Search results for “indefinite integral”

Reference table of indefinite integrals of the exponential function and its products with powers of x.

Reference table of indefinite integrals of the hyperbolic sine, cosine, tangent and cotangent.

Reference table of indefinite integrals of expressions containing the natural logarithm.

Reference table of indefinite integrals of rational functions, including quadratic denominators and partial-fraction forms.

Reference table of indefinite integrals of expressions containing square roots and other radicals.

Reference table of indefinite integrals of products of exponentials with sine and cosine.

Reference table of indefinite integrals of products of powers of x with sine and cosine.

Reference table of indefinite integrals of sine, cosine, tangent and their powers and products.

Reference table of the basic indefinite integrals of the elementary functions: power, exponential, logarithmic and trigonometric.

A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.

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

The axioms of a linear (vector) space, examples and counterexamples.

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

64th International Mathematical Olympiad. Chiba, Japan, 2023.

Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.

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.

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

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.

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