Tag
Mathematics
65 resources tagged “Mathematics” across the knowledge library.
65 items · page 3 of 3
Guide6 Aug 2026The Algebraic Approach to Constraint SatisfactionThe constraint satisfaction problem, polymorphisms as an algebraic invariant, the Feder–Vardi dichotomy conjecture, the algebraic reformulation, and the 2017 resolution.5 min readRead MoreGuide6 Aug 2026The Boolean Prime Ideal TheoremThe Boolean Prime Ideal Theorem, its equivalent formulations, its strength relative to the axiom of choice, and which results in this collection depend on it.5 min readRead MoreGuide6 Aug 2026The Center of an Algebra and Affine RepresentationThe centre of an arbitrary algebra, its first-order definition, the characterisation of algebras with Z(A) = ∇ as polynomially equivalent to modules, and the commutator programme it opened.5 min readRead MoreGuide6 Aug 2026The Compactness Theorem and its ConsequencesThe compactness theorem, its proof by ultraproducts, the standard consequences including non-standard models, and the systematic list of properties first-order logic cannot express.5 min readRead MoreGuide6 Aug 2026The Congruence Lattice Con AThe congruence lattice: completeness, algebraicity, its position inside Eq(A), simple and subdirectly irreducible algebras read off from it, and the classification of varieties by congruence conditions.4 min readRead MoreGuide6 Aug 2026The Finite Basis Problem after TarskiThe resolution of Tarski's finite basis problem, the positive finite basis theorems obtained since Baker's, and what the negative answer means for how the question is now approached.5 min readRead MoreGuide6 Aug 2026The Irredundant Basis TheoremIrredundant bases, the failure of the exchange property outside vector spaces, and the theorem that the set of irredundant basis sizes of a finitely generated algebra has no gaps.4 min readRead MoreGuide6 Aug 2026The Seventeen Open Problems: Status Then and NowThe source's seventeen open problems listed in full, grouped by section, with what can be said about their status and an explicit account of where certainty ends.6 min readRead MoreGuide6 Aug 2026The Subalgebra Lattice as an Algebraic LatticeThe subuniverse lattice as an algebraic lattice, the Birkhoff–Frink representation theorem, and why an exact characterisation closes the question for arbitrary algebras while leaving the finite case open.4 min readRead MoreGuide6 Aug 2026Three Finite Basis TheoremsThe finite basis problem, Baker's theorem for congruence-distributive varieties, and the further finite basis results in the source, with the proof strategy via bounded subdirectly irreducibles.5 min readRead MoreGuide6 Aug 2026Ultraproducts and Jonsson's LemmaUltraproducts in the algebraic setting, congruence-distributive varieties, Jónsson's lemma and its proof strategy, and the consequences for finitely generated varieties.5 min readRead MoreGuide6 Aug 2026Ultraproducts and Los's TheoremUltraproducts as reduced products over an ultrafilter, Łoś's theorem and its proof, ultrapowers and the elementary embedding, and the applications to compactness and non-standard analysis.5 min readRead MoreGuide6 Aug 2026Universal Algebra: Computation and SourcesThe sourcing policy for this collection: what is written here, what is deliberately not transcribed, the reasoning behind the split, and where to obtain current computational results and literature.6 min readRead MoreGuide6 Aug 2026Universal Algebra: Discipline OverviewStructural overview of universal algebra: what an algebra of arbitrary type is, why congruences rather than substructures carry the quotient theory, how Birkhoff's two theorems organise the subject, and how the five streams of this collection fit together.8 min readRead MoreArticle5 Aug 2026Curve Geometry for CAD: Analytic and Synthetic CurvesThe mathematics of curves in CAD: analytic versus synthetic families, explicit and parametric representation, the tangent vector and why it matters for mass properties and NC programming, and methods for creating 3D curves.7 min readRead MoreGuide4 Aug 2026Measurement and Method: Instruments, Mathematics and Experimental ScienceSeven milestones in roughly a century, after which an engineer could calculate in advance what had previously only been learned by building.8 min readRead MoreGuide4 Aug 2026The Greek Contribution: Abstract Geometry, Mechanics and the First MachinesSix milestones including the single most consequential idea in the whole set: that there are general laws and that they can be known.7 min readRead More