Mathematical Justification Research Articles

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.

საქართველოს ცალკეულ რეგიონებისათვის, ლოკალური ოროგრაფიის როლის შეფასება, ატმოსფეროს მიწისპირა ფენის ჰაერის ტურბულენტური ნაკადის დინამიკაში

There are many micro-regions on Earth in which the study of evolution hydrometeorological processes is relevant and has great practical value. Such regions include various caverns, highways, open mining areas, Hydropower plant construction areas. To study local events in such areas in order to set the assignment special theoretical and model approach is given in the paper. The mathematical justification of this approach is presented also case studies and calculation procedures are given. The materials presented in the article will explain, substantiate and give practical value to the physical and climatic features that characterize the local area and are useful for ​​interest persons involved

From adherence to slip in nanofluidics: a mathematical justification based on a drop of viscosity—the scalar case

The question of the origin of the unusually large slip length that may appear in some nanofluidic experimental settings is the object of much debate in the physics community. An idea suggested by Myers (Microfluid Nanofluid 10(5):1141–1145, 2011) is that in carbon nanotubes, this phenomenon may originate from a viscosity drop of the fluid near the wall of the tube, in a very thin region called ”depletion layer”. In order to investigate this claim mathematically, we introduce as a first step a scalar diffusion model whose coefficient of diffusion tends to zero in the depletion layer, as the thickness of this region goes to zero. Using the natural energy bound satisfied by weak solutions $$u_\varepsilon $$ of the system, we develop an adaptation of the unfolding method (Arbogast et al in SIAM J Math Anal 21:823–836, 1990; Casado-Díaz in Proc R Soc Edinb 130(A):249–276, 2000; Cioranescu et al in C R Acad Sci Paris Sér I 335:99–104, 2002), (which is related to the two-scale convergence method (Allaire in SIAM J Math Anal 23:1482–1518, 1992; Nguetseng in SIAM J Math Anal 20:608–623, 1989)), based on the introduction of a fast variable and on a compactness argument applied to the corresponding sequence of rescaled solutions in the depletion region. This fine analysis of $$u_\varepsilon $$ allows us to construct an adapted sequence of test functions and to pass to the limit in the variational formulation of the initial problem, in order to derive an effective model. We conclude that a viscosity drop may result in a change of boundary condition akin to the apparition of a slip length.

Математическое обоснование сопротивления движению автотранспорта в зависимости от состояния конструкции покрытия лесовозных дорог

The currently available methods for evaluating technical-operational characteristics and conditions of hauling roads are time- consuming. Although, they provide a sufficiently accurate description of the conditions of certain road components. Furthermore, they give the possibilities to identify ways for elimination of the constructional defects and determine the work extent to correct them. A peculiarity of forestry production is the simultaneous exploitation of a branching transport network with different types of subsidiary hauling roads, which depends on weather conditions, change of logging locations, traffic intensity, and other factors. Therefore, the available methods, which clarify the relationships between the traffic resistance and strength of the road construction, are poorly usable for rapid estimation of the technical and operational conditions in renovation planning. The purpose of the study is to improve the methodological fundamentals of traction-operational calculations for timber transportation according to the constructional state of the hauling roads. It has been determined that the calculation of the rolling resistance must include the factor of structural deformation. It is influenced by the modulus of elasticity for the surface structural materials. The resulting relationship makes it possible to increase the accuracy of traction-operational calculations at the project phase of the hauling road building. It can be included into the target function for comparing alternatives in the case of optional project planning. The mathematical aspect in the evaluation of the traffic resistance gives an opportunity to use this parameter as an indirect characteristic of the constructional strength of a road, as well as it can be a comprehensive qualitative indicator of the condition of a pavement. For citation: Prokopets V.S., Kozlov V.G., Skrypnikov A.V., Ponomareva N.G., Levushkin D.M., Borovlev Yu.A. Mathematical Justification of Vehicular Traffic Resistance in Relation to Structural Surface Conditions of Hauling Roads. Lesnoy Zhurnal = Russian Forestry Journal, 2023, no. 2, pp. 106−120. (In Russ.). https://doi.org/10.37482/0536-1036-2023-2-106-120

Application of Variational Iterations Method for Studying Physically and Geometrically Nonlinear Kirchhoff Nanoplates: A Mathematical Justification

We have proposed a development of the variational iteration method (VIM), or extended Kantorovich method, by studying physically nonlinear (FN) or geometrically nonlinear (GN) Kirchhoff nanoplates as an example. The modified couple stress theory was used for modeling size-dependent factors of the Kirchhoff nanoplates. Nested one into the other iteration procedures of the Birger method of variable elasticity parameters, of the variational iteration method (VIM), and of the Newton–Raphson method for physically nonlinear (FN) Kirchhoff nanoplates were constructed. The solution of problems for geometrically nonlinear (GN) Kirchhoff nanoplates was carried out on the basis of the variational iteration method and the Newton–Raphson method. The validity of the results was ensured by the coincidence of the results obtained via several methods of reducing partial differential equations to ordinary differential equations and via the finite difference method. The computational effectiveness of the proposed iterative procedure was demonstrated in terms of both accuracy and performance. A comparison of the results obtained showed that the variational iteration method (VIM) is the most efficient and fastest of all the methods considered both for problems with physical nonlinearity and for geometrically nonlinear problems.

Mathematical Justification of the Sports Selection of Jumpers in the Water at the Stage of Initial Training

The relevance of the issue is determined by the fact that an athlete possessing genetic and physiological abilities that were discovered in time can master a complex jumping program and execute it excellently at a high technical level. Therefore, it is important to mathematically substantiate the criteria for children enrollment to the sports schools. The Purpose of the Study is to determine indicators affecting sports readiness, to calculate their prognostic significance and to form criteria for the selection of jumpers into the water at the stage of initial training. Stuff Material and Methods. The research was organized over three stages in the period from January 2020 to December 2021, with the determination of the normative assessment of technical readiness, the conduct of an ascertaining pedagogical experiment and factor analysis, the calculation of multiple regression equations relative to the total coefficient of difficulty of arbitrary jumps and the average coefficient of the complexity of jumps. 139 boys (10,98±0,1 years; range: 9–13 years) and 96 girls (11,02±0,1 years; range: 9–13 years) beginners (schoolchildren) took part in the study. The number of athletes is divided by age groups: boys – 9 years old (n=25); 10 years (n = 30); 11 years (n = 30); 12 years (n
 = 31); 13 years old (n = 23); girls – 9 years old (n=16); 10 years (n=23); 11 years (n=18); 12 years (n=21); 13 years old (n=18). The study was organized and carried out with athletes of the Junior High School (Lviv) and Yantar Junior High School (Novoyavorivsk). During the educational and training meetings, testing and measuring of anthropometric indicators, determination of the speed of a simple sensorimotor reaction and conscious self-regulation of sensorimotor activity, research of vestibular stability were carried out. All measurements were taken before training in the morning and in the afternoon at 9:00 and 15:00. According to the Results of the Research, the indicators that have prognostic significance for the success of the sport`s improvement of young jumpers at the stage of improving their sports training have been determined. The total indicators of physical fitness, the nystagmus amplitude during and after turning to the right, the speed of a simple motor reaction have been included. The comprehensive assessment obtained by calculating multiple regression equations has a high prognostic value and is an effective method of predicting the success of the sport`s improvement of young jumpers at the stage of initial sports training. Conclusions. It is proved that the comprehensive assessment obtained by calculating multiple regression equations has a high prognostic value and is an effective method of predicting the success of the sport`s improvement of young jumpers at the initial sports training stage.

Evaluation of informativeness of mutation groups of Nigella damascena L. M2 generation

Purpose. To carry out a mathematical and statistical evaluation of the mutagenesis data of the M2 generation of Nigella damascena varieties ‘Berehynia’ and ‘Charivnytsia’ in order to identify relationships between groups of mutations and mathematical justification of their use in the process of further analysis and selection of hereditary changes of mutant plants. Methods. Seeds of the varieties ‘Berehynia’ and ‘Charivnytsia’ were treated with chemical mutagens. The mutations identified by visual observation of the plants at different stages of their growth and development were divided into eight groups. In each group of each treatment option, the frequency of mutations was calculated as a function of the type of mutagen, its concentration and exposure time. The mathematical and statistical evaluation of the results was carried out using the MS Excel analysis package and Spearman’s rank correlation. Results. The mutational difference in the M2 generation between eight groups of Nigella damascena varieties ‘Berehynia’ and ‘Charivnytsia’ under the influence of the mutagens ethyl methanesulfonate, nitrosomethylurea and the new chemical mutagen DG-2 (a complex of 3-N,N-dimethylamino sulfonane with dimethyl) was identified and evaluated. It was found that the correlation value of the mutagenic effects ranged from 0.437 to 0.752 with significance levels from 3.540 ´ 10–5 to 0.037. The following groups of mutations can be considered informative: stem, shoot and leaf structures; flower structure; colour of the corolla petals; structure of the capsule; physio­logical changes in the process of growth and development. Conclusions. The effect of the mutagens EMS, NMU in concentrations of 0.01 and 0.05% and DG-2 in concentrations of 0.5 and 0.05% was determined on Nigella damascena varieties ‘Berehynia’ and ‘Charivnytsia’ after exposure for 6 and 16 hours. Mutations in the structure of the stem, shoots and leaves, the colour of the petals of the corolla, the structure of the flower and capsule, and those causing a violation of chlorophyll synthesis should be considered informative.

Application of Duhamel's convolution integral in computer measurement systems

The paper is a new approach to the Duhamel's convolution integral. It contains a brief description of the Duhamel's convolution integral and an overview of applications of this integral. The scientific novelty presented in the work is the original algorithm for determining the frequency characteristic, together with its mathematical justification as well as with hardware and software implementation. The proposed algorithm was used in computer measurement systems enabling the determination of Bode and Nyquist characteristics of objects used in automation. The measurement system was built by the authors on the basis of the proposed algorithm. Measurements were carried out using it. During the measurements of the selected automation object, actual results were obtained and referenced to theoretical results. Then their correctness was assessed using the theory of measurement errors. In this way, the correctness of the mathematical considerations concerning the application of the Duhamel's convolution integral in computer measurement systems enabling the determination of the frequency characteristics of objects used in automation was confirmed.

Reflection and transmission analysis of an acoustic trifurcated waveguide with wave‐bearing structural discontinuity

The dilution of undesired and vandalize noise to a sustainable level is the subject of continuing research in the fields of applied mathematics, physics, and engineering. Therefore, the present study investigates the mechanism of an acoustic wave scattering, which includes the force propagation due to the incidence through region of elastic boundary condition. Both planar and nonplanar surfaces are allowed in this waveguide configuration. In this structure, the scattered fields in the regions of the inner surfaces are acoustically rigid and satisfy the standard orthogonality relation, while the fields on the upper outer surfaces are flexural nature that can be tackled using modified orthogonality relation. Thus, we intend to simultaneously take into account the behavior of the scattered field in both planar and nonplanar zones. Solving nonorthogonal eigenfunctions required the development of a modified orthogonality relation. Modified orthogonal relation and the mode‐matching (MM) technique have combined to get the solution of proposed model. The last section graphically discusses the solution for various frequencies and heights to observe the scattering behavior in all regions of the duct. In accordance with different characteristics of the waveguide material, expressions for the distribution of powers in different parts of the channel are developed and schematically depicted. A variety of mathematical and physical justifications provide strong support to results.

λ-method of C-rating models for combat readiness assessment

This paper reports on a new development of great importance to readiness managers and decision-makers. As the issues concerning the status of a Combat Unit via sub resources, considerations the λ-method is a multi-dimensional approach to readiness problems and it represents an extension of the C-rating technique as developed by (Frank et al., 1968). We present a conceptual framework for this new direction and provide a mathematical justification for the concept. We establish a non-commitment relationship between the Φ-state and the λ-factor of sub resources which are the cardinal concepts of this new approach.

Semi-spectral computational algorithm for fractional characterization of time-dependent fluid model with convergence and stability analysis

The current study is carried out to analyze the unsteady MHD flow of fractional-order Oldroyd-B nanofluid and heat transfer via a semi-spectral computational method. The governing model for MHD naturally convective fractional-order Oldroyd-B is expressed in terms of a coupled PDE system. To analyze the numerical solution novel scheme is suggested, where finite difference is used to approximate the temporal derivative whereas spatial variable is approximated via operational matrices which are computed using Chelyshkov polynomials. The proposed scheme transforms the nonlinear system of PDEs into a system of linear algebraic equations. Relevant theorems are presented to support the mathematical justification of the operational matrices-based method whereas stability analysis is being made for the proposed computational algorithm. The impact of involved physical parameters on velocity temperature, skin friction, and Nusselt number profiles are studied numerically. The impact of the magnetic field drops the velocity behavior whilst the effect of buoyancy forces on velocity is the opposite. The radiation pass through the nanofluid transmits the energy to the fluid particles and eventually causes an increase in the temperature of the fluid whereas similar behavior is noticed in heat generation. The substantial effect of the fractional parameter is noted for smaller values.

Development of a kinetic model of magnetite leaching

The object of the research is the process of magnetite leaching with nitric acid solutions, and the subject of the research is the mathematical justification of the kinetic model and the calculation of kinetic parameters. The article considers the case of leaching in the kinetic region, while magnetite is considered as a polydispersity material of spherical shape. It is proposed to use the distribution function of the number of particles N by their radius r in the form N=a∙rb, where a and b are constants. This distribution was used to derive the equation for the rate of the process W, taking into account the change in the surface of the particles depending on the degree of leaching α: W=dα/dτ=K*∙(1–α)m∙((С0(γ–α)/γ))n, where K* is the rate constant; m and n are the order of solid material and nitric acid, respectively; C0 and γ are the initial concentration of nitric acid and its stoichiometric excess. The order m is defined as m=(b+2)/(b+3), at b→∞ the order m→1. When b=0, m=2/3 is the case of the equation for a shrinking sphere. An algorithm for calculating kinetic parameters in the Excel is proposed. Experimental dependences of the degree of transformation α on time τ are approximated by a third-order equation; by differentiating the obtained equation, the values of the velocity Wexp=dα/dτ at individual points are calculated. After the logarithm of the above equation, there is the expression (γ=1): ln(Wexp)=ln(K*)+m∙ln(1–α)+n∙ln(С0(1–α)). With the help of the «LINEST» function in Excel, for a temperature of 373 K, the values of the order of m=0.93 and n=1.29 and the rate constant K*=0.08 were obtained. Calculation of kinetic parameters for different temperatures takes into account the dependence of the rate constant on temperature: ln(Wexp)=lnk0–E/R∙1/T+m∙ln(1–α)+n∙ln(C0(1–α)). As a result of the calculations, the values n=0.83; m=1.2; E/R=–10402; lnk0=25.09 were obtained. The value of the multiplier k0=exp(lnk0)=7.88∙1010, the activation energy E=–8.31∙E/R=86440 J/mol, the total reaction order n+m=2.03 was calculated. The obtained kinetic parameters were used to determine the calculated values of the rate W. The average relative error between the experimental and calculated values of the leaching rates is 10 %. The proposed method of processing experimental data using a mathematical leaching model can be used for any leaching process.

Contact force cancelation in robot impedance control by target impedance modification

AbstractA force sensorless impedance controller is proposed in this paper for robot manipulators without using force estimators. From the observation of the impedance control law, the force feedback term can be canceled if the inertia matrix in the target impedance is the same as the robot inertia matrix. However, the inertia matrix in the target impedance is almost always a constant matrix, while the robot inertia matrix is a function of the robot configuration, and hence, they may not be identical in general. A modification of the coefficient matrix for the contact force term in the target impedance is suggested in this paper to enable cancelation of the force feedback term in the impedance control law so that a force sensorless impedance controller without using force estimators can be obtained. The tracking performance in the free space phase and the motion trajectory in the compliant motion phase of the new design are almost the same as those in the traditional impedance control. Modification of the inertia matrix in the target impedance will result in small variations of the contact force which is acceptable in practical applications. For robot manipulators containing uncertainties, an adaptive version of the new controller is also developed in this paper to give satisfactory performance without the need for force sensors. Rigorous mathematical justification in closed-loop stability is given in detail, and computer simulations are performed to verify the efficacy of the proposed design.

Two-step inertial forward-reflected-backward splitting based algorithm for nonconvex mixed variational inequalities

This paper presents a modification of a recently studied Malitsky–Tam forward-reflected-backward splitting method with two-step inertial extrapolation to solve nonconvex mixed variational inequalities. We give global convergence results and nonasymptotic O(1/k) convergence rate of our proposed method under some appropriate conditions and present some mathematical justifications.

Bank credit risk management: economic and mathematical justification of improvement opportunities

Bank credit risk management: economic and mathematical justification of improvement opportunities

Assessment of the Justification Degree of Scientific Facts in the School Physics Course

The problem of the justification (validity) degree of scientific facts in the school physics course is analysed, the relevance of which is due to the need for the formation of evidence-based thinking among schoolchildren. More than 150 facts of the existence of objects, phenomena and functional dependencies expressed using formulas are written out. It is shown that the more convincing the substantiation of the factual statement is, the simpler and more accessible it is for the student. A methodology for assessing the credibility of experimental and logical-mathematical justification is proposed, and appropriate scales are developed. The method of assessing the semantic complexity of scientific facts by summing up the complexities of terms that make up the corresponding factual statement is considered. The semantic complexity, persuasiveness (or simplicity) of experimental and logical-mathematical substantiation of the selected facts are evaluated. It is established that: 1) an increase in the persuasiveness of the experimental substantiation of facts is accompanied by a decrease in their semantic complexity; 2) an increase in the persuasiveness of the experimental substantiation of formulas is associated with an increase in the persuasiveness of their logical and mathematical justification; 3) for the average values calculated for each section of physics, an increase in the persuasiveness of the experimental justification of facts leads to an increase in the persuasiveness of the logical and mathematical substantiation and a decrease in semantic complexity.

Математичне моделювання стійкості руху асиметричної дискової борони

The production necessity to expand the functional capabilities of farm machinery and tractor units (MTUs), being the most demanded in small farms due to their high energy intensity, reliability, and versatility in the production of agricultural products, is an important task that requires the application of new methodological, practical approaches, and relevant design solutions. This article discusses the application of a hitch weight compensator, installed on the frame and plow of an asymmetric disc harrow, as one of the ways to rationally redistribute the hitch weight within the MTU during pre-sowing soil tillage. The mathematical justification is provided for improving the stability of the movement of an asymmetric disc harrow and the use of wide-span agricultural units, which allows increasing the productivity of their application and reducing temporary and energy costs for pre-sowing soil cultivation. In the general case, the performed mathematical analysis allows you to justify the fact that the MTA with an installed hitch weight compensator allows you to redistribute the load in the MTA composition, which will be able to expand the functional capabilities of the MTA and stabilize the stability of the movement of the asymmetric disc harrow under the conditions of its use in farms during pre-sowing soil cultivation.

Screening of patients with cerebral aneurysms: mathematical analysis and economic justification

Demonstrate the clinical and economic feasibility of screening the population (including first-line relatives) for cerebral aneurysms using an economic and mathematical model of the RF virtual population. Mathematical modeling was carried out using an algorithm that implements a discrete Markov chain. The virtual population consisted of 145 million people (the population of the Russian Federation). Magnetic resonance angiography 3DTOF was chosen as a screening method. Virtual patients underwent preventive surgical treatment in case of detection of aneurysm during screening. The number of aneurysms in the population, the number of aneurysmal subarachnoid hemorrhage (aSAH), the cost and outcomes of treatment, and the risk of disability were calculated. In the case of screening and preventive surgical treatment of aneurysms, there is a decrease in the number of aSAH by 14.3% (37.5% in first-line relatives (RPLR), which affects the reduction in mortality due to aSAH by 14.4% (24.1% in The total number of disabled people is reduced by 1.5% (5.1% for the RPHR). A shift in the structure of disability towards greater labor and social adaptation of patients was noted. An economic analysis for the entire population showed that screening saves 7.7 billion annually rubles, including in the population consisting of RPLR - 4.9 billion rubles. The created mathematical model of the virtual population demonstrated that screening and prophylactic treatment of cerebral aneurysms makes it possible to reduce the number of aSAH and associated mortality among the entire population and in the RPLR group. The number of individuals with severe disabilities is decreasing. Thus, population screening for the detection of cerebral aneurysms may be clinically effective and cost-effective in the general population, especially in RPCR.

Аналіз способів сівби, перспективи вдосконалення

One of the most important technological measures in agricultural production is the sowing of agricultural crops. Therefore, it is important to choose a sowing method that will create optimal conditions for the germination and development of the seeds of cultivated plants. Sowing methods depend on the biological characteristics of crops (different crops are not equally demanding on soil fertility, the amount of heat, lighting, humidity, etc.). One of the main requirements for sowing methods is the creation of optimal sowing density, which ensures the most intensive growth of the assimilation leaf surface - the main factor of yield. Scientists have conducted a number of studies, developed designs of combined working bodies of sowing machines and distribution devices for transporting and distributing seed material along the width of the coulter, identified the main shortcomings and proposed ways to simplify them and improve the quality of work when sowing seeds of grain crops. However, the problem of increasing the efficiency of using the entire design width of the coulters can be successfully solved if the mathematical justification of the technological parameters of the elements of the working body is applied. The purpose of the article is to conduct an analysis of sowing methods, to carry out a literature and patent review of the constructions of combined working bodies for shelf-free tillage with simultaneous subsoil-broadcasting of grain crops or in-soil application of mineral fertilizers, analysis of the constructions of distribution devices of various authors and the efficiency of their use. If you evaluate the working bodies for transporting and distributing the material when it is placed in the soil comprehensively according to three main indicators: the simplicity of the design, its reliability and ensuring the quality indicators of the execution of the technological process, then preference should be given to the working bodies with distributors of passive action. In this case, the first two indicators are simultaneously provided. In our opinion, the issue of the quality of the execution of the technological process - the uniformity of the distribution of the material along the width of the working body is urgent, therefore it requires a deeper study and the establishment of the main groups of factors that affect this indicator. A significant drawback of most theoretical studies of the process of material distribution along the grip width of a flat-cutting working body is that the distributor is considered as a final factor and a structural element on which the main characteristic of the process depends - the uniformity of distribution, but in real conditions this indicator is significantly redistributed as a result of particle contact with soil. The considered areas of improvement of the working bodies for continuous sowing are effective, but their practical use is limited, because in some cases the design of the machines is significantly complicated, in others, the designs of the distributors themselves are too complicated, and their scope of use is limited. At the same time, with modern trends in the field of tillage, its minimization, when more and more combined tools are used, which reduce not only the cost of obtaining a crop, but also, what is no less important, the load on the soil and the environment, from an ecological point of view, it proves the necessity further improvement of the designs of combined weapons and their working bodies. Therefore, in our opinion, work on improving the structures of such working bodies and further research into the parameters and modes of their functioning are promising.

An Iterative Warping and Clustering Algorithm to Estimate Multiple Wave-Shape Functions From a Nonstationary Oscillatory Signal

Nonsinusoidal oscillatory signals are everywhere. In practice, the nonsinusoidal oscillatory pattern, modeled as a 1-periodic wave-shape function (WSF), might vary from cycle to cycle. When there are finite different WSFs, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{1},\ldots,s_{K}$</tex-math></inline-formula> , so that the WSF jumps from one to another suddenly, the different WSFs and jumps encode useful information. We present an iterative warping and clustering algorithm to estimate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{1},\ldots,s_{K}$</tex-math></inline-formula> from a nonstationary oscillatory signal with time-varying amplitude and frequency, and hence the change points of the WSFs. The algorithm is a novel combination of time-frequency analysis, singular value decomposition entropy and vector spectral clustering. We demonstrate the efficiency of the proposed algorithm with simulated and real signals, including the voice signal, arterial blood pressure, electrocardiogram and accelerometer signal. Moreover, we provide a mathematical justification of the algorithm under the assumption that the amplitude and frequency of the signal are slowly time-varying and there are finite change points that model sudden changes from one wave-shape function to another one.