Search results for “cosh”

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

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

Recommended for publication at the Journal of Algebraic Geometry; under review.

With the increasing popularity of cashless payment methods for everyday, seasonal and special expenses popular banks accumulate huge amount of data about customer operations. In the article, we report a successful application of topic modelling to extract behaviour patterns from the data. The resulting models are built with BigARTM framework: flexible and efficient tool for topic modelling. The framework allows us to experiment with various models including PLSA, LDA and beyond. Results demonstrate ability of the approach to aggregate information about behaviour patterns of different customer groups. The results analysis allows to see the topics of such people clusters varying from travellers to mortgage holders. Moreover, low-dementional embeddings of the customers, which was given with topic model, were studied. We display that the client vector representations store demographic information as well as source data. We also test for a best way of preparing data for the model with metric above in mind.

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.

A pilot study mapping ion-induced charge collection against bandgap and carrier mobility, and what it revealed about the cost of getting the time integration wrong

Cold atomic ensembles are among the most versatile tools in modern physics research. They play a key role in advancing next-generation frequency standards, qubit architectures, and quantum sensing technologies. Many experiments demand prolonged spatial confinement of atoms to facilitate extended interactions with electromagnetic fields, achievable through electric, magnetic, gravitational, or optical trapping mechanisms.

A cornerstone of atomic trapping is the magneto-optical trap (MOT), which relies on six counterpropagating laser beams and a pair of anti-Helmholtz coils to produce a radially symmetric quadrupole magnetic field with a central zero point.

Alternatively, optical dipole traps exploit the electric dipole interaction with a tightly focused, fardetuned high-power laser beam, offering weaker confinement than MOTs—typically below 1 mK. Unlike MOTs, these traps permit extremely weak optical excitation, circumventing limitations imposed by radiation pressure. Additionally, their trapping mechanism is largely insensitive to ground-state magnetic sublevels (neglecting tensor polarizability effects), enabling versatile configurations such as optical lattices.

Figure 1: Image of atoms trapped in the MOT. The atomic cloud is at the center of the image. The cloud dimensions are approximately 1 mm.

ble of confining Rb 87 atomic clouds at temperatures near 175 µ K. For dipole trapping, we adopted a farred-detuned (1012 nm) laser system, chosen for its dual functionality: we plan to use it both as a trapping beam and as one component in two-photon Rydberg excitation schemes. A further advantage of this configuration is its potential to confine Rydberg atoms while meeting the magic wavelength condition for ground-to-Rydberg transitions.

Magic-wavelength optical dipole traps offer particularly powerful advantages for cold atom manipulation. At this specific wavelength, the light shift for two atomic states becomes identical, effectively decoupling the internal atomic dynamics from the external motional states. This enables long coherence times for quantum operations while maintaining strong spatial confinement. Furthermore, such traps allow state-insensitive confinement, which is crucial for precision measurements and quantum information processing. The magic wavelength condition also facilitates efficient Rydberg excitation by providing identical trapping potentials for both ground and Rydberg states, minimizing decoherence during excitation processes.

We characterized the dipole trap intended for atom transfer, achieving a maximum depth of 3 mK under our experimental conditions—surpassing the Doppler cooling limit and ensuring efficient atomic confinement.

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.

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.

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