RefacteTest the next correction
Compare how additional terms permit a correction or rule it out.
Make first order exactMake a hidden cubic appearCoordinate repairFurther coefficient examples
Mathematical writing
Mathematics
Choose a question. Follow a complete argument.
14 worked articles, followed by 10 notes and problem collections. Each text states its background and develops its own argument; the subject groups are starting points, not a required course.
Where would you like to begin?
About the approach →Matrices and graphs
Separate a recurrence, count spanning trees, and prove a matrix identity.
Explore 6 articlesGroups and labelled paths
Turn words and edge labels into calculations with paths and loops.
Explore 4 articlesPolynomial equations and calculus
Make first-order change exact, then test what happens at the next order.
Explore 4 articlesMatrices and graphs
Separate a recurrence, count spanning trees, and prove a matrix identity. A first example →
Choose a weighted difference, derive the weight, and turn a coupled recurrence into a single update.
Background: Algebra and a second-order recurrence.
Build the incidence matrix and watch its minors select exactly the spanning trees.
Background: Matrices and determinants; Cauchy–Binet is stated and applied.
Augment walks with reserved cycles, then pair terms with equal edge products and opposite signs to prove Cayley–Hamilton.
Background: Matrix multiplication and the permutation definition of a determinant.
Multiply block matrices, derive the compatibility map and the allowed gauge map, then row-reduce.
Background: Block matrices and solving linear equations over a field.
Build a lowering operator, solve its coefficient recurrence, and construct chains of every required length.
Background: Finite-dimensional linear algebra and polynomial arithmetic; characteristic zero is explicit.
Choose the least available edge and pair terms by an involution that reverses their signs.
Background: Finite graphs, subsets, and inclusion–exclusion.
Groups and labelled paths
Turn words and edge labels into calculations with paths and loops. A first example →
Turn generators into loops, fold their shared moves, and read membership and intersections from paths.
Background: Words, inverses, and cancellation; the group language is introduced as needed.
Choose a root, propagate coordinate changes along a spanning tree, and read the remaining circuit products.
Background: Graphs, multiplication and inverses in a group.
Traverse the two-sided graph, retain both word labels, and calculate the intersection group attached to each component.
Background: Finite groups, subgroups, left and right multiplication.
Separate the total sum from the remaining labels and calculate the face rules in those coordinates.
Background: Tuples and addition in an abelian group; all operations are defined.
Polynomial equations and calculus
Make first-order change exact, then test what happens at the next order. A first example →
Choose an arithmetic in which first-order change is exact, then restore the next term to expose a lifting obstruction.
Background: Polynomial expansion and the product rule.
Change variables, eliminate a relation, and carry the residual through successive coefficients.
Background: Polynomial expansion and coefficients of a power series.
Cancel a quadratic defect, calculate the next one, and see directly why every cubic repair fails.
Background: Polynomial expansion and arithmetic modulo three.
Integrate a selected edge parameter, then track the rational or logarithmic expression left behind.
Background: One-variable integration and nonnegative improper integrals.
Notes and problems
Further arguments in algebra, arithmetic, and analysis. The full texts and downloads are linked from each page.
Algebra & geometry
When a Formal Logarithm Fails
Change coordinates one degree at a time: a quadratic repair succeeds, but a cubic obstruction remains in characteristic three. A filtered complex identifies the precise failure.
Background: Polynomial expansion and characteristic-three arithmetic suffice for the first complete calculation in §1.
Divisibility of Separated Polynomials
A free-module expansion and a diagonal remainder calculation expose when divisibility of polynomial differences forces composition through a common polynomial.
Background: Polynomial rings and basic field theory.
When a nonlinear law becomes additive
Determine exactly when a parameterized multiplication law becomes addition, construct the coordinate and its inverse, and classify the cubic term retained by every coordinate beginning with t.
Background: Polynomial multiplication and arithmetic modulo a prime.
Number theory
IBM Ponder This July 2010
Collect algebraic conjugates into an integer trace recurrence, then use a companion matrix to turn the rounding question into finite modular arithmetic.
Background: Algebraic numbers, polynomial roots, and modular arithmetic.
Extracting the First Prime Greater than m
Build a finite arithmetic selector for the first prime above a bound, using factorial congruences, greatest common divisors, and valuations.
Background: Elementary number theory, factorials, and divisibility.
Analysis & topology
A Weighting Argument for Pointwise Convergence to Convergence in Measure
Choose a positive measurable weight that changes the scale at each point, turning almost-everywhere convergence into convergence in measure for the weighted sequence.
Background: Measure theory and convergence of measurable functions.
A Note on Disjoint Decompositions of [0,1) into Closed Sets
A carefully chosen closed remainder exposes the Baire-category obstruction to a countable partition of [0,1) into sets closed in the real line.
Background: Closed sets and the Baire category theorem.
Lebesgue's Criterion for Riemann Integrability
Prove the sufficiency direction of Lebesgue’s criterion: local oscillation and a small discontinuity set give finite control of a Darboux sum.
Background: Riemann integration, compactness, and sets of measure zero.
Products Preserve Riemann Integrability
Insert one mixed term into a product difference, bound the separate oscillations, and sum over a common partition to preserve Riemann integrability.
Background: Bounded functions, upper and lower Darboux sums, and partition refinement.
Problems
Selected Problems in Discrete Mathematics and Analysis
Six problems invite different choices of what to count, track, or record: congruences, recurrences, series, and observation rules. Statements are supplied without solutions.
Background: Counting, elementary number theory, recurrences, and convergence of series.
Optional connections
Starting points ↑Each group suggests a comparison between independently readable pieces.
Turn divisibility into composition
Compare a remainder calculation with a second proof using Galois theory and different hypotheses.
Divisibility of Separated PolynomialsGalois and Frobenius companion · Tensorgami archive
Make a numerical question exact
Recurrences, congruences and valuations give different ways to obtain an exact answer.
Separate a recurrenceControl rounding in the IBM problemExtract the next prime
Pass from local control to a global statement
Compare oscillation estimates, measure bounds and Baire category, each with its own hypotheses.
Products and oscillationsLebesgue’s criterionA measurable weightDisjoint closed sets
Try a problem
Try the six problems without supplied solutions, or study how a worked example chooses its variables.
Read a construction beside an essay
Read cycle exchange beside the proof essay, or take operator chains as an independent branch beside the essay on descriptions.
Cycle exchangeWhat a New Proof Makes PossibleOperator chainsWhen a Description Becomes a Verdict