Concept guide
X={0,1,2,3}, Σ_X={∅,{0,1},{2,3},X}; Y={0,1,2}, Σ_Y={∅,{0},{1,2},Y}; Z={0,1} with Σ_Z the power set. (1) Prove by complete enumeration that f=(0↦0,1↦0,2↦1,3↦2) is measurable and g=(0↦0,1↦1,2↦1,3↦2) is not; (2) prove that a composite of measurable maps is measurable, and read φ∘f back as an indicator function; (3) prove that an indicator 1_E is measurable exactly when E∈Σ_X; (4) use m and k to show that 'checking images' can both wrongly reject and wrongly accept.
Measurable function
derivation-mko-measurable-function-01
Concept guide
The states are ordered 1, 2, 3; the transition matrix P has rows (1/2,1/2,0), (1/4,1/2,1/4), (0,1/3,2/3), and the initial law is μ_0=(1/2,1/2,0). (1) Check that P is row-stochastic and explain why column sums need not be 1; (2) prove the one-step formula μ_{n+1}=μ_nP and that the result is again a probability vector, and compute μ_1 and μ_2; (3) prove the path-probability formula and compute P(X_0=1,X_1=2,X_2=3) and P(X_0=2,X_1=3,X_2=3); (4) explain why P alone generally does not determine complete unconditional marginal and path laws.
Finite-state Markov chain
derivation-mko-finite-markov-chain-01
Concept guide
For a quadratic f with coefficients in the order [1,x,y,x²,xy,y²], prove from the difference-quotient limit that ∂f/∂x=c_1+2c_3x+c_4y and ∂f/∂y=c_2+c_4x+2c_5y. Then for c=(1,−2,3,1,2,−1) compute the partials and the gradient (4,1) at (1,2), and show that a single nonzero-h quotient (5 or 41/10) and moving both coordinates at once (limit 5) are not the x-partial.
Partial derivative
derivation-mko-partial-derivative-01
Concept guide
In ℝ² let v_1=(0,0), v_2=(6,0), v_3=(0,6). (1) Check that λ=(1/6,1/3,1/2) are convex weights and reconstruct w=(2,3); (2) show that zero weights are allowed and that two-point convex combinations are segment points; (3) prove that u=(4,4) is not a convex combination of v_1, v_2, v_3 (its only affine weights include −1/3), with a second proof by a linear-functional bound; (4) show that three weights of 0.333 do not qualify; (5) after adding v_4=(3,3), exhibit two different convex weight vectors for the same point.
Finite convex combination
derivation-mko-finite-convex-combination-01
Concept guide
On ℝ², take the linear constant-coefficient operator L(u)=u_xx+2u_xy−3u_yy+u_x and the right-hand side g(x,y)=2x+y−2. First classify L using the explicit principal-part convention a u_xx+b u_xy+c u_yy with discriminant b²−4ac; then, for the three quadratic polynomial candidates u_A=x²+xy+y², u_B=x²+2xy and u_C=x²+xy+y²+y, compute the symbolic residual R=L(u)−g exactly, decide which candidates satisfy L(u)=g on all of ℝ², and explain why u_B is still not a solution although its point residual vanishes on the whole line y=−8.
Partial differential equation
derivation-mko-partial-differential-equation-01
Concept guide
In the real vector space ℝ²: (1) prove that the closed halfspace H={(x,y) ∣ 2x−3y≤6} is convex; (2) prove that the nonempty axis-aligned box B=[−1,2]×[0,3] is convex; both proofs must hold for all p, q and all λ∈[0,1]. (3) Refute the convexity of the finite point set F={(0,0),(2,0),(0,2)} with one exact midpoint witness, and explain why checking finitely many λ, or failing to find a counterexample, cannot prove convexity.
Convex set
derivation-mko-convex-set-01
Concept guide
On the rectangle Ω=(0,2)×(0,1) declare the BVP: in the interior Δu=u_xx+u_yy=4; Dirichlet data u(0,y)=y² on the left, u(2,y)=y²+2y+4 on the right (0≤y≤1), u(x,0)=x² on the bottom and u(x,1)=x²+x+1 on the top (0≤x≤2). For the quadratic polynomial candidates u=x²+xy+y² and v=2x²+xy−2x+y, record separately the interior Poisson obligation (a symbolic Laplacian identity) and the Dirichlet obligation on all four edges (exact restriction to each whole edge, then coefficient comparison); decide which candidate satisfies this declared BVP, and explain why v fails although its interior equation holds and it matches the data at all four corners.
Boundary-value problem
derivation-mko-boundary-value-problem-01
Concept guide
(1) Prove that f(x)=2x²−3x+1 is convex on the interval I=[−1,4]: first confirm that I is convex, then prove f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y) for all x,y∈I and all λ∈[0,1], by showing that the Jensen gap of a quadratic equals aλ(1−λ)(x−y)². (2) From the same identity, the affine function ℓ(x)=3x−5 gives equality on I. (3) Refute the convexity of g(x)=−x²+x on J=[0,2] with one exact Jensen violation, and explain why a few passing samples cannot prove convexity.
Convex function
derivation-mko-convex-function-01
Concept guide
Problem P: minimize f(x)=x²−6x+10 over the feasible domain X=[0,5]. (1) Check separately that the domain and the objective are convex, so that P qualifies as a convex problem; (2) decide feasibility of the candidates x₁=1, x₂=4, x₃=−1; (3) evaluate f(1) and f(4) and compare the two feasible candidates; (4) explain why the candidate with the lower objective value is not thereby an optimal solution of P, and use P′: minimize x² over X′=(0,1], to show that a convex problem need not have a minimizer.
Convex optimization problem
derivation-mko-convex-optimization-problem-01
Concept guide
On the carrier G={0,1,2,3} a complete 4×4 Cayley table is given (the cell in row a, column b is a*b): row 0 is (1,2,3,0), row 1 is (2,3,0,1), row 2 is (3,0,1,2), row 3 is (0,1,2,3). Check closure, associativity, a unique identity and inverses in that order, then check a*b=b*a for all a, b, to decide whether (G,*) is a finite abelian group; find the orders of the elements 0 and 1 and the cyclic subgroups they generate; and use the symmetric table a∘b=|a−b| on {0,1,2} to show that a symmetric table cannot replace group qualification.
Finite abelian group
derivation-mko-finite-abelian-group-01
Concept guide
On ℝⁿ (n≥1 fixed), for all vectors u, v and all reals a, prove that ‖v‖₁=Σ|v_i| and ‖v‖_∞=max|v_i| both satisfy the three norm axioms: nonnegativity with ‖v‖=0 ⇔ v=0, ‖av‖=|a|‖v‖, and ‖u+v‖≤‖u‖+‖v‖. Then, in ℝ³, compute both norms and instances of the triangle inequality and homogeneity for the integer-coordinate vectors u=(3,−4,1) and v=(−1,2,5); finally show that φ(v)=|v₁| and ψ(v)=(‖v‖₁)² are not norms.
Real normed vector space
derivation-mko-real-normed-vector-space-01
Concept guide
(1) Prove that x_n=1/n (n≥1) is a Cauchy sequence in (ℝ, |s−t|): for every ε>0 give N such that |1/m−1/n|<ε for all m,n≥N. (2) Prove that the same sequence is still Cauchy in X=(0,1] with the restricted metric, but has no limit in X. (3) Prove that y_n=(−1)^n is not Cauchy. (4) Prove that the ultimately constant sequence z (z₁=0, z₂=7, z_n=5 for n≥3) is Cauchy. Also explain why a finite tail cannot prove the Cauchy property.
Cauchy sequence
derivation-mko-cauchy-sequence-01
Concept guide
Given the support S={(0,0),(0,1),(1,0),(1,1),(2,1)} with masses p(0,0)=1/6, p(0,1)=1/3, p(1,0)=1/4, p(1,1)=1/12, p(2,1)=1/6, verify that (S,p) is a two-dimensional finite joint distribution; from this one joint law compute the marginals of X and Y, the rectangle event P(X∈{1,2}, Y=1) and the single-pair mass P(X=2, Y=0); finally construct a different joint law with the same marginals, showing that two marginals do not in general determine the joint distribution.
Finite joint distribution
derivation-mko-finite-joint-distribution-01
Concept guide
On one exact finite joint law: (1) prove Cov(Y,X)=Cov(X,Y); (2) prove Cov(aX+b,cY+d)=ac·Cov(X,Y); (3) prove that independence of X and Y gives Cov(X,Y)=0, so a nonzero covariance refutes independence; (4) for the joint law with X uniform on −1, 0, 1 and Y=X², compute E[X], E[Y], E[XY] and Cov(X,Y)=0, and prove that X and Y are not independent.
Covariance
derivation-mko-covariance-01
Concept guide
G has vertices 1, 2, 3, 4 and edges {1,2}, {2,3}, {3,4}, {1,4}; H has vertices a, b, c, d and edges {a,c}, {b,c}, {b,d}, {a,d}. Prove that f: 1↦a, 2↦c, 3↦b, 4↦d is a graph isomorphism G→H; show that the bijection g: 1↦a, 2↦b, 3↦c, 4↦d is not an isomorphism; and prove that the path P (edges {1,2}, {2,3}, {3,4}) is not isomorphic to H, even though f sends every edge of P to an edge of H.
Graph isomorphism
derivation-mko-graph-isomorphism-01
Concept guide
On X={0,1,2,3,4} the relation R is listed completely as 13 ordered pairs: (0,0), (1,1), (2,2), (3,3), (4,4), (0,2), (2,0), (1,3), (3,1), (1,4), (4,1), (3,4), (4,3). Check reflexivity, symmetry and transitivity item by item, prove that R is an equivalence relation and emit its classes; then show that R′, obtained by removing (3,4) and (4,3), is reflexive and symmetric but not an equivalence relation.
Equivalence relation
derivation-mko-equivalence-relation-01
Concept guide
On the carrier X={0,1,2,3}, complete addition and multiplication tables ⊕ and ⊙ (addition and multiplication mod 4) are given. Prove for all pairs and triples: (X,⊕) is an abelian group; ⊙ is associative; ⊙ distributes over ⊕ on both sides; there is a unique two-sided multiplicative identity 1, and 1≠0. Note also that 2⊙2=0 does not stop it from being a unital ring.
Finite unital ring
derivation-mko-finite-unital-ring-01
Concept guide
Let (V,‖·‖) be a real normed vector space. Prove that d(x,y)=‖x−y‖ is nonnegative, separates points, is symmetric and satisfies the triangle inequality; prove that the l1 and l∞ formulas on ℝⁿ are norms, so both induce metrics; then compute d1 and d∞ exactly in ℝ³ for x=(1,−2,3), y=(4,0,−1), z=(2,5,1).
Norm-induced metric
derivation-mko-norm-induced-metric-01
Concept guide
Prove that every finite metric space (X,d) with at least one point is complete (immediate for a single point; with at least two points, the least positive distance δ forces every Cauchy sequence to be ultimately constant). On X0={0,1,3} with d(x,y)=|x−y|, compute δ=1, give an ultimately constant certificate and an explicit non-Cauchy sequence; finally prove that the subspace Y=(0,1] is not complete: x_n=1/n is Cauchy in Y but has no limit in Y.
Complete metric space
derivation-mko-complete-metric-space-01
Concept guide
Let A=[[1,2],[−1,3]] and B(x,y)=x^T A y for x, y∈ℝ². Prove separately that B is linear in the first slot and in the second slot (bilinear); evaluate exactly B(x,y)=7 and B(y,x)=−20 for x=(2,−1), y=(1,4), showing that bilinearity does not require symmetry; and prove that the black-box function f(x,y)=x1y1+(x2³−x2)y2 passes some additivity samples but is not bilinear.
Real bilinear form
derivation-mko-real-bilinear-form-01
Concept guide
Let A be the 2-by-3 integer matrix with first row (2,−1,0) and second row (1,3,4), and let T:ℝ³→ℝ², T(x)=Ax. (1) Prove that T(αu+βv)=αT(u)+βT(v) for all u,v∈ℝ³ and all reals α,β. (2) Prove that S(x)=Ax+b with b=(1,0) is not linear, and that for every b∈ℝ² the map x↦Ax+b is linear if and only if b=0. (3) Show that the black box F(x)=Ax+(x₁²−x₁,0) passes one sample test and yet is not linear.
Linear map between real vector spaces
derivation-mko-general-linear-map-01
Concept guide
Fair-die model: Ω={1,2,3,4,5,6}, each outcome with mass 1/6. Let A={2,4,6}, B={1,2,3,4}, D={1,3,5} and Z=∅. (1) Decide by the product definition whether A and B are independent, and with P(B)>0 check P(A|B)=P(A). (2) Prove that independence of A and B implies independence of A and Bᶜ, and check it numerically. (3) Decide whether the mutually exclusive events A and D are independent. (4) Decide whether A and the zero-probability event Z are independent, and explain why the conditional form cannot be used.
Event independence
derivation-mko-event-independence-01
Concept guide
Finite undirected loop-free simple graph G: V={0,1,2,3,4}, E={{0,1},{1,2},{2,3},{0,3},{1,3},{3,4}}. For the vertex sequences W₁=(0,1,2,3,4), W₂=(0,1,3,1,2), W₃=(0,2,3), W₄=(2,3,0,1,3,4), W₅=(4) and W₆=(1,1), decide whether each is a walk, its length, and whether it is a simple path; also prove that a simple path repeats no edge and that in this graph a simple path has length at most 4.
Graph walks and simple paths
derivation-mko-graph-walk-simple-path-01
Concept guide
(1) On ℤ₆={0,1,2,3,4,5} (a⊕b=(a+b) mod 6, a⊗b=(ab) mod 6), already qualified as a finite unital ring, check a⊗b=b⊗a for all 36 ordered element pairs and decide whether ℤ₆ is a finite commutative ring. (2) Show that ℤ₆ has nonzero zero divisors and that 2 is not invertible, so it is neither an integral domain nor a field. (3) In the qualified ring U of 2-by-2 upper-triangular matrices over 𝔽₂, show that several products commute and yet U is not commutative.
Finite commutative ring
derivation-mko-finite-commutative-ring-01