MASTER INDEX OF "SELECTED TOPICS" FILES

Here's the aggregate of selector lines describing the "Selected Topics" files on the index pages at this site. It's in a tabular format, one line per file, in no useful order, giving the link to the file and a one-sentence description. A simple scan with your browser will then find keywords in the file descriptions. Of course, if a file really is relevant but your keyword was not used in the file's description, you won't find it this way.

Now that several search engines are available for this site, I have removed notice of this file's existence from the pages of this site. I am including it here in case you found this place from an old link. Please replace your bookmarks with the site's main page:

http://www.math-atlas.org/welcome.html

From that page you'll be able to choose several methods of finding the right material at this site. In this way you will also get the information from the index files (e.g. subject summaries, book recommendations, external links, etc) which is not included below.

Here then are the file descriptions lifted from the various index files. I have not included links to off-site resources here, nor to crosslinks among the index pages. There may be duplicates and other errors here; I haven't edited the data much.

"Proof" of closed formulae for zeta(2n) (e.g. Sum (1/n^2) = pi^2/6 )
"The sum of first n squares can not be an perfect square, except for n=24."
"Typical" mathematical modelling problem (operations research)
(An interesting variant on computing areas of cyclic polygons is also included.
(John Baez) Historical example of use of symmetry groups in modelling in physics.
Image of a one-holed torus made only with triangles, in which all pairs of the (seven) vertices are joined by edges.
Table Of Contents from "Graphic Gems" series (v. 1-5)
Incompatibility of two "certain" conjectures (prime constellations vs pi(x+y) < pi(x)+pi(y).)
Divide a square into acute triangles
Drawing a circle without the trig functions
Euler's conjecture generalizing Fermat's Last Theorem
Factoring rational functions as composites.
Find the differential equation satisfied by a family of functions
General remarks on solving functional equations, using f( x^2/(4x-2) ) = (x-1)/x f(x) as the example.
Interpolating a function on R^2 from values at discrete points.
Modal logic: what one knows vs. believes (treated formally)
How many equilibrium configurations for the placement of N points on the sphere?
How far apart are the primes?
Parameterizing the solution set to a quadratic
Factoring the resultant of two 1-variable polynomials.
How big is the sphere from which a cap was cut? (Spherical geometry)
Optimal packings of {circles, squares,...} in {squares, ...} [Dave Boll]
Average distance between two points in a ball
Theta functions: Sum(x^(n^2)), Jacobi identity, use to solve polynomial equations;
Thue equations (homogeneous 2-variable polynomial= const)
Numerical data for many polyhedra -- pointer
Efficiently four-coloring planar graphs
Four Color Theorem and analogues to surfaces with holes.
Steiner's Theorem: straightedge and compass constructions can be accomplished with straightedge, one circle, and its center; not without that center.
Every triangle is equilateral :-)
Boys' surface (pictures and formulae)
Chromatic number of genus-g graphs, with history.
Convex hull computations: summary and pointers
Multiply all the digits (even zero); repeat until single-digit.
Generalized inverses of a matrix: definitions and applications
Number of graphs with n vertices.
Volume of a tetrahedron (in terms of sides)
Mertens' Conjecture: is sum_{n < x} \mu(n) always less than sqrt(x)? (no)
Bezout's theorem counts points (sort of).
Parameterizations of unitary operators on a Hilbert space (and thus parameterization of the unitary and orthogonal groups).
Triangularizations of tori -- how nice can they be?
Quasiperiodic tilings of Euclidean space (e.g. Penrose tiles) [Chris Hillman]
SheafHom -- software for computing
Statistical distributions of quantities derived from random points on spheres (Citations)
Easy method for a fairly good point distribution [Saff/Kuijlaars]
Interpolating a function on R^2 from values at discrete points.
Global Complex Analysis is differential topology; low-dim manifolds which are groups
Decomposing polyhedra into convex or tetrahedral pieces
Tait's conjecture: polyhedra have Hamiltonian cycles through the vertex set (false), and its connection to the four-color theorem.
Classifying cubic polynomials under the narrow equivalence relation of rotational equivalence.
Finding a curve of minimal degree in the plane which passes through a given set of points (an answer does appear, toward the end!)
Generalities on systems of polynomial equations (resolvents, Bezout's theorem)
Francis Sergeraert on his approach to computable homotopy.
General discussion regarding explicit homotopy computation.
Monotonicity of rational functions of several variables.
Speeding the convergence of a slowly converging series (via integral test).
Theta functions ( Sum a^k^2, k from 0 to n) and the Jacobi identity.
Estimating integrals of solutions to a differential equation.
A paint can which can be filled with a finite volume of paint, but which takes an infinite amount of paint to coat its sides!
Citations and pointers to computational geometry sources.
Kahan's old list of (easy!) Mathematica stumpers.
Pointer to exposition of conjugate-gradient method of optimization.
Citation for conjugate-gradient method of optimization.
CORDIC algorithms for evaluating elementary (trig. etc.) functions -- citations, summary, pointers to code.
Conjugate gradient methods of optimization.
Levenberg-Marquardt non-linear optimization.
Limitations of optimization methods; software
Citation: computation of elementary functions.
Apollonius' method of trisecting an angle.
Course notes by Charles Blair for a course in cryptography.
Abelian integrals: y"=k(y^2). Bonus Offer: article includes careful distinct between variables and functions. How to handle 2nd order ODE with no y' term.
Autonomous system of two linear differential equations.
Rolling balls: a nearly-linear system of 3 first-order differential equations.
Differential-equations-by-mail server
Integrating the solution to an ODE x'=f(x).
Classification of 4-dimensional manifolds (up to homeo- or diffeo-morphism), and applications to mathematical physics.
Additive computability -- if S_1 = {1}, S_2 = {1,2}, and S_m= S_(m-1) \union (S_(m-1) + S_(m-1)) , what's the first set S_m containing a given n?
Frankl conjecture on finite sets (open)
Traveling Salesman: citations, some code.
Historical introduction to elliptic curves.
Maple input file to force a curve with rational point into normal form.
Henri Cohen on curves with high rank.
Maple input file for reducing a curve y^2 = quartic to normal form, too.)
review of primality testing routines in commercial systems
Elliptic curve primality testing.
detailed summary of primality testing routines
Polynomial tests for primality.
technical discussion of factorization of RSA129.
Ceva's theorem (and Menelaus' theorem) on line segments associated with a triangle.
Morley's theorem about the trisected angles in a triangle.
An article describing one particular Grassmannian space.
Stabilizers of normal series are nilpotent.
Lagrange's theorem as an extension (to matrices) of Fermat's little theorem.
How many groups of order n?
ANU p-quotient program (for p-groups)
Generators and relations for the Rubik group, with an introduction to the GAP program.
Lifting homomorphisms into G/Torsion to homomorphisms into G (G abelian)
Complements of compact sets in Hilbert space are contractible.
Hotelling's method of inverting matrices (Newton's method).
Godel's theorem in a new light (lots of impressive big-number arithmetic).
List of papers classified 00A99 in the MathSciNet database
Comparative anatomy (was: what happens if you change the dimensions of a living being)
Drawing a circle on the computer.
Illustration of LLL execution on pari/gp for finding approximate algebraic identities.
Henri Cohen describes the 2nd edition of his book.
Congruence conjecture on !n = 1! + 2! + ... + n!
Calculating a product in Z[exp(2 pi i)/p].
How many triangles with all vertices lying in a square portion of Z^2? (up to similarity,...)
How many lattice points in a circle of radius r ? (pi*r^2; But error estimate = ?)
Pointers to LLL (lattice) algorithms and comparison of implementations.
Solving a^2+b^2+1=a mod 2^r.
Hugh Montgomery: software to accompany his number theory text.
Irrationality of pi.
Is there always a prime in the range...
Quartic reciprocity
Solving polynomial equations in the ring of quaternions; passing to extension rings.
How many triangles with all vertices lying in a square portion of Z^2? (up to similarity,...); this time I answered it! There's a little follow-up information, too (never posted).
Compactly generated topologies and products,
Peano curves (mappings [0,1] \mapsto [0,1]^2 (or [0,1]^n ) which are really close to being homeomorphisms.)
No homeomorphisms are possible between [0,1] and [0,1]^2.
Flexible polyhedra.
Pasting information for 1- and 2-holed tori with few cells.
realizability of polyhedral surfaces.
Another post computing volumes of polyhedra.
Factoring the polynomial x^500+x+1 modulo a 152-digit prime. (with citations)
Sturm sequences - a technique to determine how many real roots a poly has.
Old methods of root-finding: Graeffe's, Vincent's.
Groebner (Grobner) bases (bases for ideals in polynomial rings which permit rapid computations): citations + pointers for general descriptions.
Can one do number theory within the quaternions?
fortran code to determine location by triangulation; not written by me but I didn't get the author's permission to post his name.)
Determine present location from distances to three fixed points.
An example: how to put dimples in a ball?
Book citation on the statistical analysis of spherical data.
sphere.bas -- a hohum BASIC program showing how to implement a couple of the approximation procedures mentioned in the FAQ
Variations on the theme (I'm hoping that anyone who asks the question can find the form(s) of the answer they like best.)
Hough transform of data, to find patterns.
A 2nd order linear equation. After spending time on this equation I learned (a)about Bessel functions (2)to use a symbolic algebra program to solve differential equations.
A general optimization problem between the knapsack problem and sphere-packing problems.
A set-covering problem (which arose in weaving!)
A summary of factorization techniques (with citations to the literature).
A post by Wiles himself in December 1993 acknowledging difficulties with the proof. (The proof was repaired during 1994 with the assistance of Taylor, and published early in 1995 in the Annals of Mathematics.)
A rational square of the form A.A in its decimal expansion!
A Maple package that computes intersection numbers on algebraic varieties, etc.
A [dated] list of available programs for large-integer arithmetic.
A bibliography on magic squares.
A brief introduction to Abelian varieties
A calculus of variations problem: find the curve of minimal length which joins two points and includes an area of 1.
A citation on speeding up convergence of series.
A citation to the irreducibility of trinomials of the form x^a + x^b + 1
A classic question is whether one can with compass and straightedge trisect an arbitrary angle; the answer is no, but there are ways to trisect angles with a marked straightedge
A closed for is sought for a sequence defined recursively by x_{n+1}=x_n-(x_n^2)/n
A combinatorial question: how many regions result when connecting all the vertices of a regular polygon?
A computer algebra challenge: to find subfields of a certain extension of Q.
A derivation of the explicit formula for the group law .
A description of Blumberg's Theorem -- functions are always continuous on a dense set
A dissection problem: how to dissect a square into pieces with minimal perimeter.
A few near misses of solutions to the Integral Brick problem.
A free-for-all on fixed-point theorems
A generalization of Heron's formula to pentagons.
A humorous newspaper column I copied to sci.math, which uses the Wiles announcement to parody Chicago sports culture. [Newly restored link! Zorn is going to have this on the Tribune's website, so I'll jump the gun here...]
A little analytic geometry (finding the intersection of two cones.)
A little about Stein manifolds
A pointer to code for delaunay triangulation
A practical(?) application of the embedding of matrix rings M_n(C) into M_2n(R).
A question about writing a sphere as a union of compact pieces (good chance to think about what a "component" is.)
A question whose answer is "operations research"
A randomly selected response to the FAQ, "How do you solve the Traveling Salesman Problem?"
A recursively-defined sequence akin to the Bernoulli numbers: a_k = 1 - 2\sum_{j=0}^{k-1} {k\choose j} a_j
A related post on polyhedral tori.
A sample functional equation: solve f(x) + a = f( x + a*sqrt(x) )
A sample post showing the use of calculus techniques (finding the surface area of a baseball with Stokes' theorem).
A sample post showing the use of vector methods for solving 3D problems involving straight objects (lines, planes, vectors, angles, etc.)
A short discussion of the state of the art of primality testing.
A short summary of some basic data for polyhedral tori.
A shot at Metrizing topological fields (and embedding them into R).
A similar question: how many disks of radius r needed to cover the unit sphere in R^n ?
A summary of what the questions are regarding polyhedral tori.
A topological proof (!) of the infinitude of primes
A tough synthetic geometry problem
A triangle question whose solution depends on its premise of special 'adventitious' angles. (This is an example of trying to forgo the use of trigonometry when its use would be straightforward but inelegant).
ASCII-art version of the Fano plane (7 points on 7 lines)
Actually using a Stewart platform -- control system
Add n to its "opposite"; repeat until a palindrome appears. Will this end if we start with 196? (open)
Advice for numerical work on large (1000 x 1000) matrices
After I understood the geometry better I had a follow-up post giving a ruler-and-compass solution to the triangulation problem at hand.
All the standard solutions to the cubic
Almost all Galois groups are the symmetric group.
Also in the algebra department: a derivation of Heron's formula for the area of a triangle.
Among solutions of 3 x^2 + 5 y^2 = 2^(2n+1), estimate growth of min(x,y).
An assignment problem (optimally split people into groups)
An example of transformation to normal form for an elliptic curve.
An example using Pari to factor.
An application of line integrals to computing center of mass, area, etc using Green's (Stokes') theorem.
An application of covering spaces to complex analysis
An applied (?) questions which boils down to: when can an ordered topological space be embedded into R?
An elliptic curve formulation of the n = 3 case of Fermat's Last Theorem; in which quadratic extensions of Q does it have a solution?
An example from Galois theory: calculating the fixed field K(X)^G, for a certain small G.
An example of regular sequence applications in cohomology spectral sequences.
An example of two different fields, each contained in the other (up to isomorphism, of course)
An example of a recurrence relation defining a sequence growing doubly exponentially: f(n)=f(n-1)+f(n-1)f(n-2)
An example of a series expansion with very delicate convergence.
An explanation of the practical origins of the triangulation problem.
An interesting calc-1 problem: which tangent line is closest to the size of the graph?
An interesting example of matrices which satisfy the Fermat equation was found by a young boy.
An interesting example of computing Hausdorff dimension (and continued fractions!)
An interesting paradox distinguishing sets with similar topology by finding direct sums (resp direct products) in homology groups.
An interesting problem in Euclidean geometry: show that a map which sends spheres to spheres must be an isometry.
An offbeat technique (still theoretical) is the use of quantum computing.
An old list of challenge questions for Computer Algebra systems (by Richard Pavell, courtesy the REDUCE library)
An unusual question regarding p-sided figures where p is prime or pseudo-prime! (e.g. 341)
Annotated reading list for programmers (Nick Maclaren)
Announcement of p-group software.
Announcement of the sequence server at ATT
Announcement: 2^756839 - 1 is prime.
Announcement: Table of number fields (Henri Cohen)
Announcements flying through the aether when the repaired documents were circulated (1994)
Another interpretation of well-distributed one might give.
Another program for polyhedral g-holed tori.
Another proof that mathematicians have a language all their own :-)
Any planar set of area less than 1 can be translated so as to avoid lattice points.
Application of Green's theorem (Stokes' theorem) to calculating areas and center of mass of a polygon.
Application of isogeny to a question about fields.
Application of automorphic forms(!) to the question at hand.
Applications of 3D (discrete) Fourier transforms to data compression.
Applications of octonions in mathematical physics, again [John Baez]
Applications of fuzzy logic to clustering and image processing
Applications of the Schoenflies theorem
Applications to physics [John Baez]
Are C^\infty manifolds real-analytic?
Are most manifolds hyperbolic?
Are roots of polynomials over Q dense in C^n? Yes, by the Hilbert Irreducibility Theorem
Are there algebraic numbers on the unit circle besides roots of unity? (yes, many)
Are there any methods for finding closed formulas for 2-dimensional recurrence problems in general?
Aren't (continuous) bijections the same as homeomorphisms? (no) (This spawned a discussion about compactifications.)
Arranging a round-robin tournament
Asymptotic formula for number of partitions p(n).
Background and citations for the 1000 digits problem (deBruijn sequences)
Bailey/Borwein/Plouffe method of computing digits of pi.
Basic how-tos of continued fractions (applied to Pi ).
Basic pointers: netlib, gams
Basics on showing that a number is irrational.
Best proven estimates on the distribution of primes.
Book announcement: The Handbook Of Discrete And Computational Geometry, J. E. Goodman and J. O'Rourke, editors.
Buffon's needle problem.
C implementation of the sieve of Eratosthenes.
Calculating the envelope around a curve (a new curve a fixed perpendicular distance away).
Calculating the antiderivative of sin(x)^ N .
Calculating the fundamental groups of (compact, connected, orientable) surfaces.
Calculus (multivariable): how to recognize a global optimum?
Calculus: careful statment of theorem locating maxima for functions of one variable
Calculus: curve yielding equal volumes under two rotations
Calculus: do similar functions have similar derivatives?
Can a 3-dimensional polyhedron be decomposed into tetrahedra? (Not without adding interior points in general)
Can every integer be represented as a sum of three squares? How? (Summary of other sums-of-squares questions too)
Can one decompose polynomials into p = q o s + r with r small? (no).
Can one parameterize the points on an elliptic curve (e.g. y^2=x^3+x, y^2=quartic in x)?
Can one reconstruct a function knowing all integrals over balls of radius 1 ?
Can one determine whether a knot is really knotted from its projection to the plane?
Can one permute the entries in an n x m grid by just permuting the rows and columns? (No.)
Can we recognize the Borel sets among the measurable ones?
Can we determine G from the cardinalities of all conjugacy classes (no)
Can you get the sofa into the elevator and close the door?
Can you hear the shape of a drum? (no) Citations, URLs, and a summary.
Can you comb the hair on a sphere? (no)
Can you factor a linear combination of polynomials of the form a^2+b^2+1 (a, b linear in x,y)?
Can you determine analytic functions on a manifold from the values on a set with an accumulation point? (no -- read to see djr mix up results from complex analysis with real manifolds!)
Cataloguing automorphisms of a surface
Cells either die or split in two; what's the long-term outcome?
Characterization of compactness in metric spaces
Characterization of equivalence classes of quadratic curves.
Choose elements of a finite set without replacement. Probability of missing a particular one?
Citation and pointer for survey of Euclidean number rings and recent results.
Citation and pointer for use of octonions in physics
Citation for textbook (Bressoud) on factorization and primality testing.
Citation for Functional analysis for the practical man
Citation for dynamical systems (in re: Julia set of x^2+c)
Citation for PPMPQS (including its use in factoring RSA 129 -- see below).
Citation for statistical analysis of spherical data.
Citation for Mathematica primer.
Citation for chestnut: cubics with real roots are not in a real-radical extension.
Citation for convex hull and other geometric topics
Citation for decomposition H=A u B u C of hyperbolic plane with A, B, C all congruent and A congruent to B u C
Citation for solving Pell's equation (for N \not= 1)
Citation for the case n=5 of Waring's problem (Chen: g(5)=37).
Citation to Quaternionic analysis.
Citation: book of Kantor and Solodovnikov on real algebras
Citation: finding primitive roots in finite fields.
Citation: how do computers factor in Z[X]?
Citation: is an integer polynomial a product of cyclotomic polynomials?
Citation: variations of Newton's method (better around multiple roots).
Citations and pointers for computer algebra and symbolic computation techniques
Citations and pointers for latitude/longitude calculations
Citations and summary of what's possible with computational group theory.
Citations for methods of summation (Moenck, Zeilberger, Koepf, Karr, etc.)
Citations for efficient multivariate resultants.
Citations for busy-beaver, Turing-machine etc.
Citations for computing prime ideal decomposition.
Citations for implementation of simplex method
Citations for references on smooth numbers
Citations for the number field sieve
Citations for the Monty Hall problem.
Citations for the enumeration of finite topologies.
Citations for treatments of Sturm sequences and other methods of root-finding.
Citations to doublings, etc., and mention of sphere-packing.
Citations to Sloane's book. (The pointers are a little old, but new links to Sloane are below.)
Citations to literature concerning pi.
Citations, cautions to numerical fitting of polynomial to data
Citations: optimal convex decompositions of polygons
Citations: geometric modelling of botanical phenomena.
Citations: into what Euclidean spaces can a metric space be embedded?
Classic estimate of the sum of logs of first few primes.
Closed curves must have at least two points of maximal curvature.
Code to perform (4-parameter) curve fitting.
Code: sample Genetic Algorithm for optimization
Collection of logic paradoxes (liar, etc.)
Combinatorial question reducing to 2-colorability of a graph.
Comments about non-Archimedean fields.
Comments on Jacobi sums (Sum exp( - n(n+a)/2 ). )
Comparative efficiencies of sort algorithms, with C code
Compare and contrast the countability axioms
Comparing slopes of various interpretations of least-squares lines.
Comparison of assignment problem and traveling salesman problem.
Comparison of generic v. specific ranks of families of elliptic curves
Comparisons of Macsyma to its competitors.
Computing psi(x)=Gamma'(x)/Gamma(x) at rational points.
Computing square roots by hand.
Computing modular square roots.
Computing determinants of Toeplitz matrices
Computing elliptic integrals with the arithmetic-geometric mean (See also references in PI bibliography.
Computing the determinant of the Hilbert matrices.
Computing the inverse of the ill-conditioned Hilbert matrices
Computing the solution curves of predator-prey models (autonomous systems of 2 differential equations.)
Concerning linear differential equations with polynomial coefficients.
Connection between fractals and Newton's method.
Connection between Taylor series and area of images.
Connections between Hensel's lemma and Newton's method (Lit review)
Connections between knot theory and statistical mechanics (the Jones polynomial)
Connections between representations of Sym(n) and GL(n).
Connections between the Rubik's group and physics.
Consequences of strange replacements for Liebniz's formula for differentiation.
Constructing irreducible polynomials over finite fields -- Citation
Constructing exotic differentiable structures on the 7-sphere.
Constructing split extensions with certain properties.
Constructing a heptagon
Constructing a pentagon.
Constructing a finite group with three elements x, y, z having arbitrary orders and xyz=1.
Constructing the Geometric mean
Convex hull of points in the plane.
Convex hull: Preparata and Hong algorithm
Cool problem still open: Given some sticks of integral length at least n, whose total length is n(n+1)/2, can one cut them into pieces of length 1, 2, ..., n?
Coordinates of a dodecahedron
Copy of the post made announcing availability of the FAQ.
Could Fermat have had a proof of FLT? (for popular audience)
Countability of set of accumulation points.
Counting annihilating matrices over a finite field.
Counting the dimensions of magic squares and cubes.
Criteria for roots of a polynomial to be outside the unit disc.
Cryptanalysis of the WordPerfect document encryption algorithm.
Current (Oct. 1997) record in primality proving (2196 digits)
Curves and land surveying.
Dan Asimov asks about "hooples"
Decomposing a self-intersecting plane curve into simple curves
Decomposing a square and a circle into congruent (nonmeasurable!) parts
Delaunay triangulation for non-convex regions
Delicate estimates of Taylor series coefficients using contour integrals.
Densest sphere packings relation to distributing points on spheres
Deriving the equations of a torus.
Describe the motion of a Stewart platform (a triangle suspended by 6 pistons of changeable length joining the vertices to three stationary points on the ground.)
Describing solutions of autonomous systems of differential equations.
Description and application of the "Plate trick", a parlor trick giving a concrete example of a homotopy class of order 2!
Description of some optimal distributions of N points on a sphere for small N.
Description of the NFSNET network of number-factorers, and a progress report on one of the Cunningham numbers.
Descriptions of the method to put a curve in Weierstrass form.
Detailed descriptions of the elliptic curve method
Difference between compactness and closedness; different topologies on a single set.
Difference between PL and differentiable manifolds.
Differentiating the "difference" (f o g^(-1)) of two monotonic polynomials with resultants.
Discussion of FFT procedures for sizes not a power of two; pointers to implementations
Discussion of dimensions of metric spaces and their products.
Discussion of solutions to a differential equation arising in economics. Discussion includes being careful about technical requirements assumed.
Discussion of triangles whose sides are of rational length.
Distributing points on spheres with point-repulsion methods: codes, pointers.
Distributions of roots, and factorizations, of entire complex-analytic functions.
Do bees dance to mimic projections of flag manifolds? (And other math papers involving bees!)
Do subsets of R^2 and R^3 have torsion-free fundamental groups?
Do the integrals of a function over triangles determine F?
Do the integrals of a function over rectangles determine that function uniquely?
Do there exist groups with many zero cohomology groups?
Do two convex polyhedra intersect? (Use linear programming.)
Does polynomial interpolation of decreasing data yield a decreasing function? (no)
Does a given 0-1 vector have all "1"'s consecutive?
Does a particular polynomial have roots of unity among its roots?
Does a^3b^3=(ab)^3 mean G is abelian? (no)
Does integrability imply an easy asymptotic bound? (no)
Doing number theory in the ring of quaternions.
Drawing 7 regions on a torus, each touching all others.
Easy pattern to generate only prime numbers! (ha ha)
Effect of rotation on the graph of a function
Effective calculations of discriminant (etc) without having normal forms.
Efficient (recursive) methods of matrix multiplication (Strassen algorithm)
Efficient iterative computation of sqrt(x)
Eigenvalues of a circulant matrix.
Elementary statistical paradox.
Elementary proofs of the Borsuk-Ulam theorem
Elementary summary of marginal and conditional distributions
Elliptic curves must have an identity element. (Example: 3x^3+4y^3=5)
Elliptic curves with high rank? A summary
Elliptic curves with high rank? Another summary
Email with an author who had a program to "solve" Diophantine equations
Estimates of the product of the first N primes.
Estimating (x+0.5)!/x! with the Gamma function.
Evaluate sum of 1/(phi(n)sigma(n))
Evaluating an infinite sum from probability -- Sum( a^(d-N) (1-a)^N d! / (N-1)!(d-N)!, d &ge N )
Examination of (x+y+z)^3=xyz
Examination of x^2+y^3=z^6
Example of limit cycles for iterations of a map f: R \mapsto R
Example of companion matrices (to Chebyshev polynomials)
Example of a bad Newton's method problem
Example of computing the right affine change-of-variables
Example of elimination (implicitization of parameterized curve) using inexact coefficients.
Example of expressing a vector as a linear combination of two others.
Example of use of flag manifolds and counting stabilizers to enumerate orbits of subspaces
Examples of failure of local-to-global principle.
Examples of functions just barely integrable.
Examples of sequences of rationals which have different limits in different p-adic completions of Q.
Explicit Cebotarev density: which primes split well in algebraic number fields?
Expressing a rational as sums of Egyptian fractions (1/n)
Extending the Poncelet-Steiner ("no compass") theorem.
Extending the domain of the subfactorial function (hypergeometric functions)
Extension of continued fractions to complex numbers.
Extensions of that fact about 1729.
Extracting the axis of rotation from a 3x3 orthogonal matrix.
FAQ for sci.crypt.research
Factor n, n-1, n+1 where n is the order of the Monster finite simple group.
Factoring as a lifesaving activity!
Facts about primes which are "the mathematical equivalent of junk food".
Fast primality testing through 16 000 000.
Fast verification of primality from a certificate
Fastest modular multiplications (summary, lit review)
Fastest way to find nearest neighbors among finitely many points.
Find Fibonacci numbers divisible by p^2 ?
Find five integers with each xi*xj + 1 a square (open)
Finding generators for the modular group PSL(2,Z).
Finding integer points on curves (e.g. y^2=x^3+17). Mention of SIMATH.
Finding volumes of an n-dimensional polyhedron
Finding a fundamental domain for the action of a group of symmetries of a sphere
Finding a T1 space with no connected open subsets.
Finding a basis for the nullspace of an integer matrix with small entries.
Finding a manifold with boundary RP^n
Finding a polynomial whose roots are products of the roots of two other polynomials.
Finding all integral solutions to a homogeneous quadratic in 3 variables -- example.
Finding an orthogonal family of functions with vanishing derivatives at endpoints.
Finding nice (small) generators for the integers in a number field.
Finding solutions to a single multivariable homogeneous quadratic equation
Finding the furthest pair of points (in R^2): with citations
Finding the inverse Laplace transform
Finding the best rational approximation to a real number.
Fitting a curve with nonlinear parameters via GNUPLOT
Fitting data to a particular (exponential) family of curves, or, why not to have a mathematician try to do statistics.
Fluid mechanics and dimensional analysis applied to model planes!
For certain values of n one may construct a regular n-gon.
For comic relief you might want this UBASIC program I wrote to look for rational points on a certain elliptic curve with a square x coordinate. (Turns out to be none)
For triangles, you may wish to use Heron's formula
For variety, here's a sample of a trigonometric approach to determining the area of a pentagon.
Formal group examples related to Jacobian varieties
Formal groups and elliptic curves.
Formula for the equation of a curve formed by rotating the graph of a function
Formulae for the Lagrange inversion formula (Taylor series of inverse).
Frequencies of patterns in cointosses [Denis Constales]
From the sci.math FAQ: How can you chop up a ball and reassemble the parts (the Banach Tarski paradox, and related issues).
Fun examples of the "law of small numbers"
Functions in the convex hull of some functions of the form 1/(z-z_j) have roots in the convex hull of the z_j.
Functions with many negative integrals
Generalities on "finding the next term in this sequence" problems.
Generalities on convergence of series of matrices (and diagonalization)
Generalizations of Kuratowski embedding theorem to higher-dimensional complexes: citations to literature.
Generalizations of the Platonic solids to dimensions 4 and up.
Generalizing the ABC Conjecture for integers (ABC Theorem for polynomials) to n summands
Geometries with a "betweenness" relation.
Getting rational approximations using Farey sequences and continued fractions
Given 3 known points in 3-space, and the distances from each known point to an unknown point, how to determine the position of the unknown point?
Given a random ordering of k black balls and n-k white balls, what's the expected value for the length of the largest interval of black balls?
Given a Riemannian manifold with a group action on it (by isometries), is there an equivariant isometric embedding into some R^n? (yes if the manifold is compact, not necessarily if otherwise)
Given an analytic manifold, does it have an analytic embedding in some R^n? (yes)
Given many vectors in a vector space, how to find linear relations among few of them?
Given three angles as above and two points, where's the third point on the sphere making the appropriate angles? (shows the vector- and trig-calculations necessary).
Given two points in spherical coordinates, what's the angle between the rays joining them to the center of the sphere?
Good algorithm for computing minimal polynomial?
Good way to find N nearest neighbors among finitely many points.
Grassmannians are topological spaces which enumerate subspaces of a given dimension. I had a long exchange with a person seeking to randomly select subspaces in a uniform way.
Heath-Brown's theorem on primitive roots.
Help wanted on circle-through-three-points problem.
Here is a summary of the Cunningham project
Here is some information on which groups can be the groups of units for a field.
Here's a long spiel (with short punchline) on evaluating volumes of polyhedra.
Here's a theorem half-way to algebraic geometry or elliptic curves: if P is a quintic, there are 80 cubics y such that y^2-P is a perfect cube (Noam Elkies)
Homotopy groups of SO(3) (special orthogonal group).
How about fast primality checks for smallish numbers?
How can I efficiently multiply many-digit numbers?
How can one decide if a polynomial is irreducible (here, over F_p).
How can one define determinants in M_n(A) if A
How can we express a number as a sum of two squares (assuming that's possible!)
How can you check a corner for concavity.
How can you decide whether two ellipses intersect? (long use of analytic geometry and then symbolic algebra).
How common are numbers expressible as a sum of 2 squares?
How different is the real-analytic category from the C-infty category (for real manifolds)?
How do Lie groups enter the analysis of a differential equation?
How do calculators compute sin(x) (etc.)?
How do electrons distribute themselves?
How do they schedule elevators?
How do we define higher-order derivatives of multivariate functions?
How do you parameterize a curve -- i.e. how do you know that a quadratic curve in 2 variables with a rational point is in one-to-one correspondence with the rationals?
How do you compute the intersection of lines in a plane?
How do you render hidden-line objects in 3D? (BSP trees)
How do you compute the area enclosed by a polygon?
How do you decide if a point is interior
How do you find its center of mass of a polygon (program included)
How do you find the centroid of a polygon (pointer)
How does Mathematica determine primality?
How does a Taylor series behave on the circle of convergence?
How does one factor polynomials in 1 variable?
How does one factor polynomials in 1 variable? -- take 2
How hard to compute expressions of N as a sum of 4 squares?
How is geometry different in four-dimensional space?
How many normals to a surface meet at a point?
How many quadratic residues in a row mod p? (Lit review)
How many groups of order p^n? [Derek Holt]
How many tours on a hypercube? (No answer; it's the same as in the Cayley diagram of (Z/2Z)^n.)
How many homeomorphisms of an interval are there, having order 2, that is, having f(f(x))=x ?
How many finite topologies are there?
How many shuffles before a deck of cards is "random"?
How many colors needed to color a planar graph if opposite corners are considered touching (as at Four Corners USA)
How many colors to color the plane if different colors are required for points a unit distance apart?
How many integers less than n are the sum of two squares?
How many isomorphism classes of vector bundles are on the spheres?
How many lines pass through four given lines in R^3 (two; generalize?) This is Enumerative Geometry (use the Schubert calculus: 14N10).
How many solutions to x^3=2 mod p? (Class field theory) [Noam Elkies]
How many triangles are there on a Geoboard (tm)?
How many triangles with rational sides and a given rational area?
How many ways to group p people into n teams?
How many ways to select at most one item in each subset... (Generic counting problem)
How many ways to write a number as a sum of 3 squares (Citation)
How might Newton's method fail?
How to fit the best circle/ellipse to some points in the plane? (Summary, pointers, citations, code)
How to make irreducible polynomials over a finite field, of large degree
How to reduce a quartic to simpler form (Mobius or Tschirnhaus transformation)
How to generate a random variable with a given pdf
How to list all subsets of a given cardinality of a given set.
How to determine group size from the number of conjugacy classes.
How to compute eigenvectors (after the eigenvalues) for a 3x3 matrix.
How to plot circular motion on a display?
How to compute (a primitive element for) the splitting field of a polynomial? (Maple example.)
How to compute the volume of a simplex in R^n in terms of its sides.
How to compute the volume of a polyhedron? Pointers, citations, summary
How to compute the area of a collection of circles?
How to compute the convex hull of some points in the plane?
How to decide if you're inside a polygon? (pointer, citation)
How to determine whether two ellipses intersect (or: solving quartics graphically)
How to find algebraic relations approximately satisfied by real numbers. (including: LLL routine.)
How to find a good ellipse to match a cluster of points in the plane?
How to find a monotone function to fit data?
How to find the closest pair of points on two circles in R^3?
How to find the center of an ellipse with Euclidean tools? (Includes Newton's theorem on secants.)
How to fit a curve y(x)=a+b*sin(c*x+d) to data: ODRPACK
How to generate numbers with a Gaussian distribution (not a uniform one)?
How to randomly generate points on an ellipse (ellipsoid)?
How to tell if a family of polynomials is linearly independent over Q.
How to decide if you're inside a polyhedron?
Humdrum instance of Newton's method.
A general pointer to web site discussing Groebner bases etc.
Are there are other rational functions which could be used to make groups. This is essentially the study of formal groups.
"Eliptic", a public domain implementation of elliptic curves public key cryptography. [not tested]
Pointers to hexaflexagons
A couple of questions from physicists regarding the 7-dimensional sphere and other 7-manifolds.
If S(a)=a^2 and T(a)=a+1, are there two words in S and T of equal length such that w1(a)=w2(a) for some integer a? No.
If X is a fundamental domain for the usual action of Z x Z on the plane, how do we determine in which translate of X a point of the plane lies?
If all points in a space have homeomorphic neighborhoods, is the space homogeneous?
If shown one real number out of two, how can you guess whether it's the larger? (heh heh)
If the 2-sphere is written as the union of two compact pieces K and L having finitely many components, then (K intersect L) has finitely many components.
If the Galois group is solvable, one can express the roots with the standard operations. (Citations)
If a root of a cubic are rational, must sqrt(disc) be rational, too? (no)
If you want to really build these things, here are a couple of construction tips
Illustration of Galois theory as it pertains to certain values of trig functions.
Illustration of a modelling project: what to consider in the cooling of a cup of water.
Illustration of search of GAMS numerical software library.
Improvements to Newton's method
Independence of the Axiom of Foundation
Instructions for making a kaleidocycle (flexible polyhedron)
Interesting example of Galois groups of certain sextics.
Interpolation for functions of several variables.
Introduction to Dessins des enfants
Irreducibility of trinomials of the form x^a + A x^b + B
Is a a primitive root for infinitely many primes?
Is subgroup membership (e.g.) effectively computable? (yes)
Is completeness a homeomorphic invariant? (no)
Is a stably rational variety actually rational? No.
Is a cover of a cover again a cover? (no)
Is angular momentum really what keeps a bicycle up?
Is the period length for the continued fraction expansion of, say, sqrt(x^2-18) bounded independent of x? (no)
Is the Mandelbrot set measurable?
Is the smooth image of a manifold still a manifold? (no)
Is there a polyhedral torus made of equilateral triangles?
Is there a complex structure on S^6?
Is there a cross-product in R^n?
Is there a closed-form "solution" for an elliptic integral? (no)
Is there a mathematical model of color?
Is there an algebraic characterization of the real field?
Is there an easy way to calculate primitive roots?
Is there an infinite group with only finitely many conjugacy classes? (yes)
Isn't mortality rate the reciprocal of lifespan? (not quite)
It turns out the constructions we are accustomed to can be carried out using only straightedge or only compass
Iterate this procedure: multiply the nonzero digits of n together to get n'; repeat until one digit remains. What digit is it?
John Baez describes framed embeddings
Kirkman's schoolgirl problem (block design)
Kuratwoski-like conditions for embeddability of a 2-simplex into R^3?
Lame UBASIC code to give a quick distribution of points
List of Hilbert's problem.
List of huge-precision programs and
List of all groups of small order and an appeal to discount Cayley tables for their enumeration.
Listing all (restricted) partitions of n
Listing of open questions with cash rewards offered by Erdos.
Lit review and pointer for equations x^n+y^n=2*z^n [Ken Ribet]
Literature review on regular (completely-tied) tournaments; how rare are they?
Long summary of modularity and other terms used. (It's a good intro to the theory).
Long example showing how to use APECS to analyze an elliptic curve.
Looking for points on curves of the form y^2=x(x^2-d^2) (Tunnel's theorem).
Mail from Tim Chow on coloring planar graphs.
Making the space of continuous functions from R to R into a metric space
Making the set of irrational numbers into a complete metric space.
Many answers -- take your pick -- to the incessantly-asked question, "What's the (great-circle) distance between two points on a sphere (such as Earth) given their latitude and longitude (spherical coordinates): Clairaut's formula.
Maple V release 3 cannot factor 3511^2 !
Maple code to do QR decomposition of a matrix.
Matching data to a curve y=A sin(x+B)
Matrix inversion by Monte-Carlo techniques(!): citations.
Maximizing a sum of sines (of different periods) (really a question of approximating a number by rationals).
Meanwhile, I did some reading on the problem. Since I was at the time teaching a course on elliptic curves, when I found the relation of this problem to that topic I subjected my students to it. Here are some course notes clarifying details of the relationship.
Method of doing arithmetic on your fingers.
Might Euler's constant gamma be rational? (unlikely)
Might Peano curves be good for image compression?
Model the fall of an elevator.
Modeling swinging cables.
Modeling the passage of light through
More about de Bruijn sequences (citations etc)
More general curves and surfaces too, in this case ruled surfaces.
More on the historical introduction to elliptic curves.
More references on magic squares.
Multidimensional analogues of continued fractions (summary and bibliography)
Must a p-group have a non-trivial center? (not if infinite)
My answer to "What does it mean for a curve to be modular?".
Names of several people who work in this area
Near misses of the Fermat equation.
Need a parameterization of the set of orthogonal matrices
Need bounded functions with reciprocal symmetry, that is, f(x)+f(1/x)=1.
News posts from June 1993 when Wiles first announced his proof.
Nielsen fixed point theory for maps on a surface
Non-sequence-based completion of a metric space [Ron Bruck]
Normal subgroups of infinite symmetric groups including the corresponding alternating groups.
Not really math but I was intrigued by the measurement of color ("what wavelength is brown?":-) ). Citations and pointers.
Note that any variety can be described by quadratics alone.
Notice of software for computing zeta
Notice that Diamond has generalized Wiles' work on elliptic curves.
Number of regions formed joining chords of equidistant points on a circle.
Number of ways N ordered whole numbers, each no greater than N, add up to N^2-N. (A classic counting problem)
Numerically stable formula for distances on a sphere
Numerous summaries of the history of the problem and Wiles' approach to it are available on the net. I did save a copy of one such expository talk.
Obtaining the Smith normal form for matrices (or modules) over a PID.
Old (1970s) citations for triangularizability of manifolds
One makes heavy use of symmetric polynomials. Here's an application to solving a system of (trigonometric) equations
One may ask about quadratic extensions of the integers -- which are Euclidean, factorial, or PIDs.
Ongoing computer search for Mersenne primes (pointers and Call For Participation).
Open question: can every convex polyhedron be cut along edges, then laid flat without self-overlap?
Origin and scope of the instability associated with the Hilbert matrix
Other open questions worth money.
Other variations on the Poncelet-Steiner ("no compass") theorem.
Overview of dynamical systems (p(x)=kx(1-x), Feigenbaum)
Overview of options and pitfalls of (1-dimensional) interpolation.
PDF for taxicab distances between two points in a rectangle.
Pairing off ideal classes with classes of quadratic forms
Parameterization of Klein bottle, Mobius strip
Parameterizing the solutions to x^3+y^3+z^3+w^3=0
Part 1 of RSA, Inc.'s FAQ.
Peculiar set of equations equivalent to : solve x^2=-x mod n
People have looked for curves with large Sha (if you have to ask what Sha is, you don't want to know)
Perhaps we return the favor to analysis by doing analysis geometrically: a study of ellipses to decide whether some inequalities imply another.
Placing points uniformly around other shapes
Plenty of algorithms to compute pi.
Pointer and citation to solving quadratic equations in the ring of quaternions.
Pointer for trapdoor encryption procedures
Pointer for 2D interpolation.
Pointer for C++ package Range -- variable precision 'range arithmetic'
Pointer for the list of sporadic finite simple groups. (Small ones too)
Pointer to (lecture notes and) algorithms for graphs (diameter, etc)
Pointer to FAQ file for comp.graphics.algorithms
Pointer to Quickhull (convex hull in R^N).
Pointer to Game Theory Resources page
Pointer to global optimization code
Pointer to codes for optimization and linear programming.
Pointer to Richard Pinch survey of primality proving techniques.
Pointer to Busy Beaver problem.
Pointer to lecture notes on factoring and primality testing
Pointer to HOMPACK (numerically solve systems of polynomial equations).
Pointer to KALEIDO (program for regular polyhedra)
Pointer to text on Matrix Algorithms [G B "Pete" Stewart]
Pointer to Computational Algebra archives
Pointer to free Large Integer Package.
Pointer to Indiana University's "Knowledge Base" (Computer FAQs)
Pointer to Mathematica algorithms and packages.
Pointer to Mathsource (Mathematica information from Wolfram)
Pointer to Macsyma source.
Pointer to Minitab (statistical software)
Pointer to MuPad.
Pointer to Numerical Recipes site, and citation to alternative text.
Pointer to shareware versions of Numerical Recipes codes.
Pointer to optimization server (NEOS).
Pointer to symbolic-algebra information. See also math.berkeley.edu:/pub/Symbolic_Soft/Available_Systems
Pointer to graph theory software.
Pointer to Dummit's article on solving solvable quintics.
Pointer to FAQ on Binary Space Partitioning (BSP) Trees
Pointer to a Linear Programming FAQ
Pointer to a FAQ of the sci.nonlinear newsgroup (inc dynamical systems)
Pointer to a website simulating that Monty Hall paradox!
Pointer to comprehensive Operations Research page
Pointer to gallery of Archimedean solids
Pointer to implementations of bounding sphere calculations in R^n.
Pointer to list of recommended optimization software
Pointer to numerical data on 4-dimensional polytopes
Pointer to pointer(!) on simulated annealing (references, code) See also http://www.cs.cmu.edu/afs/cs/project/ai-repository/ai/areas/anneal/0.html
Pointer to sample Finite Element code.
Pointer to software for n-dimensional geometric modelling
Pointer to software for combinatorial optimization (shortest path, etc.)
Pointer to software for interpolating over a sphere
Pointer to software for the Delauney triangulation for a set of points in the plane.
Pointer to software: modelling plant growth.
Pointer to the excellent sequence server ("what sequence begins as follows...?")
Pointer, citations for global optimization
Pointer: tournament scheduling program.
Pointer: primality proving program.
Pointer: virtual polyhedra (pretty pictures).
Pointer: determining the intersection of a ray and a torus.
Pointer: free Large Integer Package
Pointer: using elliptic functions to solve the quintic
Pointers for General numerical analysis software
Pointers for connections to Logic journals.
Pointers to Delaunay triangulation codes
Pointers to TSP code
Pointers to FFT code and descriptions
Pointers to a factor by mail project.
Pointers to the Buffon needle problem and experimental evaluation of Pi
Pointers to understanding the construction of exotic differentiable structures on R^4.
Pointers to web sites for classical logical fallacies
Possibilities for analogue computers (computing by use of predictable physical systems)
Possible answers to, "what curve is the seam on a baseball"?
Post asking for alternative ways to divide polynomials (divide P1/P2 if you know the roots of P2).
Practical application of set theory axioms :-)
Program and literature review (both long) for g-holed tori with few vertices
Proof of Bertrand's postulate: there is always a prime between n and 2n.
Proof of the quadratic reciprocity theorem.
Proofs of Fermat's Last Theorem for polynomial rings (using Mason's ABC theorem or Wronskians)
Pros and cons of variant conjugate gradient methods
Proving the analogue of the Pythagorean theorem in higher dimensions.
Putting a recursive sequence into closed form (with and without Maple)
Putting an elliptic into Weierstrass canonical form
Quality of approximations of an irrational by rationals.
Questions related to an Erdos conjecture: that 4/n = 1/x + 1/y + 1/z has a solution for every natural number n.
Quick note to myself reminding myself how to execute LLL algorithm in Maple.
Quickest way to find 10 largest among a set of 100 numbers (say)?
RSA129 was a certain 129-digit number containing a coded message (as a test). Here is a call for participants, which resulted in a completed factorization.
Radius of inscribed circle in a triangle.
Randomly generating numbers to fit a specified distribution.
Randomly generating numbers to fit a specified distribution.
Randomly generating numbers to fit a specified distribution.
Rapid tests for primality (e.g. Miller-Rabin)
Read about Polyhedral versions of 1- and 2-holed tori which have a small number of vertices
Recent progress on solving polynomial systems, and multidimensional resultants; lit review. [J. Maurice Rojas]
Recent research has looked for elliptic curves over Q with high rank.
Recognizing the reciprocal function from the equation xy f(x+y) [f(x)+f(y)] = 1.
Recollections of the logic "game" WFF'N'PROOF
Reference for asymptotic expansions of integrals.
References on Euler's formula for polyhedra.
References on non-standard logics.
References on applications of the AUTOMATH system (automatic proof checker)
References to computational knot theory
Relating volumes of simplices to vertices, edges, or lengths.
Relating the complex trigonometric and exponential functions.
Relationship between Laplace and Fourier transforms?
Representations of crystallographic groups.
Representing a rotation in R^3 using rotations around only two axes
Reverse the digits and add; get to a palindrome? (open)
Review of terms for classifying first-order ODEs.
Review of the "long line" (the imposter manifold), along with some questions.
Right triangles with integer area and integer (or rational) sides (includes the "congruent number problem")
Sample algorithm to compute many digits of pi quickly.
Sample from economics: model distribution of incomes over time.
Sample modelling of geometric problem: when do truncated cones intersect?
Sample problem in representations of finite groups solved with techniques of semi-simple algebras.
Seeking sum(int(ax+b), x=1..n)
Seeking integral points on the intersections of 3 quadratics in P^4.
Seeking solutions to the set of congruences ab=c mod (a+b), bc=a mod (b+c), ca=b mod (c+a)
Show there is a prime of the form k*2^n + 1 for every odd k less than 78557 (none for k=78557).
Simple illustration: how do topological vector spaces arise in basic calculus questions?
So which manifolds are triangularizable?
Software announcement: Effective Algebraic Topology Program [Francis Sergeraert]
Software pointer: NTL: a C++ library for bignums and algebra over Z and finite fields [Victor Shoup]
Software to compute all zeros of an analytic function in a rectangle.
Solutions to x^5+y^5+z^5=w^5? (open)
Solutions to A x^p + B y^q = C z^r? (none with A=B=C=1 if p,q,r greater than 2)
Solutions to x^3 + y^3 = z^2
Solutions to generalized Fermat equation x^a+y^b=z^c.
Solve a^6 + 5(a^4)b + 6(a^2)(b^2) + b^3 = 1 in integers please.
Solve A_n = (n-1)A_{n-1} + (n-2)A_{n-2} in a closed form.
Solve this first order ODE: y'=a/y+by/x+c/sqrt(x).
Solving x^2+xy+y^2=z^2 to make nice calculus problems.
Solving x^2+y^2+1=0 mod p efficiently
Solving quadratic equations over Q and Z. (that is, studying rational conic curves).
Solving Pell's equation x^2+dy^2=N ( esp: N \not= 1 ).
Solving x^2 + y^2 = u^4, x+y = v^2 in integers.
Solving f'=f o f (or, "What to Ask When Asking About Differential Equations")
Solving {a^2+b^2=square, a^2+(2b)^2=square} by infinite descent.
Some simple computations in K-theory.
Some pointers to factorization code
Some puzzles testing linking and homotopy intuition.
Some background on the convex-hull problem (finding the points which form the "outside" of a set of points in space).
Some calculus: how to locate two circles so that the area of the intersection halves the original area?
Some comments about the equivalents of the platonic solids in dimensions greater than three.
Some comments on the constructiveness of Wiles' proof.
Some discussions about the Penrose tilings of the plane (aperiodic tilings with as few as 2 distinct shapes).
Some examples of non-Borel (measurable) sets.
Some exposure to conics (quadratics) and elliptic curves (cubics) suggests there ought to a "canonical form" for varieties in general. There isn't, but you might want to think about why.
Some further comments about diffeomorphisms of spheres and balls is also available.
Some hints on the Kuratowski "14" problem (How many sets can you make from one set A using complement and closure?)
Some information about the vertices of the dodecahedron
Some information about the edges of the dodecahedron.
Some posts using group theory to analyze symmetry in 3D (the "space groups" -- useful for classifying regular solids too.)
Some questions about the regular nonagon (nine-sided polygon).
Some references on primality proving
Some things can still be done even with a short straightedge.
Some ways of phrasing the Taniyama conjecture, an important case of which was solved by Wiles.
Sometimes what you need is really linear algebra -- in this case, describing rotations in 3D.
Source code for Hough transform
Spaces which are homeomorphic but not diffeomorphic (etc)
Square roots of the exponential function, that is, f(f(x))=exp(x).
Squares which, in base 10, are written with only two or three distinct digits.
Statement of a couple of metrization theorems
String of 1000 digits which includes all numbers 1..1000 as substrings. (Why do we do these things?)
Strong vs. weak Law of Large Numbers.
Subdivisions of the sphere corresponding to the actions of the dihedral group.
Suggested by an arrangement of numbers in a basketball tournament: solve ab = c + d, cd = a + b in integers.
Sum of two fifth powers a square?
Summaries of the Picard theorems.
Summary of information about generators and relations of the Rubik group.
Summary of Waring's problem [Kevin Brown]
Summary of Mazur's theorem on the possible torsion subgroups of E(Q)
Summary of transcendental numbers; proof of transcendence of pi, e.
Summary of methods for generating uniformly-distributed random points on a sphere [Dave Seaman].
Summary of status of Waring's problem
Summary: there are polynomial-time algorithms giving near-optimal results in the Traveling Salesman Problem.
Sums of squares of real polynomials (Hilbert's 17th problem)
Suppose L is a set in space such that all lines through 2 points in L passes through a third. Then L is collinear.
Surely a FAQ: How can you tell whether two line segments intersect?
Survey article on homotopy groups of spheres [John Baez]
System of DEs which model highly oscillatory motion on a sphere.
Table of Contents and introduction to the cryptography FAQ.
Table of some known Ramsey numbers
Tables, algorithms, citations on pi(N), the the number of primes up to N
Testing for tautologies "efficiently". (Not really possible in general).
Testing irreducibility of polynomials over finite fields - Berlekamp's algorithm (with citations).
Testing polynomials (mod p) for irreducibility.
That four-dimensional polytope with no three-dimensional analogue.
The Burnside problem: does G torsion imply G finite? (No)
The Mascheroni (no straightedge) theorem.
The soft underbelly of symbolic computation exposed!
The multigrades problem (find sets of integers whose sums are equal, sums of squares, sums of cubes,...)
The Perrin sequence (3,0,2,..., a_(n+1)=a_(n-1)+a_(n-2).)
The topological space of integers (basis=arithmetic progressions)
The rational box: still open
The Word Problem: can we decide if a specific presentation is of the trivial group? (no) [Derek Holt]
The Catalan numbers -- some recurrence relations and other formulas.
The Fermat-Torricelli point in a triangle
The Hewitt-Savage 0-1 Law of random walks on the real line.
The subset-sum problem: solving exactly (hard) or approximately (easy)
The computational complexity of knot and link problems
The darnedest series arise in applied problems. Here was a request to sum: Sum[ v^(i-1)*Exp(-Lv)*((v-1)^(j-i)*L^j)/j!(i+1) ,1 \le i \le j \lt \infinity]
The Euler-Maclaurin formula, and other suggestions for computing partial sums of Sum( f(n) ).
The Levin transform (for speeding up convergence of infinite sums), with code fragment.
The long line (non-paracompact)
The Moebius inversion function on posets
The group structure of curves over Z/pZ
The factorization of RSA129.
The crystallographic groups.
The plate trick -- a physical manifestations of a path of order 2 in pi_1(SO*(3)).
The Monster simple group, modular forms, and applications to physics.
The Tarry-Escott multigrades problem: given a positive integer n, find two sets of integers a_1, ..., a_r and b_1, ..., b_r, with r as small as possible, such that sum (a_j)^k = sum (b_j)^k for k = 1, 2, ..., n. Conjecture: r=n+1 for all n.
The Times puzzle: find rational solutions to x^3+y^3=6. (an elliptic curve)
The "sum" of two closed sets need not be closed.
The rhombic dodecahedron, use as space filler.
The original posting and my (pretty predictable) response. In this discussion the problem was purely geometrical.
The Indiana legislature's attempt to legislate the value of Pi.
The collatz (3x+1 / Hailstone) problem is "just" a Turing machine halting problem and so may be insoluble
The formula used to determine wind chill.
The four regular nonconvex polyhedra (Kepler-Poinsot)
The irreducible rational polynomials are dense in Q[X].
The joys of the number 239
The n-th prime is roughly e times the geometric mean of the ones before it.
The part of the FAQ discussing the RSA encryption scheme (the one related to factoring).
The polynomial whose only positive values are all the primes.
The set of all fractions m!/2^n is a dense subset of the real line.
The sphere-volume page from the sci.math FAQ.
The sum of the squares of the binomial coefficients
The volume of a d-dimensional sphere of radius s is pi^(d/2)/(d/2)!s^d. You might want that spelled out a little bit.
There are also deterministic (non-probabilistic) primality tests.
There is a classic but complicated formula for describing the roots of a general cubic equation. This has unusual behaviour when all three roots are real -- the so-called casus irreducibilis
Thinking about the definition of a manifold
To what extent do homotopy groups (say) determine a topological space?
Topological proof of Van der Waerden's theorem (any finite partition of the set of natural numbers leaves an arithmetic progression in at least one subset).
Topological proof of the infinitude of primes.
Transcription of planar graph "requiring 5 colors" (joked Martin Gardner)
Triangulating polygons in R^3 -- when is it even possible? (problems if knotted)
Triangulating simple, nonconvex polygons
Trisection is relevant if you wish to construct a regular nonagon (nine-sided polygon).
True or false: the reals are the only metrically complete ordered field?
Two pointers to graph-coloring sites.
Two polygons of equal area may be decomposed into congruent triangles.
Typical (but convoluted) counting problem.
Typical example (from physics) of estimating rate of growth of a series.
URLS for solution of Rubik's cube.
Under what circumstances do the Pade approximations converge to the original function?
Under what conditions do all roots have magnitude 1?
Under what conditions is an open subset of R^n contractible?
Under what conditions is there in G a subgroup isomorphic to the quotient group G/N ?
Under what conditions will a representation of a subgroup extend to a larger group? (Applications to Rubik's cube).
Under what conditions will an automorphism of a subgroup extend to a larger group?
Uniqueness of K(G,1)'s; two K(G,1)'s with the same homology.
Unusual consequence of unique factorization
Use Newton's method for sets of functions of several variables? (Yes)
Use of Sturm sequences to determine if two ellipses intersect (without actually finding intersections!)
Use of elliptic curves for factoring.
Use of generalized continued fractions to simultaneously approximate several numbers by small rationals
Use of permutation groups to determine a method for transposing nonsquare matrices in place.
Using (extensions of) Fermat's Little Theorem to generate large prime numbers (and prove they're prime)
Using multidimensional scaling to approximately embed metric data sets into the plane.
Using Tietze's theorem to extend statements of fixed-point theorems.
Using projective geometry to perform a construction meeting incidence conditions.
Using space-filling curves to compress images.
Using Sturm sequences to count real roots [in an interval]
Using alternative notions of best to decide on point placement.
Using Groebner bases to find closed-form solutions to multivariate recurrence relations and difference equations.
Using a generalized spiral to distribute points on spheres.
Using a quadratix to multisect an angle.
Using integrals to show that pi isn't 22/7.
Using the Cholesky factorization of a matrix to find an isometric embedding of a finite set of points.
Using the Intermediate Value Theorem to disallow functions with f(f(f(x)))=x
Using the saddle point method to estimate an alternating sum.
Variants of Kuratowski's theorem for positive genus: graphs which prevent embeddings to surface of genus g.
Various methods of nonlinear optimization: pointer to software, citation.
We ask (not answer) the question, "for which quadratic extensions of Q does the curve x^3+y^3=z^3 have positive rank?"
What Model Theory is not!
What about algebras over the algebraic closure of Q?
What are "holes" and what do homotopy and homology measure?
What are Formalism and Constructivism? [Robert Israel]
What are spectral sequences? [Tim Chow]
What are characteristic classes (elements of cohomology rings)?
What are Steiner systems?
What are the "cross-products" in dimension n?
What are the (other) roots of p(X)=0 in the ring M_n(F) where p is the characteristic polynomial of a matrix A?
What are the multiplicative scalar functions on matrices? (determinants...)
What are the Chebyshev polynomials and what are they good for?
What are the endomorphisms of the matrix ring M_n(R)?
What are the expected run-times of the principal factoring routines?
What are the Legendre polynomials?
What are the finite subgroups of the group of rotations in R^n?
What are the conditions on the coefficients of a polynomial for all roots to be rational?
What are the possible orders of elements in symmetric groups? -- literature review
What are the possible semigroup structures on the real line?
What are the possible orders of elements in symmetric groups? -- discussion
What are the quadratic fields with small class numbers? (literature citations)
What are the seven 1-dimensional symmetry groups?
What are these classes of problems: P, NP, NP-complete?
What can a quadratic surface in R^3 look like?
What can an algebraic surface in R^3 look like?
What can be constructed if we assume a trisector?
What closed curve in R^3 has the smallest convex hull?
What do we learn from the law of large numbers?
What does Goedel's Incompleteness Theorem say? [Theodore Hwa]
What does "NP-hard" mean?
What does factorization have to do with cryptography? (elementary).
What does Tychonoff's Theorem say? (The product of compact spaces is compact.)
What does it mean to say one set is more infinite than another?
What does it mean to select a random triangle? [Terry Moore]
What does the Invariance of Domain theorem say and how do we use it?
What functions have antiderivatives which are elementary functions ? Citations and long article by Matthew Wiener. (Includes topics in symbolic integration.) Frequently-mentioned integrands with no elementary antiderivative include exp(-x^2), sin(x)/x, x^x, sqrt(1-x^4), and many variants.
What happens when you trisect the sides of a triangle and look at the intersection of those lines? ("Marion's theorem")
What is (are) topological dimension(s)?
What is Differential Geometry; how does it differ from differential topology? May manifolds always be embedded into Euclidean space?
What is Russell's paradox (en francais)
What is Waring's problem (write each N as a sum of powers)
What is Arrow's Impossibility Theorem (there is no fair voting system)
What is Bezout's theorem (and who was Bezout?)
What is simulated annealing? (for optimization). Includes pointer to software.
What is curvature? (NB - I had suggested: the product of the eigenvalues of the local parameterization. Or something.)
What is a regular prime?
What is a Tschirnhaus transformation of a polynomial?
What is a Voronoi diagram and what is it good for?
What is a free module and what are some modules that aren't free?
What is a stably trivial fibre bundle?
What is a loop? (sort of a non-associative group)
What is an order (in ring theory)
What is the Brouwer Fixed-Point Theorem?
What is the Elliptic Curve Primality Prover and how do I get it?
What is the Euclidean algorithm for computing GCDs? [Richard Pinch]
What is the Hausdorff metric on the set of (closed) subsets of a space?
What is the Lefschetz fixed-point theorem?
What is the Poincare' sphere? The Poincare conjecture?
What is the ABC theorem for polynomial rings (or, the ABC conjecture for the ring of integers).
What is the cohomology of groups and how is it used to enumerate group extensions?
What is the Generalized Riemann Hypothesis?
What is the projective dimension of a Z[G]-module?
What is the gradient (in differential topology)?
What is the ABC conjecture?
What is the Langlands Program?
What is the compact-open topology good for?
What is the discriminant? (It's used to find multiple roots)
What is the group of units in the ring Z/mZ ? (When is it cyclic?)
What is the relation between the angles as shown above and the angles at the vertices of the resulting spherical triangle?
What is the basic idea behind splines?
What is the correct sign in the congruence ((p-1)/2)! = +-1 mod p? (cf. Wilson's theorem). Answer: depends on class number formula.
What is the representation of a group induced by a representation of a subgroup?
What is the use of paracompactness (e.g. for metrization)?
What kind of functions satisfy an anti-Lipshitz condition?
What numbers are the sum of three squares? In how many ways?
What path do flying objects really follow? (artillery and fungus spores!)
What path does a light ray take?
What really happens when you flip a coin -- couldn't it land on its edge?
What shape is a soccer ball?
What should we take for an infinite-dimensional sphere, and is it contractible?
What's a Schauder basis? Hamel basis? [Robert Israel]
What's new with the Riemann Hypothesis?
What's the Edge of the Wedge Theorem?
What's the best route from London to Edinburgh? (It's not the Traveling Salesman; it is polynomial-time.)
What's the distance between a point and a parabola? (This is essentially an elimination-theoretic description of an envelope of the parabola.)
What's the real path of a billiard ball?
What's the connection between unique factorization and the preponderance of primes of the form x^2-x+41?
What's the difference between covariant and contravariant vectors/functors?
What's the formula for windchill?
What's the homology of this bad space ? It depends on the kind of homology you use.
When every group of order n is abelian
When can cos(p*Pi/q) be expressed with real radicals?
When can a 2-variable quadratic equation be solved in integers?
When did it start to snow?
When does the number of primes less than x first exceed Li(x)? (Skewes' number)
When is right to do polynomial interpolation?
When is the sum of consecutive cubes again a cube? (Solve x*y*(x^2+y^2-1)=z^3)
Where to find group tables for all the groups with order up to N .
Where to get Mathematica information on the Web?
Where to set teeth on an elliptical gear.
Which points on a box are furthest apart (geodesic distance) -- it's not opposite corners!
Which triangular numbers are squares? (example of Pell's equation).
Which cyclotomic fields are unique factorization domains? (What is their class number?)
Which integers are the sum of three integer cubes? (unknown, e.g. n=30)
Which integers may be written as the sum of two rational cubes?
Which is larger: a^(1/3) or b^(1/3)+c^(1/3) (remarkable close calls are solutions to |(a-b-c)^3-27abc|=1)
Who says you can't get rich solving a system of ODEs?
Why cross-products exist only in dimensions 3 and 7
Why Groebner bases grow so nastily; any way around that?
Why are eigenvalues of Hermitian matrices real?
Why are there so many primes of the form n^2+n+41? (Citations)
Why do the last few digits of a^n cycle?
Why is exp(pi*sqrt(163)) so close to an integer?
Why there are no 3-dimensional real fields
Will X and its quotient space X/A have the same fundmental group under nice circumstances?
Will just a few numerical invariants characterize a group up to isomorphism? (no)
WordPerfect encryption can be easily defeated -- pointer to decryption program wpcrack (and other miscellaneous tools for cryptanalysis).
Wordlength in the Rubik group, with a URL.
Worked-out analysis of one equation worth money!
Yet another open Erdos problem.
You can't solve a 2-variable (1st order) partial differential equation unless it's closed.
[Chris Stover] - Pointer to some literature on obstruction theory.
[Daniel Henry Gottlieb] Use vector fields to prove all the classical theorems! (Gauss-Bonnet, Jordan Curve, etc.)
[Tim Chow] are there subsets of R^2 with interesting homology?
[various authors] - What is the fundamental group of the Hawaiian earring?
Online textbook in Mathematical Logic [Stefan Bilaniuk]
Some data for modelling the heating of a house
Announcement: Table of number fields
I also mentioned map-making in the FAQ. Here are some pointers to map-making tools (esp. the Mercator projections)
Pointer to Mesa, a 3-D graphics library (similar to OpenGL).
So Here's the Sphere FAQ.
Summary, pointer for Mathematical application called Optica
This is the division algebra FAQ itself.
:-)
Products of normal spaces which are not normal.
Adams method for solving ODEs (a predictor-corrector method).
Bairstow's method for finding the roots of a polynomial.
Clustering algorithms.
Conformal embeddings of Riemann surfaces into R^3.
Conformal mappings to the interior of a curve or region between two curves.
Dekker's algorithm of finding zeros of functions.
DeRham's theorem links differential forms with the underlying topology of a space.
Faa di Bruno's formula for the iterated derivatives of a composite f o g .
Gear's method for solving ODEs.
Hartogs' lemma (on removable singularities) in the theory of several complex variables.
Jensen's inequality, with an application.
Mean, median and mode viewed as minimizing total variation.
Auxiliary files for the MSC-Biographies project (mostly 20th century mathematicians by name)
Random walks on the sphere.
Random walks on the plane and in R^n.
What bearing needed, knowing starting and ending locations?
A Bayes problem: if two medical tests show negative, what the probability I'm really sick?
A little background on the Global Positioning Systems
A polynomial in two variables with two local maxima, no minima or saddle points: two mountains without a valley.
Aerodynamic study of the flights of insects. (They said it couldn't be done...)
Another go: how to describe great circles with latitude and longitude.
Applications of kernel functions to the solutions of PDEs.
Applications of Stieltjes integrals.
Basic algorithm for computing trigonometric functions with CORDIC algorithms.
Basic method of fitting data to a polynomial (uni- or multivariate) of fixed degree.
Bellows theorem: flexible polyhedra maintain their volume
Calculating volumes of intersections of polyhedra.
Calculating logarithms of the gamma function (and factorials).
Calculation of the expected number of pin-line crossings in the Buffon needle crossing problem.
Characterizing polynomials by the pointwise vanishing of a high-order derivative
Citation for software for determining positions on a non-spherical earth
Computing the volume element on GL_n(R).
Consequences of the Axiom of Choice include the Banach-Tarski paradox.
Coordinates of an icosahedron and "rhombicubeoctahedron".
Description of the truncated octahedron.
Distances and direction on a spherical earth: tutorial and QBASIC program.
Example of oscillation in Newton's method.
Example of a symbolic solution to a PDE with Maple.
Extending the Frenet vectors and formulas for curves in R^n.
Families of polynomials which commute under composition must be either powers or the Chebyshev polynomials.
Functions continuous precisely at the rationals? No (by the Baire Category Theorem).
Gauss's Theorem Egregium: the intrinsic nature of curvature.
How closely do asymptotics of the coefficients of a system of differential equation mirror the asymptotics of the solution? (not necessarily closely).
How do statisticians stay interested? :-)
How long will an oldest living person keep that title?
How many cylinders pass through five given points?
How to handle multiple-objective optimization problems?
Illustration of a space-filling curve.
Impacts of nonlinear dynamics in the financial markets.
Integral definitions of the Gamma function.
Code and references for Hankel transforms.
Just what is a polytope and how does it differ from a polyhedron? (opinions vary!)
Measures of information content of a message.
Measuring the randomness of card shuffling with group theory.
More careful fitting of an ellipse to some data points.
Numerical calculation of "special functions" (trig/exp/log/sqrt...)
One possible mathematical Who's Who: the set of people whose names are part of the Mathematics Subject Classification scheme. This is now a long file and includes preliminary biographical information sorting out over 350 mathematically prominent individuals. (Includes quite a few
Pointers for information on branching processes.
Pointers regarding stochastic differential equations.
Pointers to basic Finite Element resources.
Pointers to designing Kalman filters
Pointers to original work and tutorials on the conjugate-gradient method.
Pointers to the history of computational fluid dynamics.
Quick proof of the isoperimetric inequality (that other closed curves enclose less area than a circle of the same length).
Records and other tidbits regarding the literature (scientific as well as mathematical).
References on the Lambert W-function (defined by x=W(x)*exp(W(x)) ).
Solving delay differential equations.
Solving the Ricatti equation.
Some suggestions for implementations of Fast Fourier Transforms.
Suggested answers to "What is Control Theory?"
Summary of multidimensional scaling (dimension reduction, singular-value decomposition, rather like principal component analysis) to pick out key data attributes -- or locate cities on a map.
Summary of Runge-Kutta methods for solving ODEs.
Summary of basic methods for integrating PDEs.
Summary of the Uzawa method for optimization of convex functions.
The Casorati-Weierstrass theorem.
The cutting stock problem: how to divide line segments (or rectangles, or...) into preassigned shapes with minimal loss? (also known as bin-packing, etc.)
The Mean Value Theorem, continuity, and differentiability.
Tiling 3-space using tetrahedra and square pyramids
Triangulation: determining positions when only differences between distances are known.
Unusual four-dimensional polyhedra: the 24-cell, 120-cell, and 600-cell.
Use of orthogonal polynomials for quadrature (numerical integration a la Gauss-Legendre).
Use of the Chebyshev polynomials (or other orthogonal families) for approximating other functions.
Using linear programming to answer questions with binary variables (an example of a transportation problem).
Using the Poisson distribution to debunk numerology based on the appearance of integers in a set or real numbers.
We exclude elementary topics from this collection in general but the question "What is i^i?" is so frequently asked, it needs inclusion.
What are differential forms?
What are Grand Unified Theories (GUTs)?
What are general (non-metric) topologies good for?
What are the general themes of point-set topology? [Henno Brandsma]
What does "simplify" mean? A challenge for computer algebra systems.
What is entropy?
What is Van der Corput's Lemma (on boundedness of integral transforms)
What is the assignment problem?
What is the Moebius inversion function?
What, exactly, does continuity almost everywhere mean?
Who was "Stone"?
Who was Weibull of the Weibull distribution?
Why aggregrate errors by summing their squares? And what are the consequences of using the least-squares criterion?
You need infinitely many colors to color "maps" in R^3, even if the regions are convex subsets of R^3.
What is the connected sum of two manifolds?
Clairaut's formula: how far north does a great circle pass?
The spider-on-a-box problem: there are parameterized families of integer boxes with all three geodesic distances between opposite corners being integral too.
Reducing the search for integer points on y^2=quartic to Thue equations.
Parameterizing the family of lines tangent to two spheres (an algebraic surface).
Describing the family of reducible cubic surfaces among all cubic surfaces (hopeless?)
What are motives? And what does motivic mean?
Can we find varieties birationally equivalent to V but with more rational points?
What is Bott periodicity (homotopy groups of SO(n) and related topics).
The Ham Sandwich Theorem (Borsuk-Ulam theorem)
What is Homotopy theory all about? [John Baez]
What is the Hopf map between two spheres (of different dimensions)? [Chris Hillman]
Are there maps between a space and its loopspace? (Not usually)
What is the fundamental group and what does it have to do with knot theory?
What is the Arf invariant for mod-2 quadratic spaces (e.g. the middle dimensional homology group)
Using the language of maps between manifolds to discern whether or not a function of several variables can be "simplified".
What is the dimension of a manifold (e.g. what dimensional creatures "live on" a Klein bottle?)
Why do you get linked pieces if you cut a Moebius strip in half?
Which dimensional spheres are parallelizable?
The Morrey-Grauert theorem: any real-analytic manifold admits a real-analytic embedding into some R^n.
Curves in R^4 are unknotted; generalize?
How hard is it to distinguish among knots?
Literature highlights: Knot theory and Functional Analysis
The Smith conjecture: fixed points under periodic homeomorphisms of the sphere are unknotted.
Details of control of a Stewart platform (parallel manipulator).
Parameterizing a tubular neighborhood of a knot.
Formulae for regular polygons relating number of sides to lengths of sides, perimeter, area, and radii of inscribed and circumscribed circles.
Variations on the theme : how to position points evenly around a sphere.
Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known).
What's a 2-norm on a vector space?
Mathematics articles with some famous authors!
The Baker-Campbell-Hausdorff formula relating products in a Lie group and in its Lie algebra (and the Poincare-Birkhoff-Witt theorem).
Connections among algebraicity, centers, and the fundamental group for Lie groups.
How many points on a conic over F_p?
Linear difference equations with constant coefficients: summary of pointers
What functions have the property that their n-fold iterates are the identity? (e.g. f(f(f(f(f(x)))))=x).
Example of a curve of rank 23
Small integer values of |x^3-y^2|
Solving the functional equation f(ax+b)=cf(x)+d
Using the Jacobian of y^2=quartic to transform the curve to Weierstrass form
Generalizing the Kuratowski problem: how many sets can be generated with complement, closure and union?
Computer code to draw the Mandelbrot set.
Citations for reference materials for Maple.
Some further resources in the history of mathematics [Ken Pledger]
Connections of Dirichlet series and related topics to modularity of elliptic curves.
Example of use of the program MWRANK to examine sample elliptic curves.
Parameterize the curve where a sphere and cylinder intersect? (no)
What if a curve is not parameterizable -- just how simple can you make it?
Computing terms of the Laurent series of 1/(1-x*cot(x))
Looking for triples of numbers satisfying simultaneous Pell (i.e. quadratic) equations
Cesaro's (singular) solution to f(x)= p*f(2x) for x < 1/2, f(x)=(1-p)*f(2x-1) + p for x > 1/2.
Are there methods for symbolic summation (as for symbolic integration)?
Twist, writhe, linking numbers and applications of differential geometry to the double helix of DNA.
How is it that the "plate trick" demonstrates pi_1(SO_3) is Z/2Z ?
Deciding the class of functions to use for a fit.
Are all self-maps really contractions with respect to some metric?
When can we lift a map from S^2 to S^3 ?
Find the lines tangent to a pair of circles in the plane.
What is "casting out nines" and why does it work?
There are no nontrivial automorphisms of the field of real numbers.
Literature reviews: constructing the regular n-gon for n=257 or (allowing use of a trisector) for n=7, 13, 19, etc.
Pointer to short proof of Mascheroni's theorem (no straightedges are needed for classi