Search results for “векторные представления”

Problems for the course "Topology 2" (F. E. Vylegzhanin, Independent University of Moscow, autumn 2026), sheet 1 of 9 September 2026 — simplicial homology and chain complexes. Zeroth homology and connected components; homology of a disjoint union and of a wedge; triangulations of the circle, the sphere and the Möbius band; the join and the product of simplicial complexes; the Euler characteristic and the Morse inequalities; the number of vertices of a triangulated surface; operations on chain complexes; exact sequences; the Smith normal form and the splitting of a complex.

Circulation of a field as the flux of its curl: computing the curl, matching the orientation of the contour and the normal, circulation computed two ways.

Multiple linear regression and least squares, the pseudoinverse and the geometric meaning of least squares; the solution through the singular value decomposition; multicollinearity and the condition number; ridge regression, choosing the regularization parameter and the effective dimension; LASSO and non-smooth regularizers for feature selection; principal component analysis as a low-rank matrix decomposition; the spectral method of least squares.

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.

Геометрические векторы: определение, длина, равенство, коллинеарность и компланарность; линейные операции над векторами и их свойства; разложение вектора по коллинеарному.

The flux of a vector field through an oriented surface: the normal and the choice of side, the explicit and parametric forms, the lateral surface of a cylinder.

A contour integral as a double integral: the hypotheses of Green's theorem, orientation of the contour, area as a contour integral, closing an open arc, an annulus, a singular point inside the contour.

Integrating a vector field along a directed curve: parametrising a segment and a graph, the work of a force, path dependence, a potential field and its potential, a curl-free field in a domain that is not simply connected.

Integration with respect to surface area: the area element for an explicit and a parametric surface, pieces of planes, the sphere in spherical coordinates.

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

Разложение вектора по базису на плоскости и в пространстве, ортонормированный базис и декартова система координат, координаты вектора и линейные операции над ними, проекция вектора на ось, её линейность и проекция на вектор.

Задачные листки к курсу линейной алгебры. Институт и учебное заведение в исходных публикациях не указаны, поэтому здесь не названы.

The optimal separating hyperplane and the width of the separating strip; the hinge loss and L2 regularization; the Karush–Kuhn–Tucker conditions, the dual problem and the types of objects; kernels, rectifying spaces and the constructive building of kernels; SVM as a two-layer neural network; choosing the constant C; SVM regression; L1 regularization, Elastic Net SVM, SFM and RVM.

Singular homology — standard simplices and the inclusions of facets, singular simplices and chains, the differential; zeroth homology and path components, the homology of a point, reduced homology; chain maps and maps of homology, functoriality, the chain map induced by a continuous map; chain homotopies and the prism; the homotopy invariance theorem with a full proof via the prism operator; subcomplexes, quotient complexes and the homology of a pair.

The neuron as a linear model; implementing Boolean functions and the exclusive-OR problem; two-layer networks as universal approximators, Cybenko's theorem; the multilayer network and the backpropagation method; the adaptive gradient methods RMSProp, AdaDelta, Adam, Nadam; gradient explosion and gradient clipping; Dropout, ReLU activation functions, batch normalization, initialization and network pruning (Optimal Brain Damage).

Nonlinear regression: the Newton–Raphson and Gauss–Newton methods as a series of linear regressions; logistic regression and iteratively reweighted least squares (IRLS); the generalized additive model and backfitting; the probabilistic meaning of least squares; the generalized linear model (GLM), the exponential family of distributions and the link function; logistic regression as a special case of GLM; the median, quantile regression and robust loss functions.

The probabilistic formulation of classification: prior and posterior class probabilities, class likelihood functions, the maximum a posteriori principle; the optimal Bayes classifier and losses from errors; empirical estimation of prior probabilities and class densities; the naive Bayes classifier, features with exponential-family densities and the linear naive Bayes classifier; a survey of Bayesian classifiers: the Parzen window method, the quadratic discriminant, Fisher's linear discriminant, Gaussian mixtures with diagonal covariance matrices, and the radial basis function network.

Евклидовы и эрмитовы пространства, скалярное произведение, неравенство Коши–Буняковского и неравенство треугольника, ортогональные системы векторов.

Estimating a probability density from a sample: the maximum likelihood principle, the multivariate Gaussian density and its geometric meaning, multicollinearity and regularisation of the covariance matrix; separating a mixture of distributions, the EM algorithm as a method of simple iterations, the Gaussian mixture (GMM); the nonparametric Parzen–Rosenblatt estimate, the choice of kernel and window width; Bayesian classification theory: the generative and discriminative approaches, the optimal Bayes classifier, the Parzen window method, the quadratic discriminant, Fisher's linear discriminant, the radial basis function network and its training by the EM algorithm.