Search results for “SAGE”

59th International Mathematical Olympiad. Cluj-Napoca, Romania, 2018.

63rd International Mathematical Olympiad. Oslo, 2022.

The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.

The basic properties of determinants with brief proofs, and their use in simplifying computations.

Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.

A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.

Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.

Vector representations (embeddings) of texts and graphs: the distributional semantics hypothesis, syntagmatic and paradigmatic similarity of words, the CBOW and Skip-gram models of the word2vec program, hierarchical SoftMax on a Huffman tree, negative sampling (SGNS) and the connection of word2vec with the matrix factorisation of the PMI matrix, semantic analogies of words, the FastText model on character n-grams; texts as graphs; multidimensional scaling and the stress criterion, stochastic neighbour embedding SNE, perplexity, t-SNE; matrix factorisations of graphs, the random-walk models DeepWalk and node2vec; graph embedding methods as autoencoders, the generalised GraphEDM model, graph neural networks and message passing.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

48th International Mathematical Olympiad, 2007.

54th International Mathematical Olympiad. Colombia, 2013.

55th International Mathematical Olympiad. Cape Town, South Africa, 2014.

58th International Mathematical Olympiad. Rio de Janeiro, Brazil, 2017.

61st International Mathematical Olympiad, 2020.

Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.

Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.

Time series forecasting problems: the time series model, the forecasting horizon, the least squares method; econometrics as the main source of problems — trends, seasonalities, change points; examples of forecasting sales volumes and electricity prices; the linear autoregression model; a brief survey of forecasting methods; adaptive short-term forecasting methods: the exponential moving average and the smoothing parameter, the models of Holt, Theil–Wage, Winters with linear and exponential trend, adaptive autoregression; the tracking signal and a statistical test of model adequacy, the Trigg–Leach model; adaptive selection and adaptive composition of models, the LAVR method — local adaptation of weights with regularisation, the monotonicity constraint as a regulariser; experiments on the sales forecasting problem.