# Knowledge objects

Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.

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Knowledge object



## [Numerical approximation and error](/en/mko/mko-numerical-approximation-error/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value.

mko-numerical-approximation-error · Numerical and computational mathematics



Knowledge object



## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects.

mko-optimization-problem · Optimization and operations research



Knowledge object



## [Dimensional analysis](/en/mko/mko-dimensional-analysis/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides.

mko-dimensional-analysis · Mathematical physics



Knowledge object



## [Discrete exponential growth model](/en/mko/mko-discrete-exponential-growth/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-discrete-exponential-growth · Mathematical biology



Knowledge object



## [Compound-interest model](/en/mko/mko-compound-interest/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-compound-interest · Mathematics of economics and finance



Knowledge object



## [Mathematical representation](/en/mko/mko-mathematical-representation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-mathematical-representation · Mathematics education



Knowledge object



## [Diagnostic mathematical assessment records](/en/mko/mko-diagnostic-mathematical-assessment/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-diagnostic-mathematical-assessment · Mathematics education



Knowledge object



## [Orbits, fixed points and periodic points](/en/mko/mko-orbit-fixed-periodic/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-orbit-fixed-periodic · Dynamical systems



Knowledge object



## [Initial-value problem](/en/mko/mko-initial-value-problem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-initial-value-problem · Differential equations



Knowledge object



## [Finite sign-bracket updates in the bisection method](/en/mko/mko-bisection-method/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-bisection-method · Numerical and computational mathematics



Knowledge object



## [Linear programming](/en/mko/mko-linear-programming/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-linear-programming · Optimization and operations research



Knowledge object



## [Discrete logistic increment model](/en/mko/mko-discrete-logistic-growth/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-discrete-logistic-growth · Mathematical biology



Knowledge object



## [Linear harmonic-oscillator model](/en/mko/mko-harmonic-oscillator-model/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-harmonic-oscillator-model · Mathematical physics



Knowledge object



## [Present value and discounting](/en/mko/mko-present-value-discounting/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value.

mko-present-value-discounting · Mathematics of economics and finance



Knowledge object



## [Complex number](/en/mko/mko-complex-number/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1.

mko-complex-number · Arithmetic and number systems



Knowledge object



## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms.

mko-real-vector-space · Algebra



Knowledge object



## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element.

mko-finite-group · Algebra



Knowledge object



## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-metric-space · Topology



Knowledge object



## [Finite discrete probability distribution](/en/mko/mko-finite-discrete-probability-distribution/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one.

mko-finite-discrete-probability-distribution · Probability and statistics



Knowledge object



## [Complex modulus and conjugation](/en/mko/mko-complex-modulus-conjugate/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares.

mko-complex-modulus-conjugate · Analysis



Knowledge object



## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination.

mko-linear-combination-span · Algebra



Knowledge object



## [Subgroups and group homomorphisms](/en/mko/mko-subgroup-homomorphism/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-subgroup-homomorphism · Algebra



Knowledge object



## [Convergence in a metric space](/en/mko/mko-metric-convergence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-metric-convergence · Analysis



Knowledge object



## [Sigma-algebras and measure spaces](/en/mko/mko-sigma-algebra-measure-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D5)

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.

mko-sigma-algebra-measure-space · Analysis





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