2024 year, volume 28, issue 3 (PDF)

Drozdov I.Yu., Parfenov D.V. Accuracy of algorithms of singular value decomposition for matrices with various spectra

We continue to develop our new approach of treating singular spectrum of a matrix as a probability density function to investigate dependencies between accuracy of numerical computation of singular values and spectrum. We conduct massive numerical experiments to demonstrate such dependencies in our new suggested metrics: root-mean-square relative error and median. We present illustrative plots of such dependencies and analyze conclusiveness of these metrics.

Keywords: singular value decomposition, SVD, condition number, matrix spectrum, numerical stability.

Chestnov R.V. Jump diffusion model with mean-reverting lognormal volatility

Currently, there are numerous stochastic models built for different market concepts. However, almost all of these models are based and tested for traditional markets, while at present the cryptoasset market is gaining tremendous momentum in terms of volume and market capitalisation. For this reason, the idea arises to create such a model, which would be built specifically for the cryptoasset market, taking into account all its peculiarities and behavioural patterns. In this paper, we will try to look at some of the features that are noticeable for cryptoassets and build a stochastic model that takes them into account, and after that, we will compare it with another model that is similar in structure. It is notable right away that, similar to how traditional models are used for cryptomarkets, our model for cryptoassets will interpret the traditional market just as well. Moreover, even better than in the opposite case, because the main idea of our model is to take into account various critical events that occur with this or that asset, but by calibrating the model in the right way, we can ignore them.

Keywords: Financial mathematics, stochastic calculus, cryptoas-sets, neural networks, Feynman-Kac equation, Bates model, likelihood function.

Mironov A.M. Mathematical foundations of time series prediction

The article outlines the basic concepts and methods of prediction time series. Various mixing prediction algorithms are considered and assessments of the quality of these algorithms are provided.

Keywords: time series, prediction algorithms, mixing prediction.

Tsaregorodtsev K. D. On the associativity index of finite quasigroups

In this paper we review the results on the number of associative triples in generic quasigroups and quasigroups from restricted classes. Lower and upper bounds on the number of triples are considered, and experimental results on the associativity index of quasigroups generated by proper families of boolean functions are provided.

Keywords: associative triple, quasigroup, proper family of boolean functions.

Gasanov E.E., Khaybullin B.F. Fast algorithms for multiplication and division of natural numbers using cellular automata with locators

For multiplication and division of \[n\]-digit natural numbers, algo-rithms with complexity of order \[ n^{\log_2 3} \] and even order \[ n^{\log n} \] are known. In this paper, an algorithm for multiplying \[n\]-digit natural numbers in \[2n + 2\] cycles is proposed. Here, the digit of number a is understood as the number \[ ]\log_2 a[ \]. For division of natural numbers with remainder, an algorithm with a running time of \[3n + 8\] cycles is proposed, where n is the digit of the quotient. The proposed algorithms use two-dimensional cellular automata with locators as calculators.

Keywords: multiplication of natural numbers, division of natural numbers, cellular automata with locators.