Search results for “indefinite integral”
37 results
IMO 1994
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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Quadratic Forms
Theory
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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Convergence of the Algorithm of Additive Regularization of Topic Models
2021Journal articleI. A. Irkhin, K. V. Vorontsov
Proceedings of the Steklov Institute of Mathematics
The problem of probabilistic topic modeling is as follows. Given a collection of text documents, find the conditional distribution over topics for each document and the conditional distribution over words (or terms) for each topic. Log-likelihood maximization is used to solve this problem. The problem generally has an infinite set of solutions and is ill-posed according to Hadamard. In the framework of Additive Regularization of Topic Models (ARTM), a weighted sum of regularization criteria is added to the main log-likelihood criterion. The numerical method for solving this optimization problem is a kind of an iterative EM-algorithm written in a general form for an arbitrary smooth regularizer as well as for a linear combination of smooth regularizers. This paper studies the problem of convergence of the EM iterative process. Sufficient conditions are obtained for the convergence to a stationary point of the regularized log-likelihood. The constraints imposed on the regularizer are not too restrictive. We give their interpretations from the point of view of the practical implementation of the algorithm. A modification of the algorithm is proposed that improves the convergence without additional time and memory costs. Experiments on a news text collection have shown that our modification both accelerates the convergence and improves the value of the criterion to be optimized.
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IMO 1971
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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IMO 1975
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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IMO 1982
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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IMO 1993
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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Basis, Dimension, and Span
Theory
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
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IMO 1967
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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The Steinberg Representation
Theory
An introduction to the Steinberg representation of a finite group of Lie type — its alternating-sum construction from parabolic inductions, worked out explicitly for SL_2.
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