Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.展开更多
In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and dedu...In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.展开更多
The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the ext...The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.展开更多
The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and o...The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them.展开更多
Degenerate versions of special polynomials and numbers applied to social problems,physics,and applied mathematics have been studied variously in recent years.Moreover,the(s-)Lah numbers have many other interesting app...Degenerate versions of special polynomials and numbers applied to social problems,physics,and applied mathematics have been studied variously in recent years.Moreover,the(s-)Lah numbers have many other interesting applications in analysis and combinatorics.In this paper,we divide two parts.We first introduce new types of both degenerate incomplete and complete s-Bell polynomials respectively and investigate some properties of them respectively.Second,we introduce the degenerate versions of complete and incomplete Lah-Bell polynomials as multivariate forms for a new type of degenerate s-extended Lah-Bell polynomials and numbers respectively.We investigate relations between these polynomials and degenerate incomplete and complete s-Bell polynomials,and derive explicit formulas for these polynomials.展开更多
This paper investigates the asymptotic behavior of high-order vector rogue wave(RW)solutions for any multi-component nonlinear Schr¨odinger equation(denoted as n-NLSE)with multiple internal large parameters.We re...This paper investigates the asymptotic behavior of high-order vector rogue wave(RW)solutions for any multi-component nonlinear Schr¨odinger equation(denoted as n-NLSE)with multiple internal large parameters.We report some novel RW patterns,including nonmultiple root(NMR)-type patterns with distinct shapes such as semicircular sector,acute sector,pseudo-hexagram,and pseudo-rhombus shapes,as well as multiple root(MR)-type patterns characterized by right double-arrow and right arrow shapes.We demonstrate that these RW patterns are intrinsically related to the root structures of a novel class of polynomials,termed generalized mixed Adler-Moser(GMAM)polynomials,which feature multiple arbitrary free parameters.The RW patterns can be interpreted as straightforward expansions and slight shifts of the root structures for the GMAM polynomials to some extent.In the(x,t)-plane,they asymptotically converge to a first-order RW at the location corresponding to each simple root of the polynomials and to a lower-order RW at the location associated with each multiple root.Notably,the position of the lower-order RW within these patterns can be flexibly adjusted to any desired location in the(x,t)-plane by tuning the free parameters of the corresponding GMAM polynomials.展开更多
Economic losses and catastrophic casualties may occur once super high-rise structures are struck by low-probability but high-consequence scenarios of concurrent earthquakes and winds. Therefore, accurately predicting ...Economic losses and catastrophic casualties may occur once super high-rise structures are struck by low-probability but high-consequence scenarios of concurrent earthquakes and winds. Therefore, accurately predicting multi-hazard dynamic responses to super high-rise structures has significant engineering and scientific value. This study performed a parametric global sensitivity analysis (GSA) for multi-hazard dynamic response prediction of super high-rise structures using the multiple-degree-of-freedom shear (MFS) model. Polynomial chaos Kriging (PCK) was introduced to build a surrogate model that allowed GSA to be combined with Sobol’ indices. Monte Carlo simulation (MCS) is also conducted for the comparison to verify the accuracy and efficiency of the PCK method. Parametric sensitivity analysis is performed for a wide range of aleatory uncertainty (intensities of coupled multi-hazard), epistemic uncertainty (bending stiffness, k_(m);shear stiffness, kq;density, ρ;and damping ratio, ξ), probability distribution types, and coefficients of variation. The results indicate that epistemic uncertainty parameters, k_(m), ρ, and ξ dramatically affect the multi-hazard dynamic responses of super high-rise structures;in addition, Sobol’ indices between the normal and lognormal distributions are insignificant, while the variation levels have remarkably influenced the sensitivity indices.展开更多
Fins are extensively utilized in heat exchangers and various industrial applications as they are lightweight and can benefit in various systems,including electronic cooling devices and automotive components,owing to t...Fins are extensively utilized in heat exchangers and various industrial applications as they are lightweight and can benefit in various systems,including electronic cooling devices and automotive components,owing to their adaptable design.Furthermore,spine fins are introduced to improve performance in applications such as automotive radiators.They can be shaped in different ways and constructed from a collection of materials.Inspired by this,the present model examines the effects of internal heat generation and radiation-convection on the thermal distribution in a wetted convex-shaped spine fin.Using dimensionless terms,the proposed fin model involving a governing nonlinear ordinary differential equation(ODE)is transformed into a dimensionless form.The study uses the operational matrix with the Charlier polynomial collocation method(OMCCM)to ensure precise and computationally efficient numerical solutions for the dimensionless equation.In order to aid in the analysis of thermal performance,the importance of major parameters on the temperature profile is graphically illustrated.The main outcome of the study reveals that as the radiation-conductive,wet,and convective-conductive parameters increase,the heat transfer rate progressively improves.Conversely,the ambient temperature and internal heat generation parameters show an inverse relationship.展开更多
This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-...This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-oped using the material point method.To reduce the computational cost of Monte Carlo simulations,response surface models are created as surrogate models for the material point system to approximate its dynamic behavior.An adaptive randomized greedy algorithm is employed to construct a sparse polynomial chaos expansion model with a fixed order,effectively balancing the accuracy and computational efficiency of the surrogate model.Based on the sparse polynomial chaos expansion,sensitivity analysis is conducted using the global finite difference and Sobol methods.Several examples of structural dynamics are provided to demonstrate the effectiveness of the proposed method in addressing structural dynamics problems.展开更多
In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) i...In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.展开更多
The zero coprime system equivalence is one of important research in the theory of multidimensional system equivalence,and is closely related to zero coprime equivalence of multivariate polynomial matrices.We first dis...The zero coprime system equivalence is one of important research in the theory of multidimensional system equivalence,and is closely related to zero coprime equivalence of multivariate polynomial matrices.We first discuss the relation between zero coprime equivalence and unimodular equivalence for polynomial matrices.Then,we investigate the zero coprime equivalence problem for several classes of polynomial matrices,some novel findings and criteria on reducing these matrices to their Smith normal forms are obtained.Finally,an example is provided to illustrate the main results.展开更多
This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schröodinger equation with additive noise on a 1D bounded domain.The noise is white in time and smooth in space.We c...This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schröodinger equation with additive noise on a 1D bounded domain.The noise is white in time and smooth in space.We consider both focusing and defocusing nonlinearities,with exponents of the nonlinearityσ∈[0,2)andσ∈[0,∞),and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method.In the focusing case,our result generalizes the earlier results in[12],whereσ=1.展开更多
GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieve...GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieved by several parameters,such as polynomial terms,periodic terms,offsets,and post-seismic models.The latter contains some stochastic noises,which can be affected by detecting the former parameters.If there are not enough parameters assumed,modeling errors will occur and adversely affect the analysis results.In this study,we propose a processing strategy in which the commonly-used 1-order of the polynomial term can be replaced with different orders for better fitting GNSS time series of the Crustal Movement Network of China(CMONOC)stations.Initially,we use the Bayesian Information Criterion(BIC)to identify the best order within the range of 1-4 during the fitting process using the white noise plus power-law noise(WN+PL)model.Then,we compare the 1-order and the optimal order on the effect of deterministic models in GNSS time series,including the velocity and its uncertainty,amplitudes,and initial phases of the annual signals.The results indicate that the first-order polynomial in the GNSS time series is not the primary factor.The root mean square(RMS)reduction rates of almost all station components are positive,which means the new fitting of optimal-order polynomial helps to reduce the RMS of residual series.Most stations maintain the velocity difference(VD)within ±1 mm/yr,with percentages of 85.6%,81.9%and 63.4%in the North,East,and Up components,respectively.As for annual signals,the numbers of amplitude difference(AD)remained at ±0.2 mm are 242,239,and 200 in three components,accounting for 99.6%,98.4%,and 82.3%,respectively.This finding reminds us that the detection of the optimal-order polynomial is necessary when we aim to acquire an accurate understanding of the crustal movement features.展开更多
As a crucial component of intelligent chassis systems,air suspension significantly enhances driver comfort and vehicle stability.To further improve the adaptability of commercial vehicles to complex and variable road ...As a crucial component of intelligent chassis systems,air suspension significantly enhances driver comfort and vehicle stability.To further improve the adaptability of commercial vehicles to complex and variable road conditions,this paper proposes a linear motor active suspension with quasi-zero stiffness(QZS)air spring system.Firstly,a dynamic model of the linear motor active suspension with QZS air spring system is established.Secondly,considering the random uncertainties in the linear motor parameters due to manufacturing and environmental factors,a dynamic model and state equations incorporating these uncertainties are constructed using the polynomial chaos expansion(PCE)method.Then,based on H_(2) robust control theory and the Kalman filter,a state feedback control law is derived,accounting for the random parameter uncertainties.Finally,simulation and hardware-in-the-loop(HIL)experimental results demonstrate that the PCE-H_(2) robust controller not only provides better performance in terms of vehicle ride comfort compared to general H_(2) robust controller but also exhibits higher robustness to the effects of random uncertain parameters,resulting in more stable control performance.展开更多
Virtual coupling is a novel technology that enables trains to run closely together without physical connections through communication and automation systems.The paper addresses an adaptive polynomial approximation alg...Virtual coupling is a novel technology that enables trains to run closely together without physical connections through communication and automation systems.The paper addresses an adaptive polynomial approximation algorithm for the cooperative control of high-speed trains(HSTs)under virtual coupling.It aims to solve the cooperative tracking control problem of HST formation operations under various scenarios,including known and unknown parameters.To enable the HST formation system to achieve cooperative operation while ensuring an appropriate spacing distance,the tracking errors of displacement and speed throughout the entire operation converge to zero.The proposed control strategy focuses on adopting polynomial approximation to handle unknown parameters,which are estimated via adaptive laws.Additionally,the unknown parameters of the HSTs are estimated online through adaptive laws.Experimental results verify the effectiveness of this method.展开更多
This study systematically analyzed the spatiotemporal evolution characteristics of geomagnetic anomalies before and after the 2013 Sichuan Lushan M7.0 earthquake and the Gansu Minxian M6.6 earthquake by constructing a...This study systematically analyzed the spatiotemporal evolution characteristics of geomagnetic anomalies before and after the 2013 Sichuan Lushan M7.0 earthquake and the Gansu Minxian M6.6 earthquake by constructing a geomagnetic diurnal variation model based on Taylor polynomial fitting,combined with midnight mean values of the geomagnetic F component from China,s geomagnetic observatory network.The results reveal distinct differences in anomaly patterns,namely per-sistent positive anomalies were observed in the epicentral region of the Lushan earthquake,while significant negative anomalies characterized the Minxian earthquake zone.This differential response reveals the modulating effect of the electrical structure of the seismogenic medium on space electromagnetic disturbances,namely positive anomalies may correspond to the stage of stable stress accumulation in intact rock,while the expansion of negative anomalies may reflect an amplification of electromagnetic disturbances induced by fracture expansion.Further analysis demonstrates that both anomalies exhibit a three-stage evolutionary pattern,namely pre-seismic accumulation,co-seismic release,and post-seismic adjustment.The phase transitions in these anomalies are closely correlated with regional tectonic stress accumulation and destabilization processes.These findings not only provide new evidence for the physical interpretation of seismomagnetic precursors but also establish a theoretical foundation for developing earthquake prediction methods based on the dynamic evolution of geomagnetic anomalies.展开更多
A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symb...A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.展开更多
In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in ...In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in this paper,we prove that rooted products of some trees preserve real-rootedness of independence polynomials.In particular,we can obtain that their independence polynomials are unimodal and log-concave.展开更多
Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limi...Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limitation in the accuracy of the field profile restitution.Thus,it can deal with the guided waves in layered sandwich only when the material properties of adjacent layers do not change significantly.Despite the great efforts regarding this issue in the literature,there remain open questions.One of them is:“what is the exact threshold of contrasting material properties of adjacent layers for which this polynomial method cannot correctly restitute the roots of guided waves?”We investigated this numerical issue using the calculated guided phase velocities in 0°/φ/0°-carbon fibre reinforced plastics(CFRP)sandwich plates with gradually increasing angleφ.Then,we approached this numerical problem by varying the middle layer thickness h90°for the 0°/90°/0°-CFRP sandwich structure,and we proposed an exact thickness threshold of the middle layer for the Legendre polynomial method limitations.We showed that the polynomial method fails to calculate the quasi-symmetric Lamb mode in 0°/φ/0°-CFRP whenφ>25°.Moreover,we introduced a new Lamb mode so-called minimum-group-velocity that has never been addressed in literature.展开更多
This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree sig...This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.展开更多
文摘Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
基金Supported by the PCSIRT of Education of China(IRT0621)Supported by the Innovation Program of Shanghai Municipal Education Committee of China(08ZZ24)Supported by the Henan Innovation Project for University Prominent Research Talents of China(2007KYCX0021)
文摘In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.
基金funded by Cebu Normal University through its Research Institute for Computational Mathematics and Physics(RICMP).
文摘The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.
基金supported by the Basic Science Research Program,the National Research Foundation of Korea(NRF-2021R1F1A1050151).
文摘The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them.
基金supported by the Basic Science Research Program,the National Research Foundation of Korea(NRF-2021R1F1A1050151).
文摘Degenerate versions of special polynomials and numbers applied to social problems,physics,and applied mathematics have been studied variously in recent years.Moreover,the(s-)Lah numbers have many other interesting applications in analysis and combinatorics.In this paper,we divide two parts.We first introduce new types of both degenerate incomplete and complete s-Bell polynomials respectively and investigate some properties of them respectively.Second,we introduce the degenerate versions of complete and incomplete Lah-Bell polynomials as multivariate forms for a new type of degenerate s-extended Lah-Bell polynomials and numbers respectively.We investigate relations between these polynomials and degenerate incomplete and complete s-Bell polynomials,and derive explicit formulas for these polynomials.
基金supported by the NSFC(12471236)the Guangzhou Municipal Science and Technology Project(Guangzhou Science and Technology Plan,No.2024A04J6245)Guangdong Natural Science Foundation(2025A1515011868)。
文摘This paper investigates the asymptotic behavior of high-order vector rogue wave(RW)solutions for any multi-component nonlinear Schr¨odinger equation(denoted as n-NLSE)with multiple internal large parameters.We report some novel RW patterns,including nonmultiple root(NMR)-type patterns with distinct shapes such as semicircular sector,acute sector,pseudo-hexagram,and pseudo-rhombus shapes,as well as multiple root(MR)-type patterns characterized by right double-arrow and right arrow shapes.We demonstrate that these RW patterns are intrinsically related to the root structures of a novel class of polynomials,termed generalized mixed Adler-Moser(GMAM)polynomials,which feature multiple arbitrary free parameters.The RW patterns can be interpreted as straightforward expansions and slight shifts of the root structures for the GMAM polynomials to some extent.In the(x,t)-plane,they asymptotically converge to a first-order RW at the location corresponding to each simple root of the polynomials and to a lower-order RW at the location associated with each multiple root.Notably,the position of the lower-order RW within these patterns can be flexibly adjusted to any desired location in the(x,t)-plane by tuning the free parameters of the corresponding GMAM polynomials.
基金Dalian Municipal Natural Science Foundation under Grant No.2019RD01。
文摘Economic losses and catastrophic casualties may occur once super high-rise structures are struck by low-probability but high-consequence scenarios of concurrent earthquakes and winds. Therefore, accurately predicting multi-hazard dynamic responses to super high-rise structures has significant engineering and scientific value. This study performed a parametric global sensitivity analysis (GSA) for multi-hazard dynamic response prediction of super high-rise structures using the multiple-degree-of-freedom shear (MFS) model. Polynomial chaos Kriging (PCK) was introduced to build a surrogate model that allowed GSA to be combined with Sobol’ indices. Monte Carlo simulation (MCS) is also conducted for the comparison to verify the accuracy and efficiency of the PCK method. Parametric sensitivity analysis is performed for a wide range of aleatory uncertainty (intensities of coupled multi-hazard), epistemic uncertainty (bending stiffness, k_(m);shear stiffness, kq;density, ρ;and damping ratio, ξ), probability distribution types, and coefficients of variation. The results indicate that epistemic uncertainty parameters, k_(m), ρ, and ξ dramatically affect the multi-hazard dynamic responses of super high-rise structures;in addition, Sobol’ indices between the normal and lognormal distributions are insignificant, while the variation levels have remarkably influenced the sensitivity indices.
基金the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/308/46。
文摘Fins are extensively utilized in heat exchangers and various industrial applications as they are lightweight and can benefit in various systems,including electronic cooling devices and automotive components,owing to their adaptable design.Furthermore,spine fins are introduced to improve performance in applications such as automotive radiators.They can be shaped in different ways and constructed from a collection of materials.Inspired by this,the present model examines the effects of internal heat generation and radiation-convection on the thermal distribution in a wetted convex-shaped spine fin.Using dimensionless terms,the proposed fin model involving a governing nonlinear ordinary differential equation(ODE)is transformed into a dimensionless form.The study uses the operational matrix with the Charlier polynomial collocation method(OMCCM)to ensure precise and computationally efficient numerical solutions for the dimensionless equation.In order to aid in the analysis of thermal performance,the importance of major parameters on the temperature profile is graphically illustrated.The main outcome of the study reveals that as the radiation-conductive,wet,and convective-conductive parameters increase,the heat transfer rate progressively improves.Conversely,the ambient temperature and internal heat generation parameters show an inverse relationship.
基金support from the National Natural Science Foundation of China(Grant Nos.52174123&52274222).
文摘This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-oped using the material point method.To reduce the computational cost of Monte Carlo simulations,response surface models are created as surrogate models for the material point system to approximate its dynamic behavior.An adaptive randomized greedy algorithm is employed to construct a sparse polynomial chaos expansion model with a fixed order,effectively balancing the accuracy and computational efficiency of the surrogate model.Based on the sparse polynomial chaos expansion,sensitivity analysis is conducted using the global finite difference and Sobol methods.Several examples of structural dynamics are provided to demonstrate the effectiveness of the proposed method in addressing structural dynamics problems.
基金supported by the National Natural Science Foundation of China(12061035)the Research Foundation of Jiangxi Science and Technology Normal University of China(2021QNBJRC003)supported by the Graduate Innovation Fund of Jiangxi Science and Technology Normal University(YC2024-X10).
文摘In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.
基金Supported by the National Natural Science Foundation of China(12271154)the Natural Science Foundation of Hunan Province(2022JJ30234)the Postgraduate Scientific Research Innovation Project of Hunan Province(CX20231032)。
文摘The zero coprime system equivalence is one of important research in the theory of multidimensional system equivalence,and is closely related to zero coprime equivalence of multivariate polynomial matrices.We first discuss the relation between zero coprime equivalence and unimodular equivalence for polynomial matrices.Then,we investigate the zero coprime equivalence problem for several classes of polynomial matrices,some novel findings and criteria on reducing these matrices to their Smith normal forms are obtained.Finally,an example is provided to illustrate the main results.
基金supported by the National Key R&D Program of China(2023YFA1009200)the NSFC(11925102)the Liaoning Revitalization Talents Program(XLYC2202042)。
文摘This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schröodinger equation with additive noise on a 1D bounded domain.The noise is white in time and smooth in space.We consider both focusing and defocusing nonlinearities,with exponents of the nonlinearityσ∈[0,2)andσ∈[0,∞),and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method.In the focusing case,our result generalizes the earlier results in[12],whereσ=1.
基金supported by the National Natural Science Foundation of China(Grant Nos.42404017,42122025 and 42174030).
文摘GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieved by several parameters,such as polynomial terms,periodic terms,offsets,and post-seismic models.The latter contains some stochastic noises,which can be affected by detecting the former parameters.If there are not enough parameters assumed,modeling errors will occur and adversely affect the analysis results.In this study,we propose a processing strategy in which the commonly-used 1-order of the polynomial term can be replaced with different orders for better fitting GNSS time series of the Crustal Movement Network of China(CMONOC)stations.Initially,we use the Bayesian Information Criterion(BIC)to identify the best order within the range of 1-4 during the fitting process using the white noise plus power-law noise(WN+PL)model.Then,we compare the 1-order and the optimal order on the effect of deterministic models in GNSS time series,including the velocity and its uncertainty,amplitudes,and initial phases of the annual signals.The results indicate that the first-order polynomial in the GNSS time series is not the primary factor.The root mean square(RMS)reduction rates of almost all station components are positive,which means the new fitting of optimal-order polynomial helps to reduce the RMS of residual series.Most stations maintain the velocity difference(VD)within ±1 mm/yr,with percentages of 85.6%,81.9%and 63.4%in the North,East,and Up components,respectively.As for annual signals,the numbers of amplitude difference(AD)remained at ±0.2 mm are 242,239,and 200 in three components,accounting for 99.6%,98.4%,and 82.3%,respectively.This finding reminds us that the detection of the optimal-order polynomial is necessary when we aim to acquire an accurate understanding of the crustal movement features.
基金Supported by National Natural Science Foundation of China(Grant No.51875256)Open Platform Fund of Human Institute of Technology(Grant No.KFA22009).
文摘As a crucial component of intelligent chassis systems,air suspension significantly enhances driver comfort and vehicle stability.To further improve the adaptability of commercial vehicles to complex and variable road conditions,this paper proposes a linear motor active suspension with quasi-zero stiffness(QZS)air spring system.Firstly,a dynamic model of the linear motor active suspension with QZS air spring system is established.Secondly,considering the random uncertainties in the linear motor parameters due to manufacturing and environmental factors,a dynamic model and state equations incorporating these uncertainties are constructed using the polynomial chaos expansion(PCE)method.Then,based on H_(2) robust control theory and the Kalman filter,a state feedback control law is derived,accounting for the random parameter uncertainties.Finally,simulation and hardware-in-the-loop(HIL)experimental results demonstrate that the PCE-H_(2) robust controller not only provides better performance in terms of vehicle ride comfort compared to general H_(2) robust controller but also exhibits higher robustness to the effects of random uncertain parameters,resulting in more stable control performance.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.62203246 and 62003127)Shandong Provincial Natural Science Foundation(Grant No.ZR2024QF041)the Natural Science Foundation of Hebei Province(Grant No.F2023202060)。
文摘Virtual coupling is a novel technology that enables trains to run closely together without physical connections through communication and automation systems.The paper addresses an adaptive polynomial approximation algorithm for the cooperative control of high-speed trains(HSTs)under virtual coupling.It aims to solve the cooperative tracking control problem of HST formation operations under various scenarios,including known and unknown parameters.To enable the HST formation system to achieve cooperative operation while ensuring an appropriate spacing distance,the tracking errors of displacement and speed throughout the entire operation converge to zero.The proposed control strategy focuses on adopting polynomial approximation to handle unknown parameters,which are estimated via adaptive laws.Additionally,the unknown parameters of the HSTs are estimated online through adaptive laws.Experimental results verify the effectiveness of this method.
基金supported by a Collaborative Project of the National Natural Science Foundation of China on Technical Maintenance and Data Preprocessing of the GPS Observation Array for the Qiaojia Earthquake(No.0120603)the National Natural Science Foundation of China(No.41274079).
文摘This study systematically analyzed the spatiotemporal evolution characteristics of geomagnetic anomalies before and after the 2013 Sichuan Lushan M7.0 earthquake and the Gansu Minxian M6.6 earthquake by constructing a geomagnetic diurnal variation model based on Taylor polynomial fitting,combined with midnight mean values of the geomagnetic F component from China,s geomagnetic observatory network.The results reveal distinct differences in anomaly patterns,namely per-sistent positive anomalies were observed in the epicentral region of the Lushan earthquake,while significant negative anomalies characterized the Minxian earthquake zone.This differential response reveals the modulating effect of the electrical structure of the seismogenic medium on space electromagnetic disturbances,namely positive anomalies may correspond to the stage of stable stress accumulation in intact rock,while the expansion of negative anomalies may reflect an amplification of electromagnetic disturbances induced by fracture expansion.Further analysis demonstrates that both anomalies exhibit a three-stage evolutionary pattern,namely pre-seismic accumulation,co-seismic release,and post-seismic adjustment.The phase transitions in these anomalies are closely correlated with regional tectonic stress accumulation and destabilization processes.These findings not only provide new evidence for the physical interpretation of seismomagnetic precursors but also establish a theoretical foundation for developing earthquake prediction methods based on the dynamic evolution of geomagnetic anomalies.
基金supported by the National Natural Science Foundation of China(Nos.U21A20447 and 61971079)。
文摘A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.
基金supported by the National Natural Science Foundation of China(No.12271527)。
文摘In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in this paper,we prove that rooted products of some trees preserve real-rootedness of independence polynomials.In particular,we can obtain that their independence polynomials are unimodal and log-concave.
基金supported by the National Natural Science Foundation of China(Grant No.12102131).
文摘Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limitation in the accuracy of the field profile restitution.Thus,it can deal with the guided waves in layered sandwich only when the material properties of adjacent layers do not change significantly.Despite the great efforts regarding this issue in the literature,there remain open questions.One of them is:“what is the exact threshold of contrasting material properties of adjacent layers for which this polynomial method cannot correctly restitute the roots of guided waves?”We investigated this numerical issue using the calculated guided phase velocities in 0°/φ/0°-carbon fibre reinforced plastics(CFRP)sandwich plates with gradually increasing angleφ.Then,we approached this numerical problem by varying the middle layer thickness h90°for the 0°/90°/0°-CFRP sandwich structure,and we proposed an exact thickness threshold of the middle layer for the Legendre polynomial method limitations.We showed that the polynomial method fails to calculate the quasi-symmetric Lamb mode in 0°/φ/0°-CFRP whenφ>25°.Moreover,we introduced a new Lamb mode so-called minimum-group-velocity that has never been addressed in literature.
基金funded by the Ministry of Science and Higher Education of Kazakhstan and carried out within the framework of the project AP23488112“Development and study of a quantum-resistant digital signature scheme based on a Verkle tree”at the Institute of Information and Computational Technologies.
文摘This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.
