Aims In ecology and conservation biology,the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected...Aims In ecology and conservation biology,the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected.Moreover,comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled.For individual-based data,we treat a single,empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models:estimating the expected number of species(and its unconditional variance)in a random sample of(i)a smaller number of individuals(multinomial model)or a smaller area sampled(Poisson model)and(ii)a larger number of individuals or a larger area sampled.For sample-based incidence(presence–absence)data,under a Bernoulli product model,we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units.Methods The first objective is a problem in interpolation that we address with classical rarefaction(multinomial model)and Coleman rarefaction(Poisson model)for individual-based data and with sample-based rarefaction(Bernoulli product model)for incidence frequencies.The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample(multinomial model),a larger area(Poisson model)or a larger number of sampling units(Bernoulli product model),based on an estimate of asymptotic species richness.Although published methods exist for many of these objectives,we bring them together here with some new estimators under a unified statistical and notational framework.This novel integration of mathematically distinct approaches allowed us to link interpolated(rarefaction)curves and extrapolated curves to plot a unified species accumulation curve for empirical examp展开更多
The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spe...The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.展开更多
In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential...In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential equations under the integral inequalities.展开更多
We further consider the effect of rod strength by employing the compressible penetration model to study the effect of compressibility on hypervelocity penetration.Meanwhile, we define different instances of penetratio...We further consider the effect of rod strength by employing the compressible penetration model to study the effect of compressibility on hypervelocity penetration.Meanwhile, we define different instances of penetration efficiency in various modified models and compare these penetration efficiencies to identify the effects of different factors in the compressible model. To systematically discuss the effect of compressibility in different metallic rod-target combinations, we construct three cases, i.e., the penetrations by the more compressible rod into the less compressible target, rod into the analogously compressible target, and the less compressible rod into the more compressible target. The effects of volumetric strain, internal energy, and strength on the penetration efficiency are analyzed simultaneously. It indicates that the compressibility of the rod and target increases the pressure at the rod/target interface. The more compressible rod/target has larger volumetric strain and higher internal energy. Both the larger volumetric strain and higher strength enhance the penetration or anti-penetration ability. On the other hand, the higher internal energy weakens the penetration or anti-penetration ability. The two trends conflict, but the volumetric strain dominates in the variation of the penetration efficiency, which would not approach the hydrodynamic limit if the rod and target are not analogously compressible. However, if the compressibility of the rod and target is analogous, it has little effect on the penetration efficiency.展开更多
Recently,deep learning has been widely utilized for object tracking tasks.However,deep learning encounters limits in tasks such as Autonomous Aerial Refueling(AAR),where the target object can vary substantially in siz...Recently,deep learning has been widely utilized for object tracking tasks.However,deep learning encounters limits in tasks such as Autonomous Aerial Refueling(AAR),where the target object can vary substantially in size,requiring high-precision real-time performance in embedded systems.This paper presents a novel embedded adaptiveness single-object tracking framework based on an improved YOLOv4 detection approach and an n-fold Bernoulli probability theorem.First,an Asymmetric Convolutional Network(ACNet)and dense blocks are combined with the YOLOv4 architecture to detect small objects with high precision when similar objects are in the background.The prior object information,such as its location in the previous frame and its speed,is utilized to adaptively track objects of various sizes.Moreover,based on the n-fold Bernoulli probability theorem,we develop a filter that uses statistical laws to reduce the false positive rate of object tracking.To evaluate the efficiency of our algorithm,a new AAR dataset is collected,and extensive AAR detection and tracking experiments are performed.The results demonstrate that our improved detection algorithm is better than the original YOLOv4 algorithm on small and similar object detection tasks;the object tracking algorithm is better than state-of-the-art object tracking algorithms on refueling drogue tracking tasks.展开更多
We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient p...We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.展开更多
Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of ...Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of consecutive outcomes is generated. This can be seen as a game where m players select one such pattern each and the one whose pattern occurs first wins. We present symbolic formulas for the m probabilities of winning, and for the mean number of trials and the corresponding standard deviation to complete this game. Several numerical examples are presented, including a search for optimal strategy.展开更多
The internal energy change of ideal gas does not depend on the volume and pressure. The internal energy change of real gas has not any relation with the volume and pressure, which had been proved. If the internal ener...The internal energy change of ideal gas does not depend on the volume and pressure. The internal energy change of real gas has not any relation with the volume and pressure, which had been proved. If the internal energy change had not any relation with the volume and pressure, we could confirm the first law of thermodynamics in theory. Simultaneously, the internal energy change is the state function that shall be able to be proved in theory. If the internal energy change depended on the volume and pressure, we could not prove that the internal energy change is the state function and the chemical thermodynamics theory is right. The extended or modified Bernoulli equation can be derived from the energy conservation law, and the internal energy change, heat, and friction are all considered in the derivation procedure. The extended Bernoulli equation could be applied to the flying aircraft and mechanical motion on the gravitational field, for instance, the rocket and airplane and so on. This paper also revises some wrong ideas, viewpoints, or concepts about the thermodynamics theory and Bernoulli equation.展开更多
In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected proces...In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.展开更多
The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era,...The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.展开更多
In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1...In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model representing the wave propagation through incompressible fluids.The linearization of the wave structure in shallow water necessitates more critical wave capacity conditions than it does in deep water,and the strong nonlinear properties are perceptible.Some novel travelling wave solutions have been observed including solitons,kink,periodic and rational solutions with the aid of the latest computing tools such as Mathematica or Maple.The physical and analytical properties of several families of closed-form solutions or exact solutions and rational form function solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model problem are examined using Mathematica.展开更多
This study considers a nonlinear grey Bernoulli forecasting model with conformable fractionalorder accumulation,abbreviated as CFNGBM(1,1,λ),to study the gross regional product in the ChengYu area.The new model conta...This study considers a nonlinear grey Bernoulli forecasting model with conformable fractionalorder accumulation,abbreviated as CFNGBM(1,1,λ),to study the gross regional product in the ChengYu area.The new model contains three nonlinear parameters,the power exponentγ,the conformable fractional-orderαand the background valueλ,which increase the adjustability and flexibility of the CFNGBM(1,1,λ)model.Nonlinear parameters are determined by the moth flame optimization algorithm,which minimizes the mean absolute prediction percentage error.The CFNGBM(1,1,λ)model is applied to the gross regional product of 16 cities in the Cheng-Yu area,which are Chongqing,Chengdu,Mianyang,Leshan,Zigong,Deyang,Meishan,Luzhou,Suining,Neijiang,Nanchong,Guang’an,Yibin,Ya’an,Dazhou and Ziyang.With data from 2013 to 2021,several grey models are established and results show that the new model has higher accuracy in most cases.展开更多
An optimized nonlinear grey Bernoulli model was proposed by using a particle swarm optimization algorithm to solve the parameter optimization problem. In addition, each item in the first-order accumulated generating s...An optimized nonlinear grey Bernoulli model was proposed by using a particle swarm optimization algorithm to solve the parameter optimization problem. In addition, each item in the first-order accumulated generating sequence was set in turn as an initial condition to determine which alternative would yield the highest forecasting accuracy. To test the forecasting performance, the optimized models with different initial conditions were then used to simulate dissolved oxygen concentrations in the Guantlng reservoir inlet and outlet （China）. The empirical results show that the optimized model can remarkably improve forecasting accuracy, and the particle swarm optimization technique is a good tool to solve parameter optimization problems. What＇s more, the optimized model with an initial condition that performs well in in-sample simulation may not do as well as in out-of-sample forecasting.展开更多
This article considers three types of biological systems:the dengue fever disease model,the COVID-19 virus model,and the transmission of Tuberculosis model.The new technique of creating the integration matrix for the ...This article considers three types of biological systems:the dengue fever disease model,the COVID-19 virus model,and the transmission of Tuberculosis model.The new technique of creating the integration matrix for the Bernoulli wavelets is applied.Also,the novel method proposed in this paper is called the Bernoulli wavelet collocation scheme(BWCM).All three models are in the form system of coupled ordinary differential equations without an exact solution.These systems are converted into a system of algebraic equations using the Bernoulli wavelet collocation scheme.The numerical wave distributions of these governing models are obtained by solving the algebraic equations via the Newton-Raphson method.The results obtained from the developed strategy are compared to several schemes such as the Runge Kutta method,and ND solver in mathematical software.The convergence analyses are discussed through theorems.The newly implemented Bernoulli wavelet method improves the accuracy and converges when it is compared with the existing methods in the literature.展开更多
We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second...We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials.We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.展开更多
Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) appli...Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.展开更多
基金US National Science Foundation(DEB 0639979 and DBI 0851245 to R.K.C.DEB-0541936 to N.J.G.+4 种基金DEB-0424767 and DEB-0639393 to R.L.C.DEB-0640015 to J.T.L.)the US Department of Energy(022821 to N.J.G.)the Taiwan National Science Council(97-2118-M007-MY3 to A.C.)and the University of Connecticut Research Foundation(to R.L.C.).
文摘Aims In ecology and conservation biology,the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected.Moreover,comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled.For individual-based data,we treat a single,empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models:estimating the expected number of species(and its unconditional variance)in a random sample of(i)a smaller number of individuals(multinomial model)or a smaller area sampled(Poisson model)and(ii)a larger number of individuals or a larger area sampled.For sample-based incidence(presence–absence)data,under a Bernoulli product model,we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units.Methods The first objective is a problem in interpolation that we address with classical rarefaction(multinomial model)and Coleman rarefaction(Poisson model)for individual-based data and with sample-based rarefaction(Bernoulli product model)for incidence frequencies.The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample(multinomial model),a larger area(Poisson model)or a larger number of sampling units(Bernoulli product model),based on an estimate of asymptotic species richness.Although published methods exist for many of these objectives,we bring them together here with some new estimators under a unified statistical and notational framework.This novel integration of mathematically distinct approaches allowed us to link interpolated(rarefaction)curves and extrapolated curves to plot a unified species accumulation curve for empirical examp
基金supported by the Key Project of Chinese Ministry of Education(Grant No. 108117)National Natural Science Foundation of China (Grant No. 10871123,61071066,11171201)
文摘The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.
文摘In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential equations under the integral inequalities.
文摘We further consider the effect of rod strength by employing the compressible penetration model to study the effect of compressibility on hypervelocity penetration.Meanwhile, we define different instances of penetration efficiency in various modified models and compare these penetration efficiencies to identify the effects of different factors in the compressible model. To systematically discuss the effect of compressibility in different metallic rod-target combinations, we construct three cases, i.e., the penetrations by the more compressible rod into the less compressible target, rod into the analogously compressible target, and the less compressible rod into the more compressible target. The effects of volumetric strain, internal energy, and strength on the penetration efficiency are analyzed simultaneously. It indicates that the compressibility of the rod and target increases the pressure at the rod/target interface. The more compressible rod/target has larger volumetric strain and higher internal energy. Both the larger volumetric strain and higher strength enhance the penetration or anti-penetration ability. On the other hand, the higher internal energy weakens the penetration or anti-penetration ability. The two trends conflict, but the volumetric strain dominates in the variation of the penetration efficiency, which would not approach the hydrodynamic limit if the rod and target are not analogously compressible. However, if the compressibility of the rod and target is analogous, it has little effect on the penetration efficiency.
文摘Recently,deep learning has been widely utilized for object tracking tasks.However,deep learning encounters limits in tasks such as Autonomous Aerial Refueling(AAR),where the target object can vary substantially in size,requiring high-precision real-time performance in embedded systems.This paper presents a novel embedded adaptiveness single-object tracking framework based on an improved YOLOv4 detection approach and an n-fold Bernoulli probability theorem.First,an Asymmetric Convolutional Network(ACNet)and dense blocks are combined with the YOLOv4 architecture to detect small objects with high precision when similar objects are in the background.The prior object information,such as its location in the previous frame and its speed,is utilized to adaptively track objects of various sizes.Moreover,based on the n-fold Bernoulli probability theorem,we develop a filter that uses statistical laws to reduce the false positive rate of object tracking.To evaluate the efficiency of our algorithm,a new AAR dataset is collected,and extensive AAR detection and tracking experiments are performed.The results demonstrate that our improved detection algorithm is better than the original YOLOv4 algorithm on small and similar object detection tasks;the object tracking algorithm is better than state-of-the-art object tracking algorithms on refueling drogue tracking tasks.
文摘We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.
文摘Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of consecutive outcomes is generated. This can be seen as a game where m players select one such pattern each and the one whose pattern occurs first wins. We present symbolic formulas for the m probabilities of winning, and for the mean number of trials and the corresponding standard deviation to complete this game. Several numerical examples are presented, including a search for optimal strategy.
文摘The internal energy change of ideal gas does not depend on the volume and pressure. The internal energy change of real gas has not any relation with the volume and pressure, which had been proved. If the internal energy change had not any relation with the volume and pressure, we could confirm the first law of thermodynamics in theory. Simultaneously, the internal energy change is the state function that shall be able to be proved in theory. If the internal energy change depended on the volume and pressure, we could not prove that the internal energy change is the state function and the chemical thermodynamics theory is right. The extended or modified Bernoulli equation can be derived from the energy conservation law, and the internal energy change, heat, and friction are all considered in the derivation procedure. The extended Bernoulli equation could be applied to the flying aircraft and mechanical motion on the gravitational field, for instance, the rocket and airplane and so on. This paper also revises some wrong ideas, viewpoints, or concepts about the thermodynamics theory and Bernoulli equation.
文摘In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.
文摘The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.
文摘In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model representing the wave propagation through incompressible fluids.The linearization of the wave structure in shallow water necessitates more critical wave capacity conditions than it does in deep water,and the strong nonlinear properties are perceptible.Some novel travelling wave solutions have been observed including solitons,kink,periodic and rational solutions with the aid of the latest computing tools such as Mathematica or Maple.The physical and analytical properties of several families of closed-form solutions or exact solutions and rational form function solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model problem are examined using Mathematica.
基金Supported by the National Natural Science Foundation of China(72001181,71901184)the Sichuan Federation of Social Science Associations(SC20B122)。
文摘This study considers a nonlinear grey Bernoulli forecasting model with conformable fractionalorder accumulation,abbreviated as CFNGBM(1,1,λ),to study the gross regional product in the ChengYu area.The new model contains three nonlinear parameters,the power exponentγ,the conformable fractional-orderαand the background valueλ,which increase the adjustability and flexibility of the CFNGBM(1,1,λ)model.Nonlinear parameters are determined by the moth flame optimization algorithm,which minimizes the mean absolute prediction percentage error.The CFNGBM(1,1,λ)model is applied to the gross regional product of 16 cities in the Cheng-Yu area,which are Chongqing,Chengdu,Mianyang,Leshan,Zigong,Deyang,Meishan,Luzhou,Suining,Neijiang,Nanchong,Guang’an,Yibin,Ya’an,Dazhou and Ziyang.With data from 2013 to 2021,several grey models are established and results show that the new model has higher accuracy in most cases.
基金supported by the National Natural Science Foundation of China (Nos. 51178018 and 71031001)
文摘An optimized nonlinear grey Bernoulli model was proposed by using a particle swarm optimization algorithm to solve the parameter optimization problem. In addition, each item in the first-order accumulated generating sequence was set in turn as an initial condition to determine which alternative would yield the highest forecasting accuracy. To test the forecasting performance, the optimized models with different initial conditions were then used to simulate dissolved oxygen concentrations in the Guantlng reservoir inlet and outlet （China）. The empirical results show that the optimized model can remarkably improve forecasting accuracy, and the particle swarm optimization technique is a good tool to solve parameter optimization problems. What＇s more, the optimized model with an initial condition that performs well in in-sample simulation may not do as well as in out-of-sample forecasting.
文摘This article considers three types of biological systems:the dengue fever disease model,the COVID-19 virus model,and the transmission of Tuberculosis model.The new technique of creating the integration matrix for the Bernoulli wavelets is applied.Also,the novel method proposed in this paper is called the Bernoulli wavelet collocation scheme(BWCM).All three models are in the form system of coupled ordinary differential equations without an exact solution.These systems are converted into a system of algebraic equations using the Bernoulli wavelet collocation scheme.The numerical wave distributions of these governing models are obtained by solving the algebraic equations via the Newton-Raphson method.The results obtained from the developed strategy are compared to several schemes such as the Runge Kutta method,and ND solver in mathematical software.The convergence analyses are discussed through theorems.The newly implemented Bernoulli wavelet method improves the accuracy and converges when it is compared with the existing methods in the literature.
基金This work was supported by the National Research Foundation of Korea(NRF)Grant Funded by the Korea Government(No.2020R1F1A1A01071564).
文摘We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials.We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.
文摘Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.
