Search results for “number-theory”

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.

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

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

A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.

This article is devoted to the problem of calculating an exact upper bound for the functionals of the generalization ability of a family of one-dimensional threshold decision rules. An algorithm is investigated that solves the stated problem and is polynomial in the total number of samples used for training and validation and in the number of training samples. A theorem is proved for calculating an estimate for the functional of expected overfitting and an estimate for the error rate of the method for minimizing empirical risk on a validation set. The exact bounds calculated using the theorem are compared with the previously known quick-to-compute upper bounds so as to estimate the orders of overestimation of the bounds and to identify the bounds that could be used in real problems.

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

Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.

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

Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.

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

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.

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.

This paper introduces a new approach to estimating the text document complexity.Common readability indices are based on average length of sentences and words.In contrast to these methods, we propose to count the number of rare words occurring abnormally often in the document.We use the reference corpus of texts and the quantile approach in order to determine what words are rare, and what frequencies are abnormal.We construct a general text complexity model, which can be adjusted for the specific task, and introduce two special models.The experimental design is based on a set of thematically similar pairs of Wikipedia articles, labeled using crowdsourcing.The experiments demonstrate the competitiveness of the proposed approach.

Topic modeling is an area of text analysis actively developing over the past 20 years. A probabilistic topic model (PTM) finds a set of hidden topics from a collection of text documents. It defines each topic as a probability distribution over words and describes each document as a probability mixture of topic distributions. Learning algorithms for topic models are usually based on Bayesian inference or log-likelihood maximization. In both cases, EM-like algorithms are used. In this paper, we propose to replace the logarithm in the log-likelihood by an arbitrary smooth loss function. We prove that such a modification preserves both the structure of the algorithm and compatibility with any regularizers in terms of additive regularization of topic models (ARTM). Moreover, in the case of a linear loss, the Estep becomes much faster due to the omission of a normalization. We study combinations of the fast and usual E-steps and compare them to regularization using different number of topics in both offline and online versions of EM-algorithm. For an empirical comparison of the algorithms, we estimate perplexity, coherence, and learning time. We use an efficient parallel implementation of the EM-algorithm from the BigARTM open-source library. We show that in most cases the two-stage strategy wins, which uses fast E-steps at the beginning of iterations, then proceeds with usual E-steps.

Rapid growth of scientific publications and intensive emergence of new directions and approaches poses a challenge to the scientific community to identify trends in a timely and automatic manner. We denote trend as a semantically homogeneous theme that is characterized by a lexical kernel steadily evolving in time and a sharp, often exponential, increase in the number of publications. In this paper, we investigate recent topic modeling approaches to accurately extract trending topics at an early stage. In particular, we customize the standard ARTM-based approach and propose a novel incremental training technique which helps the model to operate on data in real-time. We further create the Artificial Intelligence Trends Dataset (AITD) that contains a collection of early-stage articles and a set of key collocations for each trend. The conducted experiments demonstrate that the suggested ARTM-based approach outperforms the classic PLSA, LDA models and a neural approach based on BERT representations. Our models and dataset are open for research purposes.