Logarithmic Series Research Articles

Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number <i>ρ </i>and a differentiable function <i>φ</i>. Then, I use <i>ρ </i>and <i>φ </i>to define the <i>T<sub>φ, ρ </sub></i>transformations on the partial sums of any convergent series. Furthermore, I prove that these <i>T<sub>φ, ρ </sub></i>transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute <i>φ </i>if one is given the convergent series of decreasing, positive terms.

ВИЗНАЧЕННЯ КУСКОВО-ЛІНІЙНОГО ТРЕНДУ НЕСТАЦІОНАРНОГО ЧАСОВОГО РЯДУ НА ОСНОВІ ІНТЕЛЕКТУАЛЬНОГО АНАЛІЗУ ДАНИХ. II. МАШИННI ЕКСПЕРИМЕНТИ ТА РОЗВ’ЯЗАННЯ ПРАКТИЧНОЇ ЗАДАЧІ

The article describes the results of the approbation of the method of constructing a piecewise-linear trend, which can have breaks at the switching points as well as be continuous at these points, i.e., represent a linear flow. An example of applying the method for constructing a linear regression with switches, which has two independent variables with a trend, is considered. The problem of spline approximation of the time series of logarithms of the number of infected people with COVID-19 in Ukraine is stated and solved. Keywords: trend, regression, switch point, spline, real-time calculation.

Extended Sibuya Distribution in Subcritical Markov Branching Processes

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t > 0 follows the shifted extended Sibuya distribution, with parameters depending on the time t > 0. The conditional limit probability is the logarithmic series distribution supported by the positive integers.

Research on properties of Bitcoin Based on GARCH model

Cryptocurrencies have become a world-class phenomenon, with governments, companies and investors facing huge challenges and opportunities. Bitcoin is the one of most widely known cryptocurrencies. People prefer to use Bitcoin as an asset to invest for profit and hedge risk than for its payment function. This is the reason for the study of bitcoin market is so important. Many scholars had used daily data on bitcoin to construct different GARCH models to analyse the volatility of its returns. This research builds a GARCH model for the logarithmic return series of the bitcoin price to understand the volatility of the bitcoin price over the experimental time horizon. The experimental results show that the price of bitcoin is more volatile and vulnerable to external shocks. The analysis suggests that Bitcoin is more clearly a speculative financial instrument and that the price of Bitcoin is highly frothy.

Fractional Scale Calculus: Hadamard vs. Liouville

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.

Changes in the market structure and risk management of Bitcoin and its forked coins

Inconsistency of consensus results in blockchain forks, which create a new financial risk. After filtering out Bitcoin’s linear, nonlinear, and lag impacts on forked coins, this study employs a bottom-up hierarchical clustering algorithm to examine the logarithmic return series for Bitcoin and its 14 forked coins from 2018 to 2021. The results indicate that the market for forked coins can be divided into three clusters: SegWit-supported forked coins, mature forked coins, and the latest forked coins. Bitcoin and the mature forked coins form a cluster, and its performance is superior to others. Although Bitcoin’s return significantly affects that of its forked coins, it does not affect the market structure. Furthermore, this study provides references for risk aversion among investors in forked coins and presents macro-level information for cryptocurrency market authorities.

On some aspects of a bivariate alternative zero-inflated logarithmic series distribution

In this paper, we discuss some important aspects of the bivariate alternative zero-inflated logarithmic series distribution (BAZILSD) of which the marginals are the alternative zero-inflated logarithmic series distributions of Kumar and Riyaz (2015. An alternative version of zero-inflated logarithmic series distribution and some of its applications. Journal of Statistical Computation and Simulation, 85(6), 1117–1127). We study some important properties of the distribution by deriving expressions for its probability mass function, factorial moments, conditional probability generating functions, and recursion formulae for its probabilities, raw moments and factorial moments. The parameters of the BAZILSD are estimated by the method of maximum likelihood and certain test procedures are also considered. Further certain real-life data applications are cited for illustrating the usefulness of the model. A simulation study is conducted for assessing the performance of the maximum likelihood estimators of the parameters of the BAZILSD.

The Characteristics of Cryptocurrency Market Volatility: Empirical Study For Five Cryptocurrency

In recent years, digital innovations especially emerged depend on Blockchain technology have caused a substantial transformation in the finance sector as in other sectors. Different financial assets have been revealed and began to be used as an investment tool along with this transformation in the markets. Cryptocurrencies that have a digital structure hold an important place among these assets. Dramatically increases in the daily transaction volume of currencies in the market have brought along different types of risks. These risks raised uncertainty on these currencies. Moreover, because cryptocurrencies are mostly used for the purpose of investment and speculation, it is important to understand the volatility movements and co-movements of cryptocurrencies and is substantially important, particularly because volatility can influence investment decisions.
 This study aims to determine the volatility transmission between cryptocurrencies to find useful answers about the volatility and the efficiency of markets. Daily logarithmic return series between 18 January 2018 – 14 February 2021 were used to analyze the volatility of five of the most common cryptocurrencies, namely Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), Ripple (XRP), IOTA by applying the RALS-ADF test, EGARCH, and DCC-GARCH models. We determined whether the market is efficient or not, and tested the existence of the asymmetric effect and volatility transmission in the market. According to our results, volatility shocks are not obtained persistent for only BTC. Furthermore, the presence of asymmetric effects and leverage effect valid for four cryptocurrencies. While asymmetric effects observed for BTC, no leverage effect has been observed during the period. We also analyzed nine pair-wise cryptocurrencies applying the DCC-GARCH model and we found that dynamic conditional correlation coefficients are statistically significant and positive for each pair.

Factorization of coefficients for exponential and logarithm in function fields

Let X be an elliptic curve or a ramifying hyperelliptic curve over Fq. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain v-adic convergence results for such exponential and logarithm series, for v a “finite” prime. As an application, we can show that the v-adic Goss L-value Lv(1,Ψ) is log-algebraic for suitable characters Ψ.

Fock Expansion for Two-Electron Atoms: High-Order Angular Coefficients

The Fock expansion, which describes the properties of two-electron atoms near the nucleus, is studied. The angular Fock coefficients ψk,p(α,θ) with the maximum possible value of subscript p are calculated on examples of the coefficients with 5≤k≤10. The presented technique makes it possible to calculate such angular coefficients for any arbitrarily large k. The mentioned coefficients being leading in the logarithmic power series representing the Fock expansion, they may be indispensable for the development of simple methods for calculating the helium-like electronic structure. The theoretical results obtained are verified by other suitable methods. The Wolfram Mathematica is used extensively.

Flexible multivariate zero to k inflated power series regression model with applications

Inflated distributions are applied in various fields, including insurance, traffic networks and survival analyses. First, they are defined by a baseline discrete distribution, and then, extra masses are added to some points of interest, called inflated points, to achieve more flexible models for data analyses. The baseline distribution is arbitrary and application dependent. Here, the rich family of power series distributions is considered as the baseline, which includes various common discrete distributions such as Poisson, negative binomial, multinomial and logarithmic series distributions. This paper deals with an extension of previous works in two directions. The former is an extension of the univariate inflated distributions to multivariate ones, and the latter is the generalization of the inflated points from the zero single point to the set . Under this setting, various inflated distributions in the literature fall into the proposed family of distributions. These extensions make the proposed model flexible and practically useful in data analyses. To do this, the problem of estimating parameters with various approaches as well as hypotheses testing is studied in detail. Multivariate‐generalized linear models with inflated multivariate discrete responses are also discussed. To assess the performance of the proposed family of inflated distributions, simulation studies are conducted, and a real data set on an Australian health survey study is also analysed.

Stock Prediction Analysis and Risk Conduction Path Research Based on EMD-LSTM Model

Stock market, government bond market and corporate bond market are important components of my country's financial market. It is of great significance to study stock market price fluctuations and their forecasting methods, and to reveal the risk transmission path between them. The first part of this paper mainly studies the construction of the index price prediction model, and makes a comprehensive comparison of the high-frequency fluctuation characteristics, long-term trend characteristics and average trend characteristics of the EMD decomposition results of the Shanghai Composite Index, the SSE Government Bond Index, and the SSE Corporate Bond Index. The LSTM prediction model and the EMD-LSTM prediction model were established for the three sequences respectively, and the prediction effects of the models were comprehensively compared by calculating the RMSE and MAPE of the predicted values of the test set. The results show that after the introduction of the EMD method, the prediction effects of the SSE Composite Index and the SSE Corporate Bond Index prediction model have been greatly improved, but the prediction effect of the SSE Government Bond Index prediction model is far inferior to that of the LSTM model. The second part of this paper mainly studies the financial market risk analysis and transmission path. First, the fluctuation risk analysis and stationarity analysis of the logarithmic return series of the three index price series are carried out. The yield series are stable. Then, the LSTM models based on different training data predict three index price series and compare the prediction results to study the similarity of the price fluctuation mechanism of the three financial markets. The results show that the price fluctuation mechanism of the stock market is similar to that of the corporate bond market, and the price fluctuation mechanism of the government bond market is more similar to the corporate bond market than the stock market. The paper conducts E-G cointegration test on three logarithmic return series that have been tested for stationarity to study the long-term equilibrium relationship between the three markets and finds that there is a cointegration relationship between the SSE Composite Index and the SSE Government Bond Index, as well as the SSE Government Bond Index and the SSE Corporate Bond Index. Finally, the Granger causality test is carried out between the sequences with cointegration relationship, and the result shows that the risk transmission mechanism between the national debt market and the corporate bond market is strong, while the risk transmission mechanism between the stock market and the bond market is weak.

Logarithmic A-hypergeometric series $${\text {I}}\! {\text {I}}$$

Logarithmic A-hypergeometric series $${\text {I}}\! {\text {I}}$$

Sudakov shoulder resummation for thrust and heavy jet mass

When the allowed range of an observable grows order-by-order in perturbation theory, its perturbative expansion can have discontinuities (as in the $C$ parameter) or discontinuities in its derivatives (as in thrust or heavy jet mass) called Sudakov shoulders. We explore the origin of these logarithms using both perturbation theory and effective field theory. We show that for thrust and heavy jet mass, the logarithms arise from kinematic configurations with narrow jets and deduce the next-to-leading logarithmic series. The left-shoulder logarithms in heavy jet mass ($\rho$) of the form $r\ \alpha_s^n \ln^{2n}r $ with $r=\frac{1}{3}-\rho$ are particularly dangerous, because they invalidate fixed order perturbation theory in regions traditionally used to extract $\alpha_s$. Although the factorization formula shows there are no non-global logarithms, we find Landau-pole like singularities in the resummed distribution associated with the cusp anomalous dimension, and that power corrections are exceptionally important.

Assessment of the Contribution of Scientific and Technological Progress to the Real GDP of Russia

The paper discusses the assessment of the contribution of scientific and technological progress to the real gross domestic product (GDP) of Russia. The instrument of this assessment is the production function of the national economy. The paper presents two variants of the production function of the Russian economy, which have successfully passed all the diagnostic procedures adopted in econometrics. In the first variant, the index of scientific and technological progress (STP) is included in the set of explanatory factors such as the level of fixed capital, the amount of live labor, the price of oil, the indicator of Russia's default, the Global Financial Crisis and sanctions of Western countries. In the second variant of the production function, there is no NTP index. The scheme of construction of the production function consists of two stages. At the first stage of the scheme, statistical models of the production function are evaluated with mandatory testing of the hypothesis of the cointegration of non-stationary time series of logarithms of Russia's GDP levels, fixed capital and living labor levels and oil price levels. At the second stage of the scheme, the stability of estimates of the production function calculated from different training samples was investigated, and the accuracy of ex-post forecasting was estimated. At the second stage of the production function construction scheme, the model with the NTP index showed higher accuracy of post-factum forecasting. This circumstance, combined with the result of testing the statistical hypothesis about the presence of scientific and technological progress in the Russian economy, made it possible to assess its contribution to real GDP: other things being equal, scientific and technological progress increases the annual real GDP of Russia by about 0,6%.

The Zipf-Polylog distribution: Modeling human interactions through social networks

The Zipf distribution attracts considerable attention because it helps describe data from natural as well as man-made systems. Nevertheless, in most of the cases the Zipf is only appropriate to fit data in the upper tail. This is why it is important to dispose of Zipf extensions that allow to fit the data in its entire range. In this paper, we introduce the Zipf-Polylog family of distributions as a two-parameter generalization of the Zipf. The extended family contains the Zipf, the geometric, the logarithmic series and the shifted negative binomial with two successes, as particular distributions. We deduce important properties of the new family and demonstrate its suitability by analyzing the degree sequence of two real networks in all its range.

Performance Evaluation Method of Online Supply Chain Finance Logistics Enterprises Based on GARCH-VAR

In recent years, China’s core enterprises have been plagued by high operating costs, difficult bank loans, and other practical problems in the supply chain. As a new financing model emerged in recent years, it has opened up new markets in finance and logistics industry and brought new ideas and vitality to all parties in the supply chain. Since the strengthening of supervision, the operation of the industry has gradually changed to the direction of normalization. The study of online loan interest rate fluctuation is helpful to understand the source and characteristics of online loan risk, which has important reference significance for China’s Internet lending market. The measurement of interest rate risk can help to monitor market risk in time and improve financial supervision. This paper takes the online loan interest rate under the background of supervision as the research object, combining theoretical analysis and empirical research to explore the characteristics of interest rate fluctuations and the reasons behind the P2P online loan, and discusses the online loan business model. Based on the influencing factors of interest rate fluctuation, this paper analyzes the main characteristics of network loan interest rate from the aspects of interest rate trend interval distribution, regional distribution, and background distribution. This paper adopts the daily closing price data of domestic R insurance company to establish the daily logarithmic return rate series of stock prices. The results show that (1) GARCH-VAR model can effectively reflect the stock price fluctuation and market risk degree of the insurance company; (2) the company’s VAR showed a trend of first rise and then slight decline, with the highest VAR in 2018; and (3) the interpretation of VAR value should be combined with financial indicators. The increase in net profit rate will also lead to the decline of VAR value in the next period. Based on this, countermeasures and suggestions for market risk management are proposed for insurance companies.

On bivariate alpha logarithmic series distribution and its properties

Here we propose a bivariate version of the alpha logarithmic series distribution (ALSD) of Kumar and Riyaz (South African Statist. J., 2014) and shown that its marginals are also ALSD. We study some of its important properties by deriving expressions for its probability mass function, factorial moments, conditional probability generating function, covariance etc. The parameters of the model are estimated by the method of maximum likelihood and the distribution has been fitted to certain real life data sets.

Ecological Characterization of Mosquitoes (Diptera: Culicidae) at the Southern Coast of Mar Chiquita Lake, Argentina.

In the southern coast of Mar Chiquita Lake, central Argentina, mosquitoes affect public health and community livelihood, since they transmit pathogens to human beings causing diseases such as malaria, filariasis, encephalitis, yellow fever, and dengue, among others, and have a negative effect on cattle farming as well. To characterize the structure of the mosquito assemblage of the region, we determined the species composition and diversity, the temporal distribution of different species, and the patterns of species richness, abundance, and diversity across seasons. We collected adult mosquitoes over a two-year period (October 2004-September 2006) by means of CDC light traps baited with CO2 from 18:00 to 08:00 h during the warm season (October-April) and from 12:00 h to 18:00 h in the cold season (May-September). A total of 71,501 individuals from 30 species were collected, with Culex Linnaeus and Aedes Meigen genera representing more than 98% of collected specimens (61.5% and 37.3%, respectively). The higher values of richness and abundance of Culicidae were registered in warm seasons compared to cold seasons. Chao1 estimates suggested that more than 90% of the species were detected in all seasons. Mosquito abundance distribution fit the logarithmic series and log-normal models. Aedes albifasciatus (Macquart), Ae. scapularis (Rondani), Culex interfor Dyar, Cx. saltanensis Dyar, and Cx. dolosus (Lynch Arribálzaga), vectors incriminated in arbovirus transmission, were abundant year-round, with Cx. saltanensis and Cx. dolosus most prevalent in cold seasons. Further studies are needed to assess the role of these species in arbovirus transmission in this region of central Argentina.

Analysis of Solutions of Some Multi-Term Fractional Bessel Equations

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation $$\sum\limits_{i = 1}^m {{d_i}{x^{{\alpha _i}}}} {D^{{\alpha _i}}}u\left( x \right) + \left( {{x^\beta } - {v^2}} \right)u\left( x \right) = 0,\,{\alpha _i} >0,\,\beta >0,$$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.