# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) [← Previous](/en/derivations/?page=8#results)[1](/en/derivations/?page=1#results) … [8](/en/derivations/?page=8#results) 9 [10](/en/derivations/?page=10#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=10#results) 299 results · Page 9 of 13 Concept guide ## [Absolute convergence and the two certificates for alternating harmonic](/en/derivations/derivation-mko-absolute-convergence-of-series-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Distinguish absolute convergence of a negative-ratio geometric series from conditional convergence of alternating harmonic. Absolute convergence of series derivation-mko-absolute-convergence-of-series-01 Concept guide ## [One data function under two parameter laws](/en/derivations/derivation-mko-statistical-estimator-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Validate a total estimator and its parameterwise expectations. Statistical estimator derivation-mko-statistical-estimator-01 Concept guide ## [Constructing a three-modulus CRT class](/en/derivations/derivation-mko-chinese-remainder-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) All integer solutions of x=2 mod 3, x=3 mod 5, x=2 mod 7 are 23+105t. Chinese remainder theorem derivation-mko-chinese-remainder-theorem-01 Concept guide ## [A spanning tree inside a square parent graph](/en/derivations/derivation-mko-spanning-tree-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Audit all vertices, edge membership, connectivity and acyclicity. Spanning tree derivation-mko-spanning-tree-01 Concept guide ## [An exact L-squared tail on a finite law](/en/derivations/derivation-mko-lp-convergence-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Compute the second error moment and give a tail for any tolerance. Lp convergence derivation-mko-lp-convergence-01 Concept guide ## [Unique decomposition and coverage in R3](/en/derivations/derivation-mko-direct-sum-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified. Direct sum of subspaces derivation-mko-direct-sum-01 Concept guide ## [Dual basis and covector change](/en/derivations/derivation-mko-dual-vector-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis. Dual vector space derivation-mko-dual-vector-space-01 Concept guide ## [Euclid and back-substitution for signed inputs](/en/derivations/derivation-mko-euclidean-algorithm-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) gcd(-84,30)=6, with (-84)*1+30*3=6. Euclidean algorithm derivation-mko-euclidean-algorithm-01 Concept guide ## [An Euler circuit with an isolated vertex](/en/derivations/derivation-mko-eulerian-trail-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Audit every edge exactly once and the nonisolated support condition. Euler trails and circuits derivation-mko-eulerian-trail-01 Concept guide ## [A homeomorphism between finite chain spaces](/en/derivations/derivation-mko-homeomorphism-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Audit a relabeling homeomorphism using every open preimage. Homeomorphism derivation-mko-homeomorphism-01 Concept guide ## [A complete fibre-pushforward sampling distribution](/en/derivations/derivation-mko-sampling-distribution-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Aggregate equal-valued outcomes of U under both theta laws. Sampling distribution derivation-mko-sampling-distribution-01 Concept guide ## [A Chebyshev tail for fair Bernoulli averages](/en/derivations/derivation-mko-weak-law-large-numbers-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Derive the WLLN from a whole-sequence IID premise. Weak law of large numbers derivation-mko-weak-law-large-numbers-01 Concept guide ## [Excluding every two-color assignment on a triangle](/en/derivations/derivation-mko-chromatic-number-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Prove chi=3 using an upper witness and complete lower-color search. Chromatic number derivation-mko-chromatic-number-01 Concept guide ## [All cliques and independent sets on three vertices](/en/derivations/derivation-mko-clique-and-independent-set-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Classify all2³ subsets to obtain omega and alpha. Cliques and independent sets derivation-mko-clique-and-independent-set-01 Concept guide ## [A bias table over the whole parameter set](/en/derivations/derivation-mko-estimator-bias-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Compute EthetaU-theta at every theta, distinct from observed error. Estimator bias derivation-mko-estimator-bias-01 Concept guide ## [Same-law second and centered moment cross-checks](/en/derivations/derivation-mko-estimator-variance-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Compute variance at all theta values and verify centered definitions. Estimator variance derivation-mko-estimator-variance-01 Concept guide ## [Fermat modulo 7 and a composite boundary](/en/derivations/derivation-mko-fermat-little-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) (-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality. Fermat's little theorem derivation-mko-fermat-little-theorem-01 Concept guide ## [All integer solutions with a negative coefficient](/en/derivations/derivation-mko-linear-diophantine-equation-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) All integer solutions of -6x+9y=15 are x=5+3t,y=5+2t, t in Z. Linear Diophantine equation derivation-mko-linear-diophantine-equation-01 Concept guide ## [Quotient by the diagonal line](/en/derivations/derivation-mko-quotient-vector-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal. Quotient vector space derivation-mko-quotient-vector-space-01 Concept guide ## [A complete six-element action on four points](/en/derivations/derivation-mko-finite-group-action-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) The explicit 24-entry table is a valid group action, but it is neither free nor transitive. Finite group action derivation-mko-finite-group-action-01 Concept guide ## [Finite maxima and unattained interval suprema](/en/derivations/derivation-mko-supremum-and-infimum-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality. Supremum and infimum derivation-mko-supremum-and-infimum-01 Concept guide ## [A finite-chain basis and every generated open](/en/derivations/derivation-mko-topological-basis-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Verify B={{a},{a,b},X} is a basis for the specified topology. Topological basis derivation-mko-topological-basis-01 Concept guide ## [Deriving a Markov bound pointwise](/en/derivations/derivation-mko-markov-inequality-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Prove the finite-law bound and compare its exact tail. Markov inequality derivation-mko-markov-inequality-01 Concept guide ## [Collision and ceiling load in a complete map](/en/derivations/derivation-mko-pigeonhole-principle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D9) Use all fibres to prove a collision and the generalized lower bound. Pigeonhole principle derivation-mko-pigeonhole-principle-01 [← Previous](/en/derivations/?page=8#results)[1](/en/derivations/?page=1#results) … [8](/en/derivations/?page=8#results) 9 [10](/en/derivations/?page=10#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=10#results) 299 results · Page 9 of 13