# 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=11#results)[1](/en/derivations/?page=1#results) … [11](/en/derivations/?page=11#results) 12 [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=13#results) 299 results · Page 12 of 13 Concept guide ## [Direct MSE without the decomposition formula](/en/derivations/derivation-mko-estimator-mean-squared-error-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Weight losses directly at every theta, leaving decomposition for later. Mean squared error of an estimator derivation-mko-estimator-mean-squared-error-01 Concept guide ## [The two-way point correspondence for slack variables](/en/derivations/derivation-mko-linear-program-standard-form-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Convert a fixed maximize/≤/nonnegative LP to equality slack form and verify both feasible-point directions and objective. Standard form of a linear program derivation-mko-linear-program-standard-form-01 Concept guide ## [Vertex covers and deletion certificates on a star](/en/derivations/derivation-mko-vertex-cover-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Audit covers edgewise and distinguish minimal from minimum. Vertex cover derivation-mko-vertex-cover-01 Concept guide ## [The correct image isomorphism for a nonsurjection](/en/derivations/derivation-mko-first-isomorphism-theorem-finite-groups-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) For f:C4 to C4,f(a)=2a mod 4, C4/{0,2} is isomorphic to {0,2}, not the full codomain C4. First isomorphism theorem for finite groups derivation-mko-first-isomorphism-theorem-finite-groups-01 Concept guide ## [Certifying the center from all commutation rows](/en/derivations/derivation-mko-finite-group-center-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself. Center of a finite group derivation-mko-finite-group-center-01 Concept guide ## [Bias-variance derivation with the cross term visible](/en/derivations/derivation-mko-bias-variance-decomposition-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Prove the decomposition symbol by symbol in one law and check both theta values. Bias-variance decomposition derivation-mko-bias-variance-decomposition-01 Concept guide ## [The weak-duality chain and a zero-gap optimality certificate](/en/derivations/derivation-mko-linear-program-duality-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Construct the full dual, check feasibility of x=(1,3), y=(1,1), and certify both optimal by zero gap. Linear-programming duality derivation-mko-linear-program-duality-01 Concept guide ## [All four cuts and an equal-value two-sided certificate](/en/derivations/derivation-mko-max-flow-min-cut-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Enumerate every s-t cut of a four-vertex network and give a feasible flow of value 5 together with a cut of capacity 5. Max-flow min-cut theorem derivation-mko-max-flow-min-cut-theorem-01 Concept guide ## [All eight subsets for minimum cover on P3](/en/derivations/derivation-mko-minimum-vertex-cover-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Classify every cover and all minimizers. Minimum vertex cover derivation-mko-minimum-vertex-cover-01 Concept guide ## [A complete finite reduction with preserved direction](/en/derivations/derivation-mko-many-one-reduction-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Check every x equivalence, composition and decider transfer. Many-one reduction derivation-mko-many-one-reduction-01 Concept guide ## [A complete contraction certificate on a finite metric](/en/derivations/derivation-mko-contraction-mapping-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Audit completeness, a unique fixed point and the error bound for a constant self-map on a three-point discrete metric. Contraction mapping theorem derivation-mko-contraction-mapping-theorem-01 Concept guide ## [A polynomial bracket and an exact intermediate value](/en/derivations/derivation-mko-intermediate-value-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Find value zero for f(x)=x^2-1 on [0,2], retaining endpoint and nonuniqueness boundaries. Intermediate value theorem derivation-mko-intermediate-value-theorem-01 Concept guide ## [An exact Lagrange remainder for a cubic](/en/derivations/derivation-mko-taylor-remainder-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) For f(x)=x^3, a=1, n=2, x=2, audit the exact remainder and a qualified derivative bound. Taylor remainder theorem derivation-mko-taylor-remainder-theorem-01 Concept guide ## [MLE with a positive-mass tie and an all-zero likelihood column](/en/derivations/derivation-mko-finite-maximum-likelihood-estimator-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Compute every argmax and a total estimator using a fixed tie rule. Maximum-likelihood estimator in a finite model derivation-mko-finite-maximum-likelihood-estimator-01 Concept guide ## [A sufficiency certificate from every fibre conditional law](/en/derivations/derivation-mko-finite-sufficient-statistic-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Audit invariance across theta on every positive-mass fibre, retaining undefined null conditions. Sufficient statistic in a finite model derivation-mko-finite-sufficient-statistic-01 Concept guide ## [All permutations and a Kahn trace for a three-vertex DAG](/en/derivations/derivation-mko-topological-ordering-dag-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) List every valid order and audit source-removal invariants. Topological ordering of a directed acyclic graph derivation-mko-topological-ordering-dag-01 Concept guide ## [The first power returning to one](/en/derivations/derivation-mko-multiplicative-order-mod-n-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) ord_8(-5)=ord_8(3)=2, with phi(8)=4 and 2 dividing 4 but not equal to it. Multiplicative order modulo n derivation-mko-multiplicative-order-mod-n-01 Concept guide ## [The direction of eventual comparison and finite prefixes](/en/derivations/derivation-mko-series-comparison-test-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Prove convergence using geometric upper comparison despite a finite prefix, and show divergent lower comparison. Comparison tests for series derivation-mko-series-comparison-test-01 Concept guide ## [Ratio limits, geometric domination and L=1](/en/derivations/derivation-mko-ratio-test-series-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Use complete rational ratios for L<1 convergence and L>1 divergence, then audit indecision at L=1. Ratio test for series derivation-mko-ratio-test-series-01 Concept guide ## [An epsilon/3 transfer through one fixed approximant](/en/derivations/derivation-mko-uniform-limit-continuity-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Prove uniform convergence of affine perturbations on [0,1] and transfer continuity step by step. A uniform limit of continuous functions is continuous derivation-mko-uniform-limit-continuity-theorem-01 Concept guide ## [Every open set in a four-point product](/en/derivations/derivation-mko-finite-product-topology-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Generate every product open from two Sierpinski factors and audit projections. Finite product topology derivation-mko-finite-product-topology-01 Concept guide ## [The full preimage condition determines the quotient topology](/en/derivations/derivation-mko-quotient-topology-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Compute a quotient topology for a three-to-two-point surjection and show an open-image failure. Quotient topology derivation-mko-quotient-topology-01 Concept guide ## [Universal paths and concatenation in a convex box](/en/derivations/derivation-mko-path-connected-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Prove [0,1]^2 path connected and audit a two-piece concatenation. Path-connected space derivation-mko-path-connected-space-01 Concept guide ## [Finding inclusion-maximal components from the complete subset table](/en/derivations/derivation-mko-topological-connected-component-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D12) Find every component of X={a,b,c}, tau={empty,{a},{b,c},X}. Connected component of a topological space derivation-mko-topological-connected-component-01 [← Previous](/en/derivations/?page=11#results)[1](/en/derivations/?page=1#results) … [11](/en/derivations/?page=11#results) 12 [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=13#results) 299 results · Page 12 of 13