March 15, 2003 O P - S F N E T Volume 10, Number 2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor: Martin Muldoon muldoon@yorku.ca The Electronic News Net of the SIAM Activity Group on Orthogonal Polynomials and Special Functions Please send contributions to: poly@siam.org Subscribe by mailing to: poly-request@siam.org or to: listproc@nist.gov o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Today's Topics: 1. Minisymposium during SIAM Annual Meeting 2. OPSFA 7 - Copenhagen 3. Meeting in honour of Tom Koornwinder 4. Sturm Colloquium 5. Preprints in xxx Archive 6. About the Activity Group 7. Submitting contributions to OP-SF NET and Newsletter Calendar of Events: 2003 May 14-17: Advances in Constructive Approximation, Nashville, Tennessee, USA 9.4 #7 June 8-12: AMS-IMS-SIAM Summer Research Conference "Spectral theory and inverse spectral theory for Jacobi operators", Snowbird, Utah, USA 10.1 #1 June 16-20: SIAM Annual Meeting, Montreal, Canada in conjunction with CAIMS 9.6 #2 10.2 #1 June 18-21: Special session on "Constructive Approximation Theory" during the AMS-RSME Joint Meeting, Seville, Spain 9.4 #4 July 7-11: 5th International Congress on Industrial and Applied Mathematics, ICIAM 2003, Sydney, Australia. 8.6 #6 July 14-26: Summer school on "Orthogonal Polynomials and Special Functions", Coimbra, Portugal 10.1 #2 August 11-16: Fourth ISAAC Congress, Toronto, Canada 9.2 #6 9.4 #5 August 11-16: Conference on Orthogonal Functions and Related Topics, Roros, Norway 10.1 #3 August 18-22: Seventh International Symposium on Orthogonal Polynomials, Special Functions and Applications, Copenhagen, Denmark 8.6 #7 9.4 #6 10.1 #4 10.2 #2 August 25-27: Special Functions, Representation Theory and Applications, Meeting in honour of Tom Koornwinder, Amsterdam 9.6 #3 10.2 #3 September 15-19: Sturm Colloquium, Geneva, Switzerland 10.2 #4 Future Plans: Dan Lozier (OP-SF NET 9.4, Topic #2) suggests a SIAM-sponsored meeting in Orthogonal Polynomials and Special Functions to be held in Washington, DC in June 2004. Topic #1 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: Francisco Marcellan Subject: Minisymposium during SIAM Annual Meeting A Minisymposium "Special Functions: Computational Methods and Applications I-II" will be held during the SIAM Annual Meeting (joint with CAIMS) to be held in Montreal, Canada, June 16-20, 2003. The speakers are as follows: Session I John P. Boyd University of Michigan, Ann Arbor, MI USA Peter Clarkson University of Kent, Canterbury, UK Jesús Sanchez Dehesa Universidad de Granada, Spain Larry Glasser Clarkson University, Potsdam NY, USA Session II Bruce Fabijonas Southern Methodist Univ., Dallas TX, USA Dan Lozier NIST, Gaithersburg MD, USA Martin Muldoon York University, Toronto, Canada Javier Segura Universidad Carlos III de Madrid, Spain Abstracts and further information will appear at the meeting web site: http://www.siam.org/meetings/an03/ Topic #2 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: OPSFA 7 - Copenhagen As announced in previous issues, the Seventh International Symposium on Orthogonal Polynomials, Special Functions and Applications will be held August 18-22 in Copenhagen, Denmark. Here are some highlights from the web site: http://www.math.ku.dk/conf/opsfa2003/ Invited plenary speakers Professor Richard Askey, University of Wisconsin, Madison, USA Professor Percy Deift, New York University, USA Professor Antonio J. Duran, University of Sevilla, Spain Professor Uffe Haagerup, University of Southern Denmark, Denmark Professor Mourad Ismail, University of South Florida, USA Professor Erik Koelink, Technical University Delft, the Netherlands Professor Masatoshi Noumi, Kobe University, Japan Professor Franz Peherstorfer, University of Linz, Austria Professor Simon Ruijsenaars, Center for Mathematics and Computer Science, the Netherlands Professor Jan Felipe van Diejen, University of Talca, Chile Professor Yuan Xu, University of Oregon, USA Invited 30-minute speakers Professor Giuseppe Dattoli, ENEA, Frascati, Italy Professor Walter Gautschi, Purdue University, West Lafayette, USA Professor Leonid Golinskii, Institute for Low Temperature physics, Kharkov, Ukraine Professor Alberto Grunbaum, University of California, Berkeley, USA Professor Sergei Khruschev, Atilim University, Ankara, Turkey Professor Arno Kuijlaars, Katholieke Universiteir Leuven, Belgium Professor Christian Krattenthaler, Université Claude Bernard Lyon-I, France Professor Vadim Kuznetsov, University of Leeds, United Kingdom Professor Mizan Rahman, Carleton University, Canada Professor Barry Simon, California Institute of Technology, Pasadena, USA Grants for young researchers We are able to support a number of young researchers. Deadline for application is March 1. Registration for the symposium The deadline for registration is April 1. If you register later than this date you must pay an additional fee of 500 DKK. The standard registration fee is 2200 DKK. The fee for students is 1500 DKK and for accompanying persons it is 1000 DKK. As of March 12, 1.00 USD = 6.74 DKK and 1.00 EUR = 7.42 DKK, approximately. Internet registration is encouraged. Hotel reservation Hotel reservation is handled by a conference bureau. Deadline for hotel reservation is June 15. Four hotels are listed with the rate per day for a single room ranging from 400 to 795 DKK. Double rooms cost about 20% more. Submission of abstracts Submission of abstracts must be done via internet or email. The deadline is May 1. Local Committee for OPSFA 7: Professor Christian Berg Ph.D. Student Jacob S. Christiansen Associate Professor Henrik L. Pedersen Symposium address: OPSFA Department of Matematics University of Copenhagen Universitetsparken 5 DK-2100 København Ø Denmark Email: opsfa@math.ku.dk Fax: +45 3532 0704 Topic #3 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OPSF NET Editor From: OP-SF NET Editor Subject: Meeting in honour of Tom Koornwinder As announced in OP-SF NET 9.6, Topic #3, a meeting "Special Functions, Representation Theory and Applications" Universiteit van Amsterdam, August 25-27, 2003 on the occasion of Tom Koornwinder's 60th birthday. The meeting web site http://staff.science.uva.nl/~jstokman/workshop.html includes information on hotels, as well as a revised list of speakers: R.A. Askey (Madison, USA), F.A. Bais (Amsterdam, The Netherlands), E.P. van den Ban (Utrecht, The Netherlands), O. Chalykh (Ithaca, USA), N. Ciccoli (Perugia, Italy), G. van Dijk (Leiden, The Netherlands), J. Faraut (Paris, France), M. Flensted-Jensen (Copenhagen, Denmark), G. J. Heckman (Nijmegen, The Netherlands), M.E.H. Ismail (Tampa, USA), A.U. Klimyk (Kiev, Ukraine), T.H. Koornwinder (Amsterdam, The Netherlands), V. Kuznetsov (Leeds, UK), I. G. Macdonald (London, UK), M. Noumi (Kobe, Japan), P. Paule (Linz, Austria), H. Rosengren (Goteborg, Sweden) J.V. Stokman (Amsterdam, The Netherlands) Topic #4 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OPSF NET Editor Subject: Sturm Colloquium >From the web site: http://theory.physics.unige.ch/sturm.html Colloquium on the occasion of the 200th anniversary of Charles François Sturm and Workshop on Sturm-Liouville theory Geneva, Switzerland September 15-19, 2003 Geneva wishes to honour the memory of Charles Sturm on the occasion of the 200th anniversary of his birth. The Physics Department and the group "History and Philosophy of Science" of the University of Geneva, in association with the SPHN (Geneva Society of Physics and Natural History), are organizing an international colloquium during the period September 15 to September 19, 2003. The colloquium will consist of two parts. In the first one historians of science will analyze the scientific work of Charles Sturm, whereas the second one will be devoted to modern and contemporary aspects of Sturm-Liouville theory. The colloquium, especially in its first part, is conceived as a preparation for the publication of the complete works of Sturm. A second volume will contain contributions to the second part. The material for the first part will be organized as follows: - Biography - Geometry (theory of polygons, elementary geometry, projective geometry, conic sections, ...) - Algebra - Analysis (differential equations, series, treatise on analysis, ...) - Optics (caustics, physiological optics) - Mechanics - Other work in physics (heat, compressibility of liquids, speed of sound in water) - Pedagogical activities, the two monographs (treatises on analysis and on mechanics). The invited speakers will participate directly in the publication of the works. Their presentation will serve as an introduction to the different parts of the volume containing Sturm's complete works. In the second part of the colloquium a few invited specialists on Sturm-Liouville theory will discuss the developments in this branch of mathematics during the 20th century, as well as their applications in the natural sciences, in particular in physics. Other participants in the colloquium will have the possibility of presenting their recent research in the framework of a workshop on Sturm-Liouville operators. Organizers: Jean-Claude Pont (part 1) Werner Amrein (part 2) Secretariat: Francine Gennai-Nicole Advisory board: Andreas Hinz, Jan Lacki, David Pearson Preliminary program Part I : Life and work (September 15-16) Analysis and assessment of Sturm's scientific work. Lectures given by Eisso ATZEMA (University of Maine, Orono) Jean DHOMBRES (EHESS, Paris) Jan LACKI (University of Geneva) Jesper LÜTZEN (University of Copenhagen) Angelo MINGARELLI (Carleton University, Ottawa) Jean-Claude PONT (University of Geneva) Hourya SINACEUR (CNRS, Paris) Part II : Sturm-Liouville Theory, Past and Present a) September 17 : Survey of the development of Sturm-Liouville theory from Sturm's death to the present. Lectures given by Don HINTON (University of Tennessee, Knoxville) Norrie EVERITT (University of Birmingham, UK) Yoram LAST (Hebrew University, Jerusalem) Barry SIMON (Caltech, Pasadena) Joachim WEIDMANN (Universität Frankfurt) b) September 18-19 : Workshop on Sturm-Liouville Theory (continued in the morning of September 20 if necessary) Conference languages : Part I : French and English Part II : English Contributions The workshop on Sturm-Liouville operators will be organized on Thursday, Sept 18 and Friday, Sept 19 (with the possibility of continuing on Saturday morning, September 20, if necessary). Those wishing to present a contribution should send a title and a short abstract before April 30, 2003, by filling and sending the appropriate form. The principal lecture room is equipped with two overhead projectors, a digital light projector (beamer) and a blackboard. Registration The meeting is planned without registration fees for all participants (thus encouraging young researchers to take part). However, for organizational purposes, we request that all prospective participants fill in and send us the registration form at the web site. Deadline for registration: June 30, 2003. The web site http://theory.physics.unige.ch/sturm.html has more information on travel, accommodation and related matters. Topic #5 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OPSF NET Editor Subject: Preprints in xxx Archive The following preprints related to the fields of orthogonal polynomials and special functions were recently posted or cross-listed to one of the subcategories of the xxx archives. See especially: http://front.math.ucdavis.edu/math.CA http://front.math.ucdavis.edu/math.CO http://front.math.ucdavis.edu/math.QA http://xxx.lanl.gov/archive/solv-int http://www.arXiv.org/abs/math.CA/0301037 Title: Orthogonality of Jacobi polynomials with general parameters Authors: A.B.J. Kuijlaars, A. Martinez-Finkelshtein, R. Orive Comments: 16 pages, 4 figures Subj-class: Classical Analysis and ODEs; Complex Variables MSC-class: 33C45 http://www.arXiv.org/abs/math.SP/0301077 Title: Sturm-Liouville operators with distributional potentials Authors: A.M. Savchuk, A.A. Shkalikov Comments: Submitted to Proceedings of Moscow Mathematical Society Subj-class: Spectral Theory; Functional Analysis MSC-class: 34B24 (Primary); 47E05; 34L20; 34L40 (Secondary) http://www.arXiv.org/abs/math.CO/0301094 Title: Linearization coefficients for orthogonal polynomials using stochastic processes Authors: Michael Anshelevich Comments: 15 pages, AMSLaTeX, 2 Postscript figures Subj-class: Combinatorics; Classical Analysis and ODEs; Operator Algebras MSC-class: Primary 05E35; Secondary 05A18, 05A30, 46L53, 60G51 http://www.arXiv.org/abs/math.CO/0301157 Title: Generalizations of some identities involving the fibonacci numbers Authors: Toufik Mansour Comments: 12 pages Subj-class: Combinatorics MSC-class: 11B39; 11B83; 05A15 http://www.arXiv.org/abs/math.CA/0301210 Title: An inequality on Chebyshev polynomials Authors: Igor Rivin Subj-class: Classical Analysis and ODEs; Combinatorics MSC-class: 41A63; 42C05 http://www.arXiv.org/abs/math.CA/0301241 Title: Symmetrized Chebyshev Polynomials Authors: Igor Rivin Comments: Enhancement of math.CA/0301210 Subj-class: Classical Analysis and ODEs; Probability Theory MSC-class: 41A63; 42C05; 05C25; 05C20; 60F05 http://www.arXiv.org/abs/math.NT/0301277 Title: Relations for Multiple Zeta Values and Mellin Transforms of Multiple Polylogarithms Authors: Jun-ichi Okuda, Kimio Ueno Comments: 29 pages Subj-class: Number Theory; Algebraic Geometry; Classical Analysis and ODEs MSC-class: 11M06; 40B05 http://www.arXiv.org/abs/hep-th/0301143 Title: The Barnes zeta-function, sphere determinants and Glaisher- Kinkelin-Bendersky constants Authors: J.S.Dowker, Klaus Kirsten Comments: 26 pages, JyTex. Minor improvements made Subj-class: High Energy Physics - Theory; Mathematical Physics; Spectral Theory http://www.arXiv.org/abs/math-ph/0301022 Title: One-parameter isospectral special functions Authors: M.A. Reyes, D. Jimenez, H.C. Rosu Comments: 12 pages Subj-class: Mathematical Physics http://www.arXiv.org/abs/math.CO/0302014 Title: Restricted even permutations and Chebyshev polynomials Authors: Toufik Mansour Comments: 20 pages Subj-class: Combinatorics MSC-class: 05A05; 05A15; 30B70; 42C05 http://www.arXiv.org/abs/math.QA/0302032 Title: On integral representations of q-gamma and q-beta functions Authors: Alberto De Sole, Victor Kac Comments: 16 pages Subj-class: Quantum Algebra http://www.arXiv.org/abs/math.PR/0302033 Title: A System of Differential Equations for the Airy Process Authors: Craig A. Tracy, Harold Widom Comments: 8 pages, no figures. Version 2 clarifies a point about the operator K Subj-class: Probability Theory; Mathematical Physics MSC-class: 60K35 http://www.arXiv.org/abs/math.AG/0302054 Title: Analytic continuation of multiple polylogarithms Authors: Jianqiang Zhao Comments: 16 pages, a sign error corrected Subj-class: Algebraic Geometry; Number Theory MSC-class: 32D15, 53C65 http://www.arXiv.org/abs/math.CA/0302084 Title: A Generalization of Euler's Hypergeometric Transformation Authors: Robert S. Maier (University of Arizona) Comments: 26 pages, psp documentclass (Cambridge Univ. Press) Subj-class: Classical Analysis and ODEs; Combinatorics MSC-class: 33C20, 33C05, 34M35, 05A19 http://www.arXiv.org/abs/math.CA/0302114 Title: A bilateral series involving basic hypergeometric functions Authors: Hjalmar Rosengren Comments: 5 pages Subj-class: Classical Analysis and ODEs; Quantum Algebra MSC-class: 33D15 http://www.arXiv.org/abs/math.CA/0302142 Title: Jacobi Polynomials from Compatibility Conditions Authors: Yang Chen (Imperial College) Mourad Ismail (University of South Florida) Comments: 11 pages 0 figures Subj-class: Classical Analysis and ODEs http://www.arXiv.org/abs/math.CA/0302251 Title: Continuous Hahn functions as Clebsch-Gordan coefficients Authors: Wolter Groenevelt, Erik Koelink, Hjalmar Rosengren Subj-class: Classical Analysis and ODEs; Representation Theory http://www.arXiv.org/abs/math.CO/0302270 Title: Abel-Rothe type generalizations of Jacobi's triple product identity Authors: Michael J. Schlosser Comments: 11 pages, amslatex Subj-class: Combinatorics; Classical Analysis and ODEs MSC-class: 33D15 (Primary) 33D67 (Secondary) http://www.arXiv.org/abs/math.CO/0302320 Title: Positivity preserving transformations for q-binomial coefficients Authors: Alexander Berkovich, S. Ole Warnaar Comments: 58 pages, AMS-LaTeX Subj-class: Combinatorics; Quantum Algebra MSC-class: Primary 33D15; Secondary 33C20, 05E05 http://www.arXiv.org/abs/math.PR/0302329 Title: A PDE for the joint distributions of the Airy Process Authors: Mark Adler, Pierre van Moerbeke Comments: 9 pages Subj-class: Probability Theory; Mathematical Physics http://www.arXiv.org/abs/math.CA/0302342 Title: Laguerre functions and representations of su(1,1) Authors: Wolter Groenevelt Comments: 19 pages Subj-class: Classical Analysis and ODEs; Representation Theory http://www.arXiv.org/abs/math.CA/0302357 Title: Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach Authors: A. B. J. Kuijlaars, W. Van Assche, F. Wielonsky Comments: 60 pages, 13 figures Subj-class: Classical Analysis and ODEs; Complex Variables MSC-class: 41A21; 30E10; 35Q15 http://www.arXiv.org/abs/math-ph/0302011 Title: Hypergeometric functions related to Schur Q-polynomials and BKP equation Author: A. Yu. Orlov Comments: Latex, 15 pages, no figures Subj-class: Mathematical Physics http://www.arXiv.org/abs/math-ph/0302046 Title: New exact solutions for polynomial oscillators in large dimensions Authors: Miloslav Znojil, Denis Yanovich, Vladimir P. Gerdt Comments: 22 pages plus 4 figures Subj-class: Mathematical Physics; Spectral Theory MSC-class: 81Q05 http://www.arXiv.org/abs/math-ph/0302051 Title: Zonal Functions on $SO(p,q)$ Groups Authors: B.A. Rajabov Comments: 6 pages Subj-class: Mathematical Physics; Group Theory http://www.arXiv.org/abs/math-ph/0302054 Title: On the uniform asymptotic expansion of the Legendre functions Authors: Nail R. Khusnutdinov Comments: 9 pages, 4 figures Subj-class: Mathematical Physics MSC-class: 35C20;33E30 http://www.arXiv.org/abs/nlin.SI/0301011 Title: General solution for Hamiltonians with extended cubic and quartic potentials Authors: C. Verhoeven, M. Musette (VUB, Brussels), R. Conte (CEA, Saclay) Comments: LaTex 2e. To appear, Theoretical and Mathematical Physics 2003 Subj-class: Exactly Solvable and Integrable Systems http://www.arXiv.org/abs/nlin.SI/0301043 Title: Painleve test and higher order differential equations Authors: Ugurhan Mugan, Fahd Jrad Comments: arxiv version is already official Subj-class: Exactly Solvable and Integrable Systems Journal-ref: J. Nonlinear Math. Phys., volume 9, no. 3 (2002) 282-310 http://www.arXiv.org/abs/nlin.SI/0302026 Title: Classical transcendental solutions of the Painlev\'e equations and their degeneration Authors: Tetsu Masuda Comments: 22 pages Subj-class: Exactly Solvable and Integrable Systems Topic #6 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: About the Activity Group The SIAM Activity Group on Orthogonal Polynomials and Special Functions consists of a broad set of mathematicians, both pure and applied. The Group also includes engineers and scientists, students as well as experts. We have around 140 members scattered about in more than 20 countries. Whatever your specialty might be, we welcome your participation in this classical, and yet modern, topic. Our WWW home page is: http://math.nist.gov/opsf/ This is a convenient point of entry to all the services provided by the Group. Our Webmaster is Bonita Saunders (bonita.saunders@nist.gov). The Activity Group sponsors OP-SF NET, which is transmitted periodically by SIAM. It is provided as a free public service; membership in SIAM is not required. The OP-SF Net Editor is Martin Muldoon (muldoon@yorku.ca). To receive the OP-SF NET, send your name and email address to poly-request@siam.org. Back issues can be obtained at the WWW addresses: http://staff.science.uva.nl/~thk/opsfnet http://www.math.ohio-state.edu/JAT/DATA/OPSFNET/opsfnet.html http://math.nist.gov/opsfnet/archive The NET provides fast turnaround compared to the printed Newsletter, also sponsored by the Activity Group, and edited by Rafael Yanez. It appears three times a year and is mailed by SIAM. Back issues are accessible at: http://www.mathematik.uni-kassel.de/~koepf/siam.html To receive the Newsletter, you must be a member of SIAM and of the Activity Group. SIAM has several categories of membership, including low-cost categories for students and residents of developing countries. For current information on SIAM and Activity Group membership, contact: Society for Industrial and Applied Mathematics 3600 University City Science Center Philadelphia, PA 19104-2688 USA phone: +1-215-382-9800 email: service@siam.org WWW : http://www.siam.org http://www.siam.org/membership/outreachmem.htm Finally, the Activity Group operates an email discussion group, called OP-SF Talk. To subscribe, send the email message subscribe opsftalk Your Name to listproc@nist.gov. To contribute an item to the discussion, send email to opsftalk@nist.gov. The archive of all messages is accessible at: http://math.nist.gov/opsftalk/archive Topic #7 ------------ OP-SF NET 10.2 ------------- March 15, 2003 ~~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Submitting contributions to OP-SF NET and Newsletter To contribute a news item to OP-SF NET, send email to poly@siam.org with A copy to the OP-SF Editor . Please note that Submissions to the Net are automatically considered for the Newsletter, and vice versa, unless the contributor requests otherwise. Contributions to OP-SF NET 10.3 should be sent by May 1, 2003. Please send your (printed) Newsletter contributions directly to the Editor: Rafael J. Yanez Departamento de Matematica Aplicada Universidad de Granada E-18071 Granada, Spain phone: +34-58-242941 fax: +34-58-242862 e-mail: ryanez@ugr.es preferably by email, and in latex format. Other formats are also acceptable and can be submitted by email, regular mail or fax. The deadline for submissions to be included in the June 2003 issue is May 15, 2003, and for the October 2003 issue is September 15, 2003. o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o OP-SF NET is a forum of the SIAM Activity Group on Special Functions and Orthogonal Polynomials. We disseminate your contributions on anything of interest to the special functions and orthogonal polynomials community. This includes announcements of conferences, forthcoming books, new software, electronic archives, research questions, job openings. o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Send submissions to: poly@siam.org Subscribe by mailing to: poly-request@siam.org or to: listproc@nist.gov Back issues can be obtained at the WWW addresses: http://staff.science.uva.nl/~thk/opsfnet http://www.math.ohio-state.edu/JAT/DATA/OPSFNET/opsfnet.html http://math.nist.gov/opsfnet/archive WWW home page of this Activity Group: http://math.nist.gov/opsf/ Information on joining SIAM and this activity group: service@siam.org o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o The elected Officers of the Activity Group (2002-2004) are: Daniel W. Lozier, Chair Walter Van Assche, Vice Chair Peter Clarkson, Secretary Francisco Marcellan, Program Director The appointed officers are: Rafael J. Yanez, Newsletter Editors Martin Muldoon, OP-SF NET editor Bonita Saunders, Webmaster o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o