Knowledge object
Numerical approximation and error
Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
Knowledge object
Optimization problem
An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects.
Knowledge object
Dimensional analysis
Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides.
Knowledge object
Discrete exponential growth model
With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n.
Knowledge object
Compound-interest model
With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n.
Knowledge object
Mathematical representation
Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement.
Knowledge object
Diagnostic mathematical assessment records
A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status.
Knowledge object
Orbits, fixed points and periodic points
Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time.
Knowledge object
Initial-value problem
An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness.
Knowledge object
Finite sign-bracket updates in the bisection method
Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width.
Knowledge object
Linear programming
Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid.
Knowledge object
Discrete logistic increment model
For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment.
Knowledge object
Linear harmonic-oscillator model
The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution.
Knowledge object
Present value and discounting
For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value.
Knowledge object
Complex number
A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1.
Knowledge object
Real vector space
A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms.
Knowledge object
Finite group
A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element.
Knowledge object
Metric space
A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality.
Knowledge object
Finite discrete probability distribution
A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one.
Knowledge object
Complex modulus and conjugation
Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares.
Knowledge object
Linear combinations and span
A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination.
Knowledge object
Subgroups and group homomorphisms
A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table.
Knowledge object
Convergence in a metric space
A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x.
Knowledge object
Sigma-algebras and measure spaces
A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive.