Elementary Functions Research Articles

Effect of active and passive techniques used in thromboembolic prophylaxis on venous flow velocity in the post-procedure period.

Studies have shown that using mechanical thromboembolic prophylaxis methods speeds up venous flow and decreases stasis. These studies examine the post-intervention period of 1-10min. The length of the effect of procedures to raise venous flow velocity cannot be determined by clinical trials. To apply mathematical techniques to estimate how long mechanical thromboembolism prophylaxis procedures will increase venous flow rate. In the survey, we examined 25 persons (poststroke patients), with an average age of 57.2 ± 6.3 years. Regarding the proportion of genders, 13 (52%) participants were male and 12 (48%) female. The peak venous blood flow velocity was measured with a HADECO BIDOP ES-100V II type Doppler ultrasound device, using an 8MHz head, in the femoral vein, at the level of the hip joint. We estimated the change of the venous blood flow velocity from the available sampled data using the method of least squares. For the calculations, we used Microsoft Excel, version Mac Excel 2019. The decrease in peak venous flow velocity can be approximated by a logarithm function. Mathematical calculations show that after active thromboembolic prophylaxis interventions, resting venous flow velocity is restored at 26.8min on the intact limb and 85.1min on the hemiparetic side. Resting flow velocity is restored in 131.9min after passive mobilization of the hemiparetic side and in 137.7min after the consensual effect. An elementary mathematical function can be used to estimate the time to recovery of peak venous flow velocity to resting state from measurements taken 15min after the intervention. Active and passive mechanical thromboembolic prophylaxis after the intervention has a longer-term effect on venous flow velocity.

Tropical curves and solitons in nonlinear integrable systems

We construct families of rationally degenerating Riemann surfaces corresponding to tropical curves with trivial weights, and give explicit formulas of the associated quasi-periodic solutions and soliton solutions as their regularized limits to the KP hierarchy and the 2D Toda lattice hierarchy. This result especially gives rise to soliton solutions in extensive classes to these nonlinear integrable systems which are expected to cover all solutions expressed by compositions of elementary functions. For these quasi-periodic solutions, we also discuss their regularity and limits to the mixtures with solitons obtained from tropical curves with nontrivial weights.

Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg–Landau equation

This research focuses on investigating an extended version of the Ginzburg–Landau (GL) equation that describes the motion of particles in a plasma. The other applications of this model can be found in optics, and other related fields. As part of our approach, we seek to discover some new wave solutions to the model, and then use the Galilean transformation in order to determine the model’s dynamical properties. After that, we use planar dynamical system theory to perform an in-depth bifurcation analysis, which gives us valuable insight into the system. By conducting thorough mathematical and bifurcation analyses, we identify the fundamental dynamics and critical points that regulate the system’s behavior. Moreover we explore the chaotic nature of the system, uncovering potential chaotic tendencies and their consequences. Further, we obtain several categories of optical solutions of the complex GL equation by utilizing new logarithmic transformations, offering valuable insight into its behavior and potential applications in optics. Our analytical technique yields closed-form solutions expressing elementary functions. In order to ensure the reliability of our findings, we rigorously validate the obtained solutions by substituting them back into the original model. Our research helps us better understand this equation’s characteristics and its relevance across a wide range of disciplines.

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

This paper presents a method for parameterizing new Lorentz spacetime coordinates based on coupled parameters. The role of symmetry in rapidity in special relativity is explored, and invariance is obtained for new spacetime intervals with respect to the Lorentz transformation. Using the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz spacetime coordinates are obtained. The descriptions of particle motion using the old and new Lorentz spacetime coordinates as applied to plane laser pulses are compared in terms of the particle kinetic energy. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and that of a plane laser pulse, rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions. The new and old Lorentz spacetime coordinates are compared with the Fermi spacetime coordinates. The proper Lorentz groups SO(1,3) with coupled parameters using the old and new Lorentz spacetime coordinates are also compared. As a special case, the application of Lorentz spacetime coordinates to a relativistic hydrodynamic system with coupled parameters in 1+1 dimensions is demonstrated.

Wave analysis and representation of fundamental solution in modified couple stress thermoelastic diffusion with voids, nonlocal and phase lags

In the present study, we explore a new mathematical formulation involving modifiedcouple stress thermoelastic diffusion (MCTD) with nonlocal, voids and phase lags. The governingequations are expressed in dimensionless form for the further investigation. The desired equationsare expressed in terms of elementary functions by assuming time harmonic variation of the fieldvariables (displacement, temperature field, chemical potential and volume fraction field). Thefundamental solutions are constructed for the obtained system of equations for steady oscillation,and some basic features of the solutions are established. Also, plane wave vibrations has beenexamined for two dimensional cases. The characteristic equation yields the attributes of waves likephase velocity, attenuation coefficients, specific loss and penetration depth which are computednumerically and presented in form of distinct graphs. Some unique cases are also deduced. Theresults provide the motivation for the researcher to investigate thermally conducted modified couplestress elastic material under nonlocal, porosity and phase lags impacts as a new class of applicablematerials.

Towards a change of variable formula for “hypergeometrization”

We will study the properties of “hypergeometrization” - an operator that acts on analytic functions around the origin by inserting two Pochhammer symbols into their Taylor series. This operator basically maps elementary functions to hypergeometric ones. The main goal is to derive several “change of variables” formulas for this operator, which can then be used to derive a large number of transformations for multidimensional hypergeometric functions.

The Shocks in Josephson Transmission Line Revisited

We continue our previous studies of the shocks in the lossy Josephson transmission line (JTL). The article consists of two parts. In the first part, we analyze the scattering of the “sound” (small amplitude small wave vector harmonic wave) on the shock wave and calculate the reflection and the transmission coefficients. In the second part, we show that the kinks, which we previously studied only in the lossless JTL, exist also in the lossy JTL and study the similarities and the dissimilarities between the shocks and the kinks there. We find that the nonlinear equation describing the weak kinks and the weak shocks can be integrated (in particular cases) in terms of elementary functions. We also show that the profile of the shock in the lossy JTL demonstrates oscillatory behavior with leading peaks resembling solitary waves if the losses are weak.

Improved honey badger algorithm based on elementary function density factors and mathematical spirals in polar coordinate systema

The Honey Badger Algorithm (HBA) is a new swarm intelligence optimization algorithm by simulating the foraging behavior of honey badgers in nature. To further improve its convergence speed and convergence accuracy, an improved HBA based on the density factors with the elementary functions and the mathematical spirals in the polar coordinate system was proposed. The algorithm proposes six density factors for attenuation states based on elementary functions, and introduces mathematical expressions of the polar diameters and angles of seven mathematical spirals (Fibonacci spiral, Butterfly curve, Rose spiral, Cycloid, Archimedean spiral, Hypotrochoid and Cardioid) in the polar coordinate system based on the density factors with the best synthesized effect to replace the foraging strategy of honey badger digging pattern in HBA. By using 23 benchmark test functions, the above improvements are sequentially compared with the original HBA, and the optimization algorithm with the best improvement, α4CycρHBA, is selected to be compared with SOA, MVO, DOA, CDO, MFO, SCA, BA, GWO and FFA. Finally, four engineering design problems (pressure vessel design, three-bar truss design, cantilever beam design and slotted bulkhead design) were solved. The simulation experiments results show that the proposed improved HBA based on the density factors with the elementary functions and the mathematical spirals of the polar coordinate system has the characteristics of balanced exploration and expiration, fast convergence and high accuracy, and is able to solve the function optimization and engineering optimization problems in a better way.

On the set of reduced states of translation invariant, infinite quantum systems

The set of two-body reduced states of translation invariant, infinite quantum spin chains can be approximated from inside and outside using matrix product states and marginals of finite systems, respectively. These lead to hierarchies of algebraic approximations that become tight only in the limit of infinitely many auxiliary variables. We show that this is necessarily so for any algebraic ansatz by proving that the set of reduced states is not semialgebraic. We also provide evidence that additional elementary transcendental functions cannot lead to a finitary description.

Static spherically symmetric solutions in new general relativity

We give a pedagogical introduction to static spherically symmetric solutions in models of New GR, both explaining the basics and showing how all such vacuum solutions can be obtained in elementary functions. In doing so, we coherently introduce the full landscape of these modified teleparallel spacetimes, and find a few special cases. The equations of motion are turned into a very nice and compact form by using the Levi-Civita divergence of the torsion-conjugate; and generalised Bianchi identities are briefly discussed. Another important point we make is that a convenient choice of the radial variable might be instrumental for success of similar studies in other modified gravity models.

Selected Mathematical Tools for Modeling in Tomography and Image Processing with Some Applications

In this theoretical paper, we propose several useful mathematical tools for conducting research in the fields of tomography and image processing. Several mathematical models are presented including analytical formulation of the Radon transform for three elementary functions. A family of block binary images for testing ideas is explained. Representation of an image using the box function is suggested. This representation allows exact computations for the Radon transform, Fourier transform, moments, and many others. This representation can also be involved in the mathematical arrangements of the iterative reconstruction algorithms of tomography. Overall, we believe that these tools are helpful for researchers in these fields. We show accurate results in several calculations on synthetic images as well the algebraic reconstruction.

RISC-V Custom Instructions of Elementary Functions for IoT Endpoint Devices

RISC-V Custom Instructions of Elementary Functions for IoT Endpoint Devices

Developing predictions for pion fragmentation functions

Exploiting crossing symmetry, the hadron scale pion valence quark distribution function is used to predict the kindred elementary valence quark fragmentation function (FF). This function defines the kernel of a quark jet fragmentation equation, which is solved to obtain the full pion FFs. After evolution to a scale typical of FF fits to data, the results for quark FFs are seen to compare favourably with such fits. However, the gluon FF is markedly different. Notably, although FF evolution equations do not themselves guarantee momentum conservation, inclusion of a gluon FF which, for four quark flavours, distributes roughly 11% of the total light-front momentum fraction, is sufficient to restore momentum conservation under evolution. Overall, significant uncertainty is attached to FFs determined via fits to data; hence, the features of the predictions described herein could potentially provide useful guidance for future such studies.

Three-dimensional quasi-static general solution for isotropic chemoelastic materials and their application

Based on potential theory, the three-dimensional quasi-static general solution for isotropic chemoelastic materials is presented in this work. Through the three-dimensional general solution, the Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived. This can serve as theoretical guidance for future engineering practices. Four functions constitute the expressions of the general solution that satisfy the harmonic functions and the quasi-static transport equation, respectively. The Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived by combining the general solution with the chemical balance boundary conditions at infinity. It can be expressed in terms of the error function and elementary functions. Finally, the numerical results are provided, as shown in the contours. These results can be used to analyze the variation law in the coupling fields of isotropic chemoelastic materials. The corresponding analysis can provide a theoretical basis for elucidating the mechanism of the chemoelastic coupling problem in further work.

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems.

Information and Communication Technology (ICT) Skills as Correlate of Business Education Lecturers’ Instructional Effectiveness in Colleges of Education in South-South Geo-Political Zone of Nigeria

This study examined ICT Skills as correlate of business education lecturers’ instructional effectiveness in Colleges of Education in South-South Geo-Political Zone of Nigeria. Six specific purposes, research questions and null hypotheses guided the study. The design chosen for the study was correlational research design. The population of the study comprised one hundred and seventy-three (173) business educators in seven (7) colleges of education in the study area. ICT Skills Test (IST) and the Instructional Effectiveness Questionnaire (IEQ) were used to collect data for the study. The instruments were validated by three experts and tested for reliability using K-R 20 for ICT skills Test (IST) and Cronbach Alpha coefficient for Instructional Effectiveness Questionnaire (IEQ). The result for ICT Skill Test using K-R 20 was 0.97 and that of Instructional Effectiveness Questionnaire, using Cronbach Alpha coefficient was 0.91. Data collected from the respondents were analysed using the Pearson Product Moment Correlation Coefficient (r) to answer the research questions, while the null hypotheses were tested using t-test associated with simple regression. Findings of the research questions revealed that word processing and database management skills have high positive correlation with business education lecturers’ instructional effectiveness. Findings of the research hypotheses indicated the following: There is a significant relationship between word processing skills and business education lecturers’ instructional effectiveness and there is a significant relationship between database management skills and business education lecturers’ instructional effectiveness. The educational implication of the study is that skills are needed to efficiently use at least the elementary functions of ICT in instructional effectiveness. Therefore, it is crucial for institutions operating business education programmes to recognize the importance of ICT skills and provide ample support and resources to their lecturers in order to ensure their instructional effectiveness in the digital era. It was recommended among others that institutions of learning through its ICT unit should offer workshops and seminars to help lecturers stay updated with the latest advancements in ICT skills such as word processing and database management skills.

SYNTHESIS OF DISCRETE AND ALGEBRAIC MODELS OF ELEMENTARY FUNCTIONS OF DATA-CONTROLLED OPERATIONS

Improvement of modern data exchange applications increases the complexity of cybersecurity. This leads to most applicable low-cost cryptographic algorithms becoming ineffective in the near future. On the other hand, CET encryption offers a great opportunity for development of the low-cost cryptography. The following article analyzes previously published results of CET-operations modeling, which serves as the foundation of CET encryption. The CET operations mentioned above use elementary functions as their basis. The results of our analysis allow to conclude that elementary functions of data-controlled operations have not been researched in the past. The primary goal of this article is to research these elementary functions of data-controlled operations and develop a method suitable for synthesis of a group of elementary functions of data-controlled operations. This can assist in automating the process of creating CET operations with defined attributes. This article proves that known discrete models of elementary functions of data-controlled operations do not represent their content and usage specifications during creation of CET operations. We suggest using discrete and algebraic presentation for modeling elementary functions data. The results of our analysis of the synthesized models of elementary functions of data-controlled operations allow us to develop a proper method of their synthesis. This method is adapted for usage in the automated systems of CET-operations modeling. We also provide examples of models of CET operations created based on elementary functions of data-controlled operations. The aforementioned method for synthesis of a group of elementary functions of data-controlled operations allows expanding possibilities for generating these elementary functions within the automated system used for research and creation of CET operations. Presented scientific results can be used for experimental modeling of CET operations, while the implementation algorithms of such operations will be defined by the operations themselves, as well as transformed data. Utilization of these operations allows modification of cryptographic algorithms controlled by encrypted data.

О комплексных операторных функциях комплексного операторного переменного

We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotangent, secant, cosecant; the main property of the exponent is proved. A complex Euler operator formula is obtained. Relations that express sine and cosine in terms of the exponent are found. For the trigonometric functions of sine and cosine, addition formulas are justified. The periodicity of the exponent and trigonometric functions of sine, cosine, tangent, cotangent is proved; reduction formulas for these functions are provided. The main complex operator trigonometric identity is obtained. Equalities connecting trigonometric and hyperbolic functions are found. The main complex operator hyperbolic identity is established. For the hyperbolic functions of sine and cosine, addition formulas are indicated. As an example of an elementary function from the family of complex operator functions under consideration, a rational function is considered, a special case of which is the characteristic operator polynomial of a linear homogeneous differential equation of n-th order with constant bounded operator coefficients in a real Banach space.

FRACTIONAL TAYLOR EXPANSION OF CLASSICAL AND LAGUERRE-TYPE FUNCTIONS

Beginning with the fractional exponential, which is frequently used in applications, fractional Taylor expansions of several elementary functions in both the ordinary and Laguerrian cases are introduced, showing that some basic formulas continue to apply to new expansions. Some graphs obtained using the computer algebra system Mathematica© are given for illustration purposes.

Рациональные коэффициенты ортогональных разложений некоторых функций

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.