Knowledge object
Quotient topology
For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
Knowledge object
Path-connected space
A space is path connected when every pair of its points is joined by a continuous map from the unit interval.
Knowledge object
Connected component of a topological space
The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed.
Knowledge object
Central limit theorem for finite-variance IID sequences
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.
Knowledge object
Strong law of large numbers for finite-valued IID sequences
The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation.
Knowledge object
Confidence-interval coverage in a finite model
Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter.
Knowledge object
Wilson theorem
An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n.
Knowledge object
p-adic valuation of integers
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.
Knowledge object
Regular expression
Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star.
Knowledge object
Pumping lemma for regular languages
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.
Knowledge object
Newton method for root finding
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.
Knowledge object
Composite trapezoidal quadrature
On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral.
Knowledge object
Gradient descent
In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses.
Knowledge object
Necessary Lagrange multiplier condition for equality constraints
At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients.