# Knowledge objects

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

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Knowledge object



## [Quotient topology](/en/mko/mko-quotient-topology/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open.

mko-quotient-topology · Topology



Knowledge object



## [Path-connected space](/en/mko/mko-path-connected-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

A space is path connected when every pair of its points is joined by a continuous map from the unit interval.

mko-path-connected-space · Topology



Knowledge object



## [Connected component of a topological space](/en/mko/mko-topological-connected-component/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed.

mko-topological-connected-component · Topology



Knowledge object



## [Central limit theorem for finite-variance IID sequences](/en/mko/mko-central-limit-theorem-finite-variance-iid/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law.

mko-central-limit-theorem-finite-variance-iid · Probability and statistics



Knowledge object



## [Strong law of large numbers for finite-valued IID sequences](/en/mko/mko-strong-law-large-numbers-finite-valued-iid/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation.

mko-strong-law-large-numbers-finite-valued-iid · Probability and statistics



Knowledge object



## [Confidence-interval coverage in a finite model](/en/mko/mko-finite-confidence-interval-coverage/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter.

mko-finite-confidence-interval-coverage · Probability and statistics



Knowledge object



## [Wilson theorem](/en/mko/mko-wilson-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n.

mko-wilson-theorem · Number theory



Knowledge object



## [p-adic valuation of integers](/en/mko/mko-p-adic-valuation-integers/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity.

mko-p-adic-valuation-integers · Number theory



Knowledge object



## [Regular expression](/en/mko/mko-regular-expression/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star.

mko-regular-expression · Mathematics of computer science



Knowledge object



## [Pumping lemma for regular languages](/en/mko/mko-pumping-lemma-regular-languages/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count.

mko-pumping-lemma-regular-languages · Mathematics of computer science



Knowledge object



## [Newton method for root finding](/en/mko/mko-newton-method/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses.

mko-newton-method · Numerical and computational mathematics



Knowledge object



## [Composite trapezoidal quadrature](/en/mko/mko-trapezoidal-quadrature/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral.

mko-trapezoidal-quadrature · Numerical and computational mathematics



Knowledge object



## [Gradient descent](/en/mko/mko-gradient-descent/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses.

mko-gradient-descent · Optimization and operations research



Knowledge object



## [Necessary Lagrange multiplier condition for equality constraints](/en/mko/mko-lagrange-multiplier-condition/?from=%2Fen%2Fknowledge%2F%3Fpage%3D13)

At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients.

mko-lagrange-multiplier-condition · Optimization and operations research





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