21. The Open Problems Project The open problems Project. This is the beginning of a project 1 to record open problemsof interest to researchers in computational geometry and related fields. http://cs.smith.edu/~orourke/TOPP/Welcome.html

Next: Numerical List of All The Open Problems Project edited by Erik D. Demaine Joseph S. B. Mitchell Joseph O'Rourke Introduction This is the beginning of a project to record open problems of interest to researchers in computational geometry and related fields. It commenced with the publication of thirty problems in Computational Geometry Column 42 [ ] (see Problems 1-30 ), but has grown much beyond that. We encourage correspondence to improve the entries; please send email to TOPP@cs.smith.edu . If you would like to submit a new problem, please fill out this template Each problem is assigned a unique number for citation purposes. Problem numbers also indicate the order in which the problems were entered. Each problem is classified as belonging to one or more categories. The problems are also available as a single Postscript or PDF file. To begin navigating through the open problems, you may select from a category of interest below, or view a list of all problems sorted numerically Categorized List of All Problems Below, each category lists the problems that are classified under that category. Note that each problem may be classified under several categories.

22. Open Problems Articles in PostScript format. http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/PROBLEMS/A_PROBLEMS.html

Problems Each entry includes a postscript version of the paper and a discussion section. The problems are available online only in postscript form. If you have solutions, references or additional comments please send them to pborwein@cecm.sfu.ca. Online Problems P95-1: Some old and new problems in approximation theory P95-2: Two problems on interpolation P96-1: A convergence problem for rational interpolants P96-4: Bivariate segment approximation and free knot splines ... P97-2: Conjectures around the Baker-Gammel-Wills conjecture

23. Open Problems In Graph Theory Involving Steiner Distance open problems involving Steiner distance. http://www.uwinnipeg.ca/~ooellerm/open_problems/index.html

Some Open Problems in Graph Theory It has been shown by Chartrand, Oellermann, Tian, and Zou that, for a tree T: diam n n This inequality does not hold for graphs in general as was shown by Henning, Oellermann, and Swart . It was shown in the same paper that for a graph G and n=3 and 4: diam n n G. It was shown by Oellermann and Tian that for a tree T: C n-1 (T) is contained in C n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-center of a graph is contained in its Steiner n-center. It was shown by Beineke, Oellermann and Pippert that if T is a tree, then M n-1 (T) is contained in M n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-median of a graph is contained in its Steiner n-median. Oellermann and Tian ). It is known that every graph is the 2-median of some graph (see Holbert ,and Hendry ). Steiner n-medians of trees have been completely characterized by

24. L-functions And Random Matrix Theory Conjectures and open problems concerning Lfunctions, focussing on the areas in which there has been recent progress using results from Random Matrix Theory. Maintained at AIM. http://www.aimath.org/WWN/lrmt/

L-functions and Random Matrix Theory This web page highlights some of the conjectures and open problems concerning L-functions, focussing on the areas in which there has been recent progress using results from Random Matrix Theory. Click on the subject to see a short article on that topic. If you would like to print a hard copy of the entire web page, you can download a postscript or pdf version. Distribution of zeros of L-functions The GUE hypothesis Correlations of zeros Neighbor spacing ... GOE and Graphs

25. Open Problems In Combinatorics open problems in Combinatorics. open problems of Paul Erdös, by Fan RK Chung; OpenProblems from the SIAM Activity Group Newsletter in Discrete Mathematics; http://www.combinatorics.net/problems/

Open Problems in Combinatorics Unsolved Combinatorial Problems by Zsolt Tuza Open Problems on Perfect Graphs , by Positivitiy Problems and Conjectures in Algebraic Combinatorics , by Richard Stanley Ten Problems in Combinatorics, Lecture given by Gian-Carlo Rota at the Conference on Combinatorics and Physics (CAP'98), held at the Los Alamos National Laboratory, August 10-12, 1998. To be announced. Problems in Signed, Gain, and Biased Graphs , compiled by Thomas Zaslavsky Some Problems in Matroid Theory , compiled by Thomas Zaslavsky Unsolved Problems in Graph Theory , compiled by Stephen C. Locke A problem on transitive permutation groups , by Peter J. Cameron A list of problems on permutation groups , by Peter J. Cameron Unsolved Mathematics Problems , compiled by Steven Finch. , by Fan R. K. Chung Open Problems from the SIAM Activity Group Newsletter in Discrete Mathematics Problems on the Enumeration of Matchings (ps file) , by James Propp

26. Bill Martin -- Open Problems Compiled by Bill Martin. http://www.uwinnipeg.ca/~martin/RESEARCH/open.html

Some Open Problems in Algebraic Combinatorics Last modified: May 4, 1999 Below, I list some of my favourite unsolved problems. But first, a few warnings. Many of these problems have been posed by other people. I will try to give proper attributions, but I am likely to miss someone's name eventually. Many of the problems I know of were posed by Chris Godsil . Secondly, the problems are all confined to areas in which I work. That is, the list is rather narrow in scope and may not seem thematic. (folklore) For a set C of q -ary n -tuples, let t denote the strength of C as an orthogonal array and let s denote the degree of C as a code. Prove that there exists a constant k such that t <= s+k . Note that no examples are known having t>s+4 (Delsarte, 1973) Do there exist non-trivial perfect codes in the Johnson graphs J(v,k) This question appears in Delsarte's thesis. He ``almost'' conjectures that the answer is NO. The strongest result to date on this is Roos's bound (see Brouwer, et al.). A recent paper by Tuvi Etzion also has nice results (most notably, that the answer is NO if v-2k is prime). I proved that the derived design of a perfect code is always a completely regular code. This gives more leverage to a number-theoretic attack. For example, there are no perfect 2-codes with v

27. Open Problems In Mathematical Physics open problems in Mathematical Physics. This page leads to a collection of significantopen problems gathered from colleagues during the academic year 1998/99. http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/

Open Problems in Mathematical Physics Quick Access: Soft phases in 2D O(N) models Quantum Hall Conductance dimensions ... IAMP This page leads to a collection of significant open problems gathered from colleagues during the academic year 1998/99. They are offered in the belief that good challenges stimulate our work, tempered by the dictum that preformulated questions should not discourage one from seeking new perspectives. All are invited to send pertinent comments, references to solutions, and contributions for this page to M. Aizenman (Editor): aizenman@princeton.edu List by Contributors By order of submission General Framework Quantum Field Theory Statistical Physics Quantum Many Body Systems Geometry and Physics Schroedinger Operators Disordered Systems Non-equilibrium Relativity and Gravitation Dynamical Systems Fluid Dynamics (Layout webmaster: aizenman@fas.harvard.edu

28. Past Open Problems From the SIAM Activity Group Newsletter in Discrete Mathematics. In PostScript. Compiled by Douglas B. West. http://www.math.uiuc.edu/~west/pcol/pcolink.html

Past Open Problems Columns - Douglas B. West From the SIAM Activity Group Newsletter in Discrete Mathematics These columns are the pre-publication input format sent to the Newsletter editor. In making this archive available more broadly, I am hoping also for input from readers. Please send me the open problems you would like to see solved! Email contributions to west@math.uiuc.edu DBW home page Eventually, we hope to establish a more flexible archive of open problems, searchable by keywords in various fields, with direct links from the problem pages to updates about full or partial solutions. Webmaster volunteers to help establish the searchable archive system are eagerly solicited! Open Problems #25 postscript , Winter-Spring 1998. Open Problems #24 postscript , all seasons, 1997. Open Problems #23 postscript , Summer-Fall 1996. Open Problems #22 postscript , Spring 1996. Open Problems #21 postscript , Winter 1996. Open Problems #20 postscript , Fall 1995. Open Problems #19 postscript , Summer 1995. Open Problems #18 postscript , Spring 1995.

29. Problems In Analysis Of Algorithms A list of open problems with updates and solutions. http://pauillac.inria.fr/algo/AofA/Problems/

PROBLEMS in ANALYSIS of ALGORITHMS PAGES Home Page Research Problems Bulletin board ... Resources Return to Analysis of Algorithms Home Page This page contains a list of interesting problems that we are aware of. You are encouraged to submit new ones by posting directly on the Bulletin Board . Typically 5 to 10 lines of TeX should be best for further editing. A digest of the main problems will be compiled here periodically by the Problem Editor Summer 97 Problem 1. Problem Solution ] By Conrado Martinez , 11-Jul-97. The depth of the j-th element in a random binary search tree of size n. Problem 2. Problem ] By Conrado Martinez , 11-Jul-97. Quicksort with median-of-three partitioning and halting on small subfiles. Problem 3. Problem Comments ] By Hsien-Kuei Hwang , 23-Jul-97. A limit distribution and zeros of a polynomial. Problem 4. Problem ] By Wojtek Szpankowski , 25-Jul-97. What is the distribution of node levels and height in digital search trees built on Bernoulli sources? Problem 5. Problem ] By Ed Coffman , 01-Aug-97. Analyse the waste in First-Fit bin-packing.

30. Ideas, Concepts, And Definitions open problems. open problems are not a source of frustration, they are asource of delight. open problems are the lifeblood of mathematics. http://www.c3.lanl.gov/mega-math/gloss/math/openpr.html

Open Problems In a community of mathematicians, an open problem is a question that no one has found the answer to. Open problems are not a source of frustration, they are a source of delight. Open problems are the lifeblood of mathematics. For you, as an individual mathematician, your own open problems are the questions you raise that you cannot answer. Anyone who does mathematics for very long soon discovers that open problems are abundant, and even more of them are generated as mathematicians think about something and ask themselves questions in an effort to understand it. One of a mathematician's hardest choices is deciding which open problems to give focused attention to and try to solve. When you raise a question for yoursef, and you cannot answer it, it becomes an open problem for you. It is a good idea to share this problem with other mathematicians friends, classmates, teachers, etc. to see if they know of a solution or have ideas about solving it. Share your open problems with MegaMath! Don't abandon your open problems just because they remain unsolved for a long time. Set them aside and think about them gently . A solution might surprise you and arrive when you least expect it. You might even dream it!

31. The RTA List Of Open Problems The RTA list of open problems. The RTA 98. We are currently workingto transform the list of open problems into a WWW service. This http://www.lsv.ens-cachan.fr/rtaloop/

The RTA list of open problems The RTA list of open problems summarizes open problems in the field of the International Conference on Rewriting Techniques and Applications (RTA). For the RTA 2002 conference, the topics of RTA were given as Applications: case studies; rule-based programming; symbolic and algebraic computation; theorem proving; functional and logic programming; proof checking. Foundations: matching and unification; completion techniques; strategies; constraint solving; explicit substitutions. Frameworks: string, term, and graph rewriting; lambda-calculus and higher-order rewriting; conditional rewriting; proof nets; constrained rewriting and deduction; categorical and infinitary rewriting. Implementation: compilation techniques; parallel execution; rewriting tools. Semantics: equational logic; rewriting logic. The RTA list of open problems was created in 1991 by Nachum Dershowitz Jean-Pierre Jouannaud and Jan Willem Klop on occasion of the RTA'91 conference. Updated lists have since been published at RTA'93, RTA'95 and RTA'98. We are currently working to transform the list of open problems into a WWW service. This effort is, at the moment, led by

32. RTA List Of Open Problems - Summary Of Problems linear constant restrictions and arbitrary constant restrictions in unificationproblems. of explicit substitution that is confluent on open terms, simulates http://www.lsv.ens-cachan.fr/rtaloop/problems/summary.html

Summary of Problems Which rewrite systems can be directly defined in lambda calculus? Investigate the properties of spectri for special classes of rewrite systems. [Solved] What is the complexity of deciding ground-reducibility? [Solved] Is it decidable whether a term is is typable in the second-order lambda 2 calculus? Does surjective pairing conservatively extend lambdabetaeta-conversion? [Solved] Is unicity of normal forms with respect to reduction a modular property of left-linear term-rewriting systems? [Solved] Is it possible to decide whether the set of ground normal forms with respect to a given (finite) term-rewriting system is a regular tree language? Is the decidability of strong sequentiality for orthogonal term rewriting systems NP-complete? Is left-sequentiality a decidable property of orthogonal systems? Has any full, finitely-generated and Church-Rosser term-rewriting system (or system with bound variables) a recursive, one-step, normalizing reduction strategy? Is unicity of normal forms a modular property of standard conditional systems? [Solved] What is the complexity of the decision problem for the confluence of ground term-rewriting systems?

33. Steiner Trees: Open Problems ganley.org The Steiner Tree Page - open problems. open problems.Of course, there are probably about a zillion open problems related http://ganley.org/steiner/open.html

ganley.org The Steiner Tree Page Open Problems Of course, there are probably about a zillion open problems related to Steiner trees, but here are a few I've thought about. Full trees Hwang's theorem allows us to construct an optimal rectilinear Steiner tree of a full set in linear time. I know of no other metric or type of graph in which computing the optimal Steiner tree of a full set is polynomial-time solvable but computing a general Steiner tree is NP-hard. Note that there isn't even a sufficiently strong analogue of Hwang's theorem for rectilinear Steiner trees in three dimensions. Multidimensional rectilinear Steiner ratio . What is the rectilinear Steiner ratio in arbitrary dimension d ? It is at least 2-1/ d , as the d -dimensional analogue of the "cross" has this ratio. It is obviously at most 2. It is generally believed that the lower bound is correct, but this hasn't been proven. Even an upper bound lower than 2 would be interesting. Rectilinear Steiner arborescence . These are Steiner-like trees on points in the (first quadrant of the) plane, in which every segment in the tree is directed left to right or bottom to top. It is unknown whether computing an RSA is NP-complete. (A good reference to start with is Rao, Sadayappan, Hwang, and Shor

34. Home Page For Constructive Approximation Open Problem Section Research problems section of the journal, edited by Peter Borwein, Albert Cohen, Ingrid Daubechies and Vilmos Totik. Includes problem statement (PostScript) and discussion. http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/CA_problems.html

Research Problems Section Edited by: Peter Borwein, Albert Cohen, Ingrid Daubechies and Vilmos Totik About the open problems section Problems available online Return to the Constructive Approximation Homepage The Brachistochrone Challenge "I, Johann Bernoulli, greet the most clever mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to earn the gratitude of the entire scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall then publicly declare him worthy of praise." Groningen, 1 January 1697 http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/CA_problems.html Modified: 06/13/1995 by pborwein@cecm.sfu.ca (Peter Borwein).

35. OPEN PROBLEMS IN COMBINATORIAL OPTIMIZATION open problems IN COMBINATORIAL OPTIMIZATION. compiled by Steve HedetniemiDepartment of Computer Science Clemson University Clemson, SC 296341906. http://www.cs.clemson.edu/~hedet/preface.html

OPEN PROBLEMS IN COMBINATORIAL OPTIMIZATION compiled by Steve Hedetniemi Department of Computer Science Clemson University Clemson, SC 29634-1906 September, 1998 Preface The Friday afternoon Algorithms Seminar at Clemson has been a rich source of interesting problems over the past twelve years, as have many discrete math conferences, including our twelve Clemson mini-Conferences on Discrete Mathematics. During this time I have compiled an on-going list of open problems, which fascinate me for their apparent difficulty, their simplicity of statement and their inherent interest. Motivated by numerous requests for an updated version of this list, I am finally taking the time to update it and make it available to anyone who would like to read or copy it. For each of the following problems I will attempt to identify the originators and the approximate time of origin. If you can provide me with any historical corrections or additions to any of the problems in this list, please let me know. You will note that I have not yet provided many literature citations; it will take a LONG time to provide all of the relevant references. I hope to be able to include more and more of this information in the future. So please consider this to be a 'working' document, which will repeatedly be updated. Most of these problems are stated in the Garey and Johnson format for NP-complete problems. Unless stated otherwise, by 'Open' I mean that the complexity of the problem is not known to me (no NP-completeness proof exists), neither does any algorithm exist for solving the problem, or no solution is known to exist.

36. Open Problems open problems in Dynamical Systems Ergodic Theory. Welcome! 9, Polygonal BilliardsSome open problems. Submitted by Pascal Hubert Serge Troubetzkoy. http://iml.univ-mrs.fr/~kolyada/opds/

Other sites with this page O pen P roblems in D ynamical S ystems E rgodic T heory Welcome! Katsiveli - 2000 Open Problems Session (Chairman - Benjamin Weiss ). New problems are being added to it. If you would like to submit some open problems to this page, please send them to me Anyone who has never had any (open) problem has never tried anything new. Anyone who has published a paper has solved a problem and maybe has some other open problems. Why not to present them at this www page? Sergiy Kolyada Geometric Models of Pisot Substitutions and Non-Commutative Arithmetic Submitted by Pierre Arnoux Corrections - 29.11.2001 Ergodic Ramsey Theory - an Update Submitted by Vitaly Bergelson ( See also: http://www.math.ohio-state.edu/~vitaly/ Dense Periodic Points in Cellular Automata Submitted by Francois Blanchard Non-Discrete Locally Compact Second Countable Groups Submitted by Sergey Gefter Martingale Convergence and Ergodic Theorems Submitted by Alexander Kachurovskii The problem is closed Entropy, Periodic Points and Transitivity of Maps Submitted by Sergiy Kolyada Lubomir Snoha Corrections - 26.10.2002

37. Interval Computations Presents a guide to Interval Arithmetic research featuring open problems, applications, generalizations, bibliography, researchers' sites, ftp site, and mailing list. http://www.cs.utep.edu/interval-comp/

Interval Computations: Introduction and Technical issues Interval Computations: Introduction ( PostScript dvi see also the following page B. Hayes, A Lucid Interval ... American Scientist , 2003, Vol. 91, No. 6, pp. 484-488 provides an intro based on round-off errors. Frequently Asked Questions (FAQ) Roundings Necessary for Interval Computations see also the following page Generalizations of Interval Arithmetic and their Applications ... Interval Notations (original suggestion, reaction) Early Papers on Interval Computations, by R. E. Moore and by Others Open Problems Software and Languages Languages for Interval Computations Interval and Related Software Interval Arithmetic in the Forte Fortran 95 Compiler see also More Information about Interval Computations Books and Courses on Interval Computations Bibliographies on Interval and Related Methods Interval Ftp Site and Interval Mailing Lists The Interval Computations Community Homepages of Interval Computations Research Centers and Special Interest Groups Personalia: Homepages of Interval Computations Researchers Reliable Computing (formerly ... an International Journal, and other journals of interest to the interval computations community Conferences: Forthcoming and Previous Honors Received by Interval Researchers Important Web Sites of Interest to the Interval Computations Community Applications Applications of Interval Computations (see also GlobSol - software for global solutions Generalizations of Interval Arithmetic and their Applications Intervals and Probability US-Based Website,

38. RISC Open Problems 12 open problems in Industrial Robotics. Although The list below ofthe 12 open problems is presented in no particular order. The http://http.cs.berkeley.edu/projects/risc/open_probs.html

12 Open Problems in Industrial Robotics Although there are countless open problems in robotics, we hope that by defining what we consider to be the 12 most important open problems in RISC robotics , effort can be concentrated in those areas. In the past year, a number of new problems have emerged. Our objective is to identify new theoretical problems with near-term relevance for industrial robotics. We welcome feedback and/or comments to this list. In addition we would like to add pointers to people currently working on particular problems so that open communication is maintained and duplicate work does not occur. Also, pointers to relevant papers should be added. We welcome submissions of work in progress in this location for feedback. For any comments or changes please contact Eric Paulos The current list came out of a workshop held at Adept by Ken Goldberg in December 1994 and as he points out, "The goal is to specify problems as crisply as possible; the list below is just a first cut." In addition, full credit for this exceptional list should go to Ken Goldberg The list below of the 12 open problems is presented in no particular order. The challenge for each problem is is to define optimality and then find an efficient algorithm or prove that no such algorithm exists.

39. Open Problems In Graph Theory Involving Steiner Distance open problems involving Steiner distance. http://www.uwinnipeg.ca/~ooellerm/open_problems/

Some Open Problems in Graph Theory It has been shown by Chartrand, Oellermann, Tian, and Zou that, for a tree T: diam n n This inequality does not hold for graphs in general as was shown by Henning, Oellermann, and Swart . It was shown in the same paper that for a graph G and n=3 and 4: diam n n G. It was shown by Oellermann and Tian that for a tree T: C n-1 (T) is contained in C n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-center of a graph is contained in its Steiner n-center. It was shown by Beineke, Oellermann and Pippert that if T is a tree, then M n-1 (T) is contained in M n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-median of a graph is contained in its Steiner n-median. Oellermann and Tian ). It is known that every graph is the 2-median of some graph (see Holbert ,and Hendry ). Steiner n-medians of trees have been completely characterized by

40. Geombinatorics: Making Math Fun Again A journal of open problems in combinatorical and discrete geometry and related areas. Includes tables of contents. http://www.uccs.edu/~asoifer/geombinatorics.html

GEOMBINATORICS ISSN 1065-7371 A journal of OPEN PROBLEMS of combinatorial and discrete geometry and related areas. "This quarterly journal, started by Alexander Soifer at the University of Colorado at Colorado Springs , specializes in geometry and combinatorics , but what really distinguishes it from the field is attitude!" Paul Kainen THE FIRST DOZEN YEARS EDITORS CONTENTS 1991 - PRESENT AUTHORS INDEX 1991 - PRESENT ... ORDER FORM "Dear Alex, Finally got around to reading the October Geombinatorics , which contains more than the usual supply of gems the bits by Vizing and Alon were worth more than a year's subscription, if it's not too crass to put a money value on great mathematics, and why should it be, art is sold for money all the time ..." Peter D. Johnson, Dec. 18, 1995 Subscriptions are only $24/year for individuals, with a slight extra charge for shipping. Articles are written in an informal style so they are more accessible to undergraduate and gifted high school students, than is common in other mathematical journals. The list of editors is distingished , and Geombinatorics is now reviewed both by MATHEMATICAL REVIEWS and ZENTRALBLATT FÜR MATHEMATIK . The Table of Contents for Vol. VIII, Issue 1 (July 1998) will give an idea of specialization.

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