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 classical geometric constructions).
Pointer to analysis of origami as an alternative to Euclidean constructions (origami allows more constructions, e.g. trisections).
An irreducible polynomial solvable in real radicals must be of degree a power of two. (Pointer)
Literature survey on Reauleaux triangles.
What happened to the Tacoma bridge?
How wavelets developed into tools for image compression (etc.): nice book review by Peter A. McCoy of Yves Meyer's book.
Pointer: Find sets of integers with equal sums of like powers.
Analyzing a system of three first-order ODEs.
Are highly symmetrical graphs always Cayley diagrams?
Determining the genus of a graph is NP-complete.
Use of resultants as an efficient alternative(!) to Grobner bases.
Writing a matrix as a linear combination of orthogonal matrices.
Optimal time for triangulating a polygon (n log n, or better!)
Can one always unwrap the surface of a polyhedron to get something flat and nonoverlapping? (open)
There are only finitely many subspaces of C^\infty(R) invariant under diff(R)
Analyzing a Somos sequence (T_n=(T_(n-1) T_(n-4) + T_(n-2) T_(n-3) )/T_(n-5) ); fascinating connections with quasiperiodicity and the elliptic curve y^2+xy=x^3+x^2-2x.
Can the sum of 2 fifth powers be a square? (Literature review of similar problems).
Describe parametrically the family of lines simultaneously tangent to three spheres.
What's the volume of the cone on a region?
Examples of bases from which the LLL routine fails to find the minimal basis.
Description of AXIOM (numerically solve systems of polynomial equations by continuation).
Any parametric families of integer solutions to x^3+y^3+z^3=1? (none known)
Can two squares sum to a fourth power? How about two consecutive squares?
Comparisons between Jacobi, Kronecker, and Legendre symbols.
What are the Stirling numbers? (symmetric functions in {1,...,n})
Approximately uniform distribution of points on surfaces using creation/destruction of points
Best statement about semiaxes of an ellipse to guarantee that it contains a lattice point? [11H]
Integrality questions concerning triangles.
Given n find M so M*5^n has no zeros in its decimal expansion.
How does data compression work?
Problems partitioning a polygon into triangles
Using Groebner bases to determine the image of a polynomial map.
Computing the TChebyshev polynomials nonsequentially.
Placing equidistant points along a spiral
Can we compute homology groups with Maple? (yes)
References on Morley's theorem.
Generate all (small) Pythagorean triples
How to describe graphs which can be 3-colored?
How to place mirrors on a sphere to create a disco mirror ball!
What is the topology induced by maps?
Elementary problem: how to enumerate subsets with distinct labels?
A surface may be presented as fibred over a curve, with fibres of different genera.
There are infinitely many rational points on the Euler ("Fermat") surface x^4+y^4+z^4=1; is there a parameterized family? (Open)
How can you tell if an algebraic surface is rational over a specific field?
A graph whose symmetry group has order exactly 3.
Representing integers as sums of (fifth) powers of rational numbers
Bogus proof of Collatz conjecture
Factoring N efficiently given a factorization of (Euler) phi(N)
Solutions of Fermat's Last Theorem (with n=3) over Q(sqrt(5)).
The hyperbolic Pythagorean theorem, a^2 + b^2 = c^2 * (1+a^2*b^2)
References for the local-to-global principle for quadratic equations (Brauer-Hasse-Noether theorem for associative algebras).
Characterizing sets of integers by their difference sets.
Maximum integer not a linear combination of a few others (the postage stamp problem).
Examples of pairs of perfect powers which sum to another perfect power.
Update on searches for a rational box.
"Freshman addition of reciprocals": 1/(A+B+C+D) = (1/A) + (1/B) + (1/C) + (1/D)
How efficiently can we test an integer to be square-free?
A continuum of problems linking Euler's conjecture to Waring's problem
Factoring polynomials over finite fields
Expressing an integer matrix as a sum of elements in a basis (essentially the game "Lights Out").
Classifying commutative subalgebras of a matrix ring.
Are Groebner bases better than resultants?
Does a set of real polynomials have a real solution? Tarski's Elimination of Quantifiers.
Recognizing a curve of genus 0
Computing powers of an element in a commutative ring.
What is the Brauer Group and how does it classify associative algebras over a field?
Tabulating orbits under the action of the symmetric group.
How to glue together dodecahedra face-to-face to make a closed loop? (freeness of an extension of the Dodecahedral group)
Finding enough invariants to distinguish non-isomorphic groups
Is there an algorithm to determine whether or not a group is trivial (or more generally, whether two are isomorphic)? -- No.
Finding all finite groups in which same order elements are in same conjugacy class. (Examples easy, proof hard!)
Recognizing complicated definite integrals as periods on a Riemann surface
SPHEREPACK, a set of FORTRAN programs that handles spherical harmonic expansions.
Defining k-fold iteration of a function where k is a real parameter: the Abel equation.
Using even distributions of points on a sphere for interpolation
Defining fractional derivatives with (Laplace) transforms.
Trisecting angles in the hyperbolic plane
Under what circumstances can we decompose a polyhedron into pieces which reassemble into another given polyhedron? (The Dehn invariant)
The maximum surface-area tetrahedron inscribed in a sphere is the regular one.
A "tetrahedral inequality": under what circumstances can six line segments be joined into a tetrahedron?
Hilbert's 13th problem: continuous functions of several variables can always be reduced to functions of one variable, and addition.
Optimization over R with general polynomial constraints is NP-hard.
How many minimum energy configurations are there for N points on a sphere?
Solving Moutard's equation d^2V/d^x + d^2V/d^2y = lambda(x,y) V. (Method of Darboux, Goursat)
The Budan-Fourier theorem to determine the maximum possible number of real roots of an equation on a given interval.
The Koeningsberg bridge problem: it is impossible to draw a path visiting each region in a certain picture without crossing certain edges twice.
Proving some functions have no elementary antiderivative
Pointer to information about (and pictures of) spherical harmonics.
The Poincare Eternal Return theorem.
Mayer reciprocity, the connection between minimum time and maximum range.
There cannot be an ordering of the complex numbers consistent with the expected rules of arithmetic.
Generators for the symplectic group.
What's an intuitive description of the Laplace transform?
What is the smallest Euclidean space into which hyperbolic k-space may be embedded?
What is a Green's function (and how is used for solving differential equations)?
Solving linear differential equations with linear coefficients.
Using the asymptotics on an ODE to determine the asymptotic behaviour of its solutions.
What type of mathematics is Information Theory?
Subalgebras of Hopf algebras.
What are the integral quaternions?
Are there (associative, distributive) rings in which the addition is not commutative?
Does the Fundamental Theorem of Algebra hold for extension rings of the reals? (Other than the complex numbers, no).
Are there (associative) Banach Algebras which are division rings? (only the finite-dimensional ones).
The division algebras as composition algebras, allowing product formulas for sums of (1, 2, 4, or 8) squares.
Some history of the "1,2,4,8" theorems.
Using Grassmannians to clarify the concept of sets of circles in the 3-sphere.
Getting up to 120 points on a sphere in a symmetrical arrangement.
How big can k non-overlapping (equal) disks be on the N-sphere?
On-line resources for the history of mathematics.
Finding symmetries of a differential equation (and its solutions), e.g. symmetries of the Lagrangian.
How to recognize complex varieties among (even-dimensional) real varieties? (That is, can two real equations be treated as one complex equation in half as many complex unknowns?)
How to find a conformal mapping between two complex domains?
Distinguishing notions of equivalence among metrics.
Separating disconnected subspaces in e.g. the Moore plane.
Invariance of Domain: any continuous bijection is a homeomorphism if the domain is an open subset of Euclidean space and the codomain is also in Euclidean space.
Minimizing "makespan" (the total time spent completing a multi-stage task).
Announcement of TOMLAB, a Matlab-based optimization environment.
What is the nearest point on an ellipse from a given point? (Example of Lagrange Multipliers)
Formal definition of the sine function (via integrals) and derivation of some of its properties.
What is an immersion? (Computations for a function defined on the projective plane).
Odd-dimensional projective spaces are orientable.
A vector field tangent to an odd-dimensional sphere must vanish somewhere.
Conway's game of rational tangles (a two-strand braid).
Fundamental group of the space of all unlabeled orthogonal frames in R^3.
Probability that N randomly-selected points on a sphere lie in a single hemisphere.
What are the odds in blackjack?
Proving the central limit theorem.
What is the Poisson distribution? (Analyzing coincidences of infrequent events)
A "Theory of symbolic summation" (the book "A = B").
Example: speeding up the convergence of sum 1/(n * (ln(n))^2 ) .
Simplifying trigonometric sums with symmetry.
Introductory remarks to the calculus of finite differences.
The Ising problem in statistical mechanics: how many configurations of a lattice with given total energy?
The Eikonal equation: describing the bending of light through an inhomogeneous medium.
Counting runs and clusters in sequences and arrays (percolation theory).
The diffusion equation (dC/dt = d^2 C / dx^2)
Sources for the history of calculus.
Axioms (and problems) defining fractional derivatives.
What corresponds to the Hessian matrix for vector-valued functions?
Convergence of infinite products
Area bounded by a Lissajou curve.
Functions whose derivatives are not continuous.
Reprise of basic terms (locally convex spaces, Frechet spaces, etc.)
Reflexive Banach spaces and the James space (not reflexive but close!)
What are Sobolev spaces? (Spaces of functions)
What does the Riesz representation theorem say?
How to find a class of distributions closed under products?
Why L^4 is more interesting than L^3.
Mathematical philately -- a bibliography.
Pointer to a collection of standard mathematics jokes.
We can only stretch the definition of mathematics so far! Some pointers on arithmonancy.
When does it make sense to use Bezier curves instead of interpolating polynomials (say)?
Is a Pade approximation the right one to use?
Isn't linear interpolation easiest? (not in multivariable settings)
Basics: how to calculate a spline-curve?
Joint distribution for Brownian motion.
General observations: what is model-fitting, if not just finding a straight line through data points?
Sample model-fitting problem (one-parameter, nonlinear): deciding what exactly is the goal.
Hypothesis testing: how can we decide whether or not something is zero?
Determining Lyapunov exponents of time series
Definition and application of Z-scores
Plancherel's theorem: the Fourier transform is an isometry.
Using Fourier analysis to characterize the sine function.
Conditions necessary for an application of Fubini's theorem (interchange order of integration).
Failure of Fubini's theorem when the integrand is not integral over the rectangle.
Finitely-additive measures on R^n (which are not countably additive); the Banach-Tarski paradox.
Proof of Fatou's Lemma (convergence a.e. of a sequence of functions implies convergence of the integrals).
The Mean Ergodic Theorem -- average values of iterates of an operator on Hilbert space.
What are lemniscatic functions? (Defined with elliptic integrals.)
The method of stationary phase for computing integrals of oscillatory functions.
Comparisons of deterministic and heuristic algorithms to decide whether a function has an elementary antiderivative (Risch algorithm, popular software, etc.)
Four squares from three integers: can ab+1, bc+1, ca+1, and abc+1 all be squares?
Two squares from two numbers: finding a and b so that a+b^2 and b+a^2 are both squares.
Deciding whether a point on an elliptic curve is the double of another point
Finding a provably complete set of integral points on an elliptic curve.
What are the p-adic numbers and how are they relevant to Wiles's proof of Fermat's Last Theorem?
What is known about the density of rational points in algebraic curves (as opposed to "mere" infinitude)?
Sets of points on the unit circle with rational interpoint distances
Solving equations in radicals when the Galois group is cyclic.
Outline of procedures for solving a general quartic.
The inverse Galois problem: showing that every finite group is the Galois group of some number field (with example for G = Z/3Z )
Examples of field-like algebraic systems (skew-fields, near-fields, quasi-fields, etc.-fields)
There are infinitely many automorphisms of the field of complex numbers.
Determining irreducibility of a multivariate polynomial by many specializations of one variable: the Hilbert Irreducibility Theorem.
General approach to algebraic solutions of polynomial equations: use of the Lagrange resolvent.
The algebraic closure of one field is then itself algebraically closed.
The Fundamental Theorem of Algebra: every polynomial with complex coefficients has a (complex) root.
Example of the fixed field under a subgroup of the Galois group ( = Sym(3) ).
Computing a Galois group (of a sextic whose group is not the symmetric group).
FAQ: list of currently developed and distributed software for symbolic mathematical applications [Long: includes specialty products as well.]
Existence of non-negative integer solutions to a set of linear equations is NP-complete.
Computing inverses without division, using Newton's method.
Runge-Kutta methods of integration.
How to select points and weights for Gauss quadrature when computing numerical integrals?
Pointer to code for computing elliptic integrals
Finite-volume method for solving partial differential equations.
Pointers to techniques for mesh generation for PDEs.
Iterative construction of a conformal mapping between two domains.
Pointer to an interval arithmetic tutorial.
Pointer to Octave, a noncommercial alternative to Matlab
Pointer to SciLab (free numerical software).
Performing arithmetic on real numbers using continued fractions.
Best way to compute numerical solutions to a quadratic polynomial.
Laguerre's method for determining zeroes of complex analytic functions (in a region).
Mesh generation with mgnet for solving Laplace's equation.
Fitting a plane to a large number of points -- numerical issues
Numerical evaluation of the error function.
Variation among implementations of the Fast Fourier Transform.
Some definitions and questions involving quasigroups, loops, groupoids, etc.
Number of semigroups of a given order?
Endomorphisms of a semigroup of linear transformations
Infinite torsion groups (all elements have finite order
Frobenius groups -- those with a subgroup H not meeting any of its conjugates.
For what n is the symmetric group Sym(n) a Frobenius group?
Free groups and free Abelian groups.
Computing and applying free products of groups.
Nielsen's theorem: subgroups of free groups are free.
Homomorphisms between the additive and multiplicative semigroup structures on the real line (e.g. the logarithm).
Schreier's method for computations in finitely-presented groups.
What is a stem cover of a group G ?
All finite simple groups can be generated by two elements; indeed for alternating (and symmetric) groups, such pairs of generators are legion.
Applications of the symmetric group to change-ringing! (permuting the order of bells being rung).
How many positions in an n-dimensional, edge-length k Rubik's cub(oid)?
Sims' algorithm, replacing generators of a group with "strong generators" so as to be able to determine group order, etc.
Listing of the axioms of ZFC for set theory (Zermelo-Fraenkl and Choice)
Equivalents of the Axiom of Choice and the Prime Ideal Theorem of Boolean algebra. (Does not include normality of linearly orderable topological spaces)
Defining cardinals in the absence of the Axiom of Choice
Defining (Dedekind) infinite in the absence of the Axiom of Choice
What are the definable real numbers? (compared to the computable or constructible ones)
"Natural" instances of cardinals greater than aleph_1 in topology, analysis, and set theory.
Weakly inaccessible cardinals (Regular limit cardinals -- equal to their own cofinality)
Ineffable cardinals
Testing tautologies in propositional logic with Macsyma
Minimal lengths of proofs (Friedman, Buss, et al)
Goedel's theorem masquerading as popular literature (Smullyan)
Really Goedel's theorem is unremarkable :-)
Goedel numbers as a form of information complexity (Chaitan)
Post's Problem: are there problems too hard to be solved by a Turing machine but not as hard as the Halting Problem?
What are primitive recursive functions?
Undecidable Diophantine Equations (Jones)
Number theory is provably hard: either EEAE is undecidable or finding all integer points on an algebraic curve is nonrecursive.
Could (say) Fermat's Last Theorem be proved from the axioms of Peano Arithmetic?
Tarski's Elimination of Quantifiers can be used to answer questions like, "Is this polynomial always positive?"
Pointers to sites on Non-Standard Analysis
Calculating rings of invariants under actions of linear groups on polynomial rings.
Proving and using Mason's ABC Theorem in polynomial rings.
Does the Fundamental Theorem of Algebra hold for (4-dimensional) commutative real algebras? (no)
If an integer polynomial takes only square values, is the polynomial a square?
Computing possible motions of a robot arm.
Using elimination to show two curves have two points of tangency
Announcement: DoCon software (Algebraic Domain Constructor)
Announcement: Singular computer algebra system for polynomial computations.
Pointers to Groebner basis source code
Relative merits of methods of factorization of polynomials in (Z/pZ)[x]
Heuristic irreducibility test for polynomials in Z[x]
Is the real locus of an algebraic variety non-empty? (calculable)
Using the Chinese Remainder Theorem: eliminating (say) modulo many small primes and lifting to get a rational elimination.
Computing the ring of invariants in a polynomial ring under the action of a linear group; e.g. the Dickson invariants of GL(n,q).
Computing the invariants of the diagonal action of Sym(3) on a polynomial ring in six variables.
Using elimination to help analyze a curve in R^3 specified by two polynomial equations
Typical geometric enumeration problem: compute all arrangements of 3 lines tangent to 3 given balls and perpendicular to each other.
Typical elimination problem: compute the (8) points of intersection of 3 perpendicular cylinders centered at the origin.
Using elimination to describe the (40) points in a particular variety of dimension zero
Testing whether polynomials have only simple roots
Is there a Moore graph? (diameter 2, degree 57)
Finding a minimal weight spanning tree (Kruskal's algorithm)
The Chinese Postman problem: is a given eulerization of a graph optimal?
References and pointers for the Traveling Salesman problem
Three utilities, three houses: K_{3,3} is not a planar graph.
The genus of the Petersen graph, complete graphs, etc.
The genus of K_{3,6} is 1; other K_{m,n}?
Applications of Kuratowski's theorem (planar graphs are those avoiding K_5 and K_{3,3}) and generalizations to higher genus.
Checking chromatic number of graphs derived from minimally-triangulated tori
Finding a 4-coloring of a planar graph is easy but checking for 3-colorability is an NP-complete problem
Mycielski's construction of graphs with high chromatic number but no small cycles
Perfect graphs (chromatic number = clique number)
The Secretary Problem (or, "How can a bachelor select a best wife?"): deciding when the best-so-far is nearly-best.
The "most rapidly increasing function": the Ackermann function
Generating linear extensions of posets.
Finding Graeco-Latin squares
Philip Hall's marriage theorem (making a matched set from a collection of sets)
Subsets of a committee meet several times... incidence problems and combinatorial geometries
Pruefer codes to count labeled trees
The Set Partitioning problem: find a collection of subsets which cover the whole set
How many ways to slice the cake (in n dimensions)?
Sperner's Theorem: the maximum number of incomparable subsets of a k-element set is C(k,k/2)
Stirling numbers of the second kind: counting ways to arrange N objects into K subsets.
Sample counting problem: how many ways of drawing balls from urns?
The Knapsack problem: finding subsets of a set of integers which add to a given total.
Ramsey Theory: complete disorder is impossible
Known values of Tamsey numbers
Ramsey numbers for 3-colored sets
Van Der Waerden's theorem on coloring coloring arithmetic progressions.
Vector spaces with periodic automorphism groups
Exploring Clifford algebras and Geometric Algebra and applications.
Applications of Clifford algebras to differential topology and physics
The determinant equals the product of the eigenvalues.
What are eigenvalues and linear transformations?
Eigenvalues of a symmetric matrix and the symmetric part of a general square matrix.
What is so ill-conditioned about the Hilbert matrix?
When is a matrix irreducible (no invariant subspaces)?
Do the parts of the Jordan Decomposition of a matrix vary continuously with the matrix? (Not really)
Kantorovich's Inequality for positive definite Hermitian matrices.
The game "Lights Out" is solved with (5 x 5) matrices over Z/2Z.
Basic code for Gauss-Jordan inversion of a square matrix.
Some references for multilinear algebra
The Moore-Penrose pseudo-inverse of a matrix.
Computing the pseudoinverse using SVD.
Using the pseudo-inverse and Tihonov regularization to solve linear systems of equations.
Using rank-revealing factorizations to determine (approximate) linear dependence cheaply and stably.
Finding linear combinations of matrices to have rank 1
Is similarity achieved over the ground field?
Some pointers on the computation of the Singular Value Decomposition of a matrix.
What is the tensor product of vector spaces?
Karhunen-Loeve procedure: picks out the dominant terms of the Singular Value Decomposition (Proper Orthogonal Decomposition, Principal Component Analysis, analysis by Empirical Eigenfunctions)
Using the Arnoldi method to find the largest (or smallest) eigenvalues of a matrix.
Hints for solving a large linear system of equations.
Gershgorian circles (for matrix eigenvalues)
Pointers to sparsity plots and other matrix software
Arnoldi (Lanczos) algorithms to determine a few largest eigenvalues of a large sparse matrix.
Solving a large sparse system of linear equations.
Norms of operators on Hilbert space.
Where is the point on a sphere determined as being a certain geodesic distance along a fixed arc away from a given point?
A simple curve-straightening transformation of (not necessarily simple or convex) polygons.
The Pasch axiom to define 2-dimensional geometry.
Pick's theorem (Area of a lattice triangle determined by number of interior lattice points): statement, citations, proofs
Construct the circle tangent to two lines and another circle
How to construct a hyperbola
Citation: Elementary proof that some angles cannot be trisected.
Much easier! Trisecting a line segment
Get your trisections here! and other unique mathematical experiences
Summary and reference to construction of regular polygons and other classic problems
Constructing the regular heptadecagon (17-sided polygon)
Computing the area of a triangle in 3-dimensional space
Pappus's Theorem: volume of a solid of revolution
How many points determine a torus?
A FAQ: Shortest distance between two lines in 3-dimensional space.
Solid angle subtended by a polygon
Trisecting an angle by motions off the plane.
Finding the largest circle which can be inscribed in a polygon?
Pointers for code for triangulating a polygon
Computing the envelope of a planar polygon (all points a given distance away).
Best code for computing Voronoi diagrams
What is the largest box contained in a general 3-dimensional shape?
What is the maximum number of pieces formed with N slices of the cake?
Helley's theorem: If several given convex sets cover R^n then n+1 convex sets cover R^n
The isoperimetric quotient: area versus circumference
Kepler's conjecture: the densest packing of balls in R^3 is the one used to stack fruit.
Loewner's theorem: there is a unique minimal-volume ellipsoid containing any given bounded set in R^n
How many shapes formed from glueing N blocks face-to-face?
How many shapes formed from glueing N squares edge-to-edge? (n-ominos)
Optimal distribution of points in a box?
How many spheres can be packed into a rectangular box?
Decomposing a square as a union of distinct squares
Penrose tilings and others.
Pointers to Grassmann geometry
How to test for being in the interior of the convex hull?
Comparing complexity of convex hull, bounding sphere, and sorting problems.
Statements in number theory whose truth is independent of the Peano axioms (e.g. the Paris-Harrington statement).
A tour of moduli spaces, stacks, PSL(2,Z), and related delights. [John Baez]
If there are disks of radius r at the lattice points in the plane, how far can you see from the origin?
Checking to see whether a (rational) polynomial is a sum of real squares.
Is a number like 3 be expressible as a sum of three cubes in infinitely many ways?
Finding subsets of a given set of integers whose sum is a multiple of N (and the Erdos-Ginzburg-Ziv theorem)
Status of the Goldbach conjecture: open
Numerical data for the Twin-Prime conjecture.
Values of Waring's numbers G(N) and g(N) for low N.
The Bell numbers (number of ways to partition n elements into nonempty subsets)
The Stirling numbers (number of ways to partition n elements into k nonempty subsets)
The Bernoulli numbers and Bernoulli polynomials (to express the sum of the first n perfect k-th powers.)
Number of partitions of n objects into subsets of bounded size
Sylvester's Theorem: smallest integer not a positive linear combination of two given integers (a "postage stamp" problem).
Counting (improper) q-adic representations of a number
Artin's conjecture: any positive nonsquare is a primitive root for infinitely many primes. (Open)
What is the complexity of the discrete log calculation?
Software announcement: ZEN, for finite fields and related topics.
Elementary remarks about solving equations mod N.
The Law of Quadratic Reciprocity
Tonelli's method of extracting square roots mod p
The S-unit equation: Solving X+Y=Z subject to X, Y, and Z having all their prime divisors in a fixed set S.
Irrationality of zeta(2) and zeta(3) and representation of zeta by integrals
Catalan's conjecture: that 8 and 9 are the only two consecutive positive perfect powers.
Mahler classification of transcendental numbers: how well can a number be approximated by algebraic numbers?
Fun with pi and e
Programs to compute decimal digits of pi
Spigot algorithms to compute individual digits of pi.
Pi is irrational -- proof by muse
Routh-Hurwitz criterion for all roots of a polynomial to have positive real part.
Schinzel's conjecture: sets of polynomials will yield simultaneous prime values i.o. unless there is an obvious reason why not.
Integer solutions to (2x+y)^n = 2(x+y)^n + y^n?
Erdos's 4/n problem (write each 4/n as a sum of three Egyptian fractions)
Extremal data when testing the ABC conjecture
Long cycles of amicable numbers
Crandall's conjecture (obvious generalization of the Collatz conjecture)
Connection between Collatz and Catalan conjectures.
Recasting the Collatz conjecture as a complex functional equation
Proofs of infinitude of primes, and unique factorization from The Book.
De la Loubere's method of creating magic squares.
Magic squares -- literature review.
Detailed description: how to create magic squares
Magic squares -- history, literature, some solutions.
Dame Ollerenshaw's enumeration of certain magic squares.
Perfect numbers -- recent literature
A formula for primes! (Mills's "M^(3^n)").
Arrange the first few integers on a circle with modest successive differences near-integrality of exp(pi sqrt(163))
How many representations of an integer as a sum of 3 squares?
The numbers sin((p/q)*Pi) are algebraic.
Predicting the Galois group of a polynomial using Cebotarev's Theorem (f splits mod p for 1/|G| of all primes p)
Enumerating imaginary quadratic fields having a given small class number
What is a class number? (e.g. counting equivalence classes of quadratic forms)
The Hasse Principle: a quadratic equation may be solved rationally if it may be solved mod p for every p (with or without solvability in the reals).
Compute class numbers by counting inequivalent quadratic forms.
Looking for positive integral solutions to as set of linear equations.
Sqrt(n) is irrational if n is not a square: history and types of proof.
How to decide if an algebraic number is a square (in which ring?...)
Bonus primes in Fermat's little theorem: when p^2 divides N^p - N.
The APR-CL primality test
Logical complexity of prime factorization (NP)
Submitting new progress for the Cunningham project
A free implementation of the Elliptic Curve method of factorization.
Factoring in polynomial rings over finite fields
Survey of techniques and literature on integer factorization
Coding advice for fast integer factorization (and GCD calculation)
Using HGCD to compute common divisors of rational polynomials
What is MPQS (Multiple Polynomial Quadratic Sieve factorization technique)
Sources and background: the Number Field Sieve for factoring.
How hard is it to compute the values of the Euler phi function
What is the Pollard "rho" method of integer factorization?
Current trends in factorization of integer polynomials
Fastest methods of primality proving?
(Pointless) primality tests using Chebyshev polynomials
Best known time estimates for integer factorization
Estimates of theta(x)=sum( log(p), p < x) and the relation to the prime number theorem.
Artin's conjecture on zeta functions of number fields.
Brun's constant counting twin primes.
Why we expect infinitely many primes of the form (10^n-7)/3 (prime for all n < 9 )
Probabilistic conjectures regarding sequences with infinitely many primes
Sums and products taken over all primes, involving Dirichlet characters
Sum of log(log(p)) over primes less than N
Table: number of primes less than 2^N, N=1, 2, ..., 32
Estimating the number of primes less than N with zeros of the zeta function.
Estimates of gaps between consecutive primes
"Prime server": ask it for the 20 000 000-th prime
Estimating sum f(p) over primes p (e.g. f(x)=1/x )
Find integers N divisible by all primes up to sqrt(N)
Connection between zeros of zeta and the error term in the PNT
Obtaining closed forms for the sum sum( 1/n^2 ) = pi/6 and similar sums
Definite integrals which lead to values of the zeta function.
Relationships between sets of numbers and their continued-fraction partial quotients (e.g. algebraic numbers, numbers with bounded quotients)
Algorithms for manipulating continued fractions -- pointers, literature.
Evaluating "regular" continued fractions with differential equations
Connection between continued fractions and the Euclidean algorithm
Citations regarding multidimensional continued fractions
Solve x^n + d y^n = c: Thue equations.
The Beal conjecture: solutions to x^n + y^m = z^r
Counterexamples to FLT (n=3) (in other rings of course)
Rational boxes with weakened conditions to be met
Angles between three vectors determine location of the vector in their "center".
Pointer for exponential Diophantine equations (e.g. 3^x+5^y=y^z+1 ).
There are infinitely many solutions to x^2 + y^3 = z^4
Summary: how many N-th powers needed to sum to another N-th power?
Various proofs of Fermat's Last Theorem for polynomial rings
Small generalization of FLT: x^n + y^n = z^m has solutions iff gcd(n,m)=1 or n=m=2.
Extreme examples concerning Hall's conjecture: no "big" solutions to y^2 = x^3 + k
Sqrt(n) is irrational if n is not a square: history and types of proof.
Subalgebras of matrix algebras.
Various methods to solve Pell's equations, with citations, special cases, etc.
Are there 7 points in the plane which are a rational distance apart?
Is there a regular n-simplex in R^n with integer coordinates?
Long summary: triangular numbers which are perfect squares and related topics
Some curves of high genus: rational solutions to y^k = f_k(x) where f_k(tan(u)) = tan(k u)/tan(u)
Location of the sun in the sky as seen from a point on earth
Explanations of various techniques for proving a number to be irrational
A collection of examples of calculator and computer errors (hardware constraints and symbolic-algebra bloopers)
Table of contents for the list of Frequently-Asked Questions in Mathematics which appears in the newsgroup sci.math.
Miscellaneous typos and errors in standard reference materials.
Review of classification categories in number theory before 1985
A list of mathematicians whose photographs appear in the literature, and a pointer to citations.
A sampling of recent work on the Riemann zeta-function and the Riemann hypothesis.
"Natural" axioms which imply the negation of Continuum Hypothesis
"Natural" example of a function with distinct one-sided limits
"Natural" smooth, nowhere analytic functions
(Tauberian) types of convergence of sequences
Subfields of number theory under pre-1986 MSC classifications
Heegaard splittings of 3-manifolds
Polynomial solutions to the 4/n problem
Tiling S^3 with 600 congruent spherical tetrahedra
Comparative strength of Axiom of Choice, Boolean Prime Ideal Theorem, Hahn-Banach Theorem, Banach-Tarski Theorem
Algorithmic Information Theory as a measure of program-size complexity
Balanced tournament designs
Converse of Lagrange's Theorem implies solvable
CRC (Cyclic Redundancy Check) efficiency
G/Z(G) is a square if Z cyclic, contains G'
H=w theorem (Myers and Serrin) on density of Sobolev spaces
Generalizations of the Riemann zeta function: Polygamma function, Hurwitz zeta, etc
Proofs of existence, uniqueness of solutions of initial value ODE problems
KC spaces (all compact subsets are closed)
Least common multiple of the first n integers, about exp(n)
NP completeness -- references
QZ algorithm for simultaneous factorization of two matrices
Iterate f(x) = 3x+1; always get a power of 2? (No)
Abel's original paper on the general quintic
Acceleration is not a contravariant tensor
Ackermann function -- pointers and code
Akima's interpolating spline
Algebraic groups resemble linear groups and finite groups
Amicable numbers (each the sum of the other's divisors)
Parallelizability in smooth and analytic categories
Three 120 degree angles inside a triangle, all six lengths integral
Polygon areas with Green's theorem (and little cancellation)
Distribution of sum (etc.) of two independent random variables
Assignment Problem: reordering matrix rows to minimize diagonals
Aurifeuillian factorizations of sums of powers
Topology of the spaces of automorphisms (invertible self-maps, in various categories) of S^3
Banach-Mazur paradox and the Axiom of Determinateness
Axiomatic treatment of infinite series
Busy Beaver references
Bernoulli inequality proves e exists
Bertrand's postulate: there is a prime between n and 2n
Big numbers used in proofs (Friedman -- strings in alphabets)
Bonse's inequality; each prime no more than sqrt(prod(previous))
Regular signed Borel measures
Borsuk-Ulam theorem: no injections from S^n to R^n
Generalization of the Borsuk-Ulam theorem: no injections from S(k,n) to R^n where S(k,n)=n-skeleton of k-cube
Minimum bounding circle of a collection of planar points
Minimum bounding sphere of a collection of points in space
Better-quasi-ordering transfinite sequences
Brauer characters of symplectic groups in characteristic p
Bunch-Parlett matrix decomposition (and counting negative eigenvalues)
Cancellation in topological categories (X^2=Y^2 =implies X=Y?)
Exponentiation behaves differently for cardinals and ordinals
Cayley graph of symmetry group of icosahedron is a soccer ball (football)
Stability of Cholesky decomposition for a large matrix
Chromatic numbers of non-planar graphs
Church-Turing thesis on computability
Non-constructibility of some extremal circle-points
Cluster points are limits of subsequences?
Cohen's models showing independence of Axiom of Choice
Computational complexity of n-coloring a planar graph?
Proving commutativity of rings from assumptions "x^n=x"
Computational complexity of eigenvalue-related problems
Perfect squares among the binomial coefficients
Continuous Newton's Method
Dissecting a cube into distinct smaller cubes
Cunningham tables as of May, 1999
Comparison of convergence in probability vs. convergence with probability one?
Deflating a matrix after some eigenvectors are found, to find orthogonal ones
De Giorgi - Nash Theorem: Continuity of solutions of linear elliptic equations
Determinant-preserving endomorphisms of End(V)
Differential Algebraic Equations (DAEs) -- systems with both differential equations and algebraic equations
Convergence of the derivatives of a convergent sequence of functions
Class field theory in case of dihedral Galois groups
Direct rotation -- shortest rotation taking one vector to another
How much did Dirichlet prove about primes in progressions?
Dirichlet series related to zeta function and arithmetic functions
Dissecting n-cubes into (minimal numbers of) simplices
Dissecting a rectangle into squares -- ratio of sides rational
Duality gap in optimization of nonlinear programs
Cyclides of Dupin (a class of surfaces in R^3)
Garden of Eden Patterns (non-successors) in Conway's Game of Life
Edge-transitive polyhedra in R^3
Ehrenfeucht-Fraisse games to establish equivalence of theories
Algebraic tools for electromagnetics
Embedding dimensions of compact smooth manifolds into Euclidean space
Enumerating all permutations of a finite set
Dissecting an equilateral triangle into incongruent equilateral triangles (can't)
L^p norms converge to L^\infty norm
Euler-Maclaurin-summation technique
Sum of values of Euler-phi(n), n through N
Exotic differentiable structures on spheres
Extending maps from submanifolds of R^k to open neighborhoods of R^k
Fejer's theorem (sequences mod 1)
Weiszfeld algorithm to solve Fermat-Weber (Steiner) facility location problem
Divisors of Fibonacci numbers
Fibonacci numbers mod p
Filon's numerical integration formula
Floyd's algorithm for finding cycles under iterates of maps
Counterexample to Fermat's Last Theorem for exponent 7 (in the ring of 7-adic integers)
What to study to understand Wiles' proof
Formal characterization of "closed-form solutions" to differential equations, integrals, and summation problems
Fixed-point-free automorphism of prime order implies nilpotent
Subgroups of free groups always have a conjugate with which they have trivial intersection
Apollonian gasket (packing the plane with non-congruent circles)
Curvature and areas of triangles in spherical/hyperbolic 2-D geometry
First appearance of Gaussian quadrature
Generalized eigenvalue problem -- complete set of eigenvectors?
Generalizing trig identities to cover functions F(x)=exp(alpha*x) with alpha some other root of unity
Ordinal analysis, infinite proofs, Gentzen proof of consistency of elementary arithmetic
Geodesics on tori and other surfaces
Gerschgorin circles method of bounding eigenvalues
Making initial guesses for local optimization procedures
Goldbach conjecture
Goldie's theorem (noncommutative fields of fractions)
References describing Groebner bases
Graffiti -- program to generate conjectures in Graph Theory
Graham's number (very big!) in Ramsey Theory
Euler cycles vs Hamiltonian cycles in a graph (when they exist)
Invariant (Haar) measures on SO(3) and SE(3) -- some summaries
Hales-Jewett theorem on large multiplayer tic-tac-toe games
Hamel bases of R over Q cannot be 'nice' (Borel, etc.)
Hartogs' theorem: multivariate analyticity follows from analyticity in each variable
Hartogs' theorem (There are always larger well-ordered sets) does not require Axiom of Choice
Heisenberg's Inequality on non-commuting operators
Hilbert's 13th problem (all multivariable functions may be obtained from univariate ones)
Hilbert's axioms for geometries
Riesz transforms and others as generalizations of the Hilbert Transform
Hirano's method root finding
Alexander's horned sphere and other wild embeddings of S^2 into S^3
Huygens' principle (no light diffusion in wave equation in odd dimensions)
Infinite product representations of functions from their zeros
Semi-infinite Gauss integration rule
Numerical solution of Fredholm integral equations of the first kind
Invariance of Domain theorems
Invisible hand algorithms for optimization
Isoperimetric inequality (circle has maximum area for fixed perimeter): definitions, pointers, proofs
What comes next after addition, multiplication, exponentiation?
Ito's formula, derivatives in stochastic processes
Non-planarity of K_{3,3} (the "three houses, three utilities" puzzle)
Karatsuba multiplication of large integers
Kiss precise -- the radii of mutually tangent spheres in R^n
Kiss precise -- the four radii of mutually tangent circles
Kolmogorov complexity of strings of symbols (relative to a fixed universal Turing machine)
Carmichael numbers: Korselt's criterion
k shortest paths in a graph
Kwapien's theorem: norms equivalent to Hilbert space
Langford problem (sequence 2 sets of first n numbers with each pair suitably far apart)
Geometric interpretation of Laplace transform as a linear map
lattice for positive Boolean functions
English alphabet -- compare frequency of letter use to encoding (Huffman, Morse, etc.)
Null geodesics, Koszul formula
Levy-Steinitz theorem: conditionally convergent sums of vectors can be rearranged to sum to many values
Proof of L'Hospital's Rule
What are Lie Groups and how are they used in mathematics and physics?
Approximation of Lipschitzian functions by C^1 functions having the same Lipschitz constant
Löb's Theorem: "This statement is provable" is provable
Löwner's Lemma (analytic function on the disc stretch circular arcs)
Lyapunov matrix equation, Sylvester matrix equation
Mahalanobis metrics to estimate linearity of data swarms
Mahler function sum(x^(2^k))
Marcinkiewicz' theorem: a probability distribution with only a finite number of non-vanishing cumulants must be Gaussian
Expected return in martingales is zero (e.g. the stock market)
Comparison of various factorizations of symmetric, positive definite matrices
Matrix inequalities for positive definite matrices (Hadamard, Szász, etc.)
Extreme-value distributions (invariant under MAX of two random variables)
Product measures on the unit square
Mercator projection -- formulae and pointers
Mertens' Conjecture
Automated proof of Pons Asinorum
Connections among Bessel, Hankel, Struve, Anger, Weber, Kelvin, and Airy functions (each a solution to a second-order linear ODE)
Multiply-perfect numbers (sum of divisors is a multiple of n)
n-categories (categories, then functors, then natural transformations,...)
Integration in R^n: Monte Carlo or grid methods better?
Series of analytic functions with non-entire coefficients
Computing derivatives numerically, using piecewise-polynomial fit, Savitsky-Golay filters or complex integration
Nowhere dense subsets of R
Construction of octonions and related real algebras
Ox -- free package for matrix computation
Upper bounds for the n-th prime number
Packing rectangles to demonstrate famous infinite sums
Papakyriakopolous' Theorems (historical)
Pappus' theorems on centroids
Parallelogram Law in a Banach space implies an inner product exists
Number of partial orders on a finite set
Poincare duality and the cup-product mapping on cohomology of a manifold
Definite integrals of functions on Riemann surfaces over closed curves may be computable
Decidability of classification questions in topology, e.g. Poincare conjecture
Poisson's summation formula
Spanning a space of functions: Stone's Approximation Theorem, etc.
Pratt's theorem: Certificates which verify primality in polynomial time
Products of Hermite polynomials as sums of Hermite polynomials
Products of prime, or irreducible, elements of a domain
Proth's primality theorem and candidate primes generalizing the Fermat primes
Ptolémée's theorem on quadrilaterals inscribed in circles
Pyber's estimate for the number of groups of order n
Pyramidal tours for the TSP
Largest quadrilateral with given sides
Convex functions and related notions
Quasiperiodical functions on R
Evaluating zeta and finding zeros
Raabe's formula (integral of log(Gamma) )
Raabe's test for convergence of a series
Probability of return in random walk on Z^n
Real line is unique uncountable complete linearly ordered set with a countable dense subset and no endpoints
Real Polar Decomposition of a real matrix as (symmetric positive semidefinite)*(orthogonal)
Regularization, error propagation, and confidence intervals
Eliminating variables from systems of ODEs using resultants
Determine locations of points given all bipartite distances
How many points in a variety of dimension zero? (Bernshtein's theorem and algorithms)
Challenge problems for root-finding algorithms
Odds of a run of given length existing in a sequence
Sarkovskii's Theorem on lengths of cycles under iterates of continuous real functions (period 3 implies chaos)
Schauder bases in Banach spaces (e.g. Haar functions)
Schreier Refinement Theorem for modular lattices (and thus to groups etc.)
Schroeder's equation (iterations of functions near a fixed point)
Paris-Harrington variation on Ramsey's theorem
Shapiro's Inequality regarding Sum( a_n/(a_{n-1}+a_{n+1}) )
Routh-Hurwitz criterion for all roots of a polynomial to be in the unit disk (left half-plane)
SNAG (Stone-Naimark-Ambrose-Godement) Theorem: construct measures corresponding to representations of LCA groups
Sociable numbers (each the sum of the divisors of the next)
Asymptotics of a quadratic recurrence (a(n) = a(n-1)-a(n-1)^2 )
Sperner's theorem on maximal collections of incomparable subsets
Star of David theorem (LCM of sets of three binomial coefficients)
Steiner systems: what they are, example of construction
Steiner-Lehmus Theorem on angle bisectors in a triangle
Unit vectors which sum to zero can be reordered to keep partial sums small
Stirling numbers and partial zeta sums
Stokes' theorem on surfaces with singularities
Strong generating sets and the Schreier-Sims algorithm
References and pointers to the Risch algorithm and symbolic integration
Algorithms for evaluating some types of power series in closed form
Symmetric product of (sets or) spaces
Stiefel manifold is not a symmetric space
Software for symmetric groups, representation theory
Tennenbaum's theorem (non-standard arithmetic is nonrecursive)
Tensor product of Z-modules (Abelian groups)
Tetrahedron interpolation using barycentric coordinates
Characterize the sine function by the magnitude of all its derivatives
Thebault's problem (circles and triangles) -- a difficult elementary problem!
Theta-functions (e.g. theta(z) = sum(exp q^(n^2) )
Thin plate splines
Humans can integrate 1/(1+tan(x)^c) on [0,pi/2] but machines cannot!
Triangulation of polyhedral domains
Non-classical tools which allow trisection of angles
Errors in tables of integrals and special functions
Independence vs undecidability
Units of the polynomial ring A[X]
Untouchable numbers (those not the sum of divisors of any other integer)
Vehicle Routing problem (partitioning workload)
Markov's theorem relationing continued fractions and c(k)=[(k+1)x]-[kx]-[x]
Frechet interpretation of derivatives (linear maps)
Whittaker functions (defined by a second-order linear ODE)
Wolstenholme's congruence (sum 1/k = 0 mod p^2)
Totally disconnected and zero-dimensional spaces
Computing values of the Riemann Zeta function
A Brauer pair (groups indistinguishable by character theory)
A knight's tour (but cannot be a magic square)
A Banach-Tarski-like decomposition of the unit interval (using countably many subsets)
A collection of examples of calculator and computer errors (hardware constraints and symbolic-algebra bloopers)
A description of the Hilbert Hotel (one room per natural number)
A list of mathematicians whose photographs appear in the literature, and a pointer to citations
A mixed bag of number-theoretic experiments from a recent issue of Mathematics of Computation
A pair of equations becomes a single equation over the Gaussian integers
A sampling of recent work on the Riemann zeta-function and the Riemann hypothesis
A simple delay differential equation
A strictly increasing function with a derivative equal to zero on a dense set
A topology finer than the infinite product topology on R^I
Abhyankhar's Lemma (ramification of primes in compositum of extensions)
Action of the Steenrod algebra on polynomial rings
Adamchik's formula for the Hurwitz zeta function
Adapting the simplex algorithm for optimization of nonlinear functions
Add random digits until a 0 appears; distribution of sums? (probability generating function)
Algebraic independence of Gamma(1/4) from other constants
Algebraic relations satisfied by elliptic functions
Algorithm to determine whether a 3-complex is homeomorphic to S^3
Algorithms to compute erf
Almost no perfect powers in arithmetic progression
An odd perfect number? (Almost! :-)
An ordered group with least upper property is the integers or reals.
An exercise in viewing one equation from many perspectives: x^2 + 7 =8 p^n with p prime
Any countable linear ordering can be embedded in the rationals.
Application of Clifford algebras to quadratic forms
Application of Lagrange Inversion Formula for power series
Application of Steiner systems to lottery analysis
Application of exterior algebras to generalize the factorization of adj(X) when X is singular
Applications of Lie theory for solving differential equations
Approximating functions with quadratics
Arithmetic progression of four terms nearly square
Arithmetic progressions of square-free integers
Arithmetic with multiplication and successor define addition
Artin, Chevalley theorem: homogeneous forms represent zero (nontrivially) if the number of variables is large enough.
Assured existence of uncountable measure zero subsets is undecidable
Astronomical Emden or Lane-Emden equation (second-order BVP)
Asymptotics of random walks on finitely generated groups
Attractor to the Lorentz chaotic ODE
Automorphisms and semidirect products
Average (or total) lengths of the paths of a tree (Weiner index)
Background on the Erdös' 4/n problem
Basic code for Cholesky decomposition
Basic comparison of root-finding methods
Basic description of Gauss quadrature
Basic topics in computing some matrix factorizations
Basic tricks for solving second-order autonomous ODEs
Basic: regression results depend on objective function being minimized
Behaviour (periodic points) of logistic map h(x)=rx(1-x) for large coefficient r
Bending of a cantilever beam
Biographical pointers about Evariste Galois
Biography of Herbert Busemann (1905-1994)
Block Scheduling Algorithm for tournaments
Books on Nevanlinna theory
Boundary layers in slightly viscous flow (numerical Navier-Stokes equations)
Boundedness of terms of a linear recurrence
Bounding volume from areas of projections
Bounding the variance of max(X,Y)
Bounds for the number of cycles in a planar graph
Bounds on the value of a polynomial
Brun's constant (sum of reciprocals of all twin primes
C, perl codes to generate lists of primes
Calculating position from three angles to fixed locations
Calculus of Variations -- generalities and applications to particle motion (to minimize action)
Can a square wheel roll smoothly (on the appropriate surface)?
Can one trisect an angle in the hyperbolic plane? (no)
Can planarity be expressed as a sentence in first-order logic? (no)
Cardinality of a set of subsets of R with all intersections small
Cauchy principal value for integrals (and why to treat it carefully)
Chain complex of differential forms
Characteristic subgroups and distinct isomorphic normal subgroups
Characterization of compact subsets of L_infinity
Characterization of nonreflexive Banach spaces by convex closed subsets
Characterization of the Airy functions Ai and Bi
Characterizing commuting pairs of ordinals (under addition and multiplication)
Characterizing eta, the order type of the rationals (as the unique countable densely packed linearly ordered set without min or max)
Chebyshev, Camp-Meidell estimates that a random variable lies near the mean
Choosing the path which maximizes average value of a function
Citation and remarks about solving linear equations in the ring of quaternions
Citations for proofs of FLT for low exponents
Citations on representing integers by quadratic forms
Citations: when are specific small numbers perfect powers mod p?
Classifying almost-complex structures in vector-spaces
Clausen and von Staudt's theorem (denominators of Bernoulli numbers)
Closed form integration: the case of algebraic integrands
Closed formula for position probabilities in random walk on a 2D square lattice
Closed-form (non-radical) solutions of quintic
Code for computing values of the gamma function
Comparing Jacobi-Davidson with Arnoldi/Lanczos methods for finding largest eigenvalues
Comparison of errors from various numerical integration algorithms
Comparison of semisimple, reductive, etc. for Lie algebras and groups
Comparison of negative binomial and hypergeometric distributions
Comparison of models for Optics (Ray optics, Geometrical optics Wave optics)
Comparison of definability through Lambda Calculus versus axiom systems
Comparison of methods for filtering noise from data
Complexity of graph-isomorphism problem (NP but probably not NP-complete)
Complexity of the 3-valued logics of Kleene, Post
Components of a representation induced from a non-normal subgroup
Computational complexity of reducing Boolean expressions (for Golomb rulers)
Computational complexity of matrix inversion
Computations of (individual) decimal digits of pi
Computer searches for the Integer Cuboid - an update
Computing intersection of two ellipses with elementary elimination
Computing Gaussian moments
Computing narrow band spectrum noises in signal processing
Computing modular square roots
Computing intersection of a torus and a circle in R^3
Computing Riemann-surface functions g_2(L),g_3(L) from an integer lattice L
Computing a double integral
Computing a Sylow subgroup and its normalizer in GL(n,p)
Computing best approximation of a positive function by positive functions from appropriate families
Computing distribution of an algebraic combination of several independent random variables
Computing locations in a simplicial complex using barycentric coordinates
Computing sqrt or quotients with very many digits of accuracy
Computing the intersection of two simplicial complexes in R^3
Construct a three-by-three magic square with square entries? (open)
Constructing minimum spanning trees
Constructing the (Boolean) ring from a Boolean algebra
Constructing the geodesic joining two points on Poincare disk
Construction of the 120-cell and 600-cell (solids in R^4)
Construction of the heptadecagon
Construction of the pentagon
Constructions of representations of SU(5), SO(5)
Constructive mathematics and the role of the Law of Excluded Middle
Constructing a non-measurable subset of R^1
Convergence of signed measures does not imply convergence of positive parts
Converting pitch and roll to Euler rotations
Convex shapes with worst packing density
Conway's generalizations of the Collatz conjecture include some undecidable problems
Copland-Erdos constant (a real number normal in all bases)
Critique of domain restrictions in display of integration results (Mathematica)
Current research trends in multilinear algebra
Current status: consecutive proper powers -- probably only {8,9} (Catalan)
Defining "measure zero" on infinite dimensional spaces (prevalence)
Definition and properties of the square-root function on (positive-definite) matrices
Definitions of analyticity do not generalize well to quaternions
Derivatives have the Darboux property (hence if increasing, are continuous)
Deriving closed-form expressions for infinite sums (experimentally) with PSLQ
Deriving solutions to quartic equations in radicals
Desargues' theorem, Pappus' theorem, and coordinatization
Descent proof of Fermat's Last Theorem for exponent 3
Description and utility of the Hessian matrix
Determinants, permanents, and immanents of a matrix
Determine a Euclidean motion from shadows of 5 moved points
Determining ellipses tangent to a collection of lines
Determining symmetry groups of a family of functions
Determining Fourier transform with Hermite functions
Differences between Fast Fourier Transform and Discrete Fourier Transform
Different types of integrals (Riemann, Lebesgue, etc.)
Different approaches in algebraic and geometric multigrid methods
Difficulties encountered by symbolic algebra problems when computing a limit
Direct correspondence between irreducible representations and conjugacy classes in finite groups
Discrete dynamical systems in R^3 showing symmetries in attracting sets
Distinct links with homeomorphic complements in S^3
Distribution of logarithm of a Poisson variable
Distribution of eigenvalues in random matrices
Do knot complements determine knot type?
Does X+X=X imply Axiom of Choice?
Does FFT require code length a power of 2?
Does every curve contain the vertices of some square? (open)
Does every group have a two-, or even finite-dimensional representation? (no)
Easton's theorem, Luzin's conjecture exponentiation of cardinals can be just about any function
Effective solutions of systems of polynomial equations (literature)
Efficient orthogonal array generation
Efficient computation of exponential of a matrix
Elementary construction of cubic splines
Elementary description of the "Law of Large Numbers"
Elementary example of separation of variables
Elliptic curves of rank=8 with torsion group=Z/2Z*Z/2Z
Elliptic curves of rank 5
Entire functions with no 2-cycles (f(f(x))=x) nor fixed points are translations
Enumerating representations of integers as sums of four squares
Equivalences among some varieties of groups generated by a single group
Estimates for the distance from one prime number to the next
Euler and the birth of analytic number theory
Evaluating a power series in closed form using Zeilberger's algorithm (EKHAD package)
Evaluating numerical integrals of oscillating functions
Every sequence of abelian groups can be the homology of some complex
Every square real matrix is the product of two symmetric matrices
Exact solution for the inviscid Burgers equation
Example -- linearizing and solving a nonlinear PDE
Example of intermediate swell of Groebner bases in elimination theory
Example of Chebyshev polynomial expansion (for ln(x+1) on [0,20])
Example of contour integration of definite integrals (here, of ln( a + sin(x) ) on [0, 2 pi])
Example of Dynamic Programming: select a subset with one linear function constant to minimize another
Example of connection between a Lie group and its Lie algebra -- SO(3)
Example of a transformation of an elliptic curve into Weierstrass form
Example of homogeneous Fredholm integral equation
Example of the Galois correspondence for infinite groups
Examples of failure of Newton's method
Expected performance of Heapsort
Exploiting the parallels between differential equations and difference equations
Extensions of local fields
Extensions of Perron-Frobenius theorem to nonnegative matrices
Factorization into rectangular matrices of size equal to rank
Failure of uniqueness of limits in non-Hausdorff spaces
Fairest algorithm to generate random permutation
Family of division algebras over a field (plus Milnor conjecture, K-theory)
Famous conjectures: existence of Hadamard matrices, projective planes, Jacobian conjecture
Fan theorem (König's Theorem): finitely branching trees with only finite paths are finite
Fermat's last theorem in different rings
Filtered acceleration of time series
Find the function u which minimizes an integral depending on u''
Find the largest square with exactly 3 interior lattice points
Find values for the 26-variable integer polynomial yielding primes
Finding inflection points from numerical data
Finding a global linearly independent complement to convex sets
Finding a symmetric matrix with prescribed characteristic polynomial
Finding all integer solutions to a quadratic equation in two variables
Finding sets of integers in short intervals, whose product is a square
Finding the Euclidean isometry best matching n pre- and post-point pairs
Finding the k largest trees in a graph
Finding the Pareto-optimal points in R^n
Fine points of distribution of primes in arithmetic progression
Fine-tuning error-correcting codes to detect human errors
Finite groups with distinct but isomorphic characteristic subgroups
Finitely presented groups with solvable word problem
First-order linear differential equation with smooth, nonanalytic solution
Fitting best planes to data points in R^3 (Deming's vs Pearson's methods)
Fitting a circle to data in the plane
Fitting a cylinder to data in R^3
Fitting a sum of (a few) exponential functions to data
Fitting a sum of Gaussian distributions to data
Fitting data to Johnson distributions
For which D is x^2 - D y^2 = +/-4 solvable?
For which D and N is x^2 - D y^2 = N solvable?
Formal solutions of the Schrödinger equation
Formulas useful for numerical evaluation of Bessel functions
Fortran 90 library for computation of special functions available
Frobenius problem -- largest total not expressible with n denominations of postage stamp
Functions not constant on a curve of critical points
Functions with an addition formula (F(x+y)=P(F(x),F(y)) P a polynomial) are elliptic functions
Galois group actions without n-cycles in action on roots of a polynomial of degree n
Gauss's Theorema egregium (curvature is intrinsic)
Gaussian curvature and sectional curvature
Gaussian integration formula on a triangle
General comparison of numerical integration methods
General linear groups over a finite field suggest why every p-subgroup should lie in a Sylow-p-subgroup
General remarks about connectivity and topology
General solution of matrix quadratic equations
Generalities on finding approximations to functions on R^n
Generalization of DeMoivre's Theorem to N-dimensional spaces
Generalization of binomial coefficient (Gamma, q-hypergeometric functions)
Generalization of Goldbach, Twin Prime conjectures (distributions of primes)
Generating random points in multidimensional polytopes
Generating random variables with a given correlation
Generator of homotopy group pi_7(O) of orthogonal group, spheres
Geometric theorem-provers
Given four points construct a square through them
Groups in which distinct conjugacy classes have distinct sizes
Groups isomorphic if equal counts of elements of each order? (No)
Groups which have a unique normal subgroup
Groups with cyclic Sylow 2-subgroups have normal 2-complements
HAKMEM Algorithms for dealing with arithmetic with continued fractions
Haar measure and rotation group SO(n)
History of manifolds :-)
History of the Cauchy-Schwarz inequality
Homotopy classes of maps between spheres
How are Fourier Transforms used?
How do sparse matrices "usually" arise?
How do Galois groups vary with the coefficients of the polynomial?
How do Lie groups and algebras fit in among the branches of mathematics?
How does (Set Theory axiom) V=L prove the Continuum Hypothesis?
How many simple groups of a bounded order?
How many groups of order 2^n (n through 8)
How many groups satisfy a particular set of relations among their generators?
How may Galois groups be computed?
How to find Q in the QR factorization?
How well do continued fractions behave for sqrt(D) ?
Identification of fundamental unit in a real quadratic number field
If X is bounded under all compatible metrics then X is compact
Illustrative examples of everywhere continuous, nowhere differentiable functions
Images of zeta along vertical lines
Implications of Axiom of (Dependent) Choice, Axiom of Determinateness, and the existence of inaccessible cardinals, for analysis (existence of non-measurable sets)
Importance of the Implicit Function Theorem
Improved efficiency of Euclid's algorithm/continued fraction algorithm (for integers or polynomials)
In any graph, an optimal traveling-salesman path is planar
Independence of the vector space axioms
Infinitude of primes in the congruence classes mod 8
Integer points on y^2 = x^4+x^3+x^2+x+1
Integer triangles with one angle measuring 120 degrees
Integers satisfying two Pellian equations (leads to curves y^2 = x^3 - D x )
Integral of product of spherical harmonics / Legendre Functions
Integral points on the elliptic curves y^2 = x^3 +- 2909 *2^k ?
Integral polynomials with the same range
Integrality of some ratios of factorials
Integrals over infinite range
Integration and interpolation over a sphere
Interpolating from scattered data in R^3
Intersection of regular expressions (and circuit analysis)
Intersection points of two cubic parametric curves
Into how many pieces may a torus be dissected with three planar cuts?
Intrinsic geometric description of parallel transport
Introduction to Riemann surfaces (through elliptic integrals)
Invariant measures (cylindrical, Wiener) on infinite-dimensional sphere
Inverses, determinants of Vandermonde-like special matrices (and the 'Advanced determinant calculus')
Inverting a discrete convolution
Is a sigma-algebra of sets also an algebra?
Is a given point inside the convex hull of a set of others?
Is the Galois group of random polynomial the symmetric group?
Is the sequence { n*sin(n) } dense in the real line?
Is there an isometric embedding of H^2 into R^4? (open!)5
Isn't the non-planarity of K5 sufficient to prove the four-color theorem? (no)
Isomorphic polynomial rings have the same dimension
Kepler's equation and cycloids
Knots and the category TOP^2 of pairs of spaces
Known cases of ABC conjecture; adjacent numbers with few divisors
LLL (and PSLQ) algorithms to find algebraic relations among a set of reals
Lagrange multipliers: nearest point on an ellipsoid to a given point
Large departures of pi(x) from Li(x)
Lay person's description of Principal Component Analysis
Lehmer evaluation of continued fractions with quotients in arithmetic prog
Length of chains in a dynamical system on Z (using operations add-one, subtract-one, and double)
Limiting approximation of elementary expected value
Limiting behaviour of subadditive functions on R
Linear relations between roots of unity
Listing of the 33 distinct topologies on a 4-element set
Local isomorphisms (covering maps) of Lie groups and the Spin groups
Locations of zeros of zeta
Long arithmetic progressions consisting only of (consecutive) primes
Lucas theorem (computing binomial coefficients mod p and p-adically)
Many ways to compute the area of a circle given 3 points on it
Maple incorporates algorithms to integrate functions from a genus-0 extension of the rational function field
Maple program to find all topologies on a (small!) finite set
Mathematical models of pricing and numerical PDEs
Matrix formulation of Hitchcock-Koopmans transportation problem
Maximal number of maximal cliques in a graph with N vertices
Maximal sets of integers with all subset sums distinct: Conway-Guy sequence
Maximal values for number of divisors function
Maximum number of primes in intervals (prime constellations)
Maximum order of elements in the symmetric group S_n
Mean of medians of sets of three trials; order statistics
Measures of quality of approximations by rationals (Farey, continued fractions)
Meijer G functions and the incomplete gamma function
Methods of describing roots of high-order polynomials; decision procedure to decide whether distinct
Methods of finding peaks in signal data
Minimal values for Euler's phi function
Minimal values for sum of prime divisors function
Minimum number of telephone calls to exchange information
Minimum total curvature of a space curve
Mixed differential-difference equations (differential and functional equations)
Modeling (the human perception of) color
Modeling modem pool usage as an M/M/K/K system
Models of Zermelo set theory which are not models of ZF
More examples of the "Law of Small Numbers"
Multi-dimensional analogues of (i.e. simultaneous) continued fractions
Multidimensional secant methods (quasi-Newton methods) for finding zeros
Multiple characterizations of positive (semi)definite matrices, and applications
Multiple integer solutions to x^3+y^3=N (the taxicab example) and the elliptic curves y^2=x^3+D
Name this cubic surface!
Near-proof of (in)consistency of Zermelo-Fraenkel set theory
New preprint server for Linear Algebraic Groups and Related Structures
New solution of the Prouhet-Tarry-Escott problem for k=11, other limitations
Nice proofs of Morley's Theorem
Nilpotent elements in group algebras
No Banach-Tarski paradox for subsets of the line or plane
No bounded analytic functions on the half-plane which vanish at integers, except zero
No graphs on n vertices with automorphism group Alt(n) (the alternating group)
No nontrivial solutions known for x^5 + y^5 + z^5 + t^5 = 0
Not all symmetric matrices have a (modified) Cholesky-factorization
Number of regions formed by diagonals in a polygon
Number, construction of independent vector fields on an n-sphere
Numbers expressible as sums of two cubes in multiple ways
Numerical calculation of arctangent function
Numerical calculation of Cauchy stresses
Numerical computation of surface minimizing a functional
Numerical evaluation of complex hypergeometric series
Numerical evaluation of Jacobi elliptic functions
Numerical evaluation of the Bessel functions
Numerical integration methods over the cube
Numerical root-finding methods appropriate with multiple roots
Numerical solution of Maxwell's equations
Numerically integrating monomials on 2- and 3-dimensional cells
Optimal lattice packings in R^n
Optimal packings of small circles in a larger one
Optimal search procedure for boundary of an infinite strip (= shortest path of width 1)
Optimal strategies in 2-player non-zero-sum games?
Orientable 3-manifolds are parallelizable
Orthogonal polynomials satisfy coupled 2-term recurrence relations
Outline of Banach-Tarski construction and generalizations
Outline of Runge-Kutta method (ODE) and Jenkins-Traub method (polynomial root-finding)
Overview of operations with wavelets
Overviews of fractional order derivatives
Partial quotients in continued fraction for Pi roughly follow the Gauss-Kusmin distribution
Piecewise-linear versions of Gauss-Bonnet curvature theorem
Pointer to Clenshaw-Curtis method of integration
Pointer to Runge Kutta 8th order methods of solving differential equations
Pointer to Traveling Salesman Challenge for code comparison
Pointer to model adjusting for magnetic field on earth
Pointer to paper on paper-folding
Pointer: codes for random number generation
Pointer: Fitting an ellipse to data in the plane
Pointer: derivation of formula for number of derangements
Pointer: status of (double-)Mersenne numbers
Pointers to Bin-packing algorithms
Pointers to reviews of computer algebra systems
Pointers to Fast Fourier Transform codes
Pointers to Erdos problems in graph theory
Pointers to evaluations of basic functions (exp, ln) on matrices
Pointers to Laplace Transform data
Pointers to proofs that pi=3.14... is irrational
Pointers to results on non-negative matrices (Perron-Frobenius theory)
Pointers, summaries of articles on history of types of integration
Pointers: Chinese Postman Problem
Points in the plane which are integral distances apart
Points in the plane with only two distances between them
Points of order 13 on elliptic curves
Polynomial with all roots on unit circle
Polynomial-time algorithm to count paths in acyclic digraphs
Polynomials taking prime values; Friedlander and Iwaniec theorem
Polynomials which are positive across the real line
Population dynamics computed via Leslie matrices
Possible torsion in elliptic curves over number fields
Power method: successive Rayleigh quotients converge to dominant eigenvalue of a matrix
Press release describing factorization of RSA-140
Primes in arithmetic progressions mod 11, and approximations of pi
Probability all roots of a random polynomial are real
Probability an event occurs (or not) in a Poisson process
Probability distribution of waiting time until n-th success: negative binomial (Pascal) distribution
Probability that adjacent cards have different values (standard deck)
Proof of Weierstrass Approximation theorem using Bernstein polynomials
Proof of the Cayley-Hamilton Theorem
Proof of the Rational Roots Theorem
Proof that Euler's constant e=2.718... is irrational
Proof that a Poisson distribution is a limit of Bernoulli distributions?
Proofs of Descartes' Rule of Signs
Properties of Bernoulli numbers
Properties of the Perrin sequence, references
Proving two curves crossing a square must intersect, using the Brouwer Fixed Point Theorem
Proving quadratic reciprocity with Stokes' Theorem
Proving orthonormality of Legendre polynomials
Proving weak estimates for Ramsey numbers R(k,k)
Quadratic Gauss sum
Quaternio terminorum and other types of logical fallacies
Question on numerical solution of the diffusion equation
Quick integer square root algorithm
Rational solutions to x^4 + y^4 = 17
Recognizing a tensor product of operators on Hilbert spaces
Reconciling different versions of the Hahn-Banach theorem; which depend on the Axiom of Choice?
Recurrence sequences with prime values at prime locations
Recursive formula for the Ramanujan Tau function
Recursive formulas for partition functions
Reduce Greechie-diagram isomorphism question to graph-isomorphism
Reducing a particular second-order PDE to the Heat Equation
Reducing relations among Appell hypergeometric functions
Reducing some non-elementary algebraic antiderivatives to elliptic integrals (Riemann surfaces)
Reeb, Hopf foliations of S^3
Reference: cluster algorithms for data points in R^3
Reference: algorithms for solving ordinary differential equations numerically
References on Catalan numbers
References to the Stable Marriages theorem
Regular, semiregular polyhedra and the disphenoid
Regularization and (pseudo-) inverses of ill-conditioned matrices
Relation of Axiom of Choice to existence of algebraic closures of a field, bases for vector spaces
Representation by the quadratic forms x^2 +- x y + y^2
Representation of integers by quadratic forms in three variables
Results on analytic functions parallel to those of algebraic geometry
Review of selected literature on twin primes
Rewriting trigonometric integrals as algebraic (elliptic) ones
Rings with isomorphic modules of different ranks (dimension shifting in K-theory)
Rodrigues formulae for Laguerre polynomials
Runge's theorem giving a constructive bound on the number of solutions to certain Diophantine equations in two variables
S-unit equation; finitude of solutions to Sum(a^{x_i})=power
Sample counterintuitive geometric probability question
Sample factorization output using MIRACL program
Sample fit to curve with nonlinear parameters
Sample numerical calculations showing non-unique positions possible for Stewart platform
Sample system of polynomial equations to solve modulo n
Sample use of Galois theory to solve a particular octic polynomial in radicals
Sensitive tests for (conditional) convergence of a real series
Separating distributions from smooth functions with operators
Sequence rapidly converging to the Arithmetic-Geometric Mean of two numbers
Sequences such as {(n*alpha, n*beta)} are dense on the torus
Sets generated from one set using closure, complement, and union
Sets of numbers with good continued-fraction approximations (Mahler, Koksma)
Shape of a rotating chain with top end fixed
Simple approach to Fermat's Last Theorem, case I, some primes
Simple example of hypergeometric distribution
Simplicity of linear groups
Small solutions to 2^n = s mod n
Smallest matrix norms match largest eigenvalue
Smallest non-associative pseudogroup
Smooth dynamical systems with symmetry over domain
Software announcement: control systems analysis toolbox for O-Matrix
Solution of cubic equations (and higher)
Solution of sample stochastic differential equation
Solution to X^2 - D Y^2 = 1 in polynomials D,X,Y
Solutions to y^q=1+x+x^2+...+x^r (Ljunggren)
Solutions to a x^2 + b y^3 = z^6 and the elliptic curves y^2=x^3+D
Solutions to x^3 + y^3 = z^3 + 1
Solutions to y^2 = x^m + 1 (Catalan)
Solutions to x^2 + 2 = y^3
Solutions to x^3 + y^2 = z^2 ?
Solutions to x^3 + y^3 = z^n ?
Solutions to x^p + y^p = p z^p ?
Solutions to x^3 + y^3 = a z^3?
Solve f o f = exp to find f
Solving PDEs with discontinuous coefficients (Richards equation)
Solving Toeplitz linear systems
Solving a tridiagonal n x n linear system
Some (mostly) algebraic expressions which are nearly integral
Some quartic equations in two variables (exercises from Rosen)
Some nested radical expressions simplify
Some nested radical expressions simplify
Some basics on Topological Vector Spaces
Some good(?) random number generators, with C code, comparisons
Some less-trivial applications of L'Hospital's Rule
Some literature on Jacques Tacquet (1612-1660)
Some primes dividing 1 + googolplex (10^(10^100))
Some thinking about a fair 3-sided coin
Sources of software for computational chemistry
Spearman rank-order correlation
Spectrum of a matrix of distances like Cayley-Menger matrix
Spherical geometry, pointers to earth-mapping
Stability of an elementary autonomous system of ordinary differential equations
Standard definition of Riemann integral
Statement of Fourier's theorem (representation by series)
Statements equivalent to normality
Statements from Calculus which are more or less equivalent to the completeness of the reals
Statistical mechanical treatment of magnetism (Ising, Heisenberg models)
Statistical tests to use for comparing populations (t-test, chi-squared)
Status of the Burnside Problem: if all elements have bounded order, when is the group finite?
Stieltjes' expansion of zeta(s) near s=1
Striking data on the Collatz conjecture
Strings not containing two identical consecutive substrings
Structure of groups of order pq^2
Structure theory for arbitrary Abelian groups
Suggested method to compensate for Gibbs' phenomenon
Sum of ln ln p for all primes less than P
Summary of CORDIC algorithm for computing trigonometric functions
Summary of Fresnel integrals
Summary: hierarchy of elementary particles (standard model)
Surface area of spherical triangle in terms of sides of the triangle
Survey of good algorithms to optimize quadratic function under linear constraints
Table of contents for the list of Frequently-Asked Questions in Mathematics which appears in the newsgroup sci.math
Taylor series convergence at endpoints, using Abel summation as Stieltjes integrals
Terminal velocity of a coin falling through air (stable approximations)
Textbooks, overview of the Lambda Calculus
The Taylor series for arcsin -- why is it so slow to converge?
The canonical line bundle over projective space
The Borel-Cantelli lemma: iterates of measure-preserving maps escape sets of finite measure
The Cantor function
The p-colorability of knots and the dihedral groups
The compact-open topology on function spaces X^Y
The Eikonal Equation in geometrical optics
The Halting Problem for Turing machines
The Inverse Galois Theory problem -- is every group a Galois group?
The Kissing numbers (numbers of congruent spheres which touch each other in lattices)
The Law of the Unconscious Statistician
The p-adic Waring's problem
The sum of n independent, identically distributed exponentials
The Robin boundary condition
The Spectral Radius Formula (for operator norm of matrices)
The subset-sum problem (minimize the sum of a set of reals)
The symmetric derivative and domains of symmetric differentiability
The tangent bundle and tensor bundles over manifolds
The worm problem -- minimal convex set containing all length-1 curves
The Golay Code and the game of Mogul
The can of paint with finite volume but infinite surface area (Gabriel's horn)
The role of the Mean Value Theorem
The unique magic hexagon
There are no four squares in arithmetic progression
There is not necessarily an optimal strategy in N-player games for N greater than 2
Three rational right triangles on the same hypotenuse whose areas are in arithmetic progression?
Three rational right triangles on the same hypotenuse whose areas sum
Tiling the plane with noncongruent equilateral triangles
Todd and the odd number 6
Topologies on a power set (Vietoris topology, Pixley-Roy topology)
Topology of space of functions with compact support -- Banach space?
Triangles with integer sides and area, two sides being consecutive squares
Triangular number which is also a square-pyramidal number
Tricks for evaluating integrals (add a parameter and differentiate with respect to it)
Tutorials on Finite Element Methods
Two different types of random walks on Z^n
Two nonisomorphic Banach spaces, each isomorphic to a subspace of the other
Types of pentagons which tessellate the plane
Typical evaluation of a definite integral with erf(x) in the integrand
Typical symbolic integration showing the need to assist faulty software (Maple)
Unbounded number of regions as intersection of convex polygons
Updating the solution to a linear system after a small change in the coefficient matrix (Sherman-Morrison-Woodbury Formula)
Use DeMoivre-Laplace limit theorem to test hypotheses of distribution
Use Groebner bases or resultants on polynomials with inexact coefficients?
Use generating function to evaluate Beta function definite integrals
Use of incomplete Beta function to predict population proportions from a sample
Use of a probability generating function to determine distribution of a sequential process
Use the t-test for comparing means
Using Newton's method to divide by using multiplication
Using Dodgson's condensation formula to find determinants of symmetric Toeplitz matrices
Using Kirchhoff's laws to solve geometric combinatorial problems
Using Savitzky-Golay filter to smooth data
Using Calculus of Variations to show straight lines are geodesics
Using Cayley-Menger determinant to determine radius of circumscribed sphere around a tetrahedron
Using Fibonacci numbers to make a geometric puzzler
Using axiom of choice to construct field automorphisms of C
Using linear optimization theorems to test for intersections of half-spaces in R^n
Using multigrid solvers for hyperbolic CFD problems
Using symbolic algebra programs to test injectivity of a function
Using the Steenrod algebra to find linear factors of a multivariate polynomial
Using the web to check the 'standard' symbolic integrals
Values of certain Dirichlet L-functions
Van der Waerden's theorem: sufficiently large collections of integers must contain long arithmetic progressions
Various methods to find (all) the zeros of a complex function
Various musings on the question, What is topology?
Visualizing the 16-cell (solid in R^4)
Weaknesses of Linear Congruential Generators as random number generators
Weight enumeration of Reed-Muller Codes
What algorithm does (should?) Maple use for Sturm sequences?
What are C*-algebras?
What are Borel subgroups of algebraic groups?
What are Lindenbaum Algebras (Boolean algebras in Model Theory)
What are meromorphic functions?
What are nilpotent groups (and why that name?)
What are non-Euclidean Geometries?
What are ruled, developable surfaces
What are Wang dominoes/tiles?
What are the sporadic simple groups?
What does it mean to choose some variables as dependent on others?
What does it means to say one cannot square the circle?
What is alignment in normed linear spaces
What is coding theory?
What is convolution (and more general integration kernels)?
What is Feigenbaum's constant 4.6692...?
What is foam geometrically?
What is Information Theory
What is knot theory
What is renormalization in mathematical physics?
What is Watson's Lemma (asymptotic expansions)
What is a graded algebra?
What is a Lusin space in topology?
What is a Monte Carlo method?
What is a porism? (Poncelet's, Steiner's)
What is a pseudomanifold?
What is does the R^2 statistic measure in a regression?
What is the "Continuum Hypothesis"?
What is the (topological) degree of a map and how is it used?
What is the 2nd derivative test for Lagrange multipliers?
What is the Singular Value Decomposition
What is the method of infinite descent?
What is the Langlands program?
What is the area of the Mandelbrot set?
What is the Prüfer manifold?
What is the Shafarevich group Sha
What is the point of an exact sequence?
What is the probability that two matrices will commute
What numbers are sums of two Egyptian fractions?
What's beyond Bochner integrals? (Pettis integral, Birkhoff integral)
What's new in Singular Integrals
When can this freely-moving linkage be constructed?
When does a manifold admit a smooth nowhere-vanishing vector field?
When is a real algebraic variety a bounded set?
When is the Borel sigma algebra on E x E the square of that on E?
Where can I find a predicate calculus editor/theorem checker?
Where do closure operators fit into category theory? (adjoints)
Where to find proofs of the Jordan Curve Theorem?
Which set-theory axioms are needed to prove there is no Banach-Tarski paradox in the plane?
Which quintics can be solved by radicals?
Which Gaussian integers are sums of two (or more) squares?
Which functions on R are derivatives?
Which graphs embed in a k-cube? (hard)
Which groups embed in their Bohr compactification?
Whitney's Theorem classifying immersions of circles into the plane
Who is Paul Cohen?
Why are Gaussian normal (or Poisson) distributions used in practice?
Why are only certain n-gons constructible?
Why is there no 3-dimensional field extension of R?
Why isn't the Risch algorithm fully implemented in computer algebra systems?
Why the x=tan(u/2) substitution works for ellipse and integration problems
Why use Pade Approximants?
Wiles on the discovery of the gap. Video available.
Will some variant of Eisenstein's Criterion always suffice to prove irreducibility? (no)
With an ordered set of n random variables, probability that all initial sums are less than expected value (about 1/sqrt(n) )
Worst-case code set for a Huffman code generator
Yet another elliptic surface
Zero sets of analytic functions must have appropriate dimensions (Weierstrass Preparation Theorem)
f(x)=2x+1 implies f^n(x) prime for some n? (not necessarily)
Solution of the astronomical Emden or Lane-Emden differential equation
Table showing the number of groups of order N for all N < 200 except N=192.
Table of best known packings of squares into squares