Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 120 minutes, as an unvalidated planning estimate. This collection contains 4 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the public mixed problems before consulting solutions. Define vertex and edge order and retain signs throughout.
Write complete proofs of the general certificates as well as the finite calculations. Signed flows are real variables; do not add unmentioned capacity restrictions.
These worked public cases support practice and revision. Saved answers are not independently verified proofs or a secure examination.
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0 of 4 tasks have recorded responses. No overall mastery score is assigned.
On vertices 1,…,7 take ordered edges e₁:1→2, e₂:2→3, e₃:3→4, e₄:4→1, e₅:2→4, e₆:5→6; vertex 7 is isolated. Let B have −1 at each edge tail and +1 at its head. Determine rank and bases or exact equation descriptions for all four spaces im B, ker B, im Bᵀ, ker Bᵀ. Derive the component-balance solvability certificate. For b=(−2,1,0,1,−3,3,0), give one flow and the full solution set. Explain why merely checking Σbᵢ=0 would be insufficient.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use vertices A,B,C,D and edges A→B, B→D, A→C, C→D in that order. Resistances are r=(1,2,2,4), all positive. Find the signed flow satisfying Bf=(−1,0,0,1) that minimizes E(f)=½Σrₑfₑ². Derive it through a potential p satisfying Rf=Bᵀp, where R=diag(r), and verify both balance and edge equations. Prove global minimality and uniqueness among all feasible real flows using an energy difference, not an unexplained optimization rule. Explain the gauge freedom in p and what can fail if a resistance is zero.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On vertices A,B,C,D,E take edges A→B, A→C and D→E. The demand is b=(−5,2,3,−4,4). Solve Bf=b, then reverse the second and third edge orientations and find the new coordinate vector describing the same physical signed flow. Prove the relation between the two incidence matrices using a diagonal sign matrix. A solver discards rows A and D and checks only B,C,E. State the exact condition under which this preserves the full system. Audit the altered demand b̃=(−6,2,3,−3,4), which still has total sum zero. Is agreement on all retained rows a certificate of a solution?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A chain 1–2–3–4 has conductances 2,3,6 and prescribed v1=3,v4=−1. Interior injections vanish. Compute all potentials/currents, boundary supply and dissipated power. Delete the middle edge: decide uniqueness and recompute. Now add a separate interior edge 5–6 of conductance 4. Classify zero injections, injections (1,−1), and injections (1,0) on this component. Explain why neither global connectedness nor merely counting boundary vertices is the right uniqueness test.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
On vertices 1,…,7 take ordered edges e₁:1→2, e₂:2→3, e₃:3→4, e₄:4→1, e₅:2→4, e₆:5→6; vertex 7 is isolated. Let B have −1 at each edge tail and +1 at its head. Determine rank and bases or exact equation descriptions for all four spaces im B, ker B, im Bᵀ, ker Bᵀ. Derive the component-balance solvability certificate. For b=(−2,1,0,1,−3,3,0), give one flow and the full solution set. Explain why merely checking Σbᵢ=0 would be insufficient.
Use vertices A,B,C,D and edges A→B, B→D, A→C, C→D in that order. Resistances are r=(1,2,2,4), all positive. Find the signed flow satisfying Bf=(−1,0,0,1) that minimizes E(f)=½Σrₑfₑ². Derive it through a potential p satisfying Rf=Bᵀp, where R=diag(r), and verify both balance and edge equations. Prove global minimality and uniqueness among all feasible real flows using an energy difference, not an unexplained optimization rule. Explain the gauge freedom in p and what can fail if a resistance is zero.
On vertices A,B,C,D,E take edges A→B, A→C and D→E. The demand is b=(−5,2,3,−4,4). Solve Bf=b, then reverse the second and third edge orientations and find the new coordinate vector describing the same physical signed flow. Prove the relation between the two incidence matrices using a diagonal sign matrix. A solver discards rows A and D and checks only B,C,E. State the exact condition under which this preserves the full system. Audit the altered demand b̃=(−6,2,3,−3,4), which still has total sum zero. Is agreement on all retained rows a certificate of a solution?
A chain 1–2–3–4 has conductances 2,3,6 and prescribed v1=3,v4=−1. Interior injections vanish. Compute all potentials/currents, boundary supply and dissipated power. Delete the middle edge: decide uniqueness and recompute. Now add a separate interior edge 5–6 of conductance 4. Classify zero injections, injections (1,−1), and injections (1,0) on this component. Explain why neither global connectedness nor merely counting boundary vertices is the right uniqueness test.
