Here we complete our work on the asymptotics of Hankel determinants studying the case wherein the entries are “ultrarapidly”-varying functions in the sense that their logarithms are rapidly varying. Moreover, the la...Here we complete our work on the asymptotics of Hankel determinants studying the case wherein the entries are “ultrarapidly”-varying functions in the sense that their logarithms are rapidly varying. Moreover, the last results in the paper highlight analogies between algebraic identities for Hankelians with special entries and asymptotic relations valid for large classes of entries.展开更多
A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman （J-S） fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the sma...A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman （J-S） fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the small perturbation parameter, and then the second-order asymptotic solution of the flow of a Newtonian fluid through a slowly varying pipe is obtained in the first perturbation strategy. Second, the viscoelastic parameter is selected as the small perturbation parameter in the second perturbation strategy to solve the asymptotic solution of the flow of a J-S fluid through a slowly varying pipe. Finally, the parameter effects, including the axial distance, the slowly varying angle, and the Reynolds number, on the velocity distributions are analyzed. The results show that the increases in both the axial distance and the slowly varying angle make the axial velocity slow down. However, the radial velocity increases with the slowly varying angle, and decreases with the axial distance. There are two special positions in the distribution curves of the axial velocity and the radial velocity with different Reynolds numbers, and there are different trends on both sides of the special positions. The double perturbation strategy is applicable to such problems with the flow of a non-Newtonian fluid through a slowly varying pipe.展开更多
The stabilization problem of systems that switch among a finite set of slowly varying linear systems with arbitrary switching frequency is discussed.It is shown that if the entries of the pointwise stabilizing feedbac...The stabilization problem of systems that switch among a finite set of slowly varying linear systems with arbitrary switching frequency is discussed.It is shown that if the entries of the pointwise stabilizing feedback gain matrix are continuously differentiable functions of the entries of the system coefficient matrices,then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small.展开更多
The evolution of Asian summer monsoon is analyzed by means of decomposition of the atmospheric circulationinto basic current and slowly varying monsoon disturbances. It is seen that the major slowly varying disturbanc...The evolution of Asian summer monsoon is analyzed by means of decomposition of the atmospheric circulationinto basic current and slowly varying monsoon disturbances. It is seen that the major slowly varying disturbances aretwo vortical couples, one located in the indian Ocean and indian Peninsula, and another in the Northern Hemisphericwestern Pacific Ocean and the East Asia. This indicates that the Asian summer monsoon consists of two branches, theIndian monsoon and the East Asian monsoon. Moreover, the analysis shows that the evolutionary processes of thesetwo vortical couples are rather independent each other, and they all can be qualitatively interpreted by the dynamicaltheory of wave packet. The different stages of summer monsoon can be very well characterized by the location and in-tensity of the two vortical couples. Besides, in particular years there exists also some quasistationary wave train andits characteristics should be further analyzed.展开更多
In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted,...In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.展开更多
The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly ...The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall.The Korteweg-de Vries(KdV)equation with varying coefficients is derived with the aid of the reductive perturbation method.By using the method of multiple scales,the approximate solutions of this equation are obtained.It is found that the unevenness of bottom may lead to the generation of socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic waves and of the amplitude,propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented.The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.展开更多
The rich literature concerning “asymptotic behavior of Hankel determinants” concerns the behavior, as the order n tends to ∞, of Hankel determinants whose entries are numbers, e.g., with a combinatorial interest or...The rich literature concerning “asymptotic behavior of Hankel determinants” concerns the behavior, as the order n tends to ∞, of Hankel determinants whose entries are numbers, e.g., with a combinatorial interest or arising as values of special classes of functions. Such determinants are numbers depending on n, playing roles in number theory, combinatorics, random matrices and the like;and mathematicians in the involved fields have been interested in their asymptotic behaviors as n goes to ∞, as previously mentioned, with no single exception to the author’s knowledge. The study carried on in the present paper treats an altogether different situation as suggested by the specification in the title “as the variable tends to +∞”. We deal with those types of Hankel determinants (purposely called Hankelians) which are special cases of Wronskians and, continuing our work on the asymptotics of Wronskians, we study the asymptotic behaviors of n-order Hankelians, whose entries involve either regularly- or rapidly-varying functions, when the variable tends to +∞. As in the study of Wronskians, the treatment of this case also needs the whole apparatus of the theory of higher-order types of asymptotic variation, but the most demanding results are not automatic corollaries of the general theory. In fact, in the study of generic Wronskians (study motivated by applications to asymptotic expansions), the entries were required to belong to one of the classes of “higher-order regular or rapid variation”;on the contrary, in the case of Hankelians, we are confronted with functions whose logarithms are either “regularly- or rapidly-varying functions”, roughly classifiable as “ultrarapidly-varying functions”, and the study requires both special devices and a number of preliminary lemmas about products and linear combinations of functions in the mentioned classes.展开更多
The object of this article is to introduce new classes of meromorphic functions associated with conic regions. Several properties like the coefficient bounds, growth and distortion theorems, radii of starlikeness and ...The object of this article is to introduce new classes of meromorphic functions associated with conic regions. Several properties like the coefficient bounds, growth and distortion theorems, radii of starlikeness and convexity, partial sums, are investigated. Some consequences of the main results for the well-known classes of meromorphic functions are also pointed out.展开更多
In this paper,the problem of time varying telecommunication delays in passive teleoperation systems is addressed.The design comprises delayed position,velocity and position-velocity signals with the local position and...In this paper,the problem of time varying telecommunication delays in passive teleoperation systems is addressed.The design comprises delayed position,velocity and position-velocity signals with the local position and velocity signals of the master and slave manipulators.Nonlinear adaptive control terms are employed locally to cope with uncertain parameters associated with the gravity loading vector of the master and slave manipulators.Lyapunov-Krasovskii function is employed for three methods to establish asymptotic tracking property of the closed loop teleoperation systems.The stability analysis is derived for both symmetrical and unsymmetrical time varying delays in the forward and backward communication channel that connects the local and remote sites.Finally,evaluation results are presented to illustrate the efectiveness of the proposed design for real-time applications.展开更多
This paper presents a copula technique to develop time-variant seismic fragility curves for corroded bridges at the system level and considers the realistic time-varying dependence among component seismic demands. Bas...This paper presents a copula technique to develop time-variant seismic fragility curves for corroded bridges at the system level and considers the realistic time-varying dependence among component seismic demands. Based on material deterioration mechanisms and incremental dynamic analysis, the time-evolving seismic demands of components were obtained in the form of marginal probability distributions. The time-varying dependences among bridge components were then captured with the best fitting copula function, which was selected from the commonly used copula classes by the empirical distribution based analysis method. The system time-variant fragility curves at different damage states were developed and the effects of time-varying dependences among components on the bridge system fragility were investigated. The results indicate the time-varying dependence among components significantly affects the time-variant fragility of the bridge system. The copula technique captures the nonlinear dependence among component seismic demands accurately and easily by separating the marginal distributions and the dependence among them.展开更多
This paper proposes a new adaptive iterative learning control approach for a class of nonlinearly parameterized systems with unknown time-varying delay and unknown control direction.By employing the parameter separati...This paper proposes a new adaptive iterative learning control approach for a class of nonlinearly parameterized systems with unknown time-varying delay and unknown control direction.By employing the parameter separation technique and signal replacement mechanism,the approach can overcome unknown time-varying parameters and unknown time-varying delay of the nonlinear systems.By incorporating a Nussbaum-type function,the proposed approach can deal with the unknown control direction of the nonlinear systems.Based on a Lyapunov-Krasovskii-like composite energy function,the convergence of tracking error sequence is achieved in the iteration domain.Finally,two simulation examples are provided to illustrate the feasibility of the proposed control method.展开更多
In this paper, adaptive neural tracking control is proposed based on radial basis function neural networks （RBFNNs） for a class of muki-input multi-output （MIMO） nonlinear systems with completely unknown control d...In this paper, adaptive neural tracking control is proposed based on radial basis function neural networks （RBFNNs） for a class of muki-input multi-output （MIMO） nonlinear systems with completely unknown control directions, unknown dynamic disturbances, unmodeled dynamics, and uncertainties with time-varying delay. Using the Nussbaum function properties, the unknown control directions are dealt with. By constructing appropriate Lyapunov-Krasovskii functionals, the unknown upper bound functions of the time-varying delay uncertainties are compensated. The proposed control scheme does not need to calculate the integral of the delayed state functions. Using Young＇s inequality and RBFNNs, the assumption of unmodeled dynamics is relaxed. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded.展开更多
This paper presents a novel approach to model and simulate the multi-support depth-varying seismic motions（MDSMs） within heterogeneous offshore and onshore sites.Based on 1 D wave propagation theory,the three-dimens...This paper presents a novel approach to model and simulate the multi-support depth-varying seismic motions（MDSMs） within heterogeneous offshore and onshore sites.Based on 1 D wave propagation theory,the three-dimensional ground motion transfer functions on the surface or within an offshore or onshore site are derived by considering the effects of seawater and porous soils on the propagation of seismic P waves.Moreover,the depth-varying and spatial variation properties of seismic ground motions are considered in the ground motion simulation.Using the obtained transfer functions at any locations within a site,the offshore or onshore depth-varying seismic motions are stochastically simulated based on the spectral representation method（SRM）.The traditional approaches for simulating spatially varying ground motions are improved and extended to generate MDSMs within multiple offshore and onshore sites.The simulation results show that the PSD functions and coherency losses of the generated MDSMs are compatible with respective target values,which fully validates the effectiveness of the proposed simulation method.The synthesized MDSMs can provide strong support for the precise seismic response prediction and performance-based design of both offshore and onshore large-span engineering structures.展开更多
The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in ...The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series;2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians;3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation;4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.展开更多
A whole impulsive control scheme of nonlinear systems with time-varying delays, which is an extension for impulsive control of nonlinear systems without time delay, is presented in this paper. Utilizing the Lyapunov f...A whole impulsive control scheme of nonlinear systems with time-varying delays, which is an extension for impulsive control of nonlinear systems without time delay, is presented in this paper. Utilizing the Lyapunov functions and the impulsive-type comparison principles, we establish a series of different conditions under which impulsively controlled nonlinear systems with time-varying delays are asymptotically stable. Then we estimate upper bounds of impulse interval and time-varying delays for asymptotically stable control. Finally a numerical example is given to illustrate the effectiveness of the method.展开更多
Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of ...Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.展开更多
文摘Here we complete our work on the asymptotics of Hankel determinants studying the case wherein the entries are “ultrarapidly”-varying functions in the sense that their logarithms are rapidly varying. Moreover, the last results in the paper highlight analogies between algebraic identities for Hankelians with special entries and asymptotic relations valid for large classes of entries.
基金supported by the National Natural Science Foundation of China(Nos.11572203 and11332006)
文摘A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman （J-S） fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the small perturbation parameter, and then the second-order asymptotic solution of the flow of a Newtonian fluid through a slowly varying pipe is obtained in the first perturbation strategy. Second, the viscoelastic parameter is selected as the small perturbation parameter in the second perturbation strategy to solve the asymptotic solution of the flow of a J-S fluid through a slowly varying pipe. Finally, the parameter effects, including the axial distance, the slowly varying angle, and the Reynolds number, on the velocity distributions are analyzed. The results show that the increases in both the axial distance and the slowly varying angle make the axial velocity slow down. However, the radial velocity increases with the slowly varying angle, and decreases with the axial distance. There are two special positions in the distribution curves of the axial velocity and the radial velocity with different Reynolds numbers, and there are different trends on both sides of the special positions. The double perturbation strategy is applicable to such problems with the flow of a non-Newtonian fluid through a slowly varying pipe.
文摘The stabilization problem of systems that switch among a finite set of slowly varying linear systems with arbitrary switching frequency is discussed.It is shown that if the entries of the pointwise stabilizing feedback gain matrix are continuously differentiable functions of the entries of the system coefficient matrices,then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small.
文摘The evolution of Asian summer monsoon is analyzed by means of decomposition of the atmospheric circulationinto basic current and slowly varying monsoon disturbances. It is seen that the major slowly varying disturbances aretwo vortical couples, one located in the indian Ocean and indian Peninsula, and another in the Northern Hemisphericwestern Pacific Ocean and the East Asia. This indicates that the Asian summer monsoon consists of two branches, theIndian monsoon and the East Asian monsoon. Moreover, the analysis shows that the evolutionary processes of thesetwo vortical couples are rather independent each other, and they all can be qualitatively interpreted by the dynamicaltheory of wave packet. The different stages of summer monsoon can be very well characterized by the location and in-tensity of the two vortical couples. Besides, in particular years there exists also some quasistationary wave train andits characteristics should be further analyzed.
基金The Project Supported by the National Natural Science Foundation of China
文摘In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.
基金Project Supported by National Natural Science Foundation of China
文摘The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall.The Korteweg-de Vries(KdV)equation with varying coefficients is derived with the aid of the reductive perturbation method.By using the method of multiple scales,the approximate solutions of this equation are obtained.It is found that the unevenness of bottom may lead to the generation of socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic waves and of the amplitude,propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented.The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.
文摘The rich literature concerning “asymptotic behavior of Hankel determinants” concerns the behavior, as the order n tends to ∞, of Hankel determinants whose entries are numbers, e.g., with a combinatorial interest or arising as values of special classes of functions. Such determinants are numbers depending on n, playing roles in number theory, combinatorics, random matrices and the like;and mathematicians in the involved fields have been interested in their asymptotic behaviors as n goes to ∞, as previously mentioned, with no single exception to the author’s knowledge. The study carried on in the present paper treats an altogether different situation as suggested by the specification in the title “as the variable tends to +∞”. We deal with those types of Hankel determinants (purposely called Hankelians) which are special cases of Wronskians and, continuing our work on the asymptotics of Wronskians, we study the asymptotic behaviors of n-order Hankelians, whose entries involve either regularly- or rapidly-varying functions, when the variable tends to +∞. As in the study of Wronskians, the treatment of this case also needs the whole apparatus of the theory of higher-order types of asymptotic variation, but the most demanding results are not automatic corollaries of the general theory. In fact, in the study of generic Wronskians (study motivated by applications to asymptotic expansions), the entries were required to belong to one of the classes of “higher-order regular or rapid variation”;on the contrary, in the case of Hankelians, we are confronted with functions whose logarithms are either “regularly- or rapidly-varying functions”, roughly classifiable as “ultrarapidly-varying functions”, and the study requires both special devices and a number of preliminary lemmas about products and linear combinations of functions in the mentioned classes.
文摘The object of this article is to introduce new classes of meromorphic functions associated with conic regions. Several properties like the coefficient bounds, growth and distortion theorems, radii of starlikeness and convexity, partial sums, are investigated. Some consequences of the main results for the well-known classes of meromorphic functions are also pointed out.
基金supported by Natural Sciences and Engineering Research Council of Canada (NSERC) Research Fellowship,Canada Research Chairs Program and University of Ottawa Research Chair Program
文摘In this paper,the problem of time varying telecommunication delays in passive teleoperation systems is addressed.The design comprises delayed position,velocity and position-velocity signals with the local position and velocity signals of the master and slave manipulators.Nonlinear adaptive control terms are employed locally to cope with uncertain parameters associated with the gravity loading vector of the master and slave manipulators.Lyapunov-Krasovskii function is employed for three methods to establish asymptotic tracking property of the closed loop teleoperation systems.The stability analysis is derived for both symmetrical and unsymmetrical time varying delays in the forward and backward communication channel that connects the local and remote sites.Finally,evaluation results are presented to illustrate the efectiveness of the proposed design for real-time applications.
基金Natural Science Foundation of China under Grant No.51808376
文摘This paper presents a copula technique to develop time-variant seismic fragility curves for corroded bridges at the system level and considers the realistic time-varying dependence among component seismic demands. Based on material deterioration mechanisms and incremental dynamic analysis, the time-evolving seismic demands of components were obtained in the form of marginal probability distributions. The time-varying dependences among bridge components were then captured with the best fitting copula function, which was selected from the commonly used copula classes by the empirical distribution based analysis method. The system time-variant fragility curves at different damage states were developed and the effects of time-varying dependences among components on the bridge system fragility were investigated. The results indicate the time-varying dependence among components significantly affects the time-variant fragility of the bridge system. The copula technique captures the nonlinear dependence among component seismic demands accurately and easily by separating the marginal distributions and the dependence among them.
基金supported by National Natural Science Foundation of China (No. 60974139)Fundamental Research Funds for the Central Universities (No. 72103676)
文摘This paper proposes a new adaptive iterative learning control approach for a class of nonlinearly parameterized systems with unknown time-varying delay and unknown control direction.By employing the parameter separation technique and signal replacement mechanism,the approach can overcome unknown time-varying parameters and unknown time-varying delay of the nonlinear systems.By incorporating a Nussbaum-type function,the proposed approach can deal with the unknown control direction of the nonlinear systems.Based on a Lyapunov-Krasovskii-like composite energy function,the convergence of tracking error sequence is achieved in the iteration domain.Finally,two simulation examples are provided to illustrate the feasibility of the proposed control method.
基金partially supported by National Natural Science Foundation of China(61290322,61273222,61322303,61473248,61403335)Hebei Province Applied Basis Research Project(15967629D)Top Talents Project of Hebei Province and Yanshan University Project(13LGA020)
基金supported by National Natural Science Foundation of China(No.61174046)
文摘In this paper, adaptive neural tracking control is proposed based on radial basis function neural networks （RBFNNs） for a class of muki-input multi-output （MIMO） nonlinear systems with completely unknown control directions, unknown dynamic disturbances, unmodeled dynamics, and uncertainties with time-varying delay. Using the Nussbaum function properties, the unknown control directions are dealt with. By constructing appropriate Lyapunov-Krasovskii functionals, the unknown upper bound functions of the time-varying delay uncertainties are compensated. The proposed control scheme does not need to calculate the integral of the delayed state functions. Using Young＇s inequality and RBFNNs, the assumption of unmodeled dynamics is relaxed. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded.
基金This work was supported by the National Natural Science Founda- tion of China （61374078） and Natural Science Foundation Project of Chongqing CSTC （cstc2014jcyjA40014）.
基金National Key R&D Program of China under Grant No.2016YFC0701108the State Key Program of National Natural Science Foundation of China under Grant No.51738007
文摘This paper presents a novel approach to model and simulate the multi-support depth-varying seismic motions（MDSMs） within heterogeneous offshore and onshore sites.Based on 1 D wave propagation theory,the three-dimensional ground motion transfer functions on the surface or within an offshore or onshore site are derived by considering the effects of seawater and porous soils on the propagation of seismic P waves.Moreover,the depth-varying and spatial variation properties of seismic ground motions are considered in the ground motion simulation.Using the obtained transfer functions at any locations within a site,the offshore or onshore depth-varying seismic motions are stochastically simulated based on the spectral representation method（SRM）.The traditional approaches for simulating spatially varying ground motions are improved and extended to generate MDSMs within multiple offshore and onshore sites.The simulation results show that the PSD functions and coherency losses of the generated MDSMs are compatible with respective target values,which fully validates the effectiveness of the proposed simulation method.The synthesized MDSMs can provide strong support for the precise seismic response prediction and performance-based design of both offshore and onshore large-span engineering structures.
文摘The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series;2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians;3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation;4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.
文摘A whole impulsive control scheme of nonlinear systems with time-varying delays, which is an extension for impulsive control of nonlinear systems without time delay, is presented in this paper. Utilizing the Lyapunov functions and the impulsive-type comparison principles, we establish a series of different conditions under which impulsively controlled nonlinear systems with time-varying delays are asymptotically stable. Then we estimate upper bounds of impulse interval and time-varying delays for asymptotically stable control. Finally a numerical example is given to illustrate the effectiveness of the method.
文摘Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.
