + a = 0. Use h to define a/2 and verify a/2 + a/2 = a. All of this makes R an abelian group with unique division by 2. Viewing R as an ordered abelian group it is easy to verify that any endomorphism is determined by where it sends 1 (inherited from I -> R). That, of course, allows us to define the multiplication. Those who heard my CTCS talk last September at Edinburgh, "Path Integrals, Bayesian Vision, and Is Gaussian Quadrature Really Good?", (there's an abstract at the same website as above) can appreciate why I'm particularly happy. I started by defining ordinary integration of continuous functions using -- without knowing it -- this coalgebra structure on I. Let C be the functor that assigns to a space X the set, C(X), of continuous real-valued functions on X. The "mean-value" function M:C(I) -> R can be characterized -- indeed, evaluated -- from its order-preservation together with what it does to constant functions and MxM C(I v I) --> C(I + I) --> C(I) x C(I) ---> R x R C(F) | | m v v M C(I) ------------------------------------> R where F:I -> I v I is the coalgebra structure and m is the midpoint operation. If C(F) is inverted then this diagram can be read as a fixed-point definition of M. (It's the unique fixed-point of an operator acting on the set of all those order-preserving maps from C(I) to R that do the right thing on constant functions.) PS Just for comparison, consider the category of posets and the functor that sends X to X;1;X. The open interval is an invariant object for this functor but it is not the final coalgebra. For that we need -- as we called it in Cats and Alligators -- Wilson space. Actually, not the space but the linearly ordered set, most easily defined as the lexicographically ordered subset, W, of sequences with values in {-1, 0, 1} consisting of all those sequences such that for all n a(n) = 0 => a(n+1) = 0 (take a finite word on {-1,1} and pad it out to an infinite sequence by tacking on 0s). PPS Given a coalgebra structure described by endo-functions d and u on X, let End(X) be the monoid of endo-functions on X. We will need to compose in the diagrammatic order: "ud" means "first u then d". Let {0,1}* be the free monoid with generators named 0 and 1 and Let M_X:{0,1}* --> End(X) be the monoid homomorphism such that M_X(0) = d and M_X(1) = u. Given x in X let L be the subset of {0,1}* consisting of those words w such that M_X(w) sends x to top in X. The elements of {0,1}* may, of course, be viewed as finite words on 0 and 1. By catenating "0." on the left of any such word we obtain a description -- in binary -- of a dyadic rational in the half-open unit interval [0,1). Define f:X -> I by sending each x to the supremum in [0,1] of the dyadic rationals, r(L), named by the words in its corresponding L. The L corresponding to d(x) can thus be obtained by taking each word in L that starts with 0 and deleting that 0. The resulting r(L) can be described by first throwing away each dyadic rational bounded below by 1/2 and then doubling each dyadic rational that remains, that is,doubling each dyadic rational bounded above by both f(x) and 1/2. The resulting supremum is thus min (1, 2f(x)). That's what f(d(x)) is. And it's also what d(f(x)) is. Same sort of argument for u. A little too easy. The recipe above does work for f but the proof that it works requires work. Note that the above argument requires -- among other things -- that r(L) be a downdeal. (And note that it never invoked the facts that d and u preserve top and bottom nor the fact that d(x) is top or u(x) is bottom for all x.) I think a return to Dedekind works best. Besides defining the "lower" set, L, define the "upper" set U as the subset of {0,1}* consisting of those words w such that M_X(w) sends x to bottom. We have the facts: w:L => w0:L and w1:L (using w:U => w0:U and w1:U ":" w:{0,1}* => w0:L or w1:U for membership) We may infer, just from w:L => w0:L and w:U => w0:U, that if a dyadic rational has a name in either L or U then it does not have a name in the other. Thus we obtain a disjoint pair of sets of dyadic rationals r(L) and r(U). For any pair of dyadic rationals x,y:[0,1) where x is _not_ in r(L) and x < y, it is the case that y is in r(U): it suffices to check, for any word w, that if w0 can be prolonged to a name of x, then w0 is not in L forcing w1 to be in U and, hence, any prolongation of w1 to be in U. It follows that r(U) is an updeal and if there's a dyadic rational in [0,1) that is in neither r(U) nor r(L) then there's only one such dyadic rational, to wit, the greatest lower bound of r(U). It follows that r(L) is a downdeal. The pair r(L) and r(U) thus forms a Dedekind cut and we can use it to name the value of f(x). The compatibility with d and u is a straightforward computation. For uniqueness, first verify that for every pair a < b in I there's a word w:{0,1}* such that M_I(w) sends a to bottom and b to top. If f,f':X -> I were both coalgebra maps, and x:X were such that f(x) < f'(x) let w be as described. Note that M_I(w0) = M_I(w) = M_I(w1). But either M_X(w0) sends x to top or M_X(w1) sends x to bottom. The first case contradicts that M_I(w0) sends f(x) to bottom. The second case contradicts that M_I(w1) sends f'(x) to top.

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Preprint: Proper maps of toposes
by moerdijk＠math.uu.nl 21 Dec '99

21 Dec '99

Dear category theorists, I would like to inform you that my preprint with Japie Vermeulen, "Proper maps of toposes", can now be obtained electronically from <http://www.math.uu.nl/people/moerdijk/preprints.html>. (This paper will eventually appear as an issue of the Memoirs of the AMS.) With best wishes for the holidays, Ieke Moerdijk.

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3 JOBS AT SAARBR"UCKEN, please distribute
by Michael Kohlhase 21 Dec '99

21 Dec '99

[We apologize, if you receive multiple copies of this] Three Open Positions Researcher at the German National Research Centre for Artificial Intelligence (DFKI) Saarbr"ucken AND Scientific Assistants at the University of Saarbr"ucken a) For the research project "Interactive Textbook" (VIL) funded by the BMBF (The German Ministry for Education and Research) we seek applications for an open researcher position at the German National Research Centre for Artificial Intelligence, http://www.dfki.de/) For a more detailed description see below (VIL position). b) Within the research project "OMEGA: A mathematical Assistant System" funded by DFG (Deusche Forschungsgemeinschaft, the German analogue of the NSF) within the Corporate Research Center SFB-378 "Resource-Adaptive Cognitive Processes" we seek applications for open positions as Scientific Assistant. For a more detailed description see below (LIMA and MBASE positions). OMEGA is a mixed-initiative system with the ultimate purpose of supporting theorem proving in main-stream mathematics and mathematics education. The current system is an integrated, distributed collection of tools for interactive theorem proving, automated proof planning, proof presentation, and storing mathematical knowledge. The project home pages are http://www.ags.uni-sb.de/~omega VIL is a new project that is designed to apply AI-techniques such as proof planning, distributed systems, web communication, user modelling, and user-adaptive interfaces to an education system for mathematics. VIL has its basis in and is tightly connected with the OMEGA project. We offer - an innovative research project, - a highly motivated and cooperative team of about ten researchers working on the system, - a stimulating and well equipped working environment, with intimate connections between the DFKI (German Research Center for Artificial Intelligence) and the university research teams, - the possibility to work on a related PhD thesis. Applications are invited from suitably qualified applicants who - have completed an above average Master's Degree in computer science, mathematics, cognitive science, or equivalent, - enjoy teamwork and have a taste for interdisciplinary work - are committed to success-oriented work Applications including a list of publications and a curriculum vitae should be forwarded to: Prof.Dr. J"org Siekmann Universitat des Saarlandes / DFKI D-66041 Saarbr"ucken E-mail submission to <melis(a)cs.uni-sb.de> or <kohlhase(a)cs.uni-sb.de> respectively is preferred. Priority will be given to applications received by February 1st. 2000 ====================================================================== Position (VIL) The applicant is expected to work on theoretical and practical aspects of mixed-initiative systems, user-adaptive learning systems, and student modelling within a maths education system. Applicants for this position should - have experience in formal reasoning, maths education systems, user-adaptive systems, and/or student modelling The position is available starting February 1. 2000 and will be for a period of one year with the option of further extension. Payment amounts to an annual salary of approximately 78 000.- DM before tax. Please, do not hesitate to contact Erica Melis <melis(a)cs.uni-sb.de> for further information. ====================================================================== Position (LIMA) The applicant is expected to work on theoretical and practical aspects of automatically understanding mathematical texts and translating them into proof plans. Applicants for this position should - have experience in formal reasoning and/or natural language understanding, = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Position (MBASE) The applicant is expected to work on theoretical and practical aspects of formalizing mathematics, building a mathematical data base, and web communication with mathematical content. Applicants for this position should - have experience in formal reasoning, automated deduction, data bases, or web tools, = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Both position are available starting January 1. 2000 and will be for a period of two years with the option of an additional year. Payment for will be according to BAT2a on the German scale for federal employees. BAT2a amounts to an annual salary of 56 000.- to 86 000.- DM before tax strongly depending on age and family status. Please, do not hesitate to contact Michael Kohlhase <kohlhase(a)cs.uni-sb.de> for further information.

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Category-theoretic generalization of modular commutator
by William H. Rowan 21 Dec '99

21 Dec '99

Hello all, I believe someone said, at the 1994 conference on Category Theory and Universal Algebra, that there was a category-theoretic generalization of the modular commutator. Can someone give me a reference about this? Bill Rowan

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CONCUR 2000: Call for Papers
by Concur2000 20 Dec '99

20 Dec '99

FIRST CALL FOR PAPERS CONCUR 2000 11th International Conference on Concurrency Theory State College, Pennsylvania, August 22-25, 2000 URL http://www.cse.psu.edu/concur2000/ E-mail concur2000(a)cse.psu.edu (apologies for multiple copies) SCOPE The purpose of the CONCUR conferences is to bring together researchers, developers and students in order to advance the theory of concurrency, and promote its applications. Interest in this topic is continuously growing, as a consequence of the importance and ubiquity of concurrent systems and their applications, and of the scientific relevance of their foundations. Submissions are solicited in all areas of semantics, logics and verification techniques for concurrent systems. Topics include (but are not limited to) concurrency related aspects of: models of computation and semantic domains, process algebras, Petri nets, event structures, real-time systems, hybrid systems, decidability, model-checking, verification techniques, refinement techniques, term and graph rewriting, distributed programming, logic constraint programming, object-oriented programming, typing systems and algorithms, case studies, tools and environments for programming and verification. PROGRAM COMMITTEE Samson Abramsky (Edinburgh University, UK) Jos C. M. Baeten (University of Eindhoven, NL) Eike Best (Oldenburg University, Germany) Michele Boreale (University of Florence, Italy) Steve Brookes (Carnegie Mellon University, USA) Luca Cardelli (Microsoft, UK) Ilaria Castellani (INRIA, France) Philippe Darondeau (INRIA, France) Thomas Henzinger (UC Berkeley, USA) Radha Jagadeesan (Loyola University, USA) Marta Kwiatkowska (University of Birmingham, UK) Dale Miller (Co-chair, Penn State University, USA) Robin Milner (Cambridge University, UK) Uwe Nestmann (BRICS, Denmark) Catuscia Palamidessi (Co-chair, Penn State University, USA) Prakash Panangaden (McGill University, Canada) John Reppy (Bell Labs, USA) Vladimiro Sassone (University of Catania, Italy) Moshe Y. Vardi (Rice University, USA) Wang Yi (Uppsala University, Sweden) ORGANIZATION Dale Miller and Catuscia Palamidessi (Co-chairs, Penn State University, USA) SUBMISSIONS Submissions will be evaluated by the Program Committee for inclusion in the proceedings, which will be published by Springer-Verlag. Papers must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. Papers (of at most 15 pages, accompanied by a one-page abstract) should preferably be submitted electronically as uuencoded PostScript files at the address given below. The mailing addresses (both postal and electronic), telephone number and fax number (if available) of the author to whom correspondence should be sent should be clearly indicated. In case of hardcopy submissions, send five copies to the address below. CALL FOR SATELLITE WORKSHOPS The CONCUR 2000 conference will host several satellite workshops, which will take place on Monday August 21 and Saturday August 26. Proposals for satellites are solicited. They should contain a brief description of the scope and organization of the workshop, and be sent to the workshop chair Uwe Nestmann <uwe(a)cs.auc.dk>. IMPORTANT DATES Deadline for submission: 3 March 2000 Notification of acceptance: 1 May 2000 Final version due: 29 May 2000 Extended deadline for satellite workshop proposals: 10 January 2000 ADDRESSES Dale Miller and Catuscia Palamidessi Computer Science and Engineering Department 220 Pond Lab, Penn State University University Park, PA 16802 Phone: +1-814-865-9505, FAX: +1-814-865-3176 URL http://www.cse.psu.edu/concur2000/ E-mail concur2000(a)cse.psu.edu

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CfP: Wksp. on Graph Transformation and Visual Modeling Techniques
by Andrea Corradini 19 Dec '99

19 Dec '99

[sorry for multiple copies] ---------------- FIRST CALL FOR CONTRIBUTIONS ----------------- Workshop on Graph Transformation and Visual Modeling Techniques <http://www.di.unipi.it/GT-VMT/> Geneva, Switzerland Saturday July 15, 2000 A Satellite Event of ICALP'2000 <http://cui.unige.ch/~icalp/> -------------------------------------------------------------------- SCOPE AND OBJECTIVES OF THE WORKSHOP Visual modelling techniques provide an intuitive, yet precise way in order to express and reason about concepts at their natural level of abstraction. The success of visual techniques in computer science and engineering resulted in a variety of methods and notations addressing different application domains and different phases of the development process. Diagrammatic languages are used, for example, during requirement specification in order to support the communication between developers and customers; but also for describing the architecture of systems and as high-level visual programming languages. However, despite the wide-spread usage of visual modelling techniques there is a lack of well-understood (and integrated) methodologies for defining their syntax and semantics. Until now, there exists no equivalent to Backus-Naur-Form which would be the notation for defining the syntax of a visual language. The same applies to type systems, deductive proof methods, operational or denotational semantics for visual modelling techniques. In fact, the situation could be compared with the state-of-the-art in programming languages about thirty years ago when the definition and implementation of languages was an ad-hoc task, and formal semantics was an exception rather than the rule. The workshop aims at bringing together scientists and researchers interested in discussing formal methodologies for the definition of syntax and semantics of visual modelling techniques. All aspects of this problem are of potential interest ranging from the concrete syntax defined, for example, by Constraint Grammars or Picture Layout Grammars, the abstract syntax specified by means of Meta Modelling or Graph Grammars, and the semantics of visual modelling techniques given, e.g., in terms of Graph Transformation, Process Algebra, Abstract State Machines, Type Theory, Logic, etc. In particular, we believe that Graph Grammars and Graph Transformation may play a central role in the definition of visual modelling techniques because they provide the graphical analogous to Chomsky Grammars and Term Rewriting which are widely used for defining the syntax and semantics of textual languages. CALL FOR CONTRIBUTIONS Authors are invited to submit extended abstracts of up to 5 (five) A4 pages. The contributions should report about ongoing research in the area of graph transformation and visual modelling techniques, especially on the syntax and semantics of visual languages according to the scope and objectives of the workshop. Contributions exploring the use of Graph Grammars and Graph Transformation Systems are particularly welcome, as well as papers which cover several aspects or integrate different formalisms for the definition of visual modelling techniques. Position papers and contributions making methodological statements are strongly encouraged. Submissions should be sent preferably in postscript format to the address <andrea(a)di.unipi.it> (Andrea Corradini) before the submission deadline. In the case electronic submission is not possible, three copies should be sent to the address: Andrea Corradini Dipartimento di Informatica Corso Italia 40 56125 - Pisa - ITALY IMPORTANT DATES Deadline for submissions: March 1, 2000 Notification of acceptance: April 3, 2000 Final version of accepted extended abstracts: April 28, 2000 PROGRAM COMMITTEE Andrea Corradini (University of Pisa, I) [co-chair] Hartmut Ehrig (Technical University Berlin, D) Wolfgang Emmerich (University College London, GB) Reiko Heckel (University of Paderborn, D) [co-chair] Dirk Janssens (University of Antwerp, B) Hans-Joerg Kreowski (University of Bremen, D) Fernando Orejas (University of Catalonia, Barcelona, E) Grzegorz Rozenberg (University of Leiden, NL) Andy Schuerr (University of the German Federal Armed Forces, Munich, D) PROCEEDINGS The abstracts of the contributions accepted for presentation will be published in a volume collecting the contributions to all satellite workshops of ICALP 2000. The volume will be published by Carleton Scientific, and it will be distributed to all ICALP participants. On the basis of the number and quality of the submissions, the Program Committee will consider the possibility of inviting submissions for a special issue of an international journal dedicated to the workshop. INVITED SPEAKERS Gregor Engels (University of Paderborn, D) Martin Gogolla (University of Bremen, D) Francesco Parisi Presicce (University of Rome, I) Mauro Pezze' (Politecnico di Milano) ACKNOWLEDGEMENTS We acknowledge the financial support by the TMR Research Network GETGRATS (General Theory of Graph Transformation Systems) and the ESPRIT Working Group APPLIGRAPH (Applications of Graph Transformation) -- Dr. Andrea Corradini Tel: +39-050887266 Dipartimento di Informatica Fax: +39-050887226 Corso Italia 40 Email: andrea(a)di.unipi.it 56125 - Pisa - ITALY WWW: http://www.di.unipi.it/~andrea

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Postgraduate Scholarships
by Ross Street 17 Dec '99

17 Dec '99

Mathematicians: Below is an advertisement for Postgraduate Fellowships in memory of Scott Russell Johnson. I should explain what the ad really means. For those people without permanent residency in Australia, Macquarie University charges fees in the order of $16000 per year for enrolment in a PhD program. There are a few scholarships at Macquarie available to cover those fees but the rules for those (and the conventions followed by the committee) make it hard for a standard good student with an excellent undergraduate record (or even with a Masters degree) to get one. For a person with permanent residency and a good record (equivalent of First Class Honours in our system), there are generally enough Australian Postgraduate Awards (APAs) to go around. An APA pays a living allowance of around $17000 per year tax free (it lasts 3 years subject to performance and is rarely extended much beyond that). What we are trying to do with the SRJ Memorial Scholarship is two-fold: (1) given that an overseas student has paid the Macquarie fees, this Scholarship will provide $15000 towards a living allowance; (2) for Australian students, this Scholarship will supplement the APA with an extra $5000 per annum. We cannot afford very many of these scholarships at one time, so they are competitive. All postgraduate students are able to get a little extra from tutoring or assignment marking. Overseas students may also be able to obtain a supplement of around $3000 per annum from our Department. Regards, Ross http://www.math.mq.edu.au/~street/ MACQUARIE UNIVERSITY New South Wales 2109 Australia Scott Russell Johnson Memorial Scholarships for PhD Research Applications are invited for Scott Russell Johnson Memorial Scholarships which are offered for full-time PhD studies in the Centre of Australian Category Theory (CoACT) at Macquarie University. Applicants should hold or expect to obtain a First class Bachelor Honours degree (or equivalent) in an appropriate field, and be eligible for admission to the degree of Doctor of Philosophy in the Division of Information and Communication Sciences in the area of Mathematics. The award will be made to the best qualified applicants on the basis of academic merit. The full Scholarship provides a living allowance of $15,000 per annum tax exempt. A supplementary scholarship, up to the maximum allowed, may be awarded to students who already hold an APA or MUPGRA; see <http://www.mq.edu.au/postgrad/schl/2000apasinfo.pdf >. The full scholarship may not be held concurrently with any other equivalent scholarship, and the Scholarship holder will be responsible for any student or other fees payable in relation to candidature. Additional funds for conference travel and thesis production may be available from CoACT. The Scholarship will be initially awarded for a maximum of three years, with the possibility of a one year extension. It is tenable only up to the date of submission of the thesis, subject to satisfactory progress. Conditions related to overseas study, suspension of award, multiple awards, employment, sick, maternity and recreation leave will be the same as those for the Australian Postgraduate Award Scheme. The Scholarship holder shall be free to publish any results arising from the research. Any publications shall acknowledge that the work was done while holding the Scholarship. Applications should be made on the application form available from Postgraduate Studies Section Level 1 Council Building Macquarie University N S W 2109 AUSTRALIA email <pgschol(a)remus.reg.mq.edu.au> no later than 31 January 2000. For further details contact Professor Ross Street by electronic mail <street(a)math.mq.edu.au> or phone <+61 (0)2 9850 8921>.

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Retirement
by Charles Wells 16 Dec '99

16 Dec '99

I am retiring from Case Western Reserve University. Future postal mail should be sent to Charles Wells 105 South Cedar Street Oberlin, OH 44074, USA and you should use only the email address and phone number below. The University has not yet told me whether they will allow me to keep my webpage at the address below, but the odds are that they will. Charles Wells, Oberlin, Ohio, USA. EMAIL: charles(a)freude.com. HOME PHONE: 440 774 1926. PROFESSIONAL WEBSITE: http://www.cwru.edu/artsci/math/wells/home.html PERSONAL WEBSITE: http://www.oberlin.net/~cwells/index.html

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