81. MathPages Wanted List Elementary unsolved problems in mathematics, listed at the MathPages archive. http://mathpages.com/home/mwlist.htm

MathPages Wanted List The twenty-five mathematical problems and questions listed below were first posted on the internet in 1995. Since that time, Problems 5, 7, 8, and 22 have been solved completely, and part of Question 12 has been answered. The other problems remain unsolved. The links in this list point to articles on the MathPages web site containing more background on each problem, and partial or related results. (1) If each "1" in the binary representation of the integer x signifies a point in the corresponding position on a linear lattice, and if x' denotes the binary digit reversal of x, prove or disprove that the equality xx' = yy' implies that x and y have the same multi-set of point-to point distances. Ref: Generating Functions for Point Set Distances Isospectral Point Sets in Higher Dimensions (2) Find an elementary proof that x^2 + y^2 and x^2 + 103y^2 cannot both be squares for non-zero integers x,y. Ref: Concordant Forms (3) Prove (or disprove) that the only solutions of ab = c (mod a+b) ac = b (mod a+c) bc = a (mod b+c) in positive coprime integers are (1,1,1) and (5,7,11). Ref: A Knot of Congruences Limit Cycles of xy (mod x+y) More Results on the Form xy (mod x+y) Permutation Loops ... String Algebra (5) In how many distinct ways can the integers through 15 be arranged in a 4x4 array such that the bitwise OR over each row, column, and diagonal is 15, and the bitwise AND over each row, column, and diagonal is 0? Ref:

82. Open Problems In Knot Theory A List of Approachable Open Problems in Knot Theory (To show it s false, it s enough to show that an open knot is tricolorable if and only if its http://www.williams.edu/go/math/cadams/knotproblems.html

A List of Approachable Open Problems in Knot Theory .ps and .pdf files also available) Suggested by Colin Adams during the Knot Theory Workshop at Wake Forest University during June 24-28, 2002. Problems. What knots with high symmetries have projections that demonstrate this symmetry? (eg. the Figure-8 knot) Find specific families of knots satisfying the property c(K_1#K_2) = c(K_1)+c(K_2), where c=c(K) is the crossing number and # means knot composition. (eg. This is known for alternating knots.) What about torus knots? [In a 2003 preprint, Yuanan Diao demonstrated that this does hold for compositions of torus knots, as well. This was also independently proved by Herman Gruber. His paper is available at arxiv.org under math.GT/0303273.] When is a knot equivalent to its inverse? (The inverse has the same projection but with an opposite orientation). (eg. the trefoil and its inverse) Hass and Lagrias proved that if you have an n-crossing projection of the trivial knot, you can turn it into a trivial projection by using no more than 2^(1,000,000,000n) Reidemeister moves. Find a better upper bound. Find a pair of non-tricolorable knots whose composition istricolorable or show that this is not possible. (To show it's false, it's enough to show that an open knot is tricolorable if and only if its closure is tricolorable.)

83. Open-Ended Math Problems OpenEnded Math Problems. GET READY, GET SET This site is for the specific purpose of preparing Middle School students for OPEN-ENDED problem solving on standardized tests. http://www.fi.edu/school/math2/index.html

Open-Ended Math Problems GET READY, GET SET... This site is for the specific purpose of preparing Middle School students for OPEN-ENDED problem solving on standardized tests. We have divided each month into the five strands from the Philadelphia math standards: Number Theory Measurement Geometry Patterns, Algebra, and Functions Data, Statistics, and Probability There are three levels of difficulty for each standard. We have written and chosen problems from different sources that lend themselves to more than one way of solving. It is our hope that if these are done on a consistent, weekly basis, the students will feel more confident and comfortable at test time. To better prepare for the tests, students should answer with a picture, diagram, or paragraph explaining the solution and how they determined their answers. We are including possible answers and rubrics for assessment Try these with your students. Let us know what you think and how they did. September Problems October Problems November Problems December Problems ... April Problems Gwenn Holtz (7th Grade) and Mary Lee Malen (6th Grade) For questions or comments, contact the

84. Mathematical Problems In various subjects, compiled by Torsten Sillke. http://www.mathematik.uni-bielefeld.de/~sillke/problems.html

Mathematical Problems Graph Theory Stable sets - The number of stable/independent sets of a graph Planar Graphs - Special embeddings of planar graphs Graph Parameter - The Rank and the Chromatic Number Cograph (P4 free Graphs) - Graphs with maximal rank Nordhaus type question - eigenvalues Number Theory Lucas-Lehmer series - Factors and Period length Partial anwers: K Brown 1 K Brown 3 Chapman Some Properties of the Lucas Sequence (2,4,14,52,194,...) of Kevin Brown Reputnics - when are they squares? (Chris Thompson) x^3 + y^3 + z^3 - Representations by sum of Cubes (Noam Elkies) FLT++ - x^r + y^s = z^t Combinatorics Hadamard Matrices - Maximizing Determinants - M. Hudelson, V. Klee, D. Larman; Largest j-Simplices in d-Cubes: Some Relatives of the Hadamard Maximum Determinant Problem (1996) GKL - P. Gritzmann, V. Klee, D. Larman; Largest j-Simplices in n-Polytopes (unpublished) A Library of Hadamard Matrices - N. J. A. Sloane Grid-avoidance problems - A collection of problems Gray Codes - Gray Codes with few Tracks Distance Sets - Sets with equal Distances register swap - or the diameter of GL(n,F2)

85. Open Problems In Data-sharing Peer-to-peer Systems - Stanford InfoLab Publicatio by N Daswani 2003 - Cited by 212 - Related articles http://dbpubs.stanford.edu/pub/2003-1

@import url(/style/auto.css); @import url(/style/print.css); @import url(/style/nojs.css); Stanford InfoLab Publication Server Home Browse by Year Browse by Project ... Create Account Open Problems in Data-sharing Peer-to-peer Systems Daswani, Neil and Garcia-Molina, Hector and Yang, Beverly Open Problems in Data-sharing Peer-to-peer Systems. Technical Report. Stanford InfoLab. (Publication Note: 9th International Conference on Database Theory (ICDT 2003) Siena, Italy, January 8-10, 2003) BibTeX DublinCore EndNote HTML Preview PDF Abstract In a Peer-To-Peer (P2P) system, autonomous computers pool their resources (e.g., files, storage, compute cycles) in order to inexpensively handle tasks that would normally require large costly servers. The scale of these systems, their "open nature," and the lack of centralized control pose difficult performance and security challenges. Much research has recently focused on tackling some of these challenges; in this paper, we propose future directions for research in P2P systems, and highlight problems that have not yet been studied in great depth. We focus on two particular aspects of P2P systems search and security and suggest several open and important research problems for the community to address. Item Type: Techreport (Technical Report) Uncontrolled Keywords: Peer-to-peer, search, security

86. Science - Math - Number Theory - Open Problems - Directory Science Central Science Math - Number Theory - Open Problems. Article by Ivars Peterson.. Observations posted by Ken Conrow to stimulate further research.. http://www.sciencecentral.com/category/492610

87. Unsolved Problem Of The Week Archive A list of unsolved problems published by MathPro Press during 1995. http://cage.ugent.be/~hvernaev/problems/archive.html

Unsolved Problem of the Week Archive Welcome to the archive for the Unsolved Math Problem of the Week Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We give a reference so that you can get more information about the topic. These problems can be understood by the average person. Nevertheless, we do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. Should you wish to discuss aspects of these problems with others, one of the newsgroups, such as sci.math , might be the appropriate forum. 3-Sep-1995 Problem 36 : Primes of the form n^n+1 27-Aug-1995 Problem 35 : Must one of n points lie on n/3 lines? 20-Aug-1995 Problem 34 : Squares with Two Different Decimal Digits 13-Aug-1995 Problem 33 : Unit Triangles in a Given Area 6-Aug-1995 Problem 32 : Can the Cube of a Sum Equal their Product 30-Jul-1995 Problem 31 : Different Number of Distances 23-Jul-1995 Problem 30 : Sum of Four Cubes 16-Jul-1995 Problem 29 : Fitting One Triangle Inside Another 9-Jul-1995 Problem 28 : Expressing 3 as the Sum of Three Cubes 2-Jul-1995 Problem 27 : Factorial that are one less than a Square 25-Jun-1995 Problem 26 : Inscribing a Square in a Curve 18-Jun-1995 Problem 25 : The Collatz Conjecture 11-Jun-1995 Problem 24 : Primes Between Consecutive Squares 4-Jun-1995 Problem 23 : Thirteen Points on a Sphere 28-May-1995

88. Open Problems In Data-Sharing Peer-to-Peer Systems Neil Daswani File Format PDF/Adobe Acrobat Quick View http://ilpubs.stanford.edu:8090/581/1/2003-1.pdf

89. Open Problems Open Problems Arithmetic, Geometry, Algebra, Calculus, Probability, Combinatorics, Number Theory For n = 1, let f(n) be the number of 1's needed to write all the integers http://www2.truman.edu/~erickson/openproblems.html

Open Problems Arithmetic, Geometry, Algebra, Calculus, Probability, Combinatorics, Number Theory Prove that n^2+1 unit squares in a plane cannot cover any square of side length greater than n. (The unit squares may be rotated. The interior of the large square, as well as its perimeter, must be covered.) Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid. The first player to occupy the vertices of a square with horizontal and vertical sides is the winner. What is the smallest n such that the first player has a winning strategy? Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid. The first player (if any) to occupy a set of n cells having no two cells in the same row or column is the winner. What is the outcome of the game given best possible play by both players? What does the expression (q^n-1)(q^n-q)(q^n-q^2)(q^n-q^3)...(q^n-q^(n-1))/n! count? If q is a prime power, then this is the number of bases of an n-dimensional vector space over a field with q elements. There are two urns. One contains five white balls. The other contains four white balls and one black ball. An urn is selected at random and a ball in that urn is selected at random and removed. This procedure is repeated until one of the urns in empty. What is the probability that the black ball has not been selected? We can show that this probability is Binomial(10,5)/2^10. What is being counted?

90. 226. Top Ten Open Problems In Physics 15 posts 3 authors - Last post Aug 4I am very much interested in the unsolved problems of Physics. I study lot of different topics in Physics and Mathematics hoping to http://www.nonequilibrium.net/225-top-ten-open-problems-physics/

91. Open Problems Cosmology 1. Why the present value of the cosmological constant is so extremely low? Formulation of the problem Related posts Disorder on the http://www.nonequilibrium.net/open-problems/

92. The Furstenburg Conjecture And Rigidity If two commuting endomorphisms of a torus are incommensurable (no power of one is a power of the other), then their joint action should be rigid. Some of the conjectures and open problems compiled by the AIM. http://aimath.org/WWN/furstenburg/

The Furstenburg Conjecture and Rigidity This web page highlights some of the conjectures and open problems concerning The Furstenburg Conjecture and Rigidity. 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 dvi postscript or pdf version. The Furstenburg Conjecture and Rigidity Statement of the Conjecture History and past results Analogous problems ... Approaches to a counterexample

93. August 2008 OPEN PROBLEMS On SYZYGIES And HILBERT FUNCTIONS Irena File Format PDF/Adobe Acrobat Quick View http://www.math.cornell.edu/~irena/papers/overview.pdf

94. Open Problems In Infinite Ergodic Theory There are many more open problems then those listed below. We are only listing (at this time) the ones relating to our own research. If you are interested in any of these http://www.math.neu.edu/~eigen/OpenProblemsInfiniteErgodicTheory.html

Open Problems in Infinite Ergodic Theory There are many more open problems then those listed below. We are only listing (at this time) the ones relating to our own research. If you are interested in any of these problems, let us know and we will put you in contact with others interested in the same. Standing Assumptions: Unless otherwise indicated, all transformations will be Ergodic, Invertible, and Measure Preserving on a Non-Atomic, Sigma-Finite Lebesgue Space of Infinite Measure. Stanley Eigen Arshag Hajian Explicit Exhaustive Weakly Wandering Sets and Sequences (a) No exhaustive weakly wandering sets or sequences are known for any of the following examples. (b) It is unknown if any of the following are of finite type. (c) It is unknown if any of the following can commute with a non measure preserving transformation. Geodesic Flows on Hyperbolic Surfaces Markov maps Hopf's map Maharam Transformations Kakutani-Parry Transformations (ergodic k-index) Inner Automorphism Random Walks on the Integers Weakly Wandering Sequences Does the collection of all Weakly Wandering Sequences completely define the transformation?

95. 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.

96. Open Problems In Data Collection Networks File Format PDF/Adobe Acrobat Quick View http://www.eecs.harvard.edu/~jonathan/pubs/sigops04.pdf

97. Josh Cooper's Math Pages : Open Combinatorics Problems Unattributed problems are either classical or I don t know where they came from. Problems marked with a smiley ( ) are, as far as I know, mine. http://www.math.sc.edu/~cooper/combprob.html

Combinatorial Problems I Like Definitions for much of the terminology can be found here or here ) are, as far as I know, mine. Any errors/updates would be welcomed. See my homepage for contact info. I recommend the following for even more great problems: The Open Problem Garden The Open Problems Project Open Problems for Undergraduates Problems in Topological Graph Theory ... Graph Coloring Problems Last Edit: November 24, 2009. Did I leave something out? Please let me know! Email lastname@math.sc.edu. Discrepancy B n th roots of unity, or even into the complex unit circle.) What if the coloring function must be completely multiplicative, i.e., f nm f n f (m) for all naturals n m ? This problem is sometimes known as "determining the discrepancy of homogeneous arithmetic progressions." Beck: of the integers 1 to n . If we color each number in this range red or blue, call the discrepancy Is it always possible to find a coloring for any three permutations so that the discrepancy is O (1)? (The best known result is O (log n ), then the discrepancy of the interval [5,2,7,4] is 2 :

98. L-functions And Random Matrix Theory Conjectures and open problems concerning L-functions, 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 and Random Matrix Theory. If you would like to print a hard copy of the whole outline, you can download a dvi postscript or pdf version. Distribution of zeros of L-functions The GUE hypothesis Correlations of zeros Neighbor spacing ... Statistics of the zeros of the derivative of xi Relationship between symplectic and odd orthogonal The Alternative Hypothesis p-adic L-functions Zeros and primes The distribution of primes ... Fractional moments Mollified mean values Mollifying a Family Long Mollifiers Moments of S(T) The integral of exp(i lambda S(t)) ... GOE and Graphs The individual contributions may have problems because converting complicated TeX into a web page is not an exact science. The dvi, ps, or pdf versions are your best bet.

99. Open Problems In Topology II - Elsevier Oct 10, 2010 Thirty open problems in the theory of homogeneous continua (JR Prajs) Part 4. Topological Algebra 38. Problems about the uniform structures http://www.elsevier.com/wps/product/cws_home/711125

100. Elsevier Collection includes problems in set theory, continua, algabraic structures, analysis, and dynamics. Page includes glossary of terms. Links are in PDF. http://www1.elsevier.com/homepage/sac/opit/toc.htm

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