Concept guide
Every derivative order of a quartic
Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2].
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Concept guide
Separating every pair in a discrete three-point space
Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair.
Concept guide
A continuous CDF with a flat interval
Construct a complete knot-based CDF and audit atomlessness.
Concept guide
Catalan convolution and a divisibility certificate
Generate C4 from C0 and check both closed forms.
Concept guide
The complete nonzero square table modulo 7
Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7.
Concept guide
An existential finite accepting NFA path
The specified NFA accepts aab despite a dead alternative branch; an epsilon loop alone does not accept.
Concept guide
Exact-order witnesses for the prime divisors
In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6.
Concept guide
A Taylor polynomial retaining the center and factorials
Construct T_2 at center a=1 for the specified cubic polynomial.
Concept guide
Compactness reduces separating neighborhoods to finitely many
Prove a compact subset K of a Hausdorff space is closed.
Concept guide
Separate limits at every infinite endpoint and singularity
Compute three explicit power-law improper integrals and distinguish principal value.
Concept guide
Radius, endpoints and complete coefficient laws
Find R for geometric coefficients 2^n, and exhibit endpoint and zero/infinity boundaries.
Concept guide
The three-valued Legendre symbol from squares
Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1.
Concept guide
The complete reachable-subset DFA of an epsilon-NFA
List every transition and prove language equivalence by prefix induction.
Concept guide
Normality of rotations from every conjugate
N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal.
Concept guide
Addition and strict positivity of the full exponential series
From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity.
Concept guide
A Fibonacci-type OGF with initial boundary terms
Derive the OGF from a recurrence on a justified convergence domain.
Concept guide
Two independent routes for Euler's criterion
For p=7,a=-4 the Legendre value is -1 and a^3=6=-1 mod 7; 2 gives +1.
Concept guide
A scan decider for an even number of ones
Give the complete machine, trace, ranking and all-input correctness.
Concept guide
Coordinate gradients, stationarity and differentiability
Compute the gradient of a specified bivariate polynomial and audit two false inferences.
Concept guide
Checking every edge and internal vertex of a feasible flow
Audit a complete flow network with antiparallel edges, a zero-capacity edge and a zero-flow vertex.
Concept guide
Kernel, fibers and injectivity from C4 to C2
For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective.
Concept guide
A quotient table checked on every representative pair
G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A.
Concept guide
Full conjugation action and three classes
The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate.
Concept guide
Standard-normal symmetry with explicit named facts
Separate Gaussian prerequisites from derived CDF symmetry.